Accepted Manuscript

SAFT- Force Field for the Simulation of Molecular Fluids 6. Binary and ternarymixtures comprising water, carbon dioxide, and n-alkanes

Olga Lobanova, Andrés Mejía, George Jackson, Erich A. Müller

PII: S0021-9614(15)00382-1DOI: http://dx.doi.org/10.1016/j.jct.2015.10.011Reference: YJCHT 4433

To appear in: J. Chem. Thermodynamics

Received Date: 24 April 2015Revised Date: 12 October 2015Accepted Date: 13 October 2015

Please cite this article as: O. Lobanova, A. Mejía, G. Jackson, E.A. Müller, SAFT- Force Field for the Simulationof Molecular Fluids 6. Binary and ternary mixtures comprising water, carbon dioxide, and n-alkanes, J. Chem.Thermodynamics (2015), doi: http://dx.doi.org/10.1016/j.jct.2015.10.011

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SAFT-γ Force Field for the Simulation of Molecular Fluids 6. Binary and ternarymixtures comprising water, carbon dioxide, and n-alkanes

Olga Lobanova1 , Andrés Mejı́a2 , George Jackson1 , Erich A. Müller1

1Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, South Kensington Campus, LondonSW7 2AZ, United Kingdom

2Departamento de Ingenierı́a Quı́mica, Universidad de Concepción, POB 160-C, Correo 3, Concepciı́on, Chile

Abstract

The SAFT-γ coarse graining methodology [E. A. Müller and G. Jackson, Ann. Rev. Chem. Biomol. Eng. 5, 405(2014)] is used to develop force fields for the fluid-phase behaviour of binary and ternary mixtures comprising water,carbon dioxide, and n-alkanes. The effective intermolecular interactions between the coarse grained (CG) segmentsare directly related to macroscopic thermodynamic properties by means of the SAFT-γ equation of state for molecularsegments represented with the Mie (generalized Lennard-Jones) intermolecular potential [V. Papaioannou, T. Lafitte,C. Avendaño, C. S. Adjiman, G. Jackson, E. A. Müller, and A. Galindo, J. Chem. Phys. 140, 054107 (2014)].The unlike attractive interactions between the components of the mixtures are represented with a single adjustableparameter, which is shown to be transferable over a wide range of conditions. The SAFT-γ Mie CG force fieldsare used in molecular-dynamics simulations to predict the challenging vapour-liquid and liquid-liquid fluid-phaseequilibria characterising these mixtures, and to study properties that are not accessible directly from the equation ofstate, such as the interfacial properties. The description of the fluid-phase equilibria and interfacial properties predictedwith the SAFT-γ Mie force fields is in excellent with the corresponding experimental data, and of comparable if notsuperior quality to that reported for the more sophisticated atomistic or united-atom models.

Keywords:

1. Introduction

Mixtures comprising water (H2O), carbon dioxide(CO2) and n-alkanes (CH3(CH2)nCH3) are of key im-portance due to their central role in the petrochemi-cal industry. The systems are relevant in range of ap-plications including enhanced oil recovery, CO2 stor-age, natural gas transportation, crude-oil processing,and water purification. An accurate description of thephase behaviour of these mixtures is crucial for the op-timal design of the corresponding chemical processes.From the modelling perspective, mixtures of water, n-alkanes, and carbon dioxide are challenging due to thedisparate nature of interactions between the componentswhich results in a highly non-ideal behaviour. Whilen-alkanes are non-polar, CO2 possesses a significantquadrupole moment, while H2O has a large dipole mo-

Email address: e.muller@imperial.ac.uk (Erich A.Müller1)

ment and also exhibits hydrogen bondings. The re-sulting mixtures are therefore highly asymmetric, bothin terms of the molecular size and the energetic inter-actions. The disparity of the interactions is responsi-ble for different types of global fluid-phase behaviourexhibited by mixtures of these species; binary mix-tures of CO2+H2O [1, 2] and n-alkanes+H2O [3, 4]are both characterized by type III behaviour (accord-ing to the classification of Scott and van Konynenburg[5, 6]), corresponding to extensive regions of liquid-liquid immiscibility and a discontinuous vapour-liquidcritical line. In type III mixtures, the higher-temperaturebranch of the vapour-liquid critical line that starts atthe critical point of the less volatile component mergeswith liquid-liquid critical line at higher pressures. Thelower-temperature branch of the vapour-liquid criticalline, which is usually much shorter, starts at the crit-ical point of the more volatile component and mergeswith the vapour-liquid-liquid three-phase line at a up-per critical end point (UCEP). Mixtures of carbon diox-

Preprint submitted to J. Chem. Thermodynamics October 16, 2015

ide with long n-alkanes also display type III behaviour,whereas type II behaviour is observed with short n-alkanes [7, 8]. Type II behaviour is characterized bya continuous vapour-liquid critical line connecting thecritical points of both pure components; the liquid-liquid critical line does not extend beyond the criticaltemperatures of either pure components, and mergeswith the three-phase line at an UCEP at lower presures.

The experimental phase diagrams of the aforemen-tioned mixtures have been discussed by numerous au-thors. In particular, Schneider and co-workers [7, 8]have undertaken comprehensive studies of binary mix-tures of n-alkane+CO2, and n-alkane+H2O systems hasbeen the focus of the work of Brunner [3]. A large bodyof experimental data is now available on the fluid-phasebehaviour of the CO2+H2O binary mixture, which hasbeen summarized in reviews by Diamond and Akinfiev[9] and by Spycher et al. [10]. The most exhaustivestudies at high temperatures and pressures were con-ducted by Todheide and Franck [1] and Takenouchi andKennedy [2], however these data sets exhibit consid-erable inconsistency. Subsequent work has not com-pletely resolved the discrepancies, although the data byTodheide and Franck [1] has been reported to exhibiterrors in the composition of up to 1 mol % [10]. Theabsence of consistent experimental data introduces a de-gree of uncertainty in the characterization of these sys-tems.

In terms of modelling, the use of cubic equations ofstate (EOSs) remains common practice in petrochemicalengineering applications [11]. However, because cubicEOSs are fundamentally appropriate only for mixturesof near-spherical non-associating components, an accu-rate description of the fluid-phase equilibria is only pos-sible if a number of empirical adjustable parameters areintroduced, thereby curtailing the predictive capabilityof the approach. As an example, Cismondi et al. [12]have accurately described the fluid-phase behaviour ofn-alkanes and carbon dioxide with a cubic EOS over awide range of thermodynamic conditions by employingforty adjustable parameters.

In studies of associating mixtures such as aqueousmixtures of n-alkanes, it is highly advantageous to usean equation of state that can directly account for associ-ation. The statistical associating fluid theory (SAFT)[13, 14], which is based on the Wertheim formalism[15, 16, 17, 18, 19, 20], allows one to take both thedirectional nature of the hydrogen-bonding interactionsand the non-sphericity (chain characteristics) of themolecules into account explicitly. A variety of vari-ants of this molecular-based EOS have been developedover the last twenty five years including: SAFT for vari-

able range interactions (SAFT-VR) [21, 22, 23, 24];SAFT for Lennard-Jones (LJ) based segments (soft-SAFT) [25]; perturbed-chain SAFT based on the struc-ture of the hard-sphere chain (PC-SAFT) [26]; and thegroup contribution versions GC-SAFT [27, 28], SAFT-γ [29, 30, 31]), and GC-SAFT-VR [32]. In this contextwe should also mention that the Wertheim theory of as-sociation can also been coupled with a cubic EOS, asin the cubic plus association (CPA) approach [33]; thisextends the applicability of traditional cubic EOS to as-sociating fluids.

The various incarnations of the SAFT EOSs havebeen successfully applied to study mixtures of rele-vance to our current work comprising water, n-alkanes,and carbon dioxide. The main distinctive features ofthe fluid-phase equilibria of these systems can be ac-curately reproduced including: the transition from typeII to type III phase behaviour exhibited by binary mix-tures of n-alkanes+CO2 with increasing carbon num-ber [34, 35, 36]; the barotropic density inversion andsolubility minima of water in carbon dioxide seen forCO2+H2O [37, 38, 39, 40, 41, 42, 43, 44, 45, 46]; andthe solubility minima of the hydrocarbon in the water-rich phase [47, 48] and the extreme immiscibility foundin n-alkane+H2O mixtures [49, 50, 51, 52, 53, 54, 55,56, 57]. There has been some controversy as to whetheran unlike “association” interaction is required betweencarbon dioxide and water to account for the change ofthe slope in solubility near the critical temperature ofCO2 [37, 58] or whether this behaviour can be repro-duced without the need for additional association inter-actions [40, 41, 42]. Explicit dipole and quadrupole mo-ments have also been incorporated into a SAFT descrip-tion of the intermolecular models for water and carbondioxide [38, 39, 59, 60, 61, 62, 63, 64], with the advan-tage of providing a more physically detailed represen-tation of the unlike interactions for the description ofmixtures.

Although algebraic equations of state are still thepreferred tool in the petrochemical community due toits computational efficiency, molecular-dynamics (MD)or Monte Carlo (MC) computer simulation techniquesprovide an insight at the microscopic level and allowone to study properties that cannot by accessed directlyvia an equation of state, such as interfacial and trans-port phenomena and the behaviour of the confined flu-ids [65, 66]. Molecular simulation essentially providesan exact numerical description of a system once theforce field has been specified [67], and as a consequencethe onus is on the development of accurate intermolec-ular potentials. The use of direct molecular simulationis becoming ever more prevalent as a consequence of

2

the steady increase of computational power [68] and theadvances in modelling techniques [69].

Mixtures containing n-alkanes, carbon dioxide, andwater have received considerable attention from amolecular simulation perspective. The quality of the de-scription of the thermophysical properties obtained bymolecular simulation depends critically on the under-laying intermolecular potential that is employed, partic-ularly the unlike energetic interactions between the var-ious components of the mixture. A variety of all-atom(AA) and united-atom (UA) classical (non-polarizable)force fields have been employed to study these mixtureswith varying degrees of success; owing to the large bodyof research we mention only some representative exam-ples of the more recent studies with a focus on fluid-phase equilibria to set our current work in the appropri-ate context.

Cui and Cummings [70] have used the Gibbs ensem-ble Monte Carlo (GEMC) simulation technique to deter-mine the vapour-liquid equilibria (VLE) of binary mix-tures of n-hexane+CO2 using the elementary physicalmodel (EPM2) of Harris and Yung [71] to representcarbon dioxide and the model developed by Siepmann,Karaborni, and Smit [72] (often referred to as SKS)for n-alkanes; the application of the common Lorentz-Berthelot (LB) geometric-mean combining rule for theunlike attractive dispersion interactions between the hy-drocarbon groups and carbon dioxide was shown to pro-vide a reasonable description of the mutual solubilitiesof the components in the coexisting phases for the ther-modynamic states studies, with the largest deviations(∼ 10%) seen for the liquid phase. In a grand canonicalMC study of the global phase fluid-phase behaviour ofn-hexadecane+CO2 using a simplified LJ united atommodel, Virnau et al. [73] have shown that the unlike in-teraction has to be reduced to ∼ 90% of the LB value inorder to represent the type III critical behaviour exhib-ited by the system.

The fluid-phase equilibria of a variety of mixturescomprising polar and non-polar species including car-bon dioxide, n-alkanes, and water represented withUA exponential-6 (EXP-6) models have been simulatedby Potoff et al. [74] using grand canonical histogram-reweighting MC; the LB combining rules employed forthe unlike molecular interactions were found to be in-appropriate in the case of mixtures comprising com-ponents with large differences in polarity. The focusof the subsequent GEMC study of Vorholtz et al. [75]was specifically on binary mixtures of carbon dioxide(represented with the EPM2 model) and water (repre-sented with the three-site simple point charge SPC [76]and SPC/E [77] models, or with the four-site transfer-

able intermolecular potential TIP4P [78] model); theuse of LB combining rules allowed for a reasonabledescription of the experimental fluid-phase equilibriaonly over a limited range of temperatures for super-critical conditions of carbon dioxide. One would notexpect the LB combining rule to provide a good de-scription of the thermodynamic properties of this non-ideal system [79], and a significant improvement canbe achieved by adjusting the unlike CO2−H2O interac-tion (corresponding to a modification of the geometric-mean rule) [80]. Potoff and Siepmann [81] have pro-posed a model for carbon dioxide within the transfer-able potentials for phase equilibria (TraPPE) frameworkto determine the VLE of mixtures containing nitrogen,n-alkanes, and carbon dioxide, using both grand canon-ical MC and GEMC simulation. Two-centre LJ modelswith point quadrupoles (2CLJQ) have also been devel-oped by Vrabec and co-workers [82, 83] to describe abroad variety of mixtures (including mixtures of carbondioxide with nitrogen or light hydrocarbons) to a highlevel of accuracy by again making use of empiricallyadjusted unlike interactions.

The more challenging determination of the solubilityof water in n-alkanes of increasing chain length (rangingfrom n-hexane to n-hexadecane and to longer polyethy-lene polymers) has been investigated by Johansson etal. [84] by GEMC simulation with the TraPPE modelfor n-alkanes [85] and the SPC model for water, us-ing an empirical adjustment of unlike interaction to pro-vide appropriate agreement with the experimental data;the inadequacy of the LB combining rule for these sys-tems has also been noted in the recent work of Ballal etal. [86]. Motivated by earlier studies [87, 88, 89, 90],Ferguson et al. [91] determined the complementary sol-ubility and molecular conformation of n-alkanes (rang-ing from ethane, C2, to n-docosane, C22) in water us-ing the TraPPE model for n-alkanes and the SPC/Emodel for water with the usual LB combining rule forthe unlike interactions; the aqueous solubilities of thehydrocarbons determined using the replica-exchangemolecular-dynamics (REMD) technique together withan incremental Widom insertion scheme were foundto be in excellent agreement with experiment up to n-decane, but did not support the break in the trend atn-undecane reported in the literature which the authorsattributed to possible experimental artifacts owing tothe very low solubilities of the longer hydrocarbons.It therefore appears that the LB recipe for the unlikewater-alkane attractive interactions is sufficient to de-scribe the solubility of n-alkanes in the water-rich phasebut not of water in the hydrocarbon-rich phase.

A comprehensive study of the CO2+H2O binary mix-

3

ture has been reported recently in a series of papersby Panagiotopoulos and co-workers for the both thefluid-phase equilibria [92, 93] and transport proper-ties [94, 95]; the performance of the various models forwater (SPC, SPC/E, TIP4P, TIP4P2005, TIP4P/�, andEXP-6) and CO2 (EPM2, TraPPE, and EXP-6) were as-sessed with the overall conclusion that none of the forcefields are capable of adequately reproducing the exper-imental fluid-phase equilibria over a broad temperatureand pressure range. This is a good example of the factthat in spite of the relatively high-level of molecular de-tail, atomistic (non-polarizable) models are limited interms of their transferability to different thermodynamicstates and/or representability of different properties tothe same accuracy. An appropriate choice of the unlikedispersion interaction between water and carbon diox-ide applicable over a range of conditions is of centralimportance in this regard.

Within the last two decades, the use of a less-detailedcoarse-grained (CG) description of intermolecular inter-actions is gaining popularity with the main purpose ofenhancing the time and length scales computationallyaccessible for molecular systems of increasing size orcomplexity. The key to the success of a CG approach isthe availability of reliable force fields. The most impor-tant requirements of CG models are their representabil-ity (of the various thermophysical properties), transfer-ability (to broad thermodynamic conditions), and ro-bustness (in the accuracy of the description of the targetproperties). The most common procedure for the devel-opment of CG force fields is to follow a “bottom-up”approach, where the unwanted degrees of freedom of amore detailed atomistic/molecular model are integratedout. For this purpose several techniques have beenemployed including iterative Boltzmann inversion [96],force matching [97, 98], and inverse Monte Carlo [99].Unfortunately, one generally does not known a prioriwhich degrees of freedom are important for the descrip-tion of a given target physical property, and the CG pa-rameter sets obtained in this manner are typically state,property, and system dependent.

By contrast, the SAFT-γ Mie “top-down” methodol-ogy [100] is followed in our current work, whereby anaccurate molecular-based EOS, namely the SAFT-γMieEOS [31], is employed to develop reliable Mie (general-ized LJ) CG force-field parameters directly from macro-scopic experimental thermodynamic properties. Theuse of a high-fidelity EOS enables the simultaneous ex-ploration of a wide parameter space for various targetproperties in order to estimate the optimal set of param-eters that provide the optimal description of the experi-mental data.

The first use of the SAFT EOS within a top-downapproach involved the parameterization of a modelof water using a Wertheim TPT1 description comple-mented with a contribution to account for the dipo-lar interactions [59]. A generic top-down approachof this type has also been employed by Vrabec andco-workers [101, 102, 103]. In this case a semi-empirical EOS was developed by correlating a largeset of volumetric and phase-equilibrium simulation datafor two-center LJ (2CLJ) models which incorporate adipole (2CLJD) or a quadrupole (2CLJQ) moment; theEOS was then used to develop reliable force fieldsfor 78 pure components and 267 binary mixtures, in-cluding carbon dioxide and n-alkanes [83]. In a se-ries of papers Elliot and co-workers have combinedMD with high-temperature thermodynamic perturba-tion theory [104, 105, 106] within SAFT/SPEEDMD(for models based on the discontinuous square-well po-tential) [107, 108, 109, 110, 111] and, more recently,the SAFT-γ platform (for models based on the Mie po-tential) [112, 113] to successfully parameterize CG in-termolecular potentials for a broad range of fluids in-cluding paraffins, olefins, aromatics, ethers, alcohols,and carboxylic acids. The PC-SAFT EOS has beenused by van Westen et al. [114] to obtain force-field pa-rameters for n-alkanes represented as chains of LJ seg-ments; however, because the direct link between the in-termolecular potential and the PC-SAFT description ispartly lost, the final parameter set had to be obtainedin an iterative manner by performing a number of ad-ditional simulations. Gross and co-workers [115, 116]have now refined the methodology for the developmentof transferable force fields for n-alkanes, olefins, andethers based on the Mie potential by employing the PC-SAFT EOS to guide grand canonical MC simulationsfor the determination of the size and energy parameters.

The main advantage of using a framework based onthe SAFT-γ Mie EOS [31] is that the link between theunderlying intermolecular potential and the thermody-namic properties (through the Helmholtz free energy)is explicitly retained [24], thereby allowing one to es-timate the CG force-field parameters reliably from themacroscopic properties of the system by direct parame-ter estimation with the analytical EOS. The SAFT-γMietop-down approach [100] has now been applied suc-cessfully to develop CG force fields for broad classesof molecular fluids including carbon dioxide [117],greenhouse gases [118], hydrocarbons [118, 119], aro-matics [120], water [121], and aqueous solutions ofnonionic [122], light-switching [123] and superspread-ing surfactants [124]. A corresponding states corre-lation can also be employed to parameterize models

4

for molecules represented as chains of Mie segmentsin cases where only limited experimental data is avail-able [125], though the procedure is not generally appro-priate to determine the intermolecular parameters be-tween the different species in mixtures. Some of theSAFT-γ CG Mie force-fields previously developed forpure components have already been used successfully tosimulate selected thermodynamic, interfacial and trans-port properties of numerous mixtures [126, 127, 128,129, 130].

In our current paper we apply the SAFT-γ Miemethodology to obtain accurate estimates of the un-like dispersion interactions for binary mixtures contain-ing n-alkanes, carbon dioxide, and water, thereby al-lowing for the global fluid-phase equilibria to be de-termined by direct MD simulation. The quality of thedescription of the mixtures is very sensitive to the spe-cific interactions employed between the CG beads of themolecules, particularly in the case of the aqueous sys-tems. Comparisons are made where possible with thecorresponding results obtained using the more molecu-lar detailed atomistic and united-atom models includingexplicit electrostatic interactions.

2. Methods and Models

2.1. SAFT-γ Mie coarse graining methodologyThe SAFT-γ Mie EOS [31] provides a direct and

robust link between the macroscopic and microscopicproperties of the fluids. Our current paper is a part of aseries of contributions on the development of the SAFT-γ CG force fields. These are based on the Mie inter-molecular potential, which can be represented as

φMie(r) = C�[(σ

r

)λr−

(σ

r

)λa], (1)

where

C(λa, λr) =(

λrλr − λa

) (λrλa

) λaλr−λa

. (2)

Here, λr and λa are the repulsive and attractive ex-ponents, respectively, controlling the softness/hardnessand the range of the attraction of the potential, σ isthe length scale, related to the average diameter of thespherical segment, and � is the energy parameter.

The force field characterizing the interactions be-tween the molecular segments in the theory and themolecular simulations are based on the same Mie po-tential. The intermolecular parameters are determinedby using the “top-down” approach, estimated by match-ing the experimental macroscopic bulk properties of the

fluid. The focus of of our work is the description of bi-nary and ternary mixtures of water, carbon dioxide, andn-alkanes. For the three pure components, the saturated-liquid density and the vapour pressure are chosen as thetarget properties in the parameter estimation procedure[117, 121, 122]. For the sake of convenience, the pa-rameter set is summarized in Table 1.

The CO2 model is represented by a single-site CGsphere based on the Mie (23-6.66) potential, withoutany additional electrostatic interactions [117]. Despitethe simplicity of the force field, with it one can accu-rately reproduce the entire phase envelope using a singleparameter set. The prediction of properties which arenot used in the parameterization procedure such as en-thalpy of vaporization, interfacial tension, supercriticaldensity, as well as the second-derivative thermodynamicproperties (thermal expansion coefficient, isobaric heatcapacity, isothermal compressibility, speed of sound,and Joule-Thomson coefficient) were found to be in avery good agreement with experimental data [117]. Ina comprehensive study, which compared the thermody-namic properties of seven different force fields for car-bon dioxide, Aimoli et al. [131] found that the SAFT-γMie force field provides a comparable accuracy to thehigher-resolution force fields, including rigid and flexi-ble atomistic three-site models.

The SAFT-γ CG Mie model of water is also basedon the mapping of a single molecule to a single CGbead [121]. A spherical isotropic potential alone is notappropriate to provide an accurate physical representa-tion of the interactions over a broad range of thermody-namic conditions as a consequence of averaging out thedirectional and electrostatic interactions of a highly po-lar and associating fluid such as water. Since the effec-tive CG interactions between water molecules are foundto vary significantly with temperature, temperature-dependent size and energy parameters are introduced.The saturation densities and vapour pressures can be ac-curately reproduced over a wide range of conditions us-ing this set of temperature-dependent parameters. Analternative parameterization is necessary when the in-terfacial properties of water are of interest owing to thelow level of resolution of the single-site model of wa-ter and the related issues of transferability with a CGprocedure of this type [121].

The homologous series of n-alkanes are representedhere as homonuclear chains of tangent Mie sphericalCG segments. The development of CG models for longn-alkanes such as n-decane (n-C10H22) and n-eicosane(n-C20H42) has already been successfully demonstratedusing the SAFT-γ Mie formalism [118]. The n-decanemolecule was represented by three and n-eicosane by

5

Table 1: Intermolecular and intramolecular potential parameters of the SAFT-γ CG Mie force field for the purecomponents: m corresponds to the number of CG segments, σ is the segment size, � is the energy parameter, and λrand λa are the repulsive and the attractive exponents, respectively, δρ % and δP % represent the percentage absoluteaverage deviations (AAD %) between the description with the SAFT-γ Mie EOS and experimental saturated-liquiddensities and vapour pressures for the temperature range from the triple point up to 90% of the critical point. Thesingle-site CG model of carbon dioxide was developed in reference [117]. The size and energy parameters σ and� of the single-site CG Mie water model are represented as polynomial function of temperature [121]: σ(T )/Å =1.262 × 10−9(T/K)3 − 8.720 × 10−8(T/K)2 − 4.554 × 10−4(T/K) + 3.119 and (�(T )/kB)/K = 1.105 × 10−5(T/K)2 −0.3077(T/K) + 586.8.

intermolecular parameterslike interactions

Component m σ/Å (�/kB)/K λr λa δρ % δP %CO2 1 3.7410 353.55 23 6.66 1.71 19.48H2O 1 σ(T ) �(T ) 8 6 0.03 0.01

n-alkanesn-C5H12 2 4.2449 310.78 15 6 0.51 0.48n-C6H14 2 4.5089 342.00 15 6 0.43 6.26n-C7H16 2 4.5736 380.87 15 6 11.46 17.24n-C8H18 3 4.2412 321.65 15 6 1.16 2.12n-C9H20 3 4.4212 344.83 15 6 0.30 3.85n-C10H22 3 4.4908 363.99 15 6 5.48 9.25n-C11H24 4 4.2212 328.18 15 6 1.11 3.04n-C12H26 4 4.3635 344.42 15 6 0.22 1.33n-C13H28 4 4.4680 354.25 15 6 0.42 2.06n-C14H30 5 4.2332 329.81 15 6 5.04 6.99n-C15H32 5 4.3286 344.85 15 6 0.71 2.13

unlike interactionsComponents i and j λr,i j λa,i j ki j

CO2 + H2O 13.00 6.31 -0.07H2O+n-alkane 10.75 6.00 0.36CO2+n-alkane 18.50 6.31 0.08

intramolecular parameterskbond/(J mol−1 Å−2) r0/Å kangle/(J mol−1 deg−2) θ/◦

61.296 × 103 σ 5.38 159.9

6

six fully flexible tangentially bonded Mie segments. Acertain degree of parameter degeneracy in terms of over-all performance is expected as a consequence of theconformal nature of the EOS description [132]. In ourcurrent work, we use an alternative CG mapping for n-alkanes developed in reference [122], where each seg-ment was taken to represent three alkyl carbon back-bone atoms and their corresponding hydrogen atoms.By applying this mapping, n-alkanes chains contain-ing multiples of three carbon units can be representeddirectly: n-C6H14, n-C9H20, n-C12H26, n-C15H32, n-C18H38, etc. A good description of the thermodynamicproperties of these alkanes is found to be provided withCG alkyl beads characterized by the Mie (15-6) poten-tial. For convenience, the exponent pair (15-6) is alsoused to represent the interactions between the CG beadsof the intervening alkanes considered here; the numberof segments m is taken to be the nearest integer of thedivision of the carbon number C by three. The size andenergy σ and � parameters are then estimated from theexperimental saturated-liquid density and vapour pres-sure of the individual alkanes following the usual SAFT-γ Mie procedure. The chosen mapping is by no meansunique, as one can postulate parameter sets that fulfilother requisites, such as being “universal” across the en-tire homologous series [119] or correlated to the criticalproperties [125].

It is important to treat the degree of flexibility ofchain molecules in an appropriate manner. In the case ofn-alkanes it has been shown that a fully flexible modelprovides an accurate description the vapour-liquid equi-libria and interfacial properties of the pure componentsystem [118]. The effect of the chain flexibility has beenexamined recently in the context of the representation ofn-alkanes with the SAFT-γ CG Mie force field [119]; arealistic description of the rigidity of the carbon back-bone is necessary to accurately represent the structuraland transport properties of the fluid. A physically real-istic treatment of the semi-flexible nature of n-alkanesis also very important in the case of aqueous mixtures,where the solubility and structure of the hydrocarbon inwater is found to be very sensitive to the degree of flex-ibility of chain molecules [122].

As with most of the other versions of the SAFT EOS,the SAFT-γ Mie EOS [31] used in our current workis based on the first-order thermodynamic perturbationtheory (TPT1) of Wertheim [133, 20]. The specific ge-ometry of the chain molecules is not predetermined atthis level of the theory, although the approximationsinherent in the TPT1 approach are expected to pro-vide a more accurate description for elongated chains.This is beneficial when one considers a relatively rigid

alkyl backbone, described with realistic harmonic bondstretching and bond-angle bending intramolecular con-tributions:

Uintra =12

∑bond

kbond(r−r0)2+12

∑angle

kangle(θ−θ0)2,(3)

where kbond and kangle are the bond and the bendingharmonic spring constants, respectively, and r0 and θ0are the corresponding distance and angle at the poten-tial minimum. The parameters for the intramolecularbonding interactions are also given in Table 1, and a de-tailed discussion of their determination is given in ref-erence [119].

Simple combining rules are applied to describe theunlike interactions between the various components inthe mixture [24]. The unlike size parameter σi j is ob-tained using the Lorentz arithmetic-mean combiningrule:

σi j =σii + σ j j

2. (4)

The unlike repulsive λr,i j and unlike attractive λa,i j ex-ponents are both determined using the following rela-tionship:

λr/a,i j − 3 =√

(λr/a,ii − 3)(λr/a, j j − 3), (5)

which is consitent with the geometric mean of thevan der Waals attractive constant [24]. The unlike en-ergy parameter �i j is represented as

�i j = (1 − ki j)

√σ3iiσ

3j j

σ3i j

√�ii� j j, (6)

where ki j is an adjustable binary-interaction parameter,with ki j = 0 taken as default to represent the com-monly employed Berthelot geometric-mean combiningrule (of the van der Waals constant). A proper prescrip-tion of the unlike energetic parameter is critical to ade-quately describe the interactions between different com-ponents [79]. Here we employ the SAFT-γ Mie EOS toestimate the ki j parameter from appropriate experimen-tal data of the given mixture in order to account for thedifferences in the attractive interactions between com-ponents. Although the binary ki j parameter of the mix-tures is adjusted to experimental fluid-phase equilibriumdata at only one temperature, the choice is transferableto various thermodynamic conditions and is thereforeapplied to represent the entire phase diagram. A tem-perature dependence is already incorporated in the likeinteractions between water molecules, which allows oneto capture the change in the interactions between the

7

components of the aqueous mixtures at different temper-atures. The final set of molecular parameters (cf. Table1), informed by the SAFT-γ Mie EOS, are used withoutmodification in MD simulations.

2.2. Molecular dynamics simulation detailsTwo / three adjacent slabs of pure components lo-

cated in a rectangular simulation box are taken as theinitial configuration of a binary / ternary mixture, re-spectively, with standard periodic boundary conditions[67]. The fluid-phase equilibria is determined using MDsimulation in the canonical ensemble (NVT ), with con-stant number of particles N, volume V , and temperatureT , using a temperature-quench algorithm [134]. Simu-lations of the liquid-liquid coexistence are initially car-ried out in the isobaric-isothermal NPT ensemble, withconstant number of particles, temperature, and pressureP, for the sole purpose of allowing the system to at-tain densities close to the equilibrium values, followedby simulations in the NVT ensemble at the establishedequilibrium box dimensions to determine the averagecoexistence properties. The temperature of the systemis fixed using a Nosé-Hoover thermostat [135, 136] withthe coupling constant of 1.0 ps, and the pressure is main-tained by means of a Parrinello-Rahman barostat with arelaxation constant of 10.0 ps. The equations of motionare integrated using the leap-frog algorithm [67] witha time step of 5 fs using the GROMACS 4.5.5 pack-age [137]. A cutoff radius of 3.0 nm is used through-out for all of the interactions between the various CGMie beads; a relatively large cutoff is necessary in or-der to retain the close correspondence with the the-ory [121]. As an initial configuration for the simula-tion of liquid-liquid equilibria (LLE), a phase separatedsystem is assembled in the rectangular simulation box,with dimensions constrained to a minimum of 6 nm inx and y direction and four times larger in the z direc-tion. The total number of particles is chosen accord-ing to the box dimensions. For the simulations of theVLE in the NVT ensemble, the z dimension is chosento be approximately three times larger than for the LLEin order to cater for the low density of the vapour. Af-ter equilibrium has been established, the configurationsare sampled for further 20 ns to determine the config-urational average of the various properties. The com-position of the fluid phases is estimated from the corre-sponding number density profiles of each component.

3. Results

For each of the binary mixtures studied, the unlike in-teraction parameter is adjusted to the fluid-phase equi-

libria at one thermodynamic state (isothermal slice) bymatching the value predicted with the SAFT-γ Mie EOSto the corresponding experimental data. It is importantto note that only a single binary adjustable parameteris required to account for the unlike energetic interac-tions between the components of the mixture over awide range of conditions. The parameters determined inthis way are used to investigate the entire fluid-phase be-haviour (including VLE, LLE, critical lines, and three-phase lines) predictably using the SAFT-γ Mie EOS.Three-dimensional phase diagrams are produced to vi-sualize the global phase behaviour. For the mixtureswith n-alkanes, the same binary ki j parameter is used todescribe the unlike interactions between water or car-bon dioxide and the alkyl beads for the components ofdifferent chain length. MD simulation of the fluid-phaseequilibria are then carried out for the mixtures describedwith the CG force fields developed with SAFT-γ MieEOS and the corresponding simulation data are com-pared critically with the experimental and theoretical re-sults.

3.1. carbon dioxide+water binary mixtureThe experimental data of Takenouchi and

Kennedy [2] for the CO2+H2O mixture at a tem-perature of T = 473 K are employed to obtain theunlike energetic interaction parameter between the twocomponents; the correlated pure-component data fromthe NIST database are also used in the analysis [138].A value of ki j = −0.07 estimated with the SAFT-γ MieEOS is found to provide the best possible agreementwith experimental data; this finding is consistent withthe study of the CO2+H2O mixture carried out byForte et al. [44] with the SAFT-VR EOS, and byNiño-Amézquita et al. [45] with the PCP-SAFT EOS.The quality of representation with the SAFT-γ MieEOS is illustrated in Figure 1. The solubilities in bothphases are reasonably well reproduced, including thesolubility minimum of water in CO2-rich phase.

The transferability of the force field can be vali-dated by employing the same parameter set at temper-atures which are not considered in the parameterizationprocedure. The adjustable parameter ki j is treated astemperature-independent and appears to be transferableto different conditions (e.g., at T = 383 K as seen in Fig-ure 1). For this temperature the phase compositions andthe solubility maximum are accurately captured at lowerpressures. At high pressures, the deviations becomemore apparent. Overall, the corresponding data ob-tained by MD simulations performed at the same ther-modynamic conditions agree well with the theoreticalpredictions.

8

0

50

100

0 0.2 0.4 0.6 0.8 1

P / M

Pa

xCO2

Figure 1: Isothermal pressure-composition (Px) slicesof the fluid-phase equilibria for the carbon diox-ide+water binary mixture at temperatures of T =383 K(red/dashed) and 473 K (black/continuous). The filledcircles correspond to experimental fluid-liquid equilib-rium data [2]. The curves represent the calculations withthe SAFT-γ Mie EOS [31] using the parameters givenin Table 1. The empty circles represent the correspond-ing predictions of the phase boundaries obtained withour SAFT-γ Mie CG force field by direct molecular-dynamics simulation.

The entire fluid-phase behaviour for the system withthe unlike interaction parameter ki j = −0.07 can bedetermined easily with the SAFT-γ Mie EOS. TheCO2+H2O mixture exhibits type III phase behaviour ac-cording to the classification of Scott and van Konynen-burg [5, 6], which is characteristic of highly immisciblefluids. The global features of the fluid-phase behaviour,which is displayed in Figure 2, is indeed dominated byextensive regions of fluid immiscibility.

The discontinuous vapour-liquid critical line can alsobe seen in the PT projection of the PT x surface shownin Figure 3. The critical line at the intermediate pres-sures is in good agreement with experimental data butthere are larger deviations at higher pressures. Thelower branch of the vapour-liquid critical line mergeswith the three-phase line in the UCEP; the three-phaseline is at pressures slightly lower than the vapour-pressure curve of pure carbon dioxide. The critical pointof the CO2 is overestimated by the SAFT-γ CG Miemodel [117] due to the mean-field nature of the the-ory, which in turn leads to a slight overestimation ofthe pressures of three-phase coexistence with the SAFT-γ Mie EOS. A clearer representation of the three-phase

200 400 600 0 0.4

0.8 0

50

100

150

P / M

Pa

T / KxCO2

P / M

Pa

Figure 2: Three-dimensional pressure-temperature-composition (PT x) surface of the fluid-phase equilibriafor the carbon dioxide+water binary mixture predictedwith the SAFT-γ Mie EOS [31] using the parametersgiven in Table 1. The continuous red curves on op-posing faces of the prism represent the vapour-pressurecurves of the pure components, the dashed blue curvesare the critical lines and the PT projection of the three-phase line, and the dashed black curves correspond toisothermal pressure-composition (Px) sections.

line and the lower branch of the critical line is illustratedin the inset of Figure 3.

The three-phase line separates the liquid-liquid equi-librium at higher pressures from the vapour-liquid equi-librium at lower pressures; at the three-phase line, twoimmiscible liquid phases coexist with the vapour phase.In Figure 4, the isothermal pressure-composition (Px)slice at a temperature of T = 298 K predicted with theSAFT-γ Mie EOS is compared to the corresponding ex-perimental data [139, 140, 141, 142, 143, 144]. Thevapour-liquid equilibrium is in very good agreementwith experiment; at the liquid-liquid equilibrium con-ditions, the concentration of carbon dioxide is slightlyunderestimated in both phases. The three-phase pres-sure is reproduced within a deviation of 0.45 MPa com-pared to the experimental value. A small vapour-liquidregion with a CO2-rich vapour in coexistence with aCO2-rich liquid phase is observed above the three-phaseline. The theory is able to capture the main featuresof the CO2+H2O mixture at this temperature, provid-ing an explanation to the solubility minimum of waterin the CO2-rich phase at higher temperatures: the sol-ubility minimum is clearly ascribed to the proximity ofthe three-phase line. At low temperatures, the solubilitycurve exhibits a sharp discontinuity at the three-phasecoexistence pressure, and as the temperature increases

9

0

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100

150

200 300 400 500 600

P / M

Pa

T / K

5

10

280 290 300 310

P / M

Pa

T / K

Figure 3: Pressure-temperature (PT ) projection of thefluid-phase equilibria for the carbon dioxide+water bi-nary mixture. The red circles correspond to the experi-mental vapour pressures of the pure components [138],the blue circles correspond to the experimental vapour-liquid critical line and three-phase line [1, 2]. Thecurves represent the description with the SAFT-γ MieEOS [31] using the parameters given in Table 1: thecontinuous red curves are the calculations for the vapourpressures of the pure components, the dashed bluecurves are the predictions for the vapour-liquid criticallines and three-phase line. The region close to the crit-ical point of pure carbon dioxide is shown enlarged inthe inset.

above the UCEP, the discontinuity disappears therebycausing a minimum at pressures close the UCEP. Thisbehaviour is described in greater detail in reference [40]

The MD simulations reproduce the theoretical predic-tions very accurately (cf. Figure 4). The three-phases– the H2O-rich liquid phase (L1), the CO2-rich liquidphase (L2), and the gaseous phase (G) – can be clearlydistinguished in the corresponding density profile andthe snapshot, presented in Figure 5. The adsorptionof CO2 at the interface between the L1 and L2 phasesis distinctly featured in the density profile. The CGmodel of H2O employed in our study represents accu-rately the volumetric properties at the expense of onlyproviding qualitative agreement for the interfacial ten-sions. For accurate values of the interfacial properties, adifferent parameter set should be employed [121]. Thepredictive capabilities of the aformentioned SAFT-γ CGMie models are accurate enough in aiding to detect er-rors and discern between conflicting experimental datasets [145].

0

5

10

15

20

0 0.02 0.04 0.96 0.98 1

P / M

Pa

xCO2

Figure 4: Isothermal pressure-composition (Px) slice ofthe fluid-phase equilibria for the carbon dioxide+waterbinary mixture at a temperature of T = 298 K. The filledcircles correspond to experimental vapour-liquid andliquid-liquid equilibria [139, 140, 141, 142, 143, 144].The dotted line represents the experimental three-phaseline. The continuous curves represent to the predictionsof the SAFT-γ Mie [31] using the parameters given inTable 1, and the dashed line is the three-phase line. Theempty circles represent the corresponding three-phaseline compositions predicted with our SAFT-γ CG Mieforce field by direct molecular-dynamics simulation.

The adsorption phenomena at the liquid-liquid in-terface can be studied via MD simulations at differ-ent temperatures and pressures. The density profiles attemperatures of T = 383 and 423 K are presented inFigure 6 at pressures of P = 8.18, 31.87, 62.30, and102.10 MPa. The density profile of H2O is not verysensitive to pressure and is therefore shown only at onepressure for clarity. CO2 exhibits a positive surface ac-tivity (dρCO2/dz = 0, d

2ρCO2/dz2 < 0), whereas H2O

does not. The surface activity of CO2 increases as thepressure increases until the limiting bulk liquid densityis reached and saturation is established. A lower sur-face activity and the widening of the interface producedby a decreased in the slope of the CO2 density curve isobserved with increasing temperature.

Apart from the interfacial behaviour, the pressure de-pendence of the bulk densities of the CO2+H2O mix-ture are of scientific interest as the mixture exhibits massbarotropy (density inversion). In this phenomenon, thesupercritical CO2 phase becomes denser than the H2Ophase at high pressures, so that two phases change theirpositions in a gravitational field. The experimental den-

10

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0 10 20 30 40 50

/ (k

g m

-3)

z / nm

L1

L2

G

Figure 5: Total density profile (yellow crosses) of thecarbon dioxide+water binary mixture at three-phase co-existence corresponding to a temperature of T = 298 Kand a pressure of P = 6.82 MPa obtained withour SAFT-γ Mie CG force field by direct molecular-dynamics simulation using the parameters given in Ta-ble 1. L1 denotes the water-rich liquid phase, L2 denotesthe carbon dioxide-rich liquid phase, G denotes the gasphase. In the snapshot of a representative configurationduring the simulation, the water molecules are displayedin red (red density profile in upper panel) and the carbondioxide molecules in black (black density profile in up-per panel).

sities of the mixture [146] are compared to the pre-dictions with the SAFT-γ Mie EOS and MD simula-tions at temperatures of T = 298, 304, 383, 423, and473 K in Figure 7. The theoretically calculated densi-ties of the CO2-rich phase are in excellent agreementwith experiment, both in the sub- and the super-criticalregion, whereas the densities in the H2O-rich phase areslightly underestimated. The pressures at which thebarotropic density inversion occurs, at a given tempera-ture, are compared to the experimental values [1] in Ta-ble 2. Taking into account that these are predictions withvery simple models using only one adjustable unlike in-teraction parameter, the agreement between theoreticalpredictions and experimental data is quite encouraging.The MD simulation data accurately reproduce the val-ues predicted with the SAFT-γ Mie EOS (and therebythe experimental data).

Table 2: Barotropic density inversion points for thecarbon dioxide+water binary mixture measured exper-imentally [1] and predicted with the SAFT-γ Mie EOS[31] using the parameters given in Table 1.

T / K PEXP / MPa PSAFT / MPa323 80 70373 125 108523 200 161

The description of the fluid-phase equilibria for theCO2+H2O mixture obtained with our SAFT-γ CGMie models can also be compared to the correspond-ing results of the atomistic simulations determined byLiu et al. [92] for some of the popular force fieldsfor H2O (SPC, TIP4P, TIP4P2005, and exponential-6)and CO2 (EPM2, TraPPE, and exponential-6) using LBcombining rules to describe the unlike pair interactions.The simulated data is used to assess the contradictoryexperimental data reported by Todheide and Franck [1]and by Takenouchi and Kennedy [2].

Two isothermal slices of the PT x fluid-phase equilib-rium surface are examined in Figure 8: one at T = 423K corresponds to the region well below the criticalpoint of water and the other to the near-critical region(T = 548 K). Due to considerable disagreement be-tween the experimental data sets, it is difficult to judgewhich model delivers the best performance. The rep-resentation with our SAFT-γ CG Mie force field is incloser agreement with the experimental data of Take-nouchi and Kennedy [2], clearly exhibiting a solubilityminimum at low pressures, which appears due to theproximity of the three-phase line at low temperatures, asdiscussed earlier. The solubility minimum is not repro-duced by any of the atomistic models. When comparedwith the experimental data of Todheide and Franck [1],the best performance is achieved by the exponential-6models for the CO2-rich phase and TraPPE/TIP4P2005model for the H2O-rich phase at T = 423 K. At hightemperatures, the exponential-6 models perform betterthan the other atomistic models: the lower part of thephase envelope can be accurately reproduced, whereasthe critical pressure is largely underestimated comparedto both sets of experimental data. The SAFT-γ Mie CGmodels exhibit a similar behaviour, with good agree-ment with experiment at low pressures but larger de-viations and an underestimation of the critical point athigh pressures. Despite the high level of molecular res-olution of the atomistic models one cannot adequatelyreproduce the fluid-phase behaviour over a wide tem-perature and pressure range: at low temperatures, the

11

solubility in the both phases cannot be reproduced si-multaneously; at high temperatures, only a qualitativedescription can be achieved with atomistic models. Onthat account, the performance of the SAFT-γ Mie CGmodels can be considered at least as good if not bet-ter than the atomistic models. The main issue with theatomistic force fields is the use of conventional LB com-bining rules for the dispersion interactions between un-like components of the mixture; an improved descrip-tion could certainly be obtained by refining the unlikeattractive interaction, though this typically requires aniterative simulation procedure. Having a direct robustsimultaneous link between theory, experiment, and sim-ulation enables one to assess the adequacy of the force-fields in describing the fluid-phase behaviour over awide range of conditions with greater confidence.

3.2. n-alkane+water binary mixtures

The approach described in the previous section isapplied to study aqueous mixtures of n-alkanes whichare characterized by broad regions of extreme liquid-liquid immiscibility. A binary interaction parameter ofki j = 0.36 for the CG energetic interaction betweenthe H2O and alkyl beads is estimated from the solubil-ity of H2O in a coexisting n-hexane-rich liquid phaseat a temperature of T = 473 K. The quality of thetheoretical representation of the experimental VLE andLLE reported in Refs. [147, 148, 149] is illustrated inFigure 9. The n-hexane+water mixture exhibits a het-eroazeotrope, where the vapour phase coexists with twoimmiscible liquid phases. The azeotropic point is accu-rately captured by the SAFT-γMie EOS both in terms ofcomposition and pressure at the given temperature. MDsimulations of our CG force field are seen to correctlyreproduce the solubility of water in hexane-rich liquidphase. The compositions of both phases in vapour-liquid equilibria are also accurately captured by the sim-ulations. The simulated values for the phase composi-tion at the azeotropic point display remarkable agree-ment with the theoretical prediction and the experimen-tal data. The corresponding density profile of the mix-ture at the temperature of T = 473 K and pressure ofP = 3.33 MPa is illustrated in Figure 10. Three phases– a H2O-rich liquid phase L1, an n-hexane-rich liquidphase L2, and a gas phase G – are clearly distinguish-able from the densities of the bulk regions, which canalso be seen in the snapshot of a representative configu-ration from the MD simulation.

The solubility of water in the n-hexane-rich liquidphase at LLE is accurately captured by the SAFT-γMie CG parameter set developed in our current work,

both by the equation of state and by molecular sim-ulation. The concentration of n-hexane in the H2O-rich liquid phase is however very low and representsa challenge for any modelling technique. Whereas theexperimental mole fractions are of the order of 10−6

(xC6H14 = 1.5 × 10−6) [147], the prediction with theSAFT-γ Mie EOS is xC6H14 = 4.8 × 10−9. Gratifyingly,the MD simulation of an equilibrium drop of n-hexanein bulk water described with our SAFT-γ Mie CG forcefield yields a solubility of xC6H14 = 3.3×10−5; the readeris referred to reference [122] for details of the proce-dure employed. The modelling techniques may requirerefinement and a more sophisticated analysis, howeverit is encouraging that the order of magnitude of the ex-perimental values is reasonably well reproduced.

Using a value of ki j = 0.36 for the unlike CG interac-tions between the H2O and alkyl beads, we extend theprediction to study the rest of the fluid-phase diagram.The family of n-alkanes+H2O mixtures is known to ex-hibit type III behaviour [3]. The extent of the liquid-liquid immiscibility gap can be visualized in the three-dimensional PT x phase diagram in Figure 11. The dis-continuity of the vapour-liquid critical line is clearlyseen in the corresponding PT projection shown in Fig-ure 12. The description of the UCEP with the SAFT-γMie EOS on the low-temperature branch of the vapour-liquid critical line is slightly overestimated compared toexperimental data due to the overprediction of the criti-cal point for n-alkanes, which is apparent from the insetof Figure 12. In heteroazeotropic systems of this type,the three-phase line is located at higher pressures thanthe vapour pressure of either of the pure components.As was already evident from Figure 9, the predictionof the three-phase line is in excellent agreement withthe experimental data. The upper branch of the vapour-liquid critical line should exhibit a minimum typical forthe systems with the gas-gas immiscibility of the secondkind [150]. However, our calculations do not capturethe change of the slope of the critical line thereby sug-gesting a gas-gas immiscibility of the first kind, char-acteristic for systems with even greater disparity of theintermolecular forces.

The interfacial profiles can be studied by MD simula-tion at pressures corresponding to the vapour-liquid andliquid-liquid equilibria. The interfacial region of the n-hexane+H2O mixture is shown in Figure 13 at a temper-ature of T = 473 K and pressures below (P = 2.16 and2.87 MPa) and above (P = 6.25 MPa) the three-phaseline. As in the mixture with CO2, in mixtures with n-alkanes, H2O does not exhibit any surface activity, i.e.,the average density profile has a monotonic hyperbolicshape. By contrast, at pressures below the three-phase

12

Table 3: Experimental bulk density ρ data at liquid-liquid equilibrium for the n-hexane+water binary mix-ture at a temperature of T = 373.15 K. The standarduncertainties of the measurments are: ρ ± 0.4kg m−3;T ± 0.1 K; and P ± 0.1 MPa.

n-hexane-rich phase water-rich phase

P/MPa ρ/(kg m−3) ρ/(kg m−3)0.675 582.196 959.3252.762 587.029 960.6845.297 592.773 962.1447.738 597.693 963.656

10.156 602.218 965.11012.625 606.434 966.57315.100 610.365 968.07417.655 614.461 969.51720.142 618.210 970.994

line, n-hexane exhibits a positive surface activity whichincreases as the pressure is increased. Above the three-phase line pressure, n-hexane reaches the bulk liquiddensity and does not show any activity at the interface.One can clearly see the progression of the adsorptionof n-hexane onto the water surface in Figure 13. Atlow pressures a small surface excess is present, whichgrows into a liquid-like film that eventually becomes abulk liquid phase at sufficiently high pressures.

Due to the lack of appropriate experimental datafor the densities of the coexisting phases of the n-hexane+H2O mixture at the conditions of interest, weperform measurements in our current work. The massdensities of the mixture are measured at a temperatureof T = 373.15 K over the pressure range P = 0.6 to20 MPa by using a stainless steel high-pressure LLE celland a DMA HP densimeter (Anton Paar GmbH, Aus-tria) with an accuracy of 5× 10−3 kg m−3. Details of theexperimental procedure are given in Appendix, and thedensities are summarized in Table 3.

From the modelling perspective, the bulk densitiescan be obtained both by using the SAFT-γ Mie EOSof by direct molecular-dynamics simulation. The calcu-lated densities of n-hexane+H2O mixture are comparedto the experimental data in Figure 14. A barotropic den-sity inversion for the n-hexane+H2O is expected onlyfor long-chain alkanes, starting from n-octacosane (n-C28H58) [3], and therefore is not observed in the case ofn-hexane. A very good agreement between the densitiespredicted by the SAFT-γMie EOS, the MD simulations,and the experiment data is observed at T = 373 K; theMD simulations are also seen to reproduce the theoreti-cal results with high accuracy at T = 473 K.

The wealth of data available for aqueous alkanesprovides a testing ground for the transferability of themodel. If the force field is transferable, the unlike inter-action parameter which has been adjusted to reproducethe fluid-phase equilibria of the n-hexane+H2O mix-ture should also provide a quantitative agreement withexperiment for mixtures involving the other n-alkanes.The experimental data for the n-dodecane++H2O mix-ture is assessed at a temperature of T = 603 K, whichis above the UCEP. The the SAFT-γ Mie EOS cap-tures both the low-pressure region of VLE and the high-pressure region of LLE (cf. Figure 15). The MD simu-lations of the system represented with the SAFT-γ CGMie force field given in Table 1 are found to repro-duce the theoretical values with reasonable accuracy forboth the VLE and LLE phase boundaries. The three-dimensional PT x fluid-phase diagram obtained with theSAFT-γ Mie EOS is presented in Figure 16. The dis-continuity of the vapour-liquid critical line can be vi-sualized in PT -projection in Figure 17. As for the n-hexane+H2O mixture, the three-phase line is in excel-lent agreement with experiment. Due to the overpre-diction of the critical point of the n-alkanes, the criticalpressure close to the critical point of n-dodecane is over-predicted and the minimum in the upper branch of thecritical line is not reproduced. Nevertheless, the param-eter set is capable of representing the main features ofthe fluid-phase equilibria and is transferable not only todifferent conditions but also to others members of thehomologous series of the n-alkanes.

3.3. n-alkane+carbon dioxide binary mixturesBinary mixtures of n-alkane and CO2 are particu-

larly interesting because the fluid-phase equilibria ex-hibits a change from type II to type III behaviour onincreasing the chain length of the n-alkane. Capturingthis feature with only one adjustable parameter for theentire homologous series is a challenging task for anyequation of state [34]. A value of the binary param-eter of ki j = 0.08 between the unlike interaction be-tween the CO2 and alkyl CG beads is found to provide agood match with experimental data of the vapour-liquidequilibrium data of n-hexane+CO2 mixture at a tem-perature of T = 313 K. The corresponding parameterset (cf. Table 1) is found to provide a good represen-tation of the fluid-phase equilibria at temperatures ofT = 353 and 393 K for pressures removed from the crit-ical point, whereas the critical point is overestimated bythe theory as can be seen from Figure 18 a). The re-sults of MD simulations performed at the given temper-atures compare well with the theory. The same force-field parameters can be used to study the fluid-phase be-

13

haviour of other members of the n-alkane homologousseries: an isothermal pressure-composition (Px) phasediagram for the long-chained n-pentadecane+CO2 mix-ture is displayed in Figure 18 b). The change of thefluid-phase behaviour compared to that of the shorter-chain alkanes is evident with the appearance of theliquid-liquid equilibrium at low temperatures (here atT ∼ 292 K). In the theoretical prediction, the liquid-liquid immiscibility persists at high pressures, whereasthe experimental results indicate that a total miscibilityis acheived for pressures above 20 MPa. The vapour-liquid equilibrium is correctly described with the SAFT-γ Mie EOS [31] using the parameters given in Table 1.At the temperatures above the critical point of pure CO2,the fluid-phase equilibrium is accurately reproduced bythe equation of state at low and intermediate pressures,the critical point of the mixture is however overesti-mated. MD simulations performed based on the sameforce field reproduce the theoretical and experimentalresults to high accuracy.

The change from type II behaviour for the short chainalkane to type III for the long chain alkanes is evidentfrom the three-dimensional representation of the fluid-phase equilibria of mixtures of CO2 with n-hexane andn-pentadecane in Figure 19 and the corresponding PTprojection in Figure 20. The phase behaviour of then-hexane+CO2 mixture is qualitatively reproduced, dis-playing the continuous vapour-liquid critical line, whichconnects the critical points of the both components, andthe liquid-liquid critical line, which merges with thethree-phase line in UCEP. Quantitatively, the vapour-liquid critical line is overestimated compared to the ex-perimental data [151]. By contrast, the discontinuousvapour-liquid critical line can be identified in the PTrepresentation of the n-pentadecane+CO2 mixture, il-lustrating type III behaviour. Starting from the criticalpoint of n-pentadecane, the PT projection of the vapour-liquid critical line exhibits a maximum and then a mini-mum with decreasing temperature until it finally mergeswith the liquid-liquid critical line. Again, only qualita-tive agreement with experiment is achieved: the exper-imental data [151] indicate that the critical line is char-acterized by a maximum at a temperature of T ∼ 385 Kand a pressure of P ∼ 23 MPa and a minimum at∼ 301 K and ∼ 12 MPa, while the critical line pre-dicted with the SAFT-γ Mie EOS has a maximum at∼ 450 K and ∼ 35 MPa and a minimum at ∼ 350 K and∼ 30 MPa.

An analysis of the behaviour of the density of then-pentadecane+CO2 mixture predicted with the SAFT-γ Mie EOS is also made by comparison with the cor-responding experimental data at different temperatures

and pressures. Mass barotropy has been observed ex-perimentally for mixtures of CO2 with n-alkane homo-logues that exhibit type III phase behaviour [152]. Theshortest alkane that exhibits a barotropic density inver-sion in mixtures with CO2 is n-tetradecane. The the-oretical studies of Segura and co-workers [153, 154]have suggested that n-alkane+CO2 mixtures charac-terized by type III behaviour exhibit both mass andmolar barotropy, whereas type IV mixtures (e.g., n-tridecane+CO2) only display molar barotropy. In ourcurrent work, the experimental densities reported forthe n-pentadecane+CO2 mixture by Tanaka et al. [155]at a temperature of T = 313 K are compared to thepredictions of the SAFT-γ Mie EOS, thereby revealingexcellent agreement (Figure 21). Further isotherms attemperatures of T = 292, 353, and 393 K are calcu-lated with the EOS and are found to be in good cor-respondence with the results of MD simulation. Massbarotropy is observed at low temperatures: inversionis predicted at T ∼ 292 K and P ∼ 9.5 MPa, and atT ∼ 313 K and P ∼ 22 MPa with the EOS.

The adsorption phenomena in the interfacial regionof the n-pentadecane+CO2 mixture is studied at sub-critical (T = 292 K) and supercritical (T = 353 K)temperatures of CO2 (cf. Figure 22). At these con-ditions, CO2 is found to be adsorbed at the interface,whereas n-pentadecane does not exhibit any surface ac-tivity. The extraordinary surface adsorption of CO2 inmixtures of long alkanes has already been examined indetail with a combination of the square-gradient theoryand SAFT [156, 128]. At a pressure of P = 5.2 MPa,the maximum adsorption of the density profile at thetemperature of T = 292 K is approximately three timeshigher than that observed at T = 353 K. The large ex-cess adsorption exhibited at the lower temperatures isan indicator of the proximity to the liquid-liquid equi-librium (Figure 18). An increase in the density of CO2in both phases and in the interfacial region is observedwith increasing pressure.

14

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z / nm

423 K

Figure 6: Density profiles at the interface of the carbondioxide+water binary mixture at temperatures of T =383 and 423 K and different pressures obtained withour SAFT-γ Mie CG force field by direct molecular-dynamics simulation using the parameters given in Ta-ble 1. The dotted black curves denote the density pro-files for water at P = 8.18 MPa, and the symbols repre-sent the density profiles for carbon dioxide at P = 8.18(brown squares), 31.87 (blue crosses), 62.30 (red stars),and 102.10 MPa (yellow pluses).

0

50

100

150

0 250 500 750 1000

P / M

Pa

/ (kg m-3)

298 K304 K383 K423 K473 K

Figure 7: Isothermal pressure-density (Pρ) slices of thefluid-phase equilibria for the carbon dioxide+water bi-nary mixture at temperatures of T = 298, 304, 383, 423,and 473 K. The filled circles correspond to experimen-tal densities for the saturated phases [146]. The curvescorrespond to the predictions with the SAFT-γ MieEOS [31] using the parameters given in Table 1, andthe empty circles are the corresponding phase bound-aries obtained with our SAFT-γ CG Mie force field bydirect molecular-dynamics simulation.

15

0

25

50

75

100

0 0.05 0.1 0.15

P / M

Pa

xCO2

423 K

exponential-6EPM2/TIP4P 2005

TraPPE/TIP4P 2005EPM2/SCP

EPM2/TIP4PSAFT- Mie CG

0

25

50

75

100

0.75 0.8 0.85 0.9 0.95 1

P / M

Pa

xCO2

423 K

0

25

50

75

100

0 0.2 0.4 0.6 0.8 1

P / M

Pa

xCO2

548 K

Figure 8: Isothermal pressure-composition (Px) slicesof the fluid-phase equilibria for the carbon diox-ide+water binary mixture at temperatures of T = 423and 548 K. The filled symbols correspond to exper-imental data of Todheide and Franck [1] (�) and ofTakenouchi and Kennedy [2] ( ). The continuous bluecurves corresponds to the predictions with the SAFT-γ Mie EOS [31] using the parameters given in Ta-ble 1, and the blue empty circles are the correspond-ing phase boundaries obtained with our SAFT-γ CGMie force field by direct molecular-dynamics simula-tion. The results of the CG simulations are comparedwith the data obtained using some popular atomisticmodels as reported in reference [92]: exponential-6(2); EPM2/TIP4P2005 (4); TraPPE/TIP4P2005 (5);EPM2/SPC (3); EPM2/TIP4P (D). The unlike-pair pa-rameters of the atomistic force fields are based on thestandard Lorentz-Berthelot combining rules.

0

5

10

15

0 0.2 0.4 0.6 0.8 1

P / M

Pa

xH2O

Figure 9: Isothermal pressure-composition (Px) slice ofthe fluid-phase equilibria for the n-hexane+water binarymixture at a temperature of T = 473 K. The filled cir-cles correspond to the experimental vapour-liquid andliquid-liquid equilibrium data [147, 148, 149]. Thecontinuous curves represent to the description with theSAFT-γ Mie EOS [31] using the parameters given inTable 1. The empty circles are the corresponding phaseboundaries obtained with our SAFT-γ CG Mie forcefield by direct molecular-dynamics simulation: the het-eroazeotropic point at a pressure of P = 3.33 MPa withcomposition xL1H2O ∼ 1.0, x

GH2O

= 0.396, and xL2H2O =0.116 is highlighted in red.

16

0

200

400

600

800

0 5 10 15 20

/ (k

g m

-3)

z / nm

L1

L2

G

Figure 10: Total density profile (yellow crosses) of then-hexane+water binary mixture at the three-phase co-existence (heteroazeotrope) corresponding to the tem-perature of T = 473 K and a pressure of P = 3.33MPa obtained with our SAFT-γ Mie CG force field bydirect molecular-dynamics simulation using the param-eters given in Table 1. L1 denotes the water-rich liquidphase, L2 denotes n-hexane-rich liquid phase, G denotesthe gas phase. The composition of the heteroazeotropeis xL1H2O ∼ 1.0, x

GH2O

= 0.396, and xL2H2O = 0.116. Inthe snapshot of a representative configuration duringthe simulation, the water molecules are displayed in red(red density profile in upper panel) and n-hexane in blue(blue density profile in upper panel).

T / K

xH2O 0

10

20

P / M

Pa

200 400 600

0 0.4

0.8

P / M

Pa

Figure 11: Three-dimensional pressure-temperature-composition (PT x) surface of the fluid-phase equilibriafor the n-hexane+water binary mixture predicted withthe SAFT-γ Mie EOS [31] using the parameters givenin Table 1. The continuous red curves on the oppositefaces of the prism represent the vapour pressures of thepure components, the dashed blue curves are the pre-dictions for the critical line and the PT projection ofthe three-phase line, and the dashed black curves cor-respond to isothermal pressure-composition (Px) sec-tions.

17

0

10

20

30

40

50

200 300 400 500 600

P / M

Pa

T / K

2

4

6

480 500 520

P / M

Pa

T / K

Figure 12: Pressure-temperature (PT ) projection ofthe fluid-phase equilibria for the n-hexane+water bi-nary mixture. The red circles correspond to the ex-perimental vapour pressures of the pure components[138], and the blue circles correspond to the experimen-tal vapour-liquid critical line and three-phase line [3].The curves represent the calculations with the SAFT-γMie EOS [31] using the parameters given in Table 1:the continuous red curves are the vapour pressures ofthe pure components, and the dashed blue curves are thevapour-liquid critical line and three-phase line. An en-larged region close to the critical point of pure n-hexaneis shown in the inset.

0

200

400

600

800

1000

-5.0 -2.5 0.0 2.5 5.0

/ (k

g m

-3)

z / nm

Figure 13: Density profiles at the interface of then-hexane+water binary mixture at a temperature ofT = 473 K and different pressures determined withour SAFT-γ Mie CG force field by direct molecular-dynamics simulation using the parameters given in Ta-ble 1. The dashed black curve denotes the density pro-file for water at P = 2.16 MPa, the symbols representthe density profiles for n-hexane at P = 2.16 (brownsquares), 2.87 (blue crosses), and 6.25 MPa (red pluses).

0

5

10

15

20

25

0 250 500 750 1000

P / M

Pa

/ (kg m-3)

373 K 473 K

Figure 14: Isothermal pressure-density (Pρ) slices ofthe fluid-phase equilibria for the n-hexane+water bi-nary mixture at T = 373 (blue/continuous) and 473 K(red/dashed). The filled circles correspond to experi-mental density data from Table 3. The curves representthe predictions with the SAFT-γ Mie [31] using the pa-rameters given in Table 1, and the empty circles repre-sent the corresponding phase boundaries obtained withour SAFT-γ CG Mie force field by direct MD simula-tion.

18

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

P / M

Pa

xH2O

Figure 15: Isothermal pressure-composition (Px) sliceof the fluid-phase equilibria for the n-dodecane+waterbinary mixture at a temperature of T = 603 K. Thefilled circles correspond to experimental vapour-liquidand liquid-liquid equilibrium data [4]. The continu-ous curves represent the predictions with SAFT-γ MieEOS [31] using the parameters given in Table 1. Theempty circles are the corresponding phase boundariesobtained with our SAFT-γ CG Mie force field by directMD simulation.

T / KxH2O

0

10

20

P / M

Pa

400

600

0 0.2 0.4 0.6 0.8 1

P / M

Pa

Figure 16: Three-dimensional pressure-temperature-composition (PT x) surface of the fluid-phase equilibriafor the n-dodecane+water binary mixture predicted withthe SAFT-γ Mie EOS [31] using the parameters givenin Table 1. The continuous red curves on the oppositefaces of the prism represent the vapour pressures of thepure components, the dashed blue curves are the predic-tions for the critical line and PT projection of the three-phase line, and the dashed black curves correspond toisothermal pressure-composition (Px) sections.

0

10

20

30

200 300 400 500 600

P / M

Pa

T / K

Figure 17: Pressure-temperature (PT ) projection ofthe fluid-phase equilibria for the n-dodecane+water bi-nary mixture. The red circles correspond to the ex-perimental vapour pressures of the pure components[138], and the blue circles correspond to the experimen-tal vapour-liquid critical line and three-phase line [3].The curves represent the predictions with the SAFT-γMie EOS [31] using the parameters given in Table 1:the continuous red curves are the vapour pressures ofthe pure components, and the dashed blue curves arethe vapour-liquid critical line and three-phase line.

19

0

5

10

15

0 0.2 0.4 0.6 0.8 1

P / M

Pa

xCO2

313 K353 K393 K

(a)

0

10

20

30

0 0.2 0.4 0.6 0.8 1

P / M

Pa

xCO2

292 K353 K393 K

(b)

Figure 18: Isothermal pressure-composition (Px) sliceof the fluid-phase equilibria of the (a) n-hexane+carbondioxide and (b) n-pentadecane+carbon dioxide binarymixtures for isotherms corresponding to temperaturesof T = 313, 353, 393 K and to T = 292, 353, 393 K,respectively. The filled circles are the experimentalvapour-liquid and liquid-liquid equilibrium data [146].The curves represent the description with the SAFT-γMie EOS [31] using the parameters given in Table 1.The empty circles are the corresponding fluid-phaseboundaries obtained with our SAFT-γ CG Mie forcefield by direct MD simulation.

200

400

0 0.2 0.4 0.6

0.8 1 0

5

10

15

P / M

PaT / KxCO2

P / M

Pa(a)

200

400

600

0 0.2 0.4 0.6

0.8 1 0

20

40

P / M

Pa

T / KxCO2

P / M

Pa

(b)

Figure 19: Three-dimensional pressure-temperature-composition (PT x) surface of the fluid-phase equi-libria of the (a) n-hexane+carbon dioxide and (b) n-pentadecane+carbon dioxide binary mixtures predictedwith the SAFT-γ Mie EOS [31] using the parametersgiven in Table 1. The continuous red curves on the op-posite faces of the prism represent the vapour pressuresof the pure components, the dashed blue curves are thecritical lines and the PT projection of the three-phaseline, and the dashed black curves correspond to isother-mal pressure-composition (Px) sections.

20

0

5

10

15

200 300 400 500

P / M

Pa

T / K

(a)

0

20

40

60

200 400 600

P / M

Pa

T / K

(b)

Figure 20: Pressure-temperature (PT ) projection of thefluid-phase equilibria of the (a) n-hexane+carbon diox-ide and (b) n-pentadecane+carbon dioxide binary mix-tures. The red circles correspond to the experimentalvapour pressures of the pure components [138], andthe blue circles correspond to the experimental vapour-liquid critical line [151]. The curves represent the de-scription with the SAFT-γ Mie EOS [31] using the pa-rameters given in Table 1: the continuous red curvesare the vapour pressures of the pure components, andthe dashed blue curves are the vapour-liquid critical lineand three-phase line.

0

10

20

30

0 250 500 750 1000

P / M

Pa

/ (kg m-3)

292 K 313 K 353 K 393 K

Figure 21: Isothermal pressure-density (Pρ) slices ofthe fluid-phase equilibria of the n-pentadecane+carbondioxide binary mixture at temperatures of T = 292,313, 353, and 393 K. The filled circles correspond toexperimental density data for the saturated phases atT = 313 K [155]. The curves represent the descriptionwith the SAFT-γ Mie EOS [31] using the parametersgiven in Table 1. The empty circles are the correspond-ing fluid-phase boundaries obtained with our SAFT-γCG Mie force field by direct MD simulation.

21

0

200

400

600

-5.0 -2.5 0.0 2.5 5.0

/ (k

g m

-3)

z / nm

292 K

0

200

400

600

-5.0 -2.5 0.0 2.5 5.0

/ (k

g m

-3)

z / nm

353 K

Figure 22: Density profiles at the interface of then-pentadecane+carbon dioxide binary mixture at tem-peratures of T = 292 and 353 K and different pres-sures determined with our SAFT-γ Mie CG force fieldby direct molecular-dynamics simulation using the pa-rameters given in Table 1. The dashed curve denotesthe density profile of n-pentadecane at a pressure ofP = 5.2 MPa, and the symbols represent the densityprofiles of carbon dioxide at P = 3.2 (brown squares),5.2 (blue crosses), and 15.4 MPa (red stars).

22

3.4. n-decane+carbon dioxide+water ternary mixture

A recent and comprehensive experimental study ofthe n-decane+CO2+H2O ternary mixture has been un-dertaken by Forte et al. [157]. As the authors pointout, a large part of the observable fluid-phase diagramis characterized by a three-phase region due to the im-miscibility of the corresponding binary systems. In ourcurrent work, we compare the experimental data withthe description obtained with the SAFT-γ Mie EOS andfrom MD simulations of the CG models. No additionalparameters need to be determined for the ternary mix-ture, as all of the binary unlike interaction parametershave been specified for the binary systems.

In Table 4 and Figure 23 we compare the experimen-tal [157] coexistence compositions of the ternary mix-ture with the predictions of the SAFT-γ Mie EOS andthe MD simulations at T = 393 K and four differentpressures removed from critical region (P = 6.02, 7.58,9.59, and 11.81 MPa). The fluid-phase behaviour isclearly dominated by a region of three-phase equilib-ria, the extent of which decreases with increasing pres-sure. The SAFT-γ Mie EOS can be seen to provide anaccurate description of the experimental measurementsof the three-phase compositions for this range of ther-modynamic conditions. Additionally the extent of thetwo-phase coexistence is well represented with the the-ory. Only very small single-phase regions can be de-tected at the margins of the ternary phase diagram. Thecompositions of the CO2- and H2O-rich phases obtainedby MD simulation of the three-phase region with thecorresponding SAFT-γ CG Mie force fields are in goodagreement with theoretical predictions; the solubility ofCO2 in the n-alkane-rich phase is slightly overestimatedfor all pressures. The overall trend of increasing solu-bility with pressure can be correctly captured by simu-lations.

The density profiles of the n-decane+CO2+H2Oternary mixture at P = 6.02 MPa are displayed in Fig-ure 24. Three phases (H2O-rich phase L1, n-decane-richphase L2, and CO2-rich phase G) can clearly be dis-tinguished from the magnitude of the total density, andare also apparent from the snapshot of the MD simula-tion. From the figure one finds that CO2 exhibits a largeadsorption peak at the interface between the H2O-richand n-decane-rich phases and a small adsorption peakat the interface between the CO2-rich and n-decane-richphases. The density and the width of the interfaces arealso found to increase with increasing pressure.

23

6.02 MPa

H2O C10H22

CO2

xH2O xCO2

xC10H220.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

7.58 MPa

H2O C10H22

CO2

xH2O xCO2

xC10H220.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

9.35 MPa

H2O C10H22

CO2

xH2O xCO2

xC10H220.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

11.81 MPa

H2O C10H22

CO2

xH2O xCO2

xC10H220.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

Figure 23: Ternary fluid-phase diagram for the n-decane+carbon dioxide+water mixture at T = 393 K and P = 6.02,7.58, 9.35 and 11.81 MPa, respectively. The filled circles are the experimental values for the three-phase equilibriumregion [157]; the continuous blue lines are SAFT-γ Mie [31] predictions of the three-phase equilibrium region, thedashed red lines are the corresponding two-phase coexistence tie-lines, and the continuous red curves the phaseboundaries. The empty circles are the phase boundaries of the three-phase region determined with our SAFT-γ MieCG force field by direct molecular-dynamics simulation using the parameters given in Table 1.

24

Table 4: Comparison of the experimental [157] fluid-equilibrium phase compositions with those predicted by theSAFT-γ Mie EOS [31] and obtained from the SAFT-gamma CG Mie force fields by direct MD simulations of then-decane+carbon dioxide+water ternary mixture at a temperature of T = 393.15 K and different pressures.

expt. SAFT MDP/ MPa Phase xexptC10H22 x

exptCO2

xexptH2O xSAFTC10H22

xSAFTCO2 xSAFTH2O

xMDC10H22 xMDCO2

xMDH2O6.02 I 4.50×10−6 0.0086 0.9914 0 0.0031 0.9969 0 0.0076 0.9924

II 0.0097 0.9542 0.0361 0.0079 0.9454 0.0467 0.0137 0.9447 0.0416III 0.6477 0.3191 0.0332 0.6598 0.3197 0.0204 0.5932 0.3757 0.0311

7.58 I 2.70×10−6 0.0107 0.9893 0 0.0038 0.9962 0 0.0074 0.9926II 0.0078 0.9605 0.0317 0.0084 0.9503 0.0413 0.0130 0.9473 0.0397III 0.5859 0.3795 0.0346 0.5924 0.3862 0.0214 0.4906 0.4751 0.0343

9.59 I 2.60×10−6 0.0127 0.9873 0 0.0046 0.9954 0 0.0065 0.9935II 0.0112 0.9636 0.0252 0.0099 0.9522 0.0379 0.0142 0.9482 0.0377III 0.4857 0.4821 0.0322 0.5135 0.4636 0.0229 0.4077 0.5527 0.0396

11.81 I 8.30×10−7 0.0145 0.9855 1.44×10−9 0.0054 0.9946 0 0.0072 0.9928II 0.0184 0.9569 0.0247 0.0131 0.9500 0.0369 0.0134 0.9481 0.0386III 0.4006 0.566 0.0334 0.4349 0.5402 0.0250 0.3670 0.5961 0.0370

25

4. Conclusions

In our work we employ the SAFT-γ CG Mie forcefield to study binary and ternary mixtures comprisingof water, carbon dioxide, and n-alkanes. Based onthe models developed for pure components and binary-interaction parameters informed by the SAFT-γ Mieequation of state, we obtain a quantitative descriptionof multicomponent, multiphase equilibria. The phasecompositions, densities, critical lines, and three-phaselines predicted by the theory are found to accuratelymatch the experimental data. In addition, the sameintermolecular parameter set is employed directly inmolecular-dynamics simulations, essentially reproduc-ing the aforementioned results but also allowing thestudy of interfacial providing a microscopic moleculartreatment of the system.

The key to the success of the SAFT-γ CG Mie proce-dure is that the EOS is based on a well defined Hamilto-nian and thus can be used to extract the molecular force-field parameters directly from the macroscopic proper-ties of the system. This top-down coarse-graining ap-proach offers a direct link between the theory, simula-tion, and experiment, thereby allowing multiscale mod-elling of the system. An advantage of the theory oversimulation or experiment is that it allows one to gain anoverview over the entire phase diagram in a very effi-cient manner. On the other hand, molecular simulationprovides microscopic information about the structuraland interfacial properties, which are not directly acces-sible to the equation of state and difficult to probe ex-perimentally.

0

200

400

600

800

1000

0 5 10 15 20

/ (k

g m

-3)

z / nm

L1

L2

G

Figure 24: Total density profile (yellow crosses) of theternary n-decane+carbon dioxide+water ternary mix-ture at T = 393 K and P = 6.02 MPa for a global com-position of xC10H22 = 0.1, xCO2 = 0.5 determined withour SAFT-γ Mie CG force field by direct molecular-dynamics simulation using the parameters given in Ta-ble 1. The water-rich liquid phase is denoted by L1, n-decane-rich liquid phase by L2, and the carbon dioxide-rich gas phase by G. In the snapshot of a representativeconfiguration during the simulation, water moleculesare displayed in red, n-decane in blue, and carbon diox-ide in grey . In the snapshot of a representative config-uration during the simulation, the water molecules aredisplayed in red (red density profile in upper panel), n-hexane in blue (blue density profile in upper panel), andcarbon dioxide in black (black density profile in upperpanel).

Acknowledgements

Olga Lobanova is very grateful to Imperial CollegeLondon for an Excellence Award in support of herPhD studies. Additional support to the Molecular Sys-tems Engineering Group from the the Engineering andPhysical Sciences Research Council (EPSRC) of theUK (grants EP/E016340 and EP/J014958), the JointResearch Equipment Initiative (JREI) (GR/M94426),the Royal Society-Wolfson Foundation refurbishmentscheme (RSWF/SUC11/RND3/EEP), and the ThomasYoung Centre (TYC-101) is gratefully acknowledged.

Appendix

26

Density measurements of the n-hexane+water mixtureThe mass densities of the n-hexane+H2O binary mix-

ture are measured at a temperature of T = 373.15 Kover the pressure range of P = 0.6 to 20 MPa by us-ing a stainless steel high-pressure liquid-liquid equi-librium cell and a DMA HP densimeter (Anton PaarGmbH, Austria) with an accuracy of 5 × 10−3kg m−3.The experimental procedure for determining densitiesis as follows. First the densimeter is calibrated by usingtwo reference fluids, ultra high purity nitrogen (UHP= 99.995%) and ultra-pure water. After degasificationin an ultrasonic bath (model 1210, Branson, Inc.), ap-proximately, 20 cm3 of each pure fluid is introducedin the high-pressure cell. The chamber has two ori-fices (one at the top and the other at the bottom) and itis equipped with appropriately sealed borosilicate glasswindows, which allow visualization of the inner spaceduring operation. The cell is heated to the desired tem-perature by means of electric band heaters operated bya Watlow temperature controller model TC-211-K-989(USA). The temperature of the sample in the vesselis measured by means of a K-type thermocouple, andmaintained constant to wi