AMBER-ii: New Combining Rules and Force Field ... - ACS Publications

Article pubs.acs.org/JPCB

AMBER-ii: New Combining Rules and Force Field for Perfluoroalkanes Alexei Nikitin,*,† Yury Milchevskiy,† and Alexander Lyubartsev‡ †

Engelhardt Institute of Molecular Biology, Russian Academy of Sciences, Moscow 119991, Russia Division of Physical Chemistry, Department of Materials and Environmental Chemistry, Stockholm University, Stockholm SE-10691, Sweden

‡

S Supporting Information *

ABSTRACT: A molecular mechanics force field of the AMBER/OPLS family for perfluoroalkanes, noble gases, and their mixtures with alkanes has been proposed. We had to abandon the traditional Lorentz−Berthelot combining rules for the Lennard-Jones potential to be able to uniformly describe these substance classes and their mixtures. Instead, the Waldman-Hagler rules developed for noble gases were used for all of these elements except hydrogen. Hydrogen is considered to be a particular substance to which the usual Lorentz−Berthelot rules are applied. The proposed rules have little effect on the organic chemistry of H, C, N, and O elements but make it compliant with the chemistry of heavy elements. Because of assigning a relatively high partial charge of −0.37e to fluorine atoms, the new force field reproduces the mutual insolubility of higher liquid alkanes and perfluoroalkanes.

1. INTRODUCTION The chemistry of perfluoroalkanes stands apart in organic chemistry; however, completely fluorinated alkanes are amazingly similar to normal ones by their properties and largely comply with the classical organic chemistry.1,2 This similarity is due to the low polarity of perfluoroalkanes despite the extreme electronegativity of fluorine, which can be attributed to a very low dipole (and quadrupole) moment of symmetrically substituted alkanes as a result of the tetrahedral orientation of carbon bonds. As their molecular weight grows, both alkanes and perfluoroalkanes share similar properties. Low-molecular-weight gaseous substances are followed by light hydrophobic fluids, oils, and finally high-molecular-weight solid polymers such as polyethylene and Teflon.3 Perfluoroalkanes have a number of interesting and practically significant properties. For instance, the conformational properties differ from hydrocarbon chains. Trans-conformation is not typical for perfluoroalkanes. Commonly, the energy minimum lies in a slightly shifted anticonformation, which gives rise to the helicity of long polymeric chains.1 The ortho-conformation exists, which is impossible in normal alkanes.4,5 Perfluoroalkanes are good solvents for gases such as noble gases and oxygen.6 Being hydrophobic, both classes of substances are infinitely mixable with each other only for low-molecular-weight representatives roughly up to C6.7,8 Fluorinated polymers have numerous applications because of their piezoelectric properties.9,10 These interesting properties deserve adequate modeling; however, this faces a number of problems. Because the substances are nonpolar, their properties substantially depend on weak dispersion interactions. The calculation of such interactions is among the most complex © 2015 American Chemical Society

quantum chemical tasks. For perfluoroalkanes this is further complicated by several factors. First, it is particularly important to adequately consider the correlation of electrons in the calculations of fluorine (as well as noble gases). While MP2 level calculations usually suffice for alkanes, for perfluoroalkanes more sophisticated methods are desirable. Second, the calculation cost further increases because fluorine is a heavier element than hydrogen and thus requires the application of a larger basis sets. As a result, the calculation time of similar tasks for alkanes and perfluoroalkanes differs by several orders of magnitude. On the one hand, fine dispersion effects should be calculated; on the other hand, the amount of computations for the quantum-mechanical task is much higher. High-quality calculations for the most interesting perfluoroalkanes that are fluid under normal conditions are hardly possible. Molecular mechanics modeling of perfluoroalkanes was performed many times, and several corresponding force fields have been published.4,11−14 Although the assigned job was usually done, important problems remain. For instance, the transferability of such fields, particularly, to bioorganic chemistry encounters a number of difficulties. First, the abnormally high gas solubility is hard to model. At the same time, this property is of both theoretical and practical interest (for instance, perfluorochemical blood substitutes).15−17 The immiscibility of certain fluids also finds no explanation in simple models.18 At the same time, this property underlies unique features of fluorine-containing surface-active comReceived: July 26, 2015 Revised: October 23, 2015 Published: October 26, 2015 14563

DOI: 10.1021/acs.jpcb.5b07233 J. Phys. Chem. B 2015, 119, 14563−14573

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The Journal of Physical Chemistry B

weakened in the second one. Results generated using rules of the same group are similar; therefore, only the Lorentz− Berthelot and Waldman−Hagler rules are considered here. Lorentz−Berthelot rules:

pounds. For instance, a combination of alkane and perfluoroalkane tails makes it possible to generate not naturally occurring trilayer analogs of biological bilayer membranes.19,20 Song et al.18 considered both problems: the interaction with noble gases and the mutual immiscibility. They were guided by the experimental data, in particular, by the second cross virial coefficients of methane, carbon tetrafluoride, and noble gases. Song et al. tried to modify the OPLS4 force field to correctly describe the interaction between all three substance classes. The elementary solution is not to rely on the combining rules (geometric mean for OPLS) but to introduce special (decreased) parameters for the fluorine−hydrogen interaction. While this solution works, it is not satisfactory because of the special rule for fluorine. The introduction of this rule has no background. The transfer to different models will likely require additional rules for the fluorine interaction with other elements. Fluorine becomes a mystically particular element. The second approach consisted of the selection of combining rules and parameter values providing for the desired decrease in the H−F interaction. Indeed, Song et al. found that the Fender−Halsey, Kong, Waldman−Hagler, and Halgren rules provide for the desired result. The Waldman−Hagler rules developed for noble gases were the best choice. At the same time, the values of the Lennard-Jones parameters that produced perfect results for alkanes, perfluoroalkanes, and their mixtures proved to be inapplicable for the interaction between noble gases and alkanes. Thus, the attempts to build a common force field for all three substance groups based on the available combining rules face challenges. The desired force field can be generated if two considerations are taken into account: (1) If it comes to exceptional elements, it is more natural to recognize hydrogen rather than fluorine. Indeed, this is the first element with just a single electron and variety of other specific features. Suffice it to recall the hydrogen bond. In contrast, a remarkable feature of fluorine is high electronegativity. (2) High electronegativity of fluorine leads to substantial bond polarization. At the same time, the absence of a strong electrostatic field around perfluoroalkane molecules is due solely to the symmetric arrangement of fluorine atoms; however, the electrostatic field can be significant in the short range. Song et al.18 tested the effect of partial charges of OPLS atoms and concluded that their effect is practically absent; however, the OPLS charge of −0.12e on fluorine looks underestimated. Setting the value to −0.2e or higher critically improves the situation. Now electrostatics provides for the dominant effect, which can explain the immiscibility of certain fluids. Let us discuss how these two considerations can be used to construct an appropriate force field.

1 σij = (σii + σjj) 2 εij = εiiεjj

(1)

Waldman−Hagler rules: σij =

6

σii 6 + σjj 6 2

⎛ σ 3σ 3 ⎞ ii jj ⎟ εiiεjj εij = 2⎜⎜ 6 6⎟ σ ⎝ ii + σjj ⎠

(2)

where σij is size parameter in angstroms and εij is well depth parameter in kilocalories per mole. The effect of combining rules can be visualized by the crosssecond virial coefficients of volatile substances. Figure 1 shows

Figure 1. Correlation of observed27 (abscissa) and calculated (ordinate) cross second virial coefficients B12/cm3·mol−1 of CH4, CF4, He, Ne, Ar, Kr, and Xe. Calculations are made using three types of combining rules: green triangles, Lorentz−Berthelot rules; blue circles, Waldman-Hagler rules; and brown squares, proposed in this paper mixed rules (AMBER-ii) (see text). The force field parameters were taken from the Song et al. supplementary tables.18

the correlation plot between experimental27 cross-second virial coefficients for CH4, CF4, He, Ne, Ar, Kr, and Xe and the coefficients from the models generated using three types of combining rules: green triangles, Lorentz−Berthelot rules; blue circles, Waldman-Hagler rules; and brown squares, Waldman− Hagler rules in the absence of hydrogen or Lorentz−Berthelot rules if hydrogen is one of atoms. The plot demonstrates that weak interactions (right upper part) of compounds with elements of the first and second periods are described by different rules with roughly the same accuracy. The lower part of the plot demonstrates that both Lorentz−Berthelot and Waldman−Hagler rules yield incorrect results for the interactions with heavy elements (Kr and Xe).

2. THEORETICAL BACKGROUND 2.1. Choice of the Combining Rules. There are two groups of combining rules: (1) geometric mean both for sigma and epsilon (in OPLS) and Lorentz−Berthelot21,22 rules (in AMBER) and (2) specific rules such as Fender−Halsey,23 Kong,24 Waldman−Hagler,25 and Halgren.26 The first group adequately describes bioorganic chemistry and is widely used in it. The second one is good for noble gases but also finds application in bioorganic simulations. These groups differ in that the parameters of dissimilar atoms are considered as almost an additive in the first group, while the interaction between substantially different atoms is considered to be significantly 14564


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irrelevant effects. We propose to use tetrahedrane and its analogues with Si, Ge, and Sn; see Figure 3.

Interestingly the errors signs are opposite for the different rules. We assume that the Waldman−Hagler rules are adequate for all elements except hydrogen. At the same time, the Lorentz− Berthelot rules became popular because organic compounds used in calculations nearly always contain hydrogen and usually contain elements of the second and, more rarely, third periods. The upper part of the plot demonstrates that the Lorentz− Berthelot rules are adequate in such circumstances. Accordingly, we propose to consider hydrogen as a special substance for which the Lorentz−Berthelot rules are applicable. This practice is common in biomolecular force fields (AMBER,28,29 OPLS,30 CHARMM31) that use either the Lorentz−Berthelot rules or the geometric mean. For other elements, the Waldman−Hagler rules are applicable. This primarily agrees with the data on noble gases and with certain force fields (for instance, AMOEBA32). This approach is shown with brown rectangles on the Figure 1. This illustration substantiates the idea of special rules for hydrogen; however, it is desirable to independently demonstrate their adequacy. Until now, we have considered only methane as hydrogen-containing substance. For instance, let us assume that methane itself possesses a certain overlooked property that underlies such results. To understand this issue, we will pay a special attention to hydrogen. 2.2. Are Lorentz−Berthelot Rules Applicable to Hydrogen? Let us consider second cross virial coefficients of molecular hydrogen (H2) and noble gases. The Figure 2

Figure 3. Spatial configuration of noble gases (blue balls) around tetrahedrane and its analogues, used to demonstrate the applicability of the new rules to carbon and its analogs.

Because of the tetrahedral symmetry, these substances have neither dipole nor quadrupole moments, which minimizes electrostatic effects. We will consider their interaction with neon and krypton, which obviates electrostatic effects from the other side as well. Tetrahedrane was also selected because it has an open carbon surface, while the hydrogen is far enough to have no effect comparable to that in methane. Furthermore, tetrahedral models are easy to calculate using quantum chemistry software that can make use of symmetry.33 As a result, the calculation is reduced to three atoms: carbon analog, hydrogen, and noble-gas atom. The noble-gas parameters were set as in the modeling of second virial coefficients.18 The hydrogen parameters were taken from AMBER-ii.34 The parameters of carbon and analogs were selected to reproduce the geometry and intermolecular interactions in complex with krypton, as obtained using quantum chemical calculations pbe0-D3/def2-qzvpp. Two sets of parameters were generated for the Lorentz−Berthelot rules alone and for the Lorentz−Berthelot rules for hydrogen and the Waldman−Hagler rules for other elements. Then, the models with these parameters were tested on the complexes with neon. The results are shown on the Figure 4. One can see that the hybrid parameters describe the interaction energy much better than the Lorentz−Berthelot rules. The next test was carried out for the interaction of H2, N2, O2, F2, P2, S2, Cl2, As2, Se2, and Br2 with neon and krypton. As previously, initially the models of krypton interaction were generated, and then neon was substituted for krypton in the test. The geometry of complexes was computed using pbe0-D3/ def2-qzvpp. The interaction energy was evaluated at the LPNO-CEPA/CBS(aug-cc-pVQZ) level.35 In Figure 5, one can see that the Lorentz−Berthelot and hybrid rules produce similar results for elements of the second period, while the hybrid rules are much more accurate for elements of the third and fourth periods. The least accurate results were obtained for S2 and Se2. The complication here is that they are biradicals similar to molecular oxygen. 2.4. Test with Chlorine. To not be limited by a single halogen, specifically considering that fluorine is often credited

Figure 2. Comparison of experimental27 and calculated cross second virial coefficients B12/cm3·mol−1 of molecular hydrogen (H2) and noble gases. The force-field parameters were taken from the Song et al. supplementary tables.18

presents the experimental data27 (at 300 K) as well as the data from models using the Lorentz−Berthelot and Waldman− Hagler rules. As anticipated, the Lorentz−Berthelot rules reproduce the experiment. Thus, such behavior is inherent for hydrogen but is not due to its particular chemical environment as, for example, in methane. 2.3. Are Waldman−Hagler Rules Applicable to Other Elements? The Waldman−Hagler rules were developed specifically for noble gases.25 At the same time, their applicability to other heavy elements deserves corroboration. Carbon and its analogs are of particular interest. Because no appropriate data on second virial coefficients are available, we will turn to quantum chemistry. The application of combining rules can be illustrated by model compounds with insignificant 14565


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Figure 6. Comparison of experimental27 and calculated cross second virial coefficients B12/cm3·mol−1 of carbon tetrachloride with noble gases, nitrogen, and methane.

Figure 4. Energy of intermolecular interaction: tetrahedrane analog− neon (kcal/mol). The force-field parameters of the noble gases were taken from the Song et al. supplementary tables.18 The force-field parameters of C, Si, Ge, and Sn were taken to reproduce interactions in complex with krypton, as obtained using quantum-chemical calculations pbe0-D3/def2-qzvpp; see the Supporting Information.

applied to all other elements, all considered elements are adequately described. The efficiency of this approach can be illustrated by our AMBER-i force field, where it was first employed.36 Here new combining rules are applied for the development of the force field for perfluoroalkanes compatible with that for alkanes developed in the first part.34 The application of hybrid combining rules does not affect the alkane force field, and the force field described in the first part remains unaltered. 2.5. Electrostatics Influence. The acceptance of new combining rules gives us a new framework for force-field development uniformly describing both light and heavy elements. Separate parametrization of the density and evaporation heat for liquid hexane and perfluorohexane described in more detail later gave us a force field with clearly improved properties. One of properties hardly reproducible in a mechanical model is excessive mixing enthalpy of alkanes and perfluoroalkanes.18 Figure 7 shows the mixing enthalpy ΔHmix for hexane and perfluorohexane. The calculation was carried out according to

Figure 5. Energy of intermolecular interaction: diatomic molecule− neon (kcal/mol).

ΔHmix = (1 − xC6F14)H0 + xC6F14H1 − Hx

where Hx is the enthalpy of the mixture of molar ratio x. The mixing enthalpy for a 50% mixture is shown with circles and

with abnormal properties, let us consider the interaction of carbon tetrachloride with several gaseous substances. The Figure 6 presents the second cross virial coefficients for CCl4.27 The hybrid rules correspond best to the experimental data. The Lorentz−Berthelot rules also give similar result except for helium, where they are the worst. The particular role of hydrogen is clearly seen here. Although the properties of helium and hydrogen are similar, the Waldman−Hagler rules are inapplicable to hydrogen. A pronounced disagreement with the experiment for the Waldman−Hagler rules is also observed for methane, which further supports particular behavior of hydrogen-containing compounds. The greatest difference from the experiment is observed for nitrogen for all combining rules. It should be noted that an electroneutral two-site model of nitrogen was used here, while nitrogen has a notable quadrupole moment. (A three-site model may be a better choice.) Thus, neither Lorentz−Berthelot nor Waldman−Hagler rules can be used alone to describe systems including both hydrogen and heavy elements. At the same time, if hydrogen is considered as a particular substance and the Lorentz−Berthelot rules are used for it while the Waldman−Hagler rules are

Figure 7. Mixing enthalpy of hexane and perfluorohexane mixtures (kcal/mol). X axis: perfluorohexane content. 0 or −0.37e partial charges on the fluorine atoms. 1 ns MD of 144 molecules. Experimental data are from ref 38 14566


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The Journal of Physical Chemistry B Table 1. van der Waals Parameters for Fluorine AMBER94 [1] OPLS-AA [2] Mod−OPLS [3] AMBER-ii MP2/6-311+G(2d,p) MP2/cc-pVTZ MP2/cc-pVQZ MP2/cc-pV5Z extrapolated to MP2/cc-pV6Z MP2/aug-cc-pVTZ MP2/aug-cc-pVQZ extrapolated to MP2/aug-cc-pV5Z MP2/daug-cc-pVTZ MP2/daug-cc-pVQZ extrapolated to MP2/daug-cc-pV5Z MP4/aug-cc-pVTZ MP4/aug-cc-pVQZ extrapolated to MP4/aug-cc-pV5Z LPNO-CEPA_1/cc-pVQZ DLPNO-CCSD(T)/cc-pVQZ F12/RI-CCSD(T)/cc-pVDZ F12/RI-CCSD(T)/cc-pVTZ F12/RI-CCSD(T)/cc-pVQZ extrapolated to F12/RI-CCSD(T)/cc-pV5Z LPNO-CEPA_1/aug-cc-pVTZ LPNO-CEPA_1/aug-cc-pVQZ extrapolated to LPNO-CEPA_1/aug-cc-pV5Z a

R* (Å)

ε (kcal/mol)

1.7500 1.6556 1.6276 1.5800 1.6230 1.6055 1.6333 1.6330 1.6327 1.6078 1.5888 1.5748 1.5602 1.5806 1.6086 1.5825 1.5777 1.5743 1.6434 1.6537 1.5663 1.5667 1.5878 1.6032 1.6179 1.5984 1.5871

0.0610 0.0530 0.0608 0.0802 0.1119 0.0870 0.0788 0.0752 0.0714 0.1071 (0.0675)a 0.1010 (0.0728) 0.0966 (0.0767) 0.1662 (0.0688) 0.1310 (0.0772) 0.0826 (0.0887) 0.1285 0.1201 0.1139 0.0728 0.0683 0.1340 0.1307 0.1055 (0.0966) 0.0877 0.0983 (0.0561) 0.0868 (0.0610) 0.0803 (0.0639)

Counterpoise-corrected values in parentheses. bR*: equilibrium distance; ε: van der Waals well depth.

Separate combining rules for hydrogen (Lorentz−Berthelot) combined with the Waldman−Hagler rules for other elements as well as the adjustment of the partial charge on fluorine atoms make it possible to construct a force field simultaneously describing alkanes, perfluoroalkanes, and noble gases. Because the models rely on simple combining rules, there are grounds to believe that the proposed model can become a general model for organic compounds. The Lorentz−Berthelot rules are widely used for the modeling of biomolecules with elements having generally similar Lennard-Jones parameters. The proposed model has little impact on them. At the same time, the Waldman−Hagler rules provide the basis for the extension of the AMBER/OPLS force fields to heavy elements.

triangles for the Lorentz−Berthelot rules and the AMBER-ii rules in the models with no account for electrostatics, respectively. The result slightly improved, although it is still far from the experiment. Similar results are obtained when partial charges on fluorine are set to −0.12e (OPLS4 and Song et al.18); however, these charges seem to be underestimated. We defined potential derived charges (PDCs) as with the AMBER force field.29 The PDC method works poorly for perfluorinated compounds due to the low electrostatic field around the molecules with symmetrically arranged fluorine atoms. Depending on the compound and calculation method, the resulting values vary from −0.2 to −0.4e but never drop below −0.2e (natural charges37 CF4 at PBE0/Def2-TZVPP level are −0.298e). It can be shown that the charges in this range improve the situation with excessive mixing enthalpy. Because the PDC algorithm does not allow the charge on fluorine to be unambiguously specified, we propose to consider the partial charge as an adjustable parameter that should be set to adequately reproduce experimental data on excessive mixing enthalpy. This value was found to be equal to −0.37e. This case is shown by pink squares on the plot. One can see that this approach adequately reproduces experimental data (blue diamonds on the plot). It is of interest that at the −0.37e charge both the Lorentz−Berthelot rules (brown asterisk on the plot) and hybrid rules produce the values close to experimental ones. Thus, according to our model the nonadditivity of mixing enthalpy for hexane and perfluorohexane is due to the electrostatic factor.

3. COMPUTATIONAL METHODS Molecular mechanical calculations were carried out on the Ascalaph and Abalone programs.36 Comparisons of molecular conformations always included geometry optimization. Molecular dynamics simulations were accelerated with NVIDIA GeForce GTX 560 Ti and GeForce GTX 670 video cards. A four-level r-RESPA integrator39 was used in simulations. Bonded interactions were computed with a 0.25 fs step and nonbonded interactions were computed with steps 0.5 fs at 3 Å cutoff, 1 fs at 6 Å cutoff, and 2 fs up to the half-box length cutoff in periodic boundary conditions. We used Andersen thermostat40 and Berendsen barostat41 with relaxation time τ = 2 ps with pressure and energy correction for potential cutoff.42 As it is common in the AMBER family of the force fields, LennardJones 1−4 interactions (of atoms separated by three covalent bonds) were scaled by factor 0.5, and 1−4 electrostatic 14567


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neopentane (0.035 kcal/mol).34 As a result, the epsilon of fluorine became equal to 0.0802 kcal/mol after the adjustment for the thermodynamic properties of fluid perfluoroalkanes, which agrees well with quantum-chemical calculations. R* became equal to 1.58 Å, which is also in agreement with quantum chemistry. 4.1.2. C−F Bond Length. A long time ago the developers of AMBER9448 acknowledged the problems related to the C−F bond parametrization. The C−F bond length significantly depends on the number of fluorine atoms at carbon. AMBER-ii makes no attempt to specify a certain average level; instead all four cases are considered separately (Table 2).

interactions were scaled by factor 0.83. The minimum time of molecular dynamics simulation is 1 ns. Quantum-chemical computations were carried out on ORCA,35 NWChem,33 and Firefly programs43,44 in the Ascalaph36 graphical environment. Free energies of solvation were calculated by the expanded ensemble method45 coupled to the Wang−Landau algorithm46 using the MDynaMix v.5.2.5 package.47

4. RESULTS AND DISCUSSION 4.1. Force Field for Fluoroalkanes (Conformational Properties). The development of a force field for fluorinated compounds is complicated by several factors apart from the modification of the combining rules. First, higher calculation accuracy is required for fluorine than for carbohydrates to achieve the same quantum chemical accuracy. Considering a higher proportion of heavy atoms (fluorine) in perfluorinated alkanes, high-quality quantum-chemical calculations are a challenge. Hence, the accuracy of quantum chemical data is slightly decreased compared with that obtained for the parametrization of alkanes.34 Second, while the C−H bond length remains virtually constant, the length of the C−F substantially depends on the environment.48 Disregarding this circumstance makes it impossible to obtain the thermodynamic properties of fluorine compounds corresponding to the experiment. In this context, we had to introduce a number of environment-dependent parameters for the C−F bond. Third, accurate evaluation of the partial charges in perfluorinated compounds is complicated by the weakness of the electrostatic field around perfluoroalkanes despite the extremely high electronegativity of fluorine, which is due to the symmetrical arrangement of fluorine atoms. The partial charge values obtained by the PDC method considerably vary depending on the procedure parameters. That is why the partial charges on fluorine were an adjustable parameter aimed at good thermodynamic properties of the modeled fluids. 4.1.1. Lennard-Jones Parameters for Fluorine. Unfortunately, the parameters of the Lennard-Jones potential are hard to obtain from quantum-chemical calculations. Accordingly, quantum chemistry is used to obtain preliminary results, after which the parameters are adjusted to reproduce the thermodynamic properties of liquid substances as described later (the OPLS approach49,30,50,4). Assuming that the fluorine parameters are transferable, they were initially defined using the van der Waals interaction between two fluorine molecules. The molecules were arranged perpendicular to each other, and their geometry was optimized using the quantum-chemical methods listed in the Table 1. The AMBER/OPLS force fields set fluorine epsilon equal to 0.05 to 0.06 kcal/mol, while it ranges from 0.07 to 0.13 kcal/mol according to quantum-chemical calculations. Our best quantum-chemical value is 0.09 kcal/mol. This discrepancy between the classical and quantum chemical evaluations can result from the assumption that carbon is the same in perfluoroalkanes and normal alkanes, which is not a good approximation. Similarly to substituted alkanes,34 the epsilon of carbon should decrease as the degree of substitution on carbon increases. In this respect, perfluoroalkanes are analogous to neopentane. Because the AMBER/OPLS force fields do not provide for such decrease, they had to compensate it by decreasing the epsilon of fluorine to reproduce the thermodynamic properties of fluid perfluoroalkanes. At the same time, AMBER-ii sets the epsilon of carbon as low as that in

Table 2. C−F Bond Length in Accordance with the Quantum-Chemical Calculations MP2/6-311+G(2d,p)

R0(C−F) (Å)

CF4 CHF3 CH2F2 CH3F

1.322 1.338 1.361 1.391

4.1.3. Torsional Potential. Fluoroalkanes have interesting conformational properties. In alkanes, the trans-conformation is the most energetically favorable, while it is the gauche conformation in difluoroethane. Figure 8 shows the torsional potentials calculated using quantum and classical mechanics with the parameters shown in Table 3. In the case of perfluoroalkanes, the minimum energy is also reached not in trans-conformation but in a closely spaced anticonformation. The second local minimum is gauche conformation. In contrast with alkanes, there is the third local ortho near 90°.5 Watkins and Jorgensen4 introduced fairly hard torsional potentials with the V1 up to 8 kcal/mol to describe the conformation of perfluoroalkanes in the OPLS-AA force field. Here we use a single term torsion potential acting on all CF*CF2-CF2-CF* and CF*-CF2-CF2-F torsional rotations (but not on F-CF*-CF*-F), which, mediated by nonvalent interactions of F and C atoms, provides relative energies of the local minima in a reasonable good agreement with highlevel quantum-chemical calculations, as shown in Tables 4 and 5. A substantial deviation from quantum-chemical calculations is observed only for the gauche−gauche−gauche conformation of perfluorohexane; however, this should have no significant impact on the results of the fluid modeling because this conformation has a relatively high energy and will not be present in significant concentration. We would like to stress once more that quantum-chemical calculations of perfluorinated compounds do not give accurate results, so no attempts were made to achieve full correspondence with them. In our opinion the obtained results are within the error of quantum-chemical calculations performed, which can be judged from the differences in energies obtained by different quantum-chemical methods (Tables 4 and 5). All developed parameters are available in the Supporting Information in a form directly applicable in the MDynaMix,47 Ascalaph, and Abalone software.36 4.2. Adjustment to Experimental Thermodynamic Data. As is common in the OPLS method,49,30,50,4 the Lennard-Jones parameters were defined according to the experimental data on the density and evaporation heat of 14568


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Figure 8. Torsional potential of (a) 1,2-difluoroethane and (b) perfluorobutane. Comparison of the quantum-mechanical and our molecular mechanical potential: energy (kcal mol−1) versus dihedral angle (deg).

Table 3. Torsional Parameters Type1

Type2

Type3

Type4

V(n)/2 (kcal/mol)

path no.

**

CT

**

CF3/ CF2 CF3/ CF2

CF2

Alkanes ** 9 Perfluoroalkanes ** 9

CF2

F

CF CF CF CF

CF CF CF CF

F

** F F F

CT

1

Table 6. Thermodynamic Properties of Liquids Computed in AMBER-ii and Compared with Experimental Dataa,b

gamma (deg)

1.45

n

0.0

3

1.45

0.0

3

0.0

0.0

3

0.0 0.0 0.0 180.0

3 3 2 1

Vicinal Difluoroalkanes ** 9 1.45 F 1 0.7 F 1 1.4 F 1 0.9

C6H14 C6F14

MP2/6-311+G(2d,p) LPNO-CEPA_1/CBS(aug-cc-pVQZ)// MP2/aug-cc-pVTZ AMBER-ii a

gauche

ortho

transa

0 0

0.33 0.536

1.75 1.404

0.19 0.177

0

0.34

0.93

0.32

ρexp (g/cm3)

ρ error (%)

ΔHcalc (kcal/mol)

ΔHexp (kcal/mol)

ΔH error (%)

0.6604 1.6707

0.655c 1.6717d

0.82 −0.12

7.605 7.534

7.58c 7.51d

0.28 0.29

1 ns MD of 144 molecules. bρ: liquid density; ΔH: heat of vaporization, ΔH = Einter + RT. cExperimental data from ref 51. d Experimental data from ref 4. a

In contrast with OPLS, new combining rules considering hydrogen as a unique element were used. Partial charges on fluorine atoms were selected to accurately describe the interaction between alkanes and perfluoroalkanes. Excess mixing enthalpy of hexane and perfluorohexane shown in Figure 7 demonstrates good correspondence to experimental data in the whole concentration range. 4.3. Properties Not Used as Target Data in the Parametrization. It is of interest to explore other properties of the model not involved in the parametrization. Primarily, does our model demonstrate that low-molecular alkanes can infinitely mix with perfluoroalkanes in liquid phase while highmolecular ones separate from each other? It is common knowledge that n-CkF2k+2 and n-CkH2k+2 can infinitely mix at k < 7 and separated at higher values under normal conditions. Figure 9 shows excessive mixing enthalpy of n-C8F18 and nC8H18. After reaching a certain value (0.5 kcal/mol), no further mixing enthalpy increase is observed and a plateau is established. This can be interpreted as separation. (The point at the 0.375 ratio drops out because n-C8F18 crystallized, apparently due to the small model volume.)

Table 4. Perfluorobutane: Comparison of Conformation Energy Differences (kcal mol−1) Relative to the antiConformation anti

ρcalc (g/cm3)

In trans-conformation it is a saddle point rather than minimum.

pure hexane and perfluorohexane. Initial guess for the parameters was determined from quantum-chemical calculations (Table 1), and they were finally fine-tuned by fitting to the experimental thermodynamical data. Comparison of the computed and experimental values of density and evaporation heat is given in Table 6.

Table 5. Perfluorohexane: Comparison of Conformation Energy Differences (kcal mol−1) Relative to the anti-anti-antiConformationa MP2/6-311+G(2d,p) RI-MP2/cc-pVQZ RI-MP2/aug-cc-pVTZ LPNO-CEPA_1/aug-cc-pVTZ//RI-MP2/aug-cc-pVTZ LPNO-CEPA/1(TCutPairs5e‑6)/aug-cc-pVTZ//RI-MP2/aug-cc-pVTZ AMBER-ii a

aaa

aga

ggg

aoa

0 0 0 0 0 0

0.65 0.68 0.667 0.623 0.700 0.82

2.32 2.73 2.418 2.626 2.592 7.55

1.87 1.89 1.818 1.643 1.706 1.94

a, anti-conformation; g, gauche- conformation; o, ortho-conformation. 14569


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returned to the simulation box according to the periodic boundary conditions to show that alkane molecules diffuse unlike crystallized perfluoroalkane. Snapshots of the mixed system in Figures 10 and 11 illustrate the appearance of phase separation for longer molecules. More rigorous evidence of phase separation is obtained from the free energies of insertion of C8F18 and C8H18 molecules into the pure fluids and their 50% mixture. The free energy of the mixture proved to be higher on 2.9 kJ/mol than the total free energy of pure solvents (Table 7). Thus, the 50% mixture is thermodynamically unstable, and separate alkane-rich and perfluoroalkane-rich phases should exist. We tested whether the new force field provided realistic description of dynamical properties by evaluating the selfdiffusion coefficients of perfluoroalkanes in gas and liquid phases. The results are given in Table 8. In the literature we found experimental self-diffusion coefficients for perfluoroalkanes only in a gaseous phase, for which our result for C6F14 fits well. Our self-diffusion coefficients are also close but 20−25% below those estimated from MD simulations by McCabe et al.53 In that work, the viscosity of perfluoroalkanes was also computed and a very good agreement with experimental viscosity data was found.53 Because viscosity and diffusion usually have similar scaling behavior, we can expect that our force field may overestimate viscosity of short liquid perfluoroalkanes by 20−25%. 4.4. Compatibility with Other Force Fields. The force field described here is an extension of the AMBER-ii force field previously proposed.34 The carbohydrate portion of AMBER-ii remains unaltered. It is also compatible with the AMBER/ OPLS family of force fields. The new extended force field introduces the Waldman−Hagler combining rules for elements other than hydrogen. This has no impact on our previous results because previous rules remain in effect for the hydrogen−carbon interaction, and the Waldman−Hagler rules have also no effect on carbon considering that its radius is constant (see eqs 1 and 2). The new rules have an effect on all other elements; however, this effect is minor for “bio-elements” because their radii and dispersion properties are similar. In this context, the application of the new rules to current force fields requires only a minor reparameterization for other second raw elements if any at all. This is illustrated by modeling of several substances composed of C, H, N, and O using both the proposed and Lorentz−Berthelot rules. Because the properties of elements of the second period are similar to those of carbon, the choice of the rules has a minor effect. Indeed, the changes in the thermodynamic properties were as low as 2% or less (Tables 9 and 10); however, the new rules may have a significant impact on heavy elements. This will be the subject of further analysis. The goal of this work was to propose an approach bringing force fields for fluorinated compounds to a form compatible with popular and efficient bioorganic force fields.

Figure 9. Calculated excess enthalpy of n-C8F18 and n-C8H18 mixtures (kcal/mol). X axis: perfluorooctane content. Four ns MD of 128 molecules.

A mixture of n-C9F20 and n-C12H26 demonstrates more or less clear separation. One can see separate layers after 6 ns in Figure 10.

Figure 10. Mixture of n-C9F20 and n-C12H26 was separated in 6 ns. Individual layers are clearly visible, which is not seen for more lowmolecular substances. Figure shows a periodic box of doubled size (eight boxes in space).

C6 and C8 are given for reference. The simulations were shortened so that the diffusion (that is, average distances passed by the molecules during simulation time) roughly corresponds to the same level. Perfluoroalkane n-C10F22 was found to be crystallized in the model, which corresponds to its natural behavior. This is the first perfluoroalkane whose melting temperature (36 °C52) is higher than normal conditions. Figure 11 shows a periodic box of doubled size (eight boxes in space). The molecules were not

5. CONCLUSIONS A force field for perfluoroalkanes has been proposed. At the same time, new combining rules have been proposed for the Lennard-Jones parameters. The new parametrization makes it possible to simultaneously describe the interaction between alkanes, perfluoroalkanes, and noble gases. In particular, this allows partial mutual solubility of fluid alkanes and perfluoroalkanes to be modeled. Experimentally defined properties such as second virial coefficient of gaseous substances as well as

Figure 11. Mixture of n-C10F22 and n-C10H22 after 10 ns. Perfluoroalkane crystallized. 14570


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The Journal of Physical Chemistry B Table 7. Free Energy of Molecule Insertion into Pure Solvent or Mixturea μ (kJ/mol)

composition μF (C8F18 into 50% mixture) μF0 (C8F18 into C8F18) μH (C8H18 into 50% mixture) μH0 (C8H18 into C8H18) mixture (μF + μH) − pure compounds (μF0 + μH0) a

−13.20 −13.61 −17.35 −19.82 −30.55

± ± ± ± −

0.80 0.82 0.38 0.39 −33.43 = 2.88

10 ns expanded ensemble MD: 128 molecules, 40 subensembles, 12 Wang−Landau iterations, soft transition potential with offset 2 Å.

Table 8. Self-Diffusion Coefficients of Perfluoroalkanes in the Proposed Force Fielda proposed by the model D × 10−5 (cm2 s−1)

published data D × 10−5 (cm2 s−1)

2.16 ± 0.2 0.93 ± 0.1d 3094 ± 300e

2.62c 1.20c 3220f

b

C4F10 liquid, 298 K C6F14 liquid, 298 K C6F14 gas, 298 K, 0.253 atm a

Calculated from the Einstein relationship as implemented in the Abalone program.36 b10 ns simulation of 216 molecules. cMolecular-mechanical explicit-atom (EA) model.53,54 The comparison was made with computational data since no experimental ones are available to our knowledge. d10 ns simulation of 144 molecules. e6 ns simulation of 144 molecules in the gas phase, Bussi−Donadio−Parrinello thermostat at τ = 0.02 ps. Berendsen barostat at τ = 0.01 ps. fExperimental data.55

■

Table 9. Effect of Combining Rules on the Density of Liquids Containing C, H, N, and O (g/cm3)

acetamide methanol ethanol toluene p-cresol

T (K)

Lorentz−Berthelot rules (AMBER99)

proposed combining rules

experimental data

373.15 298.15 298.15 298.15 303.15

0.989 0.742 0.780 0.829 0.987

0.980 0.740 0.777 0.829 0.981

0.981 0.786 0.785 0.865 1.025

* Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b07233. The force field, molecular models and configuration files in the format of Ascalaph software.36 (ZIP)

■

acetamide methanol ethanol toluene p-cresol

T (K)

proposed combining rules

experimental data

494.3 298.15 298.15 298.15 303.15

12.18 6.69 9.85 8.29 12.24

11.92 6.65 9.78 8.28 12.10

13.40 8.95 10.11 8.84 11.14

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Table 10. Effect of Combining Rules on the Enthalpy of Vaporization of Liquids Containing C, H, N, and O (kcal/ mol)a,b Lorentz−Berthelot rules (AMBER99)

ASSOCIATED CONTENT

S

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work has been carried out with the financial support of the Program of the Presidium of the Russian Academy of Sciences for Molecular and Cellular Biology and the Russian Foundation for Basic Research (Grant No. 14-04-01269) and Swedish Research Council (Vetenskapsrådet). We are grateful to Nikita Vassetzky for rendering the paper into English.

■

a

Thermodynamic properties of liquids do not substantially change with the combining rules used. b2 ns MD of 125 molecules.

REFERENCES

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