Optimized Mie Potentials for Phase Equilibria: Application to Branched

Article pubs.acs.org/jced

Optimized Mie Potentials for Phase Equilibria: Application to Branched Alkanes Jason R. Mick,† Mohammad Soroush Barhaghi,† Brock Jackman,‡ Loren Schwiebert,‡ and Jeffrey J. Potoff*,† †

Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, Michigan 48202, United States Department of Computer Science, Wayne State University, Detroit, Michigan 48202, United States

‡

S Supporting Information *

ABSTRACT: A transferable united-atom (UA) force field based on Mie potentials is presented for branched alkanes. The performance of the optimized Mie potential parameters is assessed for 32 branched isomers of butane, pentane, hexane, heptane, and octane using grand canonical histogram-reweighting Monte Carlo simulations. For each compound, vapor−liquid-coexistence curves, vapor pressures, heats of vaporization, critical properties, and normal boiling points are predicted and compared to experiment. Experimental saturated liquid densities and critical temperatures are reproduced with a median absolute average error of 0.6%, while vapor pressures are reproduced with a median absolute average error of 2.2%. Calculations performed with the TraPPE and NERD force fields produce median absolute average errors for saturated liquid densities and vapor pressures of 1.3−1.8% and 14.3−23.5%, respectively. Binary phase diagrams predicted by the Mie potentials for argon+neopentane, methane+neopentane, and ethane+isobutane are in close agreement with experiment.

1. INTRODUCTION Branched alkanes have diverse industrial applications as fuel additives,1−3 lubricants,4,5 and waxes. Compared to linear isomers containing the same number of carbon atoms, branching results in higher boiling points,6 and enhanced viscosities,5 traits valuable to lubricant applications. Branching of the carbon chain lowers freezing points;6 increases flash points;7 delays ignition in the negative temperature coefficient and low temperature regions;8,9 reduces burn rates;10 and increases knock resistance,11 properties that are useful for fuel additives. Other properties, such as the heat of combustion12 and density, can be optimized for weight-restricted fuel applications and other specialized industrial uses via the selection of specific isomers.13 Aviation fuels, for example, are composed of a variety of linear and branched alkanes from C7 to C18.14 The development and use of synthetic fuels, such as S-8, requires detailed knowledge of thermophysical properties and vapor−liquid equilibria for branched alkanes.15,16 While experiments are the most accurate method for the determination of vapor−liquid equilibrium properties, thermal decomposition at higher temperatures and the difficulty of separating similar isomers make such measurements expensive and less reliable for these compounds.1,17−19 Computational methods, such as molecular based equations of state and computer simulation, provide alternative routes to the prediction of phase equilibria. For example, a number of variants of statistical associating fluid theory (SAFT) have been used to predict the phase diagrams of linear alkanes (up to n© XXXX American Chemical Society

C48H98) and their binary mixtures. These efforts include perturbed chain theory (PC-SAFT),20 potentials of variable range (SAFT-VR),21 the use of soft potentials (soft-SAFT),22 the inclusion of a crossover term to improve predictions in the critical region (SAFT-VRX),23,24 and the use of Mie potentials (SAFT-γ-Mie).25 To date, SAFT has been used to predict vapor−liquid equilibria for a number of branched alkanes, which include; isobutane, isopentane, isohexane,26 neopentane, 2,2-dimethylbutane, 2,3-dimethylbutane, 3-methylpentane, and 2-methylhexane.20 Achieving accurate results, however, required the optimization of unique parameters for each compound.20,26 Atom-based computer simulations provide another route to the prediction of vapor−liquid equilibria. While more computationally intensive than equation of state approaches, computer simulation also provides microscopic information that may be used to elucidate how atomic-level interactions drive physical properties and phase behavior.27,28 Numerous force fields have been developed specifically for the prediction of vapor−liquid equilibria for linear and branched alkanes. These include the transferable potentials for phase equilibria (TraPPE);29−31 Nath, Escobedo, and de Pablo revised (NERD);32,33 and the optimized potentials for phase equilibria (OPPE).34,35 The combination of 12−6 Lennard-Jones potential and united-atom Received: December 14, 2016 Accepted: April 25, 2017

A

DOI: 10.1021/acs.jced.6b01036 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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by Mie potentials for noble gases to MP2/aug-cc-PVTZ ab initio calculations.39 For all calculations, the potential was truncated at 10 Å and analytical tail corrections were applied for the energy (eq 6) and pressure (eq 7).

approximation used in these models, however, limits their overall predictive capability. While each force field is able to provide reliable predictions of saturated liquid densities for alkanes, vapor pressures are overpredicted significantly. This problem remains even when the transferability requirement is relaxed and molecule specific parameters are regressed. For example, NERD’s branched alkane description comprises a total of ten different united-atom site definitions, versus two for TraPPE, but provides little improvement in overall performance. Using a Mie potential to represent interactions between united-atoms has been shown to produce significant improvements in the simultaneous reproduction of saturated liquid densities and vapor pressures for n-alkanes,36 alkenes,37 alkynes,38 perfluorocarbons,36 and noble gases.39,40 Therefore, in this work the Mie potentials are extended to branched alkanes. Parameters for two new interaction sites are introduced: CH and C. These sites are optimized to reproduce experimental saturated liquid and vapor densities, vapor pressures and heats of vaporization for a wide range of compounds. The transferability of the optimized parameters to different bonding environments is assessed and the limitations of the optimized models are discussed. The models are further validated through the calculation of binary mixture vapor− liquid equilibria. Radial distribution functions are used to provide insight into the effect of repulsion exponent on the observed microscopic structure of the liquid phase.

k

Ulrc =

i=0 j=0

k

Plrc =

⎞⎛ nij ⎞6/(nij − 6) ⎟⎜ ⎟ 6 ⎟⎠⎝ 6 ⎠

(4)

εiiεjj

To determine repulsion exponents for cross interactions, an arithmetic average was used.36,37,39 nij = (nii + njj)/2

3⎤

⎛ σij ⎞ ⎥ − 3⎜⎜ ⎟⎟ ⎥ ⎝ rij ⎠ ⎥⎦

(7)

pseudoatom

εi/kb(K)

σi(Å)

ni

Ar39 Ar (Vrabec)44 CH436 CH336 CH236 CH (CN ≤ 4, S/L) CH (CN > 4, S/L) CH (generalized) C (CN ≤ 4, S/L) C (CN > 4, S/L) C (generalized)

122.10 116.79 161.00 121.25 61.00 15.00 14.00 15.00 1.45 1.20 1.20

3.405 3.3952 3.740 3.783 3.990 4.700 4.700 4.600 6.100 6.200 6.100

13 12 14 16 16 16 16 16 16 16 16

taken from our previous work for n-alkanes and used without modification.36 Two sets of new parameters were optimized in this work for C and CH groups. The first set was generalized parameters for CH and C sites. These parameters were optimized to yield a single set of parameters that produced phase diagrams in close agreement with experimental data for a broad range of compounds. While an improvement over existing force fields, the primary weakness of the generalized parameters was that the phase behavior predicted for small molecules, such as neopentane and isobutane, deviated significantly from experiment. Therefore, a second set of specialized parameters was developed to provide improved accuracy for smaller molecules, while also improving the accuracy for larger compounds. Through simultaneous optimization of parameters for a large number of compounds, it was determined that compounds with similar backbone length (defined as the number of pseudoatoms CN in the longest contiguous stretch of bonded sites between terminal sites) had similar optimized parameters. Parameters were then optimized independently for “short” (CN ≤ 4) and “long” (CN > 4) molecules. The three parameter options, referred to as generalized (GEN), short molecule optimized, and long molecule optimized (S/L), are mutually exclusive and for each molecule topology only one of the three options should be selected. As in prior work on n-alkanes, bond lengths were fixed at 1.54 Å.36 For 2,2,3,3-tetramethylbutane, a highly strained isomer of octane, the central C−C bond length was set to 1.58 Å based on gas-phase electron diffraction data.45 This elongated bond length should be used for all compounds where a quaternary carbon is bonded to another quaternary carbon.

For the 12−6 potential, Cn reduces to the familiar value of 4. Parameters governing interactions between unlike interaction sites were determined using the Lorentz−Berthelot combining rules.41,42

εij =

⎞⎛ σij ⎞ ⎟⎜ ⎟ 3 ⎟⎠⎜⎝ rij ⎟⎠

(nij − 3)

Table 1. Nonbonded Parameters for Noble Gases and Alkanes

(2)

(3)

⎡ ⎛ nij 4 ⎢⎛ 3 ⎞ πCnεijσij3ρi Nj⎢⎜ ⎟⎜⎜ ⎝ ⎠ 3 ⎢⎣ 2 ⎝ nij −

All nonbonded parameters used in this work are listed in Table 1. Parameters for CH4, CH3, and CH2 pseudoatoms were

(1)

σij = (σii + σjj)/2

k

∑∑ i=0 j=0

where rij, εij, and σij are the separation, well depth, and collision diameter, respectively, for the pair of interaction sites i and j. The constant Cn is a normalization factor used such that the minimum of the potential remains at −εij for all nij. ⎛ nij Cn = ⎜⎜ ⎝ nij −

3⎤ ⎡⎛ ⎞(nij − 3) ⎛ nij − 3 ⎞⎛ σij ⎞ ⎥ 2π ⎢ σij Cnεijσij3ρi Nj⎢⎜⎜ ⎟⎟ −⎜ ⎟⎜⎜ ⎟⎟ ⎥ (nij − 3) ⎝ 3 ⎠⎝ rij ⎠ ⎥ ⎢⎣⎝ rij ⎠ ⎦

(6)

2. FORCE FIELD The n-6 Mie potential is defined as ⎡⎛ ⎞nij ⎛ ⎞6 ⎤ σij σij U (rij) = Cnεij⎢⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥⎥ r ⎝ rij ⎠ ⎦ ⎣⎝ ij ⎠

k

∑∑

(5)

The use of an arithmetic average of repulsion exponents for the combing rule is consistent with past optimization efforts for n-alkanes, perfluoroalkanes,36 alkenes,37 alkynes,38 and noble gases39 using Mie potentials. This is supported by recent work by Stiegler and Sadus, who examined the effect of combining rules on the physical properties predicted by nonidentical potentials,43 and comparisons of interaction energies predicted B



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rotational barriers determined from HF/6-31+g (d, p) ab initio calculations via a relaxed potential energy scan. In these calculations, the dihedral angle of interest was fixed and the remaining degrees of freedom were optimized. Ab initio calculations were performed in Gaussian 09.48

Angle bending was described using a harmonic potential kθ (θ − θ0)2 (8) 2 where θ is the measured bond angle and kθ = 62 500 K/rad2 was the force constant. The equilibrium bond angles are given in Table 2. Equilibrium bond angles and bending constants are Ubend =

3. SIMULATION METHODOLOGY Vapor−liquid coexistence curves, vapor pressures, and heats of vaporization were determined from histogram-reweighting Monte Carlo simulations in the grand canonical ensemble.49−51 Simulations were performed with GOMC version 1.9.52 For short compounds (e.g., neopentane, isobutane) a cubic cell size of 30 × 30 × 30 Å3 was used. For longer compounds (e.g., 2methylhexane) a cubic cell size of 35 × 35 × 35 Å3 was used. Generation of the initial configuration was performed using Packmol,53 while psfgen was used to generate coordinate (*.pdb) and connectivity (*.psf) files.54 A move ratio of 30% displacements, 10% rotations, and 60% molecule transfers was used. Due to the large number of molecule transfers, adequate sampling of conformation space was achieved without the need for additional regrowth moves. The coupled-decoupled configurational-bias Monte Carlo (CBMC) algorithm was used to improve sampling efficiency during the simulation.30 For all simulations, the following CBMC parameters were used: 100 angle trials, 10 dihedral trials, 10 trials for the initial site, and 4 trials for the growth of subsequent sites. For simulations near each molecule’s normal boiling point, the number of CBMC trials was increased to 15 and 10 for the growth of the initial and secondary sites, respectively. Acceptance rates for molecule insertions in liquid phase simulations were between 0.3% and 3% depending on molecule type, chemical potential, and temperature. Using GOMC, simulations of 3,3-dimethylhexane (nch,LJ(first) = 10, nch,LJ = 8, nch,tor = 20, nch,bend = 100) for the vapor phase, near critical point bridge and liquid phase required 156, 362, and 790 s, respectively, per 1 million Monte Carlo steps on a single core of an Intel i5 3.4 GHz CPU. To generate the phase diagram predicted by each parameter set, 9 to 10 simulations were performed; one simulation bridging the gas and liquid phases near the critical temperature, two in the gas phase, and 6 to 7 liquid simulations. For isobutane series compounds 4 × 106 Monte Carlo steps (MCS) were used for equilibration, followed by a data production period of 4 × 107 steps. For neopentane series compounds 5 × 106 MCS were used for equilibration, followed by a data production period of 2.5 × 107 steps. Histogram data were collected as samples of the number of molecules in the simulation cell and the nonbonded energy of the system. Samples were taken on an interval of 200 MCS. Average values for each quantity were determined from five independent sets of simulations, where each simulation was started with a different random number seed. Statistical uncertainties were determined as the standard deviation for each data set, and correspond to a 90% confidence level based on a t-distribution. The heat of vaporization (ΔHV) was calculated from the energies and molar volumes in each phase

Table 2. Equilibrium Bond Angles for Branched Alkanes

a

bend

θ0(degrees)

CHxCH2CHy CHxCH−CHy CHxC−CHy CH3C−Ca CH3C−CH3a

114 112 109.47 111 107

For 2,2,3,3-tetramethylbutane only.

the same as those given in the optimized potentials for liquid simulation united atom (OPLS-UA) force field.46 The one exception is 2,2,3,3-tetramethylbutane, where unique bond angles were used due to its highly strained structure compared to other branched alkanes. For 2,2,3,3-tetramethylbutane, equilibrium bond angles for CH3−C-CH3 and CH3−C-C, were calculated via ab initio geometry optimization in vacuum with Hartree−Fock theory and the 6-31+g(d,p) basis set. The angles predicted from ab initio calculations were found to be in close agreement with experimental gas-phase electron diffraction results.45 Dihedral energies were represented by a cosine series Utors =

∑ cn[1 + cos(nφ − δn)]

(9)

n=0

where φ is the dihedral angle, cn are dihedral force constants, n is the multiplicity, and δn is the phase shift. The cosine series accounts for the total rotational barrier, and no 1−4 LennardJones interactions were included in the model. Fourier constants are listed in Table 3, and were taken from the OPLS-UA force field46,47 when available. Missing Fourier constants for the CHx-CH-C−CHy, and CH3−C-C−CH3 rotational barriers were determined by fitting to reproduce Table 3. Torsional Parameters (in K) for Branched Alkanes torsion

n

cn

δn

CHx(CH2)(CH2)CHy

1 2 3 0 1 2 3 0 1 2 3 3 3 0 3 6 12

335.03 −68.19 791.32 −251.06 428.73 −111.85 441.27 −251.06 428.73 −111.85 441.27 461.29 1369 2950.91 −2047.06 529.25 150.16

0 π 0 0 0 π 0 0 0 π 0 0 0 0 0 π 0

CHx(CH2)(CH)CHy

CHx(CH)(CH)CHy

CHx(CH2)(C)CHy CHx(CH)(C)CHy CH3(C)(C)CH3

ΔHV = (UV − UL) + P(VV − VL)

(10)

where P is the saturation pressure, UV, UL, VV, and VL are the energy per mole, and molar volumes of the vapor and liquid phases, respectively. The critical temperature TC and density ρC for each model were calculated by fitting the saturated liquid and vapor densities to the law of rectilinear diameters55 C


Journal of Chemical & Engineering Data ρliq − ρvap 2

= ρC + A(T − TC)

Article

ane, and 3,4-dimethylhexane. This collection of compounds is referred to as the isobutane series for convenience. Eight isomers of octane and shorter compounds containing exclusively C sites exist. Five of these were selected for use in parameter optimization. These compounds included: neopentane, 2,2-dimethylbutane, 2,2-dimethylpentane, 2,2-dimethylhexane, and 3,3-dimethylhexane, and are referred to as the neopentane series in the remainder of the article. Reference experimental data used in the model optimization process were taken from the NIST Chemistry WebBook6 for isobutane, neopentane, and 2-methylbutane, while data for the remaining compounds were taken from Smith and Srivastava.59 For both the isobutane and neopentane series, molecules were selected to represent a variety of molecule sizes and bonding environments for the CH and C sites. The goal of the parameter optimization process was to determine Mie potential parameters for each compound that reproduced the saturated liquid densities and vapor pressures to within 1% and 3%, respectively, of experiment while also being transferable between different compounds. A preliminary heuristic-based search indicated that, similar to CH3 and CH2 parameters for n-alkanes, a repulsive exponent of ni = 16 provided an overlapping region of good results across the various compounds. With this optimal region identified and the repulsive exponent selected, an exhaustive grid-based search of the surrounding parameter space (σi, εi) was performed. For each compound in the isobutane series, 49 parameters sets were evaluated, spaced on 0.1 Å increments along σi to give 7 values on the range 4.4 ≤ σi ≤ 5.0 Å and spaced on 0.5 K increments along εi to give 7 values on the range 14.0 ≤ εi ≤ 17.0 K. For the neopentane series, between 126 and 182 parameter sets were evaluated over the range of 5.90 ≤ σi ≤ 6.50 Å and 0.7 ≤ εi ≤ 2.0 K. For the C group, additional exploratory calculations were performed around σi = 3.91 Å and εi = 17.0 (taken from the NERD force field)33 to confirm the final parameters were a true global optimum. The parameter space for the neopentane series was explored on increments 0.05 Å for σi and 0.1 K for εi. Phase diagrams were produced for each parameter set, and used to calculate an average error score, Si. A cumulative average error score for each series was calculated for a specific parameter selection, averaged over the scores for individual compounds. A scoring function was used to evaluate the performance of specific set parameters for each compound in the training set.

(11)

and to the density scaling law for the critical temperatures56 ρliq − ρvap = B(T − TC)β

(12)

where β = 0.325 is critical exponent for Ising-like fluids in three dimensions.57 A and B were constants fit to the saturated vapor and liquid densities. A two-step process was used to determine the critical temperatures. An estimate of the critical temperature predicted by the model was produced by fitting eq 11 and eq 12 to phase coexistence data on the range 0.7TC,expt to 0.9TC,expt. The estimated critical point was used to set the range of temperatures (TC,est − 125 ≤ T ≤ TC,est − 25 K) that were used to determine the true critical point predicted by each model. This approach ensured that the temperature range used in the calculation was close enough to the true critical point to provide an accurate application of eqs 11 and 12, while avoiding finite size effects near the critical point. Critical pressures PC and boiling points TNBP were calculated by fitting vapor pressure data on the range 0.64TC,sim to 0.95TC,sim, to the Clausius−Clapeyron equation ln P = −

ΔHV +C RT

(13)

where P is the vapor pressure, ΔHV is the heat of vaporization, R is the gas constant, and C is a constant. Pressure composition diagrams for all binary mixtures were determined from NPT Gibbs ensemble Monte Carlo simulations,58 except for ethane+isobutane binary mixtures with the NERD and TraPPE force fields, where NVT GEMC simulations were used. Calculations were performed on systems containing 2000 molecules. Simulations were equilibrated for 1 × 108 MCS, and production data were taken from a second 1 × 108 MCS simulation. Statistical uncertainties were determined from three independent sets of simulations, where each simulation was initiated with a different random number seed. The distribution of Monte Carlo moves was 69% displacement, 10% rigid body rotations, 1% volume exchange, and 20% molecule exchange. Radial distribution functions were determined from isobaric−isothermal Monte Carlo simulations, which were performed on cubic boxes containing 1000 molecules. Simulations were performed at the experimental normal boiling point. All simulations were equilibrated for 1 × 108 MCS, and production data were taken from a second 1 × 108 MCS simulation. Radial distribution functions were calculated using VMD.54

Si =

n

n

n

n

1 [0.6135 ∑ Err(ρL (Tj)) + 0.0123 ∑ Err(ρV (Tj)) n j=0 j=0 + 0.2455 ∑ Err(PV(Tj)) + 0.0245 ∑ Err(ΔHV(Tj))

4. RESULTS AND DISCUSSION 4.1. Parameter Optimization. To optimize n-6 potential parameters for branched alkanes, molecules containing exclusively CH or the C type sites were selected. Experimental saturated liquid density data for isomers of nonane and higher molecular weight branched alkanes were scarce, therefore, compounds were selected from isomers of butane, pentane, hexane, heptane, and octane for use in the parameter optimization process. In total there were 20 isomers of butane, pentane, hexane, heptane, and octane which contained only the CH sites. Of these, six were selected for evaluation during the parameter optimization, which included: isobutane, 2-methylbutane, 2-methylhexane, 2,3-dimethylbutane, 2,5-dimethylhex-

j=0 n

+ 0.0613 ∑

j=0

d(Err(ρL (Tj))) dT

j=0 n

∑ j=0

d(Err(ρV (Tj))) dT n

+ 0.0123 ∑ j=0

+ 0.0061 n

+0.0245 ∑ j=0

d(Err(ΔHV(Tj))) dT

d(Err(PV(Tj))) dT

] (14)

where D


Journal of Chemical & Engineering Data Err(X(Tj)) =

XSIM(Tj) − Xexp(Tj) Xexp(Tj)

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optima varied slightly for different molecules, and that the shift in the optima was correlated with the length of the carbon backbone. This illustrates a limitation in the transferability of parameters for the C and CH pseudoatoms to different bonding environments. Therefore, it was decided to split compounds into groups of short (CN ≤ 4 backbone carbon atoms) and long (CN > 4 backbone carbon atoms). By doing this, it was possible to identify transferable parameters for the C and CH pseudoatoms which produced results for the training set, excluding 3,3-dimethylhexane, that were on average within 0.6% and 2.8% of experiment for the saturated liquid densities and vapor pressures, respectively. In addition to these optimized parameters, referred to as the S/L model, a generalized set of parameters for C and CH groups are presented that are expected to produce reasonable results for any compound (on average within 1.4% and 8.0% of experimental saturated liquid densities and vapor pressures, respectively), but are of slightly lower accuracy than the specialized models presented for the short and long compounds. Nath et al. also recognized that a single set of Lennard-Jones parameters had limited transferability for these compounds.33 They hypothesized that the electron cloud around methyl groups varied with the length of the molecule, and developed specialized methyl parameters for branched alkanes with a backbone containing four carbon atoms or less. This included isobutane, 2-methylbutane, neopentane, and 2,2-dimethylbutane. To test this hypothesis, MP2/aug-cc-PVTZ ab initio calculations were performed in Gaussian 09 to determine the electron density and electrostatic potential (ESP) isosurfaces for CH3 bonded to C in 2,2-dimethylbutane and 2,2dimethylpentane. These results, presented in Figure 2, show that the length of the molecule has a negligible effect on the size or shape of the isosurfaces around CH3 groups attached to a C group, while there are small changes to the ESP around the terminal CH3 group furthest from the C group. This suggests that a single set of parameters for the CH3 group should be transferable to a broad range of branched alkanes. The small change in the isosurface around the CH3 pseudo atom furthest from the C group coincides with the transition from four to five carbon atoms in the backbone. In the Mie potentials, this is compensated for by having different C and CH parameters for short and long branched alkanes. Comparison of the optimized Mie parameters to the 12−6 Lennard-Jones parameters of the TraPPE30 and GROMOS60 force fields, listed in Table 4, reveals similar trends for sp3 hybridized groups, with σC > σCH > σCH2 > σCH3 and εCH3 > εCH2 > εCH > εC. As shown in Figure 3, each pseudoatom also overlaps with the nuclei of the bonded carbon atoms. The increase in σi with a decreasing number of hydrogen atoms per pseudoatom results from increased overlap of electron clouds from neighboring carbon atoms that replace hydrogen atoms at the branch point. When reduction in the number of hydrogen atoms per pseudoatom occurs due to a change in hybridization state, an opposite trend is observed: σCH3(sp3) > σCH2(sp2) > σCH(sp) > σC(sp). On the surface, it would seem that these results are in conflict with the parameters of the anisotropic united atom (AUA) force fields,34,35,61 since the optimized AUA parameters for sp3 hybridized groups follow an opposite trend: σCH3 > σCH2 > σCH. The AUA force fields differ from the common united-atom representation by incorporating a shift in the force centers of pseudoatoms away from the location of the atom nuclei. This

*100 (15)

The scoring function was weighted to emphasize the accurate reproduction of saturated liquid densities and vapor pressures. The error in the saturated liquid density received the largest weight as it is known to high precision experimentally for many compounds. Reproduction of experimental liquid densities is critical to the accurate determination of condensed phase physical properties and microscopic structure. The vapor pressure is also important as it is required for accurate predictions of the binary mixture phase behavior. However, experimental vapor pressures determined by different research groups may vary by up to 1%,27 and the relative uncertainties of vapor pressures predicted by histogram reweighting methods in this work were approximately 1.5% near a reduced temperature of 0.6. Hence it is assigned a weight that is 40% of the weight assigned to the saturated liquid density. Errors in the heat of vaporization and saturated vapor density and differential factors with respect to temperature were assigned smaller weights. These factors are used to differentiate between parameter sets with similar overall average absolute error. Values of the scoring function, averaged over all compounds used in the parameter optimization, are presented as heat maps in Figure 1. Heat maps of the scoring function (Si) for individual compounds are shown in Figure S1 of the Supporting Information. Examination of the heat maps for the individual compounds showed that the location of the

Figure 1. Top panel: heat map of average error scores for isobutane series compounds (2-methylbutane, 2-methylpentane, 2-methylhexane, 2,3-dimethylbutane, 2,5-dimethylhexane, and 3,4-dimethylhexane). Bottom panel: heat map of average error scores for neopentane series compounds (2,2-dimethylpropane 2,2-dimethylbutane, 2,2dimethylpentane, 2,2-dimethylhexane, and 3,3-dimethylhexane). Red depicts the best fit to experimental data,6,59 blue depicts the worst fit. E



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Figure 2. Electrostatic potential isosurfaces (ESP) in the plane formed by hydrogen atoms attached to methyl groups in 2,2-dimethylbutane (top) and 2,2-dimethylpentane (bottom). Blue line denotes the van der Waals surface defined at an electron density of 0.001 au.

the repulsive region where substantial differences exist between the various repulsion exponents. 4.2. Pure Fluid Vapor−Liquid Equilibria. A. Short/Long Molecule Optimized Potentials. Deviations of the saturated liquid densities and vapor pressures predicted by the short/long (S/L) molecule optimized potentials for all compounds are listed in Tables 5 and 6, respectively. Vapor−liquid coexistence curves, vapor pressures, and heats of vaporization for selected compounds used in the training set are presented in Figure S2. Saturated liquid densities for isobutane and neopentane series compounds are reproduced to within 0.6% average absolute deviation (AAD) of experiment, except for 3,3-dimethylhexane, where predictions were within 1.6% AAD. Maximum deviations in the predicted liquid densities for the isobutane and neopentane series were less than 2.2% and 2.1%, respectively, for all compounds in the training set. For most compounds, the largest deviations were due to overprediction of the liquid density near the critical point. Comparison of the predicted saturated liquid densities for 2,2-dimethylhexane and 3,3dimethylhexane illustrate a limitation in the transferability of the potential parameters. For 2,2-dimethylhexane, saturated liquid densities were predicted with an AAD of 0.4%, while for 3,3-dimethylhexane, saturated liquid densities were underpredicted by 1.6% AAD. Predicted heats of vaporization were in close agreement with experiment, with AADs ranging from 1.5% (neopentane) to 5.8% (3,3-dimethylhexane). To evaluate model performance comprehensively, phase diagrams for the 21 remaining isomers of butane through octane were predicted. Coexistence densities and vapor pressures for these isomers are shown in Figure S3. For the full set of 32 branched isomers of octane and lighter alkanes, tabulated vapor−liquid coexistence numerical data are listed in Tables S13−S44 of the Supporting Information, while detailed errors with respect to experiment are given for each property in Tables S1−S12. The S/L optimized parameters reproduced saturated liquid densities to within 0.6% AAD of experiment for 17 of the 32 compounds, and to within 2.0% AAD for 26 of 32 compounds. Vapor pressures were predicted to within 3% AAD

Table 4. Comparison of Pseudoatom Diameters for Mie, TraPPE, GROMOS 45A3, and AUA Force Fields GROMOS 45A360

Mie36−38

TraPPE30,62

AUA61

site

σi (Å)

σi (Å)

σi (Å)

δi(Å)

σi (Å)

3

3.783 3.99 4.7 6.1 3.705 3.81 3.57 2.875

3.75 3.95 4.68 6.4 3.675 3.73

3.61 3.46 3.36

0.216 0.384 0.646

3.748 4.070 5.019 6.639

3.48 3.32

0.295 0.414

CH3 (sp ) ethane CH2 (sp3) propane CH (sp3) isobutane C (sp3) neopentane CH2 (sp2) ethene CH (sp2) propene CH (sp) ethyne C (sp) propyne

shift in the force center δ, however, does follow the same trend as the united-atom force fields: δCH > δCH2 > δCH3.61 Increasing δ has an effect that is similar to increasing σ; both methods increase the exclusion volume occupied by the pseudoatoms. Therefore, both UA an AUA models predict an identical trend with the exclusion volume of CH > CH2 > CH3. The Mie potential provides significantly more flexibility than the standard 12−6 Lennard-Jones potential, however, alteration of the repulsion exponent has a small impact on the optimal value of σi. The largest changes occur in the optimal value for εi, which increases with increasing values of the repulsion exponent. Even when using an extreme value of n = 50 for the repulsion exponent, the optimal value of σC was reduced by 4). In addition, a fully generalized set of CH and C parameters were presented that produce reliable results for all compounds, albeit with approximately twice the deviation from experiment as the optimized potentials. The performance of the S/L optimized and generalized models was assessed via calculation of the vapor−liquid coexistence curves for 32 branched isomers of butane through octane. These results were compared to experiment, and the predictions of the TraPPE and NERD force fields. While all force fields provided comparable predictions of saturated liquid densities, the optimized force field predicted vapor pressures that were an order of magnitude more accurate than either TraPPE or NERD. Predictions of the pressure composition behavior for three binary mixtures using the optimized Mie potential were in close agreement with experiment. Given the diversity of bonding environments, developing transferable parameters for the CH and C pseudoatoms was substantially more challenging than for linear alkanes, and a number of limitations in the model were observed. For 2,2,3trimethylbutane and 2,2,3,3-tetramethylbutane phase diagrams were shifted to higher temperatures, and the reproduction of

tinguishable from each other. At 367 K, the Mie potentials predicted results that were in close agreement with experiment, however, for 323 K, substantial deviations were observed. Experimental data show that the solubility of argon in neopentane is expected to decrease with temperature.69 However, simulations predict little change in the solubility of argon in neopentane between 323 and 367 K. The molecular basis for the deviation of simulation from experiment at 323 K is unclear. The choice of combining rules could be a suspect, however, numerous prior calculations for mixtures of similar compounds represented by Mie potentials using Lorenz-Berthelot combining rules have produced pressure−composition diagrams that were in close agreement with experiment.36,37,39 The ethane+isobutane (Figure 5) and methane+neopentane (Figure 6) mixtures presented in this work were also in close agreement with experiment, as was the 367 K isotherm for argon+neopentane. Therefore, we are left with the unsatisfying conclusion that there exists a component of the argon+neopentane interaction that remains unidentified, or there is a flaw in the experimental data.

5. CONCLUSIONS In this work, optimized Mie potentials for branched alkanes were developed. During the optimization process, it was J



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Figure 6. Pressure−composition diagram for methane+neopentane at 298.15 K. Data are represented by experiment (black lines),69 optimized Mie potentials (red symbols), TraPPE29−31 (blue symbols), NERD33 (green symbols). Statistical uncertainties are provided in the Supporting Information and are smaller than the symbol size.

Figure 4. Radial distribution functions for neopentane at 298 K. Top panel: total radial distribution function predicted from experiment (red circles),63,64 optimized Mie potentials (solid black line), and TraPPE29−31 (dashed green line). Bottom panel: pairwise radial distribution functions for C−C, C−CH3, and CH3−CH3 interactions; Mie potentials (solid lines); TraPPE (dashed lines). Predictions of the NERD33 potentials are similar, but have been omitted for clarity.

Figure 7. Pressure−composition diagram for argon+neopentane at 323.1569 and 367.30 K.70 Data are represented by experiment (black lines), optimized Mie potentials (red symbols), TraPPE29−31 (blue symbols), NERD33 (green symbols). Statistical uncertainties are provided in the Supporting Information and are smaller than the symbol size.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.6b01036. Tabulated liquid and vapor densities, vapor pressures, heats of vaporization and compressibility factors, and pressure−composition data for all compounds, mixtures, and models studied in this work; topology and parameter files for use in GOMC for all force fields studied in this work (PDF) Topology file for Mie general model (TXT) Topology file for Mie optimize model (TXT) Topology file for NERD (TXT) Topology file for TraPPE (TXT) Parameter file for Mie general model (TXT) Parameter file for Mie optimize model (TXT) Parameter file for NERD (TXT)

Figure 5. Pressure−composition diagram for ethane+isobutane binary system for 311.25 ≤ T ≤ 394.04K. Data are represented by experiment (black lines),71 optimized Mie potentials (red symbols). Statistical uncertainties are provided in the Supporting Information and are smaller than the symbol size.

experimental vapor−liquid coexistence data was poor. These issues were also observed in the NERD and TraPPE models, and therefore appear to be a fundamental limitation in the transferability of parameters for united-atom force fields. After an exhaustive search of parameter space, it is clear that resolving this issue will require the optimization of unique parameters for C for this specific bonding environment. K



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Parameter file for TraPPE model (TXT)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: jpotoff@wayne.edu; Tel: 313 577 9357. ORCID

Jeffrey J. Potoff: 0000-0002-4421-8787 Funding

Financial support from the National Science Foundation ACI1148168 is gratefully acknowledged. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Many of the computations in this work were performed with resources from the Grid Computing initiative at Wayne State University.

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REFERENCES

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Optimized Mie Potentials for Phase Equilibria: Application to Branched

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