molecular-level simulations of chemical reaction ...molecular-level simulations of chemical reaction...
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Fluid Phase Equilibria 272 (2008) 18–31
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Fluid Phase Equilibria
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olecular-level simulations of chemical reaction equilibrium for nitric oxideimerization reaction in disordered nanoporous carbons
artin Lísal a,b,∗, Paolo Cosoli c, William R. Smithd, Surendra K. Jaine, Keith E. Gubbinse
E. Hála Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals of the ASCR, v.v.i., 165 02 Prague 6, Czech RepublicDepartment of Physics, Faculty of Science, J. E. Purkinje University, 400 96 Ústí n. Lab., Czech RepublicMolecular Simulation Engineering (MOSE) Laboratory, Department of Chemical, Environmental and Raw Materials Engineering (DICAMP),niversity of Trieste, 34127 Trieste, ItalyFaculty of Science, University of Ontario Institute of Technology, Oshawa, ON L1H7K4, CanadaCenter for High Performance Simulation and Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, NC 27695-7905, USA
r t i c l e i n f o
rticle history:eceived 23 June 2008eceived in revised form 30 July 2008ccepted 30 July 2008vailable online 8 August 2008
eywords:isordered nanoporous carbonacroscopic adsorption model
eaction Ensemble Monte Carloeaction equilibriumeactive adsorption isotherm
a b s t r a c t
We report a molecular-level simulation study of the effects of confinement on chemical reaction equi-librium for the NO dimerization reaction, 2NO� (NO)2, in disordered nanoporous carbons. We usethe Reaction Ensemble Monte Carlo (RxMC) method [W.R. Smith, B. Tríska, J. Chem. Phys. 100 (1994)3019–3027; J.K. Johnson, A.Z. Panagiotopoulos, K.E. Gubbins, Mol. Phys. 81 (1994) 717–733] to investigatethe effects of temperature and bulk pressure on the reaction conversion in three models of disorderednanoporous carbons obtained from sucrose in equilibrium with a vapor reservoir. Atomistic models ofthe carbons used [S.K. Jain, R.J.-M. Pellenq, J.P. Pikunic, K.E. Gubbins, Langmuir 22 (2006) 9942–9948]were constructed using the Hybrid Reverse Monte Carlo method, differing by the processing conditionsused in the preparation of the corresponding real material. In addition to the RxMC simulations, wetest conventional macroscopic adsorption models, such as the Langmuir–Freundlich, multisite Langmuir,vacancy solution and ideal adsorption solution models, in connection with the ideal-gas model for the
vapor reservoir to model the reaction equilibrium. Pure fluid adsorption isotherms needed as input tothe macroscopic models for mixture adsorption are generated using the Gibbs Ensemble Monte Carlo orGrand Canonical Monte Carlo simulations. We analyze the effects of the confinement, temperature andbulk pressure on the NO dimerization reaction equilibrium in terms of the reactive adsorption isotherms.The RxMC simulations and thermodynamic modeling show that the sucrose-based carbons substantiallyincrease the conversion of NO to (NO)2 with respect to the vapor reservoir, where the conversion is lesspdpwamm
than a few percent.
. Introduction
The behavior of chemical reactions in confinement spans aide range of scientific and engineering interest, including catalystevelopment and the study of nanoporous materials. For exam-le, in the design of heterogeneous carbon supports (which areften highly nanoporous materials), confinement strongly affects
he adsorbed phase, which in turn influences reaction equilibriand kinetics.A chemical reaction confined to a nanoscale environment canave different outcomes compared to the same reaction in the bulk
∗ Corresponding author at: E. Hála Laboratory of Thermodynamics, Institute ofhemical Process Fundamentals of the ASCR, v.v.i., 165 02 Prague 6, Czech Republic.el.: +420 220 390 301; fax: +420 220 920 661.
E-mail address: [email protected] (M. Lísal).
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378-3812/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2008.07.015
© 2008 Elsevier B.V. All rights reserved.
hase. For example, the nanopore phase generally has a higherensity than the corresponding bulk phase; Le Chatelier’s princi-le predicts that this results in an increase in yield for reactions inhich there is a decrease in the total number of moles. Conversely,drop in yield occurs in reactions for which the total number ofoles increases. Furthermore, some components of the reactiveixture are selectively adsorbed on the solid surfaces, also affecting
he reaction equilibrium. In addition, molecular orientations can betrongly influenced by proximity to a solid surface, which can alsoffect the equilibrium relative to that in the bulk phase. Finally,hase transitions such as capillary condensation have a strong
nfluence on the reaction equilibrium conversion in the nanopores
1].Kaneko et al. [2] were the first to experimentally study confinedeaction equilibria. They used magnetic susceptibility measure-ents and observed a large increase in conversion for the NO
imerization reaction in activated carbons with pores of slit widths
e Equi
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rom 0.8 to 0.9 nm with respect to the vapor phase. In the vaporhase at low pressures, the system is almost completely monomeric3,4]. The magnetic susceptibility results suggest that more than8 mol% of the adsorbed NO molecules at 298 K form the dimers5]. More recent measurements of the NO dimerization reaction iningle-wall carbon nanotubes of a diameter of 1.36 nm by trans-ission infrared spectroscopy confirmed Kaneko’s findings. The
ransmission infrared spectroscopy results indicate that the maxi-um amount of adsorbed NO monomer is less than 5 mol% in the
ange of 103–136 K [6].Laboratory challenges have necessitated the development of
hermodynamic and molecular simulation predictive capabilitieso complement the experiments. Thermodynamic approaches pre-ict reaction equilibria by minimizing the Gibbs free energy andy requiring that the total number of atoms of each element con-tituting the chemically reacting species is conserved [7]. Suchhermodynamic approaches require accurate equations of state forhe reactive mixture.
Molecular simulations can provide unique insight intoolecular-level phenomena which thermodynamic models
annot. A method with recent success in predicting the equilib-ium behavior of reactions under non-ideal environments is theeaction Ensemble Monte Carlo (RxMC) method [8,9]. The RxMCas been applied to reactions confined in nanoporous solids orear solid surfaces, reactions at high temperatures and/or highressures, reactions in solution and at phase interfaces; see theecent review [10]. The RxMC technique uses Monte Carlo (MC)ampling to directly simulate pre-defined forward and reverseeaction events in a simulation, yielding direct estimate of thequilibrium composition and other thermodynamic propertiesf a reacting mixture. The RxMC approach incorporates readilyvailable external knowledge of the ideal-gas reaction properties.nother molecular-level approach involves using an analyticaleactive potential to describe the bond breaking and bond for-ation events that occur during the simulation of a chemically
eactive system. The quality of the results depends heavily upon theccuracy of the potential. The reactive potential can be developedy parameterizing detailed experimental data about the reactionnergetics or by performing a comprehensive set of electronictructure calculations of the relevant chemical events. Once aeactive potential is made available, classical molecular dynamicsimulations of the system can be performed, yielding importantnformation about the reaction dynamics and, with long simulationimes, provide estimates of the equilibrium composition [11,12].
In this paper, we continue the studies of Turner et al. [13] andf Lísal et al. [14] on the NO dimerization reaction, 2NO� (NO)2, inodel carbon slit nanopores by focusing on the effects of confine-ent in disordered nanoporous carbons obtained from sucrose. TheO dimerization reaction is interesting for a number of reasons. The
eaction is an exothermic, thermodynamically driven reaction inhich there is a decrease in the total number of moles. The reaction
s important in atmospheric chemistry as well as in the human bodyhere it regulates blood pressure. Moreover, predicting the effects
f confinement on NO dimerization is critical to pollution abate-ent, since activated carbons are commonly used for the removal
f nitrogen oxides from auto exhaust and industrial effluent gastreams. In summary, the investigation of NO dimerization conver-ion as a function of temperature, pressure and adsorbent structures important for the understanding of the underlying phenomenanvolved in these processes and for determining optimal process
arameters [15,16].Carbons such as sucrose coke are disordered nanoporous mate-ials with heterogeneous pore structures. Due to their excellenturface activity, they have found widespread application in indus-ry for purification and separation of gas and liquid streams and
iTptc
libria 272 (2008) 18–31 19
s catalyst supports. Due to their disordered nature, the detailedicrostructure cannot be deduced from experimental techniques
uch as X-ray diffraction or high-resolution transmission electronicroscopy. Therefore, reconstruction methods such as reverse MC,
n which a 3D atomistic model is constructed that is consistent withset of experimental data, have became widely used [17,18]. The
hree models of the sucrose-based carbons investigated were con-tructed using a Hybrid Reverse MC (HRMC) procedure, in whichhe algorithm attempts to simultaneously minimize the error in thearbon–carbon pair correlation function and the configurationalnergy of the system. The carbon models differ by the temperaturesnd the atmosphere used in preparation of the corresponding realaterial [19,20].Since the previous studies on the confined reactive NO system
13,14] were limited to the ordered nanoporous carbon models theain goal of this work is to study the effects of the disorder on
he NO dimerization reaction equilibrium and compare the resultsn the disordered carbons with those in the ordered carbons. Inddition, we aim to test conventional thermodynamic models forrediction of confined reaction equilibria. This paper is organizeds follows. The simulation methodology is briefly described in Sec-ion 2, along with the simulation details. The molecular modelsor the relevant fluid–fluid and solid–fluid interactions are pre-ented in Section 3. The macroscopic adsorption models employedre described in Section 4. Results are discussed in Section 5, andection 6 gives our conclusions.
. Simulation methodology
In this work, we used the RxMC method to simulate the reac-ion equilibrium of the 2NO� (NO)2 reaction in a model disorderedanoporous carbon in equilibrium with a vapor reservoir. We alsotilized the RxMC method to determine the vapor pressure of theulk 2NO� (NO)2 system. In addition, we determined the adsorp-ion isotherms for pure NO and (NO)2 using either the Gibbsnsemble MC (GEMC) [21] or Grand Canonical MC (GCMC) [22]imulations.
The RxMC simulation of a vapor-confined phase system at fixedystem temperature T and fixed bulk pressure P establishes theollowing phase and reaction equilibria conditions:
vNO = �s
NO (1)
v(NO)2
= �s(NO)2
(2)
(NO)2− 2�NO = 0 (3)
here �i is the chemical potential of component i, and the super-cripts v and s denote the vapor and confined phases, respectively.ote that Eq. (3) can be written in either phase.
In the RxMC simulation, the bulk and confined phases were rep-esented by cubic simulation boxes in which the minimum imageonvention, periodic boundary conditions and a cut-off equal toalf the box size were applied. The RxMC method consisted ofcombination of four types of MC steps: (i) particle displace-ents/reorientations in the bulk and confined phases, (ii) volume
hanges for the bulk phase, (iii) inter-phase NO particle transfers,nd (iv) forward and reverse reactions moves in the bulk and con-ned phases. It should be mentioned that only one species (NO orNO)2) needs to be transferred between the phases since the reac-ion moves were carried out in the both phases. We chose NO since
ts transfer is easier than (NO)2 transfer due to the particle size.he particular MC steps were accepted with appropriate transitionrobabilities that are summarized together with other implemen-ation details in Appendix A. At low P, where the vapor reservoiran be treated as an ideal-gas mixture, the simulation of the bulk
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hase can be avoided. It results in substantial reduction of com-uter time. The volume changes and the inter-phase transfers werehen replaced by the GCMC insertion/deletion of NO particles gov-rned by the ideal-gas chemical potentials of the vapor reservoir;ee Appendix A for details.
The vapor pressure of the bulk 2NO� (NO)2 system was alsoetermined using the RxMC method by performing RxMC simula-ions for the vapor–liquid system at fixed T. The simulation strategyas analogous to the RxMC strategy for the vapor-confined phase
ystem, but the pressure cannot be specified in advance. Rather, thequality of pressure in the vapor and liquid phases was achievedy correlated volume changes as in the GEMC simulation of pureubstances [23]; see Appendix A for details.
Finally, pure fluid adsorption isotherms were predicted by theEMC simulation for the vapor-confined phase system at fixed (T,) [21]. The GEMC simulation performed a combination of threeypes of MC steps: (i) particle displacements/reorientations in theapor and confined phases, (ii) volume changes for the bulk phase,nd (iii) inter-phase particle transfers; see Appendix A for details.gain, at low P to reduce computer time the simulation of the vaporhase was replaced by the GCMC insertion/deletion of particlesoverned by the ideal-gas chemical potential of the vapor reservoir22].
The RxMC simulations of the vapor-confined phase systemere initiated by randomly placing NO molecules into the sim-lation boxes. The initial number of NO molecules was chosenuch that a statistically reasonable number of molecules (∼500)ere present in the vapor phase once the system had equili-rated. The long-range corrections for the configurational energynd pressure were included in the case of the vapor phase, assuminghat the radial distribution function is unity beyond the cut-ff radius [24]. The long-range correction for the configurationalnergy was ignored in the case of the confined phase becausef the anisotropic structure of the solid [25]. The RxMC simula-ions were organized in cycles as follows. Each cycle consistedf four steps: nD particle displacements/reorientations moves, nVolume moves, n� reaction moves and nT inter-phase NO trans-ers. The four types of moves were selected at random with fixedrobabilities, chosen such that the ratio nD:nV:n�:nT in each cycleas N : 1 : N : N, where N was about 10–20% greater than the
quilibrium number of molecules during a simulation run. Thecceptance ratios for displacement/reorientation moves, and forolume changes, were adjusted to be approximately 30%. After anquilibration period of ∼5 × 104 cycles, we generated (1 − 2) × 105
ycles to accumulate averages of the desired quantities. The GEMCnd GCMC simulations were performed in an analogous way ashe RxMC simulations. The precision of the simulated data wasalculated using block averages, with 5000 cycles per block. Inddition to ensemble averages of the quantities of direct inter-st, we also carefully monitored the convergence profiles of thehermodynamic quantities as the system traversed phase space26].
. Molecular models
.1. Fluid–fluid interaction
To be able to compare the results in the disordered carbonsith the previous results in the ordered carbons [13,14] we keep
o employ the model proposed by Kohler et al. [27] to describe
uid–fluid interactions in the mixture of NO and (NO)2. The modelreats NO as a single Lennard–Jones (LJ) sphere and (NO)2 as awo-site LJ molecule, with bond length l equal to the experimentalalue of 0.2237 nm. For the monomer, the model uses the LJ energyarameter ε/kB = 125.0 K and the LJ size parameter � = 0.31715 nm;pSatp
libria 272 (2008) 18–31
B is Boltzmann’s constant. The individual LJ parameters for eachite in the dimer are the same as those for the monomer. Dueo the weak dipole and quadrupole moments of the molecules,he model neglects electrostatic forces. Although Kohler’s mod-ls treat NO and (NO)2 in a simple way and more realistic modelsan be considered [28], Kohler’s models reproduce quite well ther-odynamic properties of the NO/(NO)2 mixture in the gas phase
see below) as well as in the liquid phase (see, e.g., Fig. 8 inef. [9]).
.2. Solid–fluid interaction
The models for disordered carbons used in this work wereeveloped in Refs. [19,20] and in the following, we only pro-ide a brief description of the models. We studied three typesf sucrose-based carbons denoted by cs400, cs1000 and cs1000a.he cs400 sample (density ∼ 1 g/cm3) was prepared by pyrolyz-ng pure sucrose (C12H22O11) up to 400 ◦C. The cs400 sampleas then heated at 1000 ◦C under nitrogen flow with a heating
ate of 4 ◦C/min to obtain the cs1000 sample (density ∼ 1.5 g/cm3).he cs1000a sample (density ∼ 0.7 g/cm3) was obtained by heat-ng the cs1000 sample in an atmosphere of CO2 for 20 h29].
The atomistic models for these samples were built using theRMC method in a box size of 2.5 nm. In the first route, the atomisticodels were constructed by considering only carbon atoms. For
ll carbon structures, the number of carbon atoms correspondedo the experimental density, as measured by mercury porosime-ry. In the second route, the atomistic models were constructedy considering the carbon and hydrogen atoms. The number ofarbon and hydrogen atoms present was obtained from the den-ity and composition data. The H/C ratio was 0.53 for cs400, 0.15or cs1000 and 0.091 for cs1000a. The presence of all other het-ro atoms, such as oxygen, was neglected since the compositionata obtained from experiments revealed that the actual mate-ials have very small amount of the hetero atoms. For examplehe oxygen content in cs1000a sample is 0.0087 O/C molar ratio.oth routes fit the experimental carbon–carbon pair correlation
unction, the only experimental available structural information,uite well but the local chemistry is different for the modelsbtained from the first route and the second route as observedrom the neighbor distribution, bond angle distributions, and ringtatistics. The first route leads to simplified representations of theucrose-based carbons (model S) while the second route resultsn more realistic representations of the sucrose-based carbonsmodel R).
The HRMC method simultaneously minimized the error inhe carbon–carbon pair correlation function (which is an aver-ge quantity) and the configurational energy of the system. Theonfigurational energy of the system was calculated using theeactive empirical bond-order potential of Brenner [30] developedor hydrocarbons. The HRMC method further used the simulatednnealing minimization algorithm to avoid some local minima,lthough this technique cannot guarantee location of the globalinimum.The pore size distributions of the models S and models R are
hown in Fig. 1. The models R for cs400 and cs1000 have smallerores as compared to the models S, since the amount of hydrogenresent in those samples is significant. However, cs1000a sampleontains very small amount of hydrogen; thus, the pores and the
ore size distribution are almost the same in the case of the modeland model R. Also, the overall porosity of the models S is largers compared to the models R for the cs400 and cs1000 samples. Inhe case of cs1000a sample, the model S has only slightly higherorosity as compared to the model R.
M. Lísal et al. / Fluid Phase Equi
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ig. 1. Pore size distribution of the three disordered carbon structures: (a) model Snd (b) model R.
We represented carbon and hydrogen as LJ spheres with carbonotential parameters εCC/kB = 28 K and �CC = 0.336 nm, and hydro-en potential parameters εHH/kB = 15.08 K and �HH = 0.242 nm [19].e then modeled the solid–fluid interactions by a LJ potential with
he potential parameters for unlike interactions obtained from theorentz–Berthelot mixing rules [24].
To examine the carbon structures in more detail in relation tohe adsorbates, a common way is to evaluate the free volume of theores V free
i with respect to the adsorbates. We computed values offreeNO and V free
(NO)2for particular carbon structures by means of the
onnolly surface algorithm [31]. The radii of probes used in theonnolly algorithm corresponded to spheres of volumes equal tohe volumes of NO and (NO)2, as evaluated using the LJ � and larameters. The resulting values of V free
NO and V free(NO)2
are listed in
able 1. Differences between V free and V free are quite small due to
NO (NO)2he limited difference in the occupancy volume of NO and (NO)2.xamining the free volume differences between the models S andit can be noticed how the presence of H atoms and consequente-arrangement in pore shapes affect the free volume availability.
able 1ree volumes of the pores, V free
NO and V free(NO)2
, for particular disordered carbon struc-
ures as evaluated by means of the Connolly surface algorithm [31]
cs400 cs1000 cs1000a
Model S Model R Model S Model R Model S Model R
freeNO (nm3) 5.2 2.9 3.6 2.3 9.4 9.2free(NO)2
(nm3) 4.8 2.5 3.3 2.0 9.2 9.0
he volume of the simulation box is 15.625 nm3.
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libria 272 (2008) 18–31 21
. Thermodynamic modeling
In this section, we provide a brief description of macroscopicdsorption models that were employed for the correlation of theimulated adsorption isotherms of pure NO and (NO)2, and for therediction of adsorption isotherms of the 2NO� (NO)2 system.
.1. Langmuir–Freundlich adsorption model
The Langmuir–Freundlich (LF) model [32] is a modification ofhe most widely used and cited Langmuir model [33]; for a puredsorbate it is written as:
≡ n0
n0,∞ = bPa
1 + bPa(4)
here � is the fractional coverage, n0 is the loading (the numberf moles of adsorbate per mass of adsorbent), n0,∞ is the maximumoading, and a and b are LF coefficients. Values of n0,∞, a and b werebtained by fitting the pure fluid adsorption isotherms. Note thator a = 1, the LF model reduces to the Langmuir model.
Extension of LF model to mixture adsorption was proposed byips [34], and is given by:
i ≡ ni
n0,∞i
= biPaii
1 +∑s
i=1biPaii
(i = 1, 2, . . . , s) (5)
here �i is the fractional coverage of species i, ni is the loading ofpecies i, n0,∞
iis the maximum loading for pure species i, Pi = yiP is
he partial pressure of species i, yi is its mole fraction in the vaporeservoir, ai and bi are LF coefficients for pure species i, and s is theotal number of species.
.2. Multisite Langmuir adsorption model
The multisite Langmuir (MSL) model [35] is an extension of theangmuir model; for a pure adsorbate it is written as:
= �
b(1 − �)a (6)
here a and b are MSL coefficients. Similarly as in the case of the LFodel, values of n0,∞, a and b were determined by fitting the pure
uid adsorption isotherms; for a = 1, the MSL model reduces to theangmuir model. It should be mentioned that Eq. (6) is only validf the adsorbate–adsorbate interactions are neglected.
The MSL adsorption model for mixture adsorption is then giveny:
i = �i
bi
(1 −
∑si=1�i
)ai(i = 1, 2, ..., s) (7)
here ai and bi are the MSL coefficients for pure species i. A keyequirement for this model is that the quantity ain
0,∞i
must be equalor all adsorbates to satisfy the space balance (surface area or poreolume) within the adsorbent. The condition ain
0,∞i
= const. alsoakes the MSL model, in contrast to the LF model, thermodynam-
cally consistent [36].
.3. Vacancy solution adsorption model
We further employed the vacancy solution model combined
ith the Flory–Huggins activity coefficient equations (FH-VSM)37]. The FH-VSM for a pure adsorbate can be written as:
= n0,∞
b
�
1 − �exp
(˛2
v�
1 + ˛v�
)(8)
2 e Equilibria 272 (2008) 18–31
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Table 2Vapor pressures, P*, of pure NO and (NO)2, and of the 2NO� (NO)2 system,the ideal-gas standard free-energy change �G0, and the equilibrium constantK ≡ exp[−�G0/(RT)]; R is the universal gas constant and T is the absolute temperature
T (K) P* (bar) �G0 (kJ mol−1) K
NO (NO)2 2NO� (NO)2
110 5.39 0.00756 0.676110 3.965 0.13099 × 10−1
125 13.5 0.0474 3.6030 6.032 0.30164 × 10−2
140 27.7 0.20 12.68 8.123 0.93191 × 10−3
The values of P* for pure NO and (NO)2 were determined from the equations of statefsgm
imw
4
cstgyr
N
y
ir
�
wpressure and �0
i(T, P0) is the molar standard chemical potential of
species i at T and P0. Values of �0i(T, P0) are typically obtained from
thermochemical tables such as the TRC or JANAF tables [41,42]; seeAppendix A for details. As a result, we obtain a non-linear equation
2 M. Lísal et al. / Fluid Phas
here ˛v and b are the FH-VSM coefficients. The parameter ˛v
ccounts for the adsorbate-vacancy interactions caused by non-deality in the adsorbed phase. Similarly as in the previous cases,alues of n0,∞, b and ˛v were obtained by fitting the pure adsorp-ion isotherms and for ˛v = 0, the FH-VSM reduces to the Langmuir
odel.In the case of mixture adsorption, the equation for the distribu-
ion of adsorbate i between the ideal vapor reservoir and adsorbentccording to the FH-VSM model is given by:
i = xi�in
n∞n0,∞
i
bi
exp(˛vi)1 + ˛vi
exp
{(n0,∞
i− n∞
n− 1
)
ln[(
1 − n
n∞
)�v
]}(i = 1, 2, ..., s) (9)
here xi is the mole fraction of species i in the adsorbent, � i and �v
re the Flory–Huggins activity coefficients of species i and vacancy,espectively, n is the loading of the mixture, n∞ is its maximumoading, and ˛vi and bi are FH-VSM coefficients for pure species i.xpressions for � i and �v and other details regarding the FH-VSMan be found elsewhere [37,38].
.4. Ideal adsorbed solution model
For completeness, we also tested the performance of the idealdsorbed solution (IAS) model [39] for prediction of mixturedsorption. The model is based on solution thermodynamics andhe basic thermodynamic equations for the IAS model are the sames those for vapor–liquid equilibria. The equilibrium between thedeal vapor phase and the ideal adsorbed phase can be expressedor each species i as:
i = xiP∗i (�) (i = 1, 2, . . . , s) (10)
here P∗i(�) is the vapor pressure corresponding to the solution
emperature and the solution spreading pressure �. Spreading pres-ure is not a measurable quantity, but at low pressures it can beerived from the isothermal Gibbs adsorption isotherm for a puredsorbate i as:
�iA
RT=
∫ P∗i
0
n0i(P)
PdP (11)
here n0i(P) is the pure fluid adsorption isotherm, A is the surface
rea of adsorbent and R is the universal gas constant. Therefore, cal-ulation of � again requires knowledge of the pure fluid adsorptionsotherms. In this work, we employed the LF model for n0
i(P) in Eq.
11). Prediction of mixture adsorption is then obtained by carryingut the mixing process at constant spreading pressure
1 = �2 = . . . = �s (12)
nd n and ni are obtained as:
1n
=s∑
i=1
xi
n0i[P∗
i(�)]
(13)
i = xin (i = 1, 2, . . . , s) (14)
It should be emphasized that the purpose of this work is notomprehensive testing and screening of various adsorption mod-ls. Rather, we tried to use the most commonly employed modelso explain and justify the simulation results. The LF and MSL mod-
ls are simple surface-layering models while the IAS model andSM are quite general and, in principle, applicable to both orderednd disordered adsorbants. A comprehensive screening and test-ng of various adsorption models such as the potential theoryf multicomponent adsorption by Shapiro and Stenby [40], andFot
or LJ and two-site LJ fluids, respectively [43,44].The values of P* for the 2NO� (NO)2ystem were obtained from RxMC simulations. The simulation uncertainties areiven in the last digits as subscripts. The values of �G0 and K were evaluated byeans of the TRC thermochemical table [41], and Eqs. (A-7)–(A-10).
nvestigation whether models based on a surface monolayeringechanism or on pore volume-filling postulate are more adequateill be published elsewhere.
.5. Ideal-gas model for vapor reservoir
The models for mixture adsorption establish the phase equilibriaonditions (1) and (2). The reaction equilibrium condition (3) can beolved in either phase. In our case, it is convenient to solve Eq. (3) inhe vapor phase since the vapor reservoir can be treated as an ideal-as mixture in the T and P ranges investigated. The compositionNO and y(NO)2
can then be obtained by simultaneously solving theeaction equilibrium (3) and conservation of mass conditions
NO = N0NO − 2, N(NO)2
= N0(NO)2
+ (15)
NO = NNO
NNO + N(NO)2
, y(NO)2= 1 − yNO (16)
In Eqs. (15) and (16), Ni is the number of molecules of species, the superscript 0 denotes the initial state and is the extent ofeaction. In the case of the ideal-gas, �i is given [7] by:
i(T, P) = �0i (T, P0) + RT ln
(yi
P
P0
)(17)
here R is the universal gas constant, P0 is the standard-state
ig. 2. Mole fraction of (NO)2, y(NO)2, as a function of temperature T and pressure P
btained from the ideal-gas model (lines) and the RxMC simulations (points); P* ishe vapor pressure of the 2NO� (NO)2 system.
e Equi
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M. Lísal et al. / Fluid Phas
or
xp
(−�G0
RT
)−
(N0NO + N0
(NO)2− )(N0
(NO)2+ )
(N0NO − 2)
2
P0
P= 0 (18)
here �G0 = �0(NO)2
− 2�0NO is the ideal-gas standard free-energy
hange for the reaction; see Appendix A for details. The value of s then used to evaluate yNO and y(NO)2
by means of Eqs. (15) and16).
. Results and discussion
.1. Vapor reservoir
We performed calculations at three temperatures, T = {110, 125,
40}K, and pressures P up to the vapor pressure P*. Values of Pi*or pure NO and (NO)2 were evaluated using equations of state forJ and two-site LJ fluids, respectively. The equations of state repre-ent available simulation data within the statistical uncertainties43,44]. Values of P* for the 2NO� (NO)2 system were determinedig. 3. Simulated adsorption isotherms of pure NO together with correlations by theangmuir–Freundlich (LF) and multisite Langmuir (MSL) models, and by the vacancyolution model (FH-VSM) at temperatures (a) 110 K, (b) 125 K, and (c) 140 K in thease of the cs1000a structure (model R); n0 is the loading and P is the bulk pressure.
tpdd
Fbvip
libria 272 (2008) 18–31 23
y the RxMC simulations. All values of P*, together with values ofhe equilibrium constant K ≡ exp[−�G0/(RT)], are summarized inable 2. Note that the values of P* for (NO)2 are about three ordersf magnitude smaller than those for NO.
At the T’s and P’s considered, conversion of NO to (NO)2 in theapor reservoir is very small and in addition, the vapor reservoir cane treated as an ideal-gas mixture. This is demonstrated in Fig. 2here we show y(NO)2
as a function of T and P obtained from thedeal-gas model and the RxMC simulations. We see that the simu-ation results match the ideal-gas predictions, except for the higher
points (P > 10 bar) at T = 140 K, where the simulation results are–10% higher than these for the ideal-gas values. However, even inuch cases we can safely neglect non-ideality in the vapor reservoirnd assume the fugacity coefficients to be unity within prediction ofixture adsorption by the macroscopic models. Fig. 2 further shows
hat y(NO)2increases with P at fixed T due to Le Chatelier’s princi-
le. The y(NO)2values also increase with T at fixed P/P* although K
ecreases with increasing T as a result of P* increases more than Kecreases with increasing T.
ig. 4. Simulated adsorption isotherms of pure (NO)2 together with correlationsy the Langmuir–Freundlich (LF) and multisite Langmuir (MSL) models, and by theacancy solution model (FH-VSM) at temperatures (a) 110 K, (b) 125 K, and (c) 140 Kn the case of the cs1000a structure (model R); n0 is the loading and P is the bulkressure.
24 M. Lísal et al. / Fluid Phase Equilibria 272 (2008) 18–31
Table 3The maximum loading n0,∞ and LF coefficients a and b, see Eq. (4), for pure NO and (NO)2; T is the absolute temperature
Structure T (K) NO (NO)2
n0,∞ (mol kg−1) a b (bar−1) n0,∞ (mol kg−1) a b (bar−1)
cs400 110 13.08 0.577 9.50 7.63 0.312 143.98Model S 125 13.22 0.504 3.15 8.34 0.182 11.43
140 12.78 0.558 1.69 8.04 0.257 12.20
cs400 110 7.16 0.514 8.76 4.32 0.295 56.74Model R 125 7.28 0.454 3.23 4.06 0.364 49.33
140 6.99 0.523 1.88 4.17 0.346 14.79
cs1000 110 7.35 0.373 9.46 4.22 0.287 390.34Model S 125 7.21 0.395 5.57 4.18 0.407 421.01
140 7.10 0.418 3.23 4.52 0.211 18.98
cs1000 110 4.95 0.295 7.77 2.55 0.789 112003Model R 125 5.29 0.222 3.14 2.56 0.372 201.47
140 5.01 0.286 2.62 2.79 0.192 13.22
cs1000a 110 33.40 1.219 4.74 20.96 1.112 54988Model S 125 32.33 1.187 1.04 20.14 1.530 94191
140 31.50 1.122 0.35 20.17 1.123 353.14
cM
5
ptpatsuwwb
tt
i
fetedoet
V
Tu
TT
S
cM
cM
cM
cM
cM
cM
s1000a 110 34.24 1.053odel R 125 33.44 1.015
140 32.84 0.969
.2. Pure adsorption isotherms
Figs. 3 and 4 show the simulated adsorption isotherms ofure NO and (NO)2 for T = {110, 125, 140}K, and P up to Pi*, inhe case of the cs1000a structure (model R) as a typical exam-le, together with the correlations by the LF and MSL models,nd by the FH-VSM. The pure fluid adsorption isotherms forhe other structures exhibited similar behavior and they are nothown here. The fitted coefficients for the adsorption modelssed are listed in Tables 3–5. The coefficients of the FH-VSMere omitted for the cs400 and cs1000 structures since theyere not used in prediction of reactive adsorption isotherms (see
elow).Results of the simulations and correlations for pure adsorp-ion isotherms revealed that: (i) all adsorption isotherms are ofype I [45], (ii) the values of loading n0
idecrease with increas-
ng T at fixed P/P∗i, (iii) the ratio of n0,∞
(NO)2to n0,∞
NO is about 0.6
(ott
able 4he maximum loading n0,∞ and MSL coefficients a and b, see Eq. (6), for pure NO and (NO
tructure T NO
n0,∞ (mol kg−1) a b
s400 110 14.08 3.004odel S 125 14.06 3.059
140 13.61 2.723
s400 110 7.90 3.681odel R 125 7.90 3.695
140 7.49 2.983
s1000 110 7.67 4.079odel S 125 7.41 3.427
140 7.48 3.625
s1000 110 5.48 6.470 2odel R 125 5.78 7.895
140 5.43 5.912
s1000a 110 33.79 0.956odel S 125 31.87 0.761
140 30.90 0.799
s1000a 110 32.27 0.743odel R 125 32.65 0.972
140 32.59 1.121
3.24 20.65 1.118 495580.89 20.46 1.008 1262.10.34 20.29 1.040 197.50
or all structures, and (iv) the n0i
values are lower for the mod-ls R in comparison with these for the models S. In addition,he (NO)2 adsorption isotherms show very high values of loadingven at low P’s as a result of the high affinity between the (NO)2imer and the carbon structures. This is caused by the presencef two interaction centers that are able to enhance the bindingffect between the (NO)2 dimer and the framework of the struc-ures.
The ratios n0,∞(NO)2
/n0,∞NO � 0.6 do not correlate with the ratios
free(NO)2
/V freeNO (steric effect), that range from 0.86 to 0.99; see Table 1.
he lower n0,∞(NO)2
/n0,∞NO values with respect to the V free
(NO)2/V free
NO val-es indicate significant influence of non-spherical shape of the
NO)2 molecule (l/� = 0.705) on the adsorption behavior. On thether hand, the lowering of the adsorption capacity observed inhe models R with respect to the models S correlates well withhe V freeiratios between the models S and models R; see Table 1.
)2; T is the absolute temperature
(NO)2
(bar−1) n0,∞ (mol kg−1) a b (bar−1)
177.45 8.03 5.267 1.04 × 108
32.39 8.12 5.297 2.62 × 106
6.89 8.08 4.589 9.64 × 104
333.70 4.79 6.071 1.00 × 107
60.81 4.70 6.193 7.71 × 105
10.40 4.54 4.921 2.12 × 104
3004.1 4.62 6.772 3.04 × 109
313.40 4.57 5.557 3.04 × 107
77.67 4.62 5.869 7.42 × 106
3782 3.28 10.81 4.82 × 109
7888.1 3.35 13.62 6.38 × 109
453.73 3.15 10.19 8.64 × 107
4.19 21.44 1.49 2.49 × 104
0.96 20.74 1.17 2.27 × 103
0.36 20.63 1.20 2.45 × 102
4.93 20.84 1.15 2.09 × 104
1.05 21.10 1.50 2.24 × 103
0.41 21.69 1.68 2.98 × 102
M. Lísal et al. / Fluid Phase Equilibria 272 (2008) 18–31 25
Table 5The maximum loading n0,∞ and FH-VSM coefficients ˛v and b, see Eq. (8), for pure NO and (NO)2; T is the absolute temperature
Structure T NO (NO)2
n0,∞ (mol kg−1) ˛v b (bar−1) n0,∞ (mol kg−1) ˛v b (bar−1)
cs1000a 110 34.16 0.51 159.50 21.30 2.22 9.94 × 105
Model S 125 32.73 −0.93 × 10−2 37.40 20.60 0.43 × 10−2 3.43 × 104
140 32.19 −0.18 × 10−2 12.56 20.18 −0.43 × 10−2 4.59 × 103
cM
TR
5
iPi
FaRddsato
ecpi
s1000a 110 33.85 −0.66odel R 125 33.40 0.90
140 33.01 1.15
his lowering is due to the smaller pores exhibited by the models[19,20].
.3. Reactive adsorption isotherms
Before presenting and discussing our results for the adsorptionsotherms of the 2NO� (NO)2 system for T = {110, 125, 140}K, andup to P*, for the cs400, cs1000 and cs1000a structures, we show
n Fig. 5 an example of the performance of the macroscopic mod-
ig. 5. Simulated and predicted adsorption isotherms for the 2NO� (NO)2 systemt temperature 125 K in the case of the cs1000a structure, (a) model S and (b) model; ni is the loading of species i and P is the bulk pressure. The filled and open symbolsenote NO and (NO)2, respectively. The red, green, black and blue lines represent pre-iction by the Langmuir–Freundlich and multisite Langmuir models, by the vacancyolution model and by the ideal adsorbed solution model, respectively. The solidnd dashed lines correspond to NO and (NO)2, respectively. (For interpretation ofhe references to color in this figure legend, the reader is referred to the web versionf the article.)
scIsspdo
2srdwtmiaw[
fiadttittiTncstd
dPstduatTin
348.12 20.88 0.56 4.26 × 105
44.44 20.36 1.15 4.69 × 104
17.35 19.89 0.70 6.06 × 103
ls to predict the reactive adsorption isotherms at T = 125 K in thes1000a structure. We see from Fig. 5 that all models are able toredict quantitatively (MSL and IAS models, and FH-VSM) or qual-
tatively (LF model) the adsorption behavior of the 2NO� (NO)2ystem. Similar conclusions except for the FH-VSM and IAS modelan be drawn for the other structures. Although the FH-VSM andAS model work quite well for the cs1000a structure at all T’s con-idered, their predictions are rather poor for the cs400 and cs1000tructures (not shown here). Therefore in the following, we presentredictions of the LF and MSL models for all structures, and pre-ictions of the FH-VSM and IAS model for the cs1000a structurenly.
Figs. 6–8 present the simulation adsorption isotherms for theNO� (NO)2 system, together with predictions by the macro-copic models mentioned above. The most striking feature of theeaction adsorption isotherms is the substantial enhancement ofimerization in all structures in comparison with the vapor phase,here the conversion is less than a few percent. The dimeriza-
ion enhancement is also captured by the macroscopic adsorptionodels. Such findings are in qualitative agreement with exper-
mental results of Kaneko et al. [2,5] for the activated carbonsnd of Byl et al. [6] for the single-wall carbon nanotubes, andith the previous simulations results for the carbon slit nanopores
13,14].As in the case of the pure fluid adsorption isotherms, Figs. 6–8
urther show that the models R exhibit lower adsorption capac-ty in comparison with the models S. Differences in the reactivedsorption behavior between the models S and models R reflectifferent local chemistry (the neighbor distribution, bond angle dis-ributions, and ring statistics) possessed by these models. Effects ofhe local chemistry are so significant that even qualitative trendsn the reactive adsorption behavior are not always the same forhe models S and models R; see, e.g., Fig. 6c where nNO is higherhan n(NO)2
over the entire P range for the models R, but n(NO)2s higher than nNO for the models S over almost all the P range.he differences in the reactive adsorption behavior are less pro-ounced for the cs1000a structure since the cs1000a structureontains small amount of hydrogen. It indicates that explicit con-ideration of hydrogen atoms in the carbon models is importanto properly model the NO dimerization reaction equilibrium in theisordered carbons.
Furthermore, there is competition between monomers andimers for adsorption sites when P increases. With increasing, dimers push monomers away from adsorption sites and theyimultaneously enhance adsorption in the structures. As a result,he nNO values, after reaching a maximum at very low P, slightlyecrease with increasing P; in contrast the n(NO)2
values contin-ously increase with increasing P. In contrast to the pure fluid
dsorption isotherms, T has a significant influence on the reac-ion adsorption behavior due to the strong dependence of K on; K decreases with increasing T as can be seen in Table 2. Anncrease in T results in a decrease of dimer conversion, i.e., theNO values increase with T, while the n(NO)2values decrease with
26 M. Lísal et al. / Fluid Phase Equilibria 272 (2008) 18–31
Fig. 6. Simulated and predicted adsorption isotherms for the 2NO� (NO)2 system at temperatures 110 K (circles), 125 K (triangles), 140 K (diamonds) in the case of the cs400structure, model S (left column) and model R (right column); ni is the loading of species i and P is the bulk pressure. The filled and open symbols denote NO and (NO)2,respectively. The red and green lines represent predictions by the Langmuir–Freundlich and multisite Langmuir models, respectively. The solid and dashed lines correspondto NO and (NO)2, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
M. Lísal et al. / Fluid Phase Equilibria 272 (2008) 18–31 27
Fig. 7. Simulated and predicted adsorption isotherms for the 2NO� (NO)2 system at temperatures 110 K (circles), 125 K (triangles), 140 K (diamonds) in the case of cs1000structure, model S (left column) and model R (right column); ni is the loading of species i and P is the bulk pressure. The filled and open symbols denote NO and (NO)2,respectively. The red and green lines represent predictions by the Langmuir–Freundlich and multisite Langmuir models, respectively. The solid and dashed lines correspondto NO and (NO)2, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
28 M. Lísal et al. / Fluid Phase Equilibria 272 (2008) 18–31
Fig. 8. Simulated and predicted adsorption isotherms for the 2NO� (NO)2 system at temperatures 110 K (circles), 125 K (triangles), 140 K (diamonds) in the case of cs1000astructure, model S (left column) and model R (right column); ni is the loading of species i and P is the bulk pressure. The filled and open symbols denote NO and (NO)2,respectively. The red, green, black and blue lines represent prediction by the Langmuir–Freundlich and multisite Langmuir models, by the vacancy solution model and by theideal adsorbed solution model, respectively. The solid and dashed lines correspond to NO and (NO)2, respectively. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of the article.)
e Equi
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Fod
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6
io
M. Lísal et al. / Fluid Phas
(as expected in reactions where the total number of molesecreases).
We can conclude that the MSL model is able to quantitativelyredict reaction adsorption isotherms for the cs1000a structure andemi-quantitatively those for the cs400 and cs1000 structures. TheF model gives poorer results than the MSL model; nevertheless,he LF model always captures the qualitative trends in ni versus P.
As already mentioned, the FH-VSM works quite well for thes1000a structures, but it gives very poor predictions for the cs400nd cs1000 structures. By close inspection of the ˛v coefficientsf the FH-VSM [see Eq. (8)], which characterize the non-idealitiesf the 2NO� (NO)2 system in particular structures, we found thatehavior of the 2NO� (NO)2 system in the cs400 and cs1000tructures is far more non-ideal than behavior of this systemn the cs1000a structure. This suggests that the Flory–Hugginsctivity model cannot capture highly non-ideal behavior of theNO� (NO)2 system in the cs400 and cs1000 structures. Similaronclusion can be drawn for the IAS model, which neglects non-dealities in the adsorbed phase. It also explains why prediction ofhe reactive adsorption behavior in the cs400 and cs1000 struc-
ures by the MSL and LF models worsens with respect to the samerediction in the cs1000a structures.Previous simulation studies of the 2NO� (NO)2 system wereerformed for ordered carbons represented by slit models [13,14].
ig. 9. (a) Mole fraction of (NO)2 dimers and (b) average fluid density as a functionf temperature T in the carbon slit of widths H = 0.8 and 1.0 nm [14], and in theisordered carbon structures studied in this work at a bulk pressure of 0.16 bar.
mmiRdtdtNat
pvbcddtbdtifKicda
aaacitt
A
RoN1
libria 272 (2008) 18–31 29
t is therefore interesting to compare reaction conversion betweenhe ordered and disordered carbons. Fig. 9 shows dimerization con-ersion and average fluid density in carbon slits of widths H = 0.8nd 1.0 nm [14], and in the disordered carbon structures inves-igated in this work. Fig. 9 suggests that based on Le Chatelier’srinciple the lowering of dimerization conversion in the disorderedarbons (except for the cs1000a structure at T = 110 K) with respecto the ordered carbons is caused by lower values of in the dis-rdered carbons with respect to the ordered carbons. The valuesf (except for the cs1000a structure) do not change significantlyith T and decrease of dimerization conversion with T is a result
f the strong dependence of K on T. In the case of the cs1000atructure, temperature dependence of dimerization conversion cor-elates with temperature dependence of .
. Conclusions
The influence of confinement on chemical reaction equilibriumn disordered nanoporous materials was studied in detail by meansf Reaction Ensemble Monte Carlo simulations and conventionalacroscopic adsorption models for the 2NO� (NO)2 system inodel disordered carbon structures obtained from sucrose. Atom-
stic models of the carbons were constructed using the Hybrideverse Monte Carlo method in conjunction with experimentalata. The effects of temperature, bulk pressure, and carbon struc-ures on the reaction adsorption behavior have been reported,iscussed and compared with the corresponding simulations forhe ordered porous carbons. This work also provides insights intoO dimerization reaction equilibrium in the vapor phase anddsorption behavior for pure NO and (NO)2 in the examined struc-ures.
A large increase in dimerization conversion in the adsorbedhase was found with respect to dimerization conversion in theapor phase. However, the increase was lower than this exhibitedy ordered carbons due to a lower fluid density in the disorderedarbons with respect to the ordered porous carbons. The enhancedimerization is due to the combined effects of the increased fluidensity in the adsorbed phase with respect to the vapor phase andhe preferential adsorption of the (NO)2 dimer in the disordered car-on structures. Analogously to bulk phase behavior, the extent ofimerization decreases with increasing temperature. The impact ofhe bulk pressure on the reaction conversion is quite dramatic ands accompanied by competition between monomers and dimersor adsorption sites. The magnetic susceptibility measurements ofaneko et al. [2,5] in the activated carbons, and the transmission
nfrared spectroscopy experiments of Byl et al. [6] in the single-wallarbon nanotubes have found a larger increase in NO dimerizationue to confinement than the previous simulation studies [13,14]nd this work.
Finally, simple and robust macroscopic adsorption models suchs the multisite Langmuir and Langmuir–Freundlich models wereble to predict reasonably well the highly non-ideal reactiondsorption behavior of the system; the vacancy solution modelombined with the Flory–Huggins activity coefficient model anddeal adsorbed solution model were able to accurately predict onlyhe moderately non-ideal reaction adsorption behavior of the sys-em.
cknowledgments
This research was supported by the Grant Agency of the Czechepublic (Grant No. 203/08/0094), by the Grant Agency of Academyf Sciences of the Czech Republic (Grant No. IAA400720710), by theational Research Programme “Information Society” (Projects No.ET400720507 and No. 1ET400720409), by the Grant Programme
3 e Equi
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ai
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wpT
P
w∑o
ssi
P
2t(tai
P
a(aa
P
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wip
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wb
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f Academy of Sciences of the Czech Republic “Nanotechnologyor Society” (Project No. KAN400720701), by the Natural Sciencesnd Research Council of Canada (Grant No. OGP1041), and by theational Science Foundation of the USA. (Grant No. CTS-0626031);alculations were carried out on the SHARCNET network. Theuthors wish to thank Jan Krejcí for performing some of the routineimulations.
ppendix A
This appendix gives details concerning the transition probabili-ies for particular MC moves in RxMC, GEMC and GCMC simulations,ogether with other implementation details.
The particle displacements/reorientations and volume changesre implemented in the usual way [22,24]; the transition probabil-ty k → l for particle displacements/reorientations is
Dkl = min[1, exp(−ˇ�Ukl)] (A-1)
here ˇ = 1/(kBT), kB is Boltzmann’s constant, T is the absolute tem-erature and �Ukl = Ul − Uk is the change in configurational energy.he transition probability k → l for volume changes is
Vkl = min
{1, exp
[−ˇ�Ukl − ˇP(Vl − Vk) + N ln
Vl
Vk
]}(A-2)
here P is the bulk pressure, V is the system volume, and N =si=1Ni is the total number of particles; Ni is the number of particles
f type i and s is the total number of chemical species.The inter-phase particle transfers involve randomly choosing
pecies i and the donor box a, and subsequently transferring thatpecies to a random position in the recipient box b. The correspond-ng transition probability k → l is given [23] by:
Tkl = min
[1,
NaiVb
(Nbi
+ 1)Vaexp(−ˇ�Ua
kl − ˇ�Ubkl)
](A-3)
The reaction moves for the NO dimerization reaction,NO� (NO)2, first consist of a random selection of reaction direc-ion, forward or reverse. In the forward direction, two NO moleculesreactants) are chosen at random, and an attempt is made to simul-aneously replace the NO molecule by a (NO)2 molecule (product)nd to delete the other NO molecule from the system. The attempts accepted with probability given by [8,9]
Fkl = min
[1,
�
V
NNO(NNO − 1)N(NO)2
+ 1exp(−ˇ�Ukl)
](A-4)
In the reverse direction, a (NO)2 molecule (product) is chosent random, and an attempt is made to simultaneously replace theNO)2 molecule by a NO molecule (reactant) and to randomly insertnother NO molecule (reactant) into the system. The attempt isccepted with probability given by [8,9]
Rkl = min
[1,
V
�
N(NO)2
(NNO + 1)(NNO + 2)exp(−ˇ�Ukl)
](A-5)
n Eqs. (A-4) and (A-5), � is the ideal-gas quantity defined by:
= 1ˇP0
exp
(−�G0
RT
)(A-6)
here �G0 is the ideal-gas standard free-energy change for the
eaction, R is the universal gas constant, and P0 is the standard-stateressure (taken to be 1 bar).For the NO dimerization reaction, �G0 is given by:
G0 = �0(NO)2
− 2�0NO (A-7)
[
[[
libria 272 (2008) 18–31
here �0i(T, P0) is the molar standard chemical potential of species
at T and P0. The dependence of �0i(T, P0) on P0 is usually sup-
ressed, and �0i(T) may be written as:
0i (T) = h0
i (T) − Ts0i (T) (A-8)
here the molar enthalpy, h0i(T), and the molar entropy, s0
i(T), may
e expressed as:
0i (T) = �Hfi(Tr) +
∫ T
Tr
c0pi(T)dT (A-9)
0i (T) = s0
i (Tr) +∫ T
Tr
c0pi
(T)
TdT (A-10)
In Eqs. (A-9) and (A-10), �Hfi(Tr) is the enthalpy of formationf species i at the reference temperature Tr (taken to be 298.15 K),0i(Tr) is the absolute entropy at Tr, and c0
piis the ideal-gas heat
apacity. The most accurate values for these properties are availablen thermochemical tables such as the TRC or JANAF tables [41,42],
hich are constructed using the most accurate available partitionunction data. In this work, values of �G0 at different temperaturesere evaluated by means of the TRC thermochemical table data
41], and Eqs. (A-7)–(A-10).At low pressures, the vapor reservoir is treated as an ideal-
as mixture and the simulation of the bulk phase is replaced byhe GCMC insertion/deletion of particles governed by the ideal-as chemical potentials of the vapor reservoir. The insertion andeletion of particle i are accepted with probabilities given by [22]
+1kl
= min
[1,
VˇPi
Ni + 1exp(−ˇ�Ukl)
](A-11)
nd
−1kl
= min[
1,Ni
VˇPiexp(−ˇ�Ukl)
](A-12)
espectively. In Eqs. (A-11) and (A-12), Pi = yiP is the partial pressuref species i and yi is its mole fraction in the vapor reservoir.
In the RxMC simulation for the vapor pressure of theNO� (NO)2 system, the equality of pressure in the vapor and
iquid phases is achieved by correlated volume changes, and theransition probability k → l for the correlated volume changes [23]s
Vkl = min
{1, exp
[−ˇ�Ua
kl − ˇ�Ubkl + Na ln
(Va
k+ �V
Vak
)
+Nb ln
(Vb
k− �V
Vbk
)]}(A-13)
here �V is a random volume change.
eferences
[1] E. Santiso, K.E. Gubbins, Mol. Simulat. 30 (2004) 699–748.[2] K. Kaneko, N. Fukuzaki, K. Kakei, T. Suzuki, S. Ozeki, Langmuir 5 (1989) 960–965.[3] H.L. Johnston, H.R. Weiner, J. Am. Chem. Soc. 56 (1934) 625–630.[4] E.A. Guggenheim, Mol. Phys. 10 (1966) 401–404.[5] Y. Nishi, T. Suzuki, K. Kaneko, J. Phys. Chem. B 101 (1997) 1938–1939.[6] O. Byl, P. Kondratyuk, J.T. Yates Jr., J. Phys. Chem. B 107 (2003) 4277–4279.[7] W.R. Smith, R.W. Missen, Chemical Reaction Equilibrium Analysis: Theory and
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[8] W.R. Smith, B. Tríska, J. Chem. Phys. 100 (1994) 3019–3027.
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9396–9409.
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