Letter pubs.acs.org/JPCL
Cite This: J. Phys. Chem. Lett. 2018, 9, 4765−4770
Influence of Electronic Polarization on the Structure of Ionic Liquids Jesse G. McDaniel*,† and Arun Yethiraj‡ †
School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, United States Department of Chemistry, University of Wisconsin, 1101 University Avenue, Madison, Wisconsin 53706, United States
J. Phys. Chem. Lett. Downloaded from pubs.acs.org by WASHINGTON UNIV on 08/08/18. For personal use only.
‡
ABSTRACT: The liquid structure and electrical screening ability of ionic liquids are fundamentally intertwined. The molecular nature of the charge carriers means that screening distances of external fields depend sensitively on the ion packing and structure of the ionic liquid. In this work, we quantitatively illustrate how the liquid structure itself is directly modulated by electrostatic screening conditions. In particular, electronic polarization fundamentally relaxes long-range ion structuring in asymmetric ionic liquids such as [BMIM+][BF−4 ], with the influence propagating to short-range ion−ion correlation. A consequence of the exact Stillinger−Lovett second moment condition is that, at fixed density, any pairwise-additive, nonpolarizable force field will necessarily predict artificially enhanced long-range ion structuring. This is because the screening condition is set by the infinite-frequency dielectric response. There is no adhoc fix: One has to use polarizable force fields to correctly reproduce the optical dielectric constant. Our illustration of this fundamental effect significantly clarifies interpretation of previous work comparing property prediction using polarizable and nonpolarizable force fields. where ϵ(k) is the wave-vector-dependent dielectric function. The dielectric function depends on the electrical susceptibility of the electrolyte, which is related to the ion correlation functions through fluctuation−dissipation theory.5,8 The SL sum rule thus sets an asymptotic limit on the correlation functions or their Fourier space analogues, the ion structure factors. Of central importance to the present work is modification of the originally derived sum rules due to electronic polarization by its contribution of an infinite-frequency, electronic dielectric response. The second-moment condition incorporating electronic polarization effects was derived by Caillol et al.1,2,9 and takes the form
A
s electrolytes, the liquid structure and dynamics of roomtemperature ionic liquids (ILs) are intimately connected with their electrical properties. The molecular ions entirely serve as both the charge carriers and molecular constituents of the liquid, leading to fundamental coupling conditions on both static and dynamic ion correlation.1−5 For application in electrochemical systems, it is natural to investigate how the specific molecular components of the IL alter the liquid structure, which in turn affects the electrical properties. The purpose of this Letter is to emphasize that the converse perspective is also necessary: Specifically, we illustrate how asymptotic electrical screening conditions dictate both the long- and intermediate-range structure of ILs. To our knowledge, this work represents the first quantitative evaluation of the Stillinger−Lovett (SL) second moment condition in room-temperature ILs, which directly illustrates the connection between ion structure and electrostatic screening. Electrostatic “sum rules”5 rigorously quantify electrostatic screening in liquids. These conditions asymptotically relate electrostatic quantities to the ion−ion structural correlation functions; in other words, the sum rules are constraints that the correlation functions must satisfy. The first sum rule is a charge-neutrality condition.5 The second sum rule was first derived in the seminal work of Stillinger and Lovett6,7 and is known as the second-moment or SL screening condition. The SL screening condition results from assuming that over sufficiently long length scale an electrolyte will perfectly screen any electrostatic perturbation.5,7 The SL perfect screening condition is most easily expressed in Fourier space as lim ϵ(k) = ∞
ϵ −1 S(k) 4π =1− ∞ lim ϵ∞ kBT |k|→ 0 k 2
where S(k) is the charge correlation structure factor5 that for an isotropic system depends on only the magnitude of the wave vector. The charge correlation structure factor is given as an ensemble average of Fourier space charge density operators, namely S(k) =
© XXXX American Chemical Society
1 ⟨ρ ̂(k)ρ ̂( −k)⟩ V
(3)
where N
ρ̂(k) =
∑ qieik·r
i
i=1
(4)
Received: July 6, 2018 Accepted: August 4, 2018 Published: August 4, 2018
(1)
|k|→ 0
(2)
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The Journal of Physical Chemistry Letters
which results in enhanced conductivity, and subtly alters the liquid structure, most notably anion−anion pairwise correlations. The enthalpy of vaporization reduction is actually due to modulation of ion pair interactions in the gas phase, rather than a liquid phase effect,16 and thus will not be discussed further. While the structural and dynamic effects of polarization have been well-established, clear explanation and interpretation of these physical trends has so far remained somewhat elusive. Notably, the density and cohesive energy of the IL are not significantly altered by polarization,17 and explicit polarization does not alter the rotational dynamics of the cation nor the short time behavior (≲ 0.5 ps) of the autocorrelation functions or translational dynamics.18 We show that these subtle structural effects that in turn modulate ion diffusion are a direct consequence of the electrostatic screening condition. Important context has been provided by the meticulous study of Bedrov and co-workers,18 in which detailed comparisons of property predictions between polarizable and nonpolarizable force fields were presented for several different ILs. These authors developed a nonpolarizable force field based on FM to an explicitly polarizable force field, which thus implicitly (and statically) captured polarization interactions and energetics. Interestingly, the FM nonpolarizable model predicted nearly identical counterion spatial distribution functions compared to the polarizable model. Additionally, cation−cation and cation−anion radial distribution functions (RDFs) were in good agreement between force fields, yet differences remained in predicted anion−anion RDFs as well as transport properties. For a variety of ILs, three different nonpolarizable models showed systematic deviations from the polarizable simulation predictions, with the nonpolarizable models predicting a greater degree of anion−anion structuring. Identical qualitative conclusions were made by Yan et al.,19,20 this time for a nitrate-based IL. Again, the nonpolarizable model showed greater anion−anion structuring, while cation− cation and cation−anion short-range correlations remained very similar, and the dynamic properties predicted by the polarizable model were significantly enhanced; similar observations have also been made by others.10,21 Local interactions cannot account for the discussed structural and dynamical effects. For ILs such as [BMIM+][BF−4 ], polarization results in induced dipole moments of the bulkier cations, but the smaller anions do not substantially polarize. Therefore, anion−anion interactions are not directly modulated by polarization, and thus the broadening of anion− anion RDF peaks at ∼6−8 Å length scales18−20 is an indirect effect. This indirect effect cannot be attributed to differences in local cation−anion structure/interactions, as proved by the FM study of Bedrov et al.18 Instead, long-range Coulomb interactions are responsible for the modulation of the local anion−anion structure within the IL. The polarization effect is then explained by the infinite-frequency electronic dielectric response in eq 2, which significantly relaxes the anion−anion structural correlation by altering the limit of the SL screening conditions. The primary purpose of the present work is to quantitatively illustrate the fundamental influence of electronic polarization on IL properties directly resulting from the SL screening condition. We use molecular dynamics (MD) simulations to study the prototypical IL [BMIM+][BF−4 ], allowing direct comparison to previous investigations of polarization effects in this and related systems.10,18−22 By directly comparing with
is the Fourier transform of the charge density operator. The infinite-frequency dielectric response, ϵ∞, is the adiabatic electronic response of the system; without electronic polarization, there is no mechanism for instantaneous response; thus, ϵ∞ = 1, and the right-hand side of eq 2 collapses to unity and recovers the originally formulated SL sum rules.5,6 We note that the contribution of polarization to ϵ∞ significantly affects the dielectric spectrum of ILs, as examined by Schroder et al.10,11 We will refer to eq 2 as the SL screening condition for the rest of this paper. We emphasize that the SL screening condition leads to a fundamental influence of electronic polarization on the structure (correlation functions) of ILs. Through dynamic screening (ϵ∞ > 1), polarization relaxes the long-range structural correlations in the liquid. Note that in the limit of an infinite dynamic response (ϵ∞ → ∞), the right-hand side of eq 2 would collapse to zero, and there would be no requisite long-range ion structure as the electronic response would sufficiently screen any external perturbation. The SL screening condition thus implies that the structure of ILs is directly altered by their electronic properties, without resorting to any discussion of specific ion interaction strength. We believe that this fundamental relation has been overlooked in previous studies of IL structure/property relationships, and we attempt to illustrate its significance. An important consequence of the SL screening condition is that computer simulations with and without explicit treatment of electronic polarization will fundamentally differ in their prediction of ion correlation and liquid structure. Polarizable force fields explicitly incorporate adiabatic electronic response, ϵ∞ > 1, while for nonpolarizable force fields ϵ∞ = 1. Thus, the predicted asymptotic ion structure of such simulations will necessarily differ as a consequence of eq 2. There is no “simulation fix” to compensate for this liquid structure alteration when polarization is omitted; any pairwise additive force field, even if attempting to implicitly account for polarization using modified charges or force matching (FM), will have ϵ∞ = 1 and thus predict artificially enhanced ion correlation at long range. We note that this work focuses on room-temperature ILs such as [BMIM+][BF−4 ], which exhibit asymmetry in the size and/or shape of cations and anions. Asymmetry is important as it provides a mechanism for changes in structural ordering of ionic functional groups at fixed system density, e.g., different packing motifs of the hydrophobic cation side chains. While our conclusions should also apply to molten metal halide salts, the magnitude of the polarization effect may vary due to the smaller polarizabilities and higher symmetry of the ions. Indeed, polarization significantly alters the structure of certain metal halide salts12,13 but not others;29 for a comprehensive discussion, we refer the reader to two excellent review articles.14,15 We believe that the results of previous simulation work on ILs is significantly clarified by consideration of the SL screening condition. It is thus worth summarizing some of the most notable prior findings concerning the influence of electronic polarization on the IL structure and dynamics. The role of electronic polarization has been both postulated based on observed discrepancies between experiment and predictions of computer simulations utilizing nonpolarizable force fields and elucidated based on careful comparison of polarizable and nonpolarizable simulations. It has been established that polarization significantly reduces the enthalpy of vaporization of ILs, accelerates the ion diffusion (diffusion coefficients), 4766
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similar methodology as the particle-mesh Ewald (PME) approach,25 as will be discussed in more detail in subsequent work. The long-range (small-wave-vector) electrostatic interactions in [BMIM+][BF4−] are fundamentally altered by electronic polarization through the SL screening condition. In Figure 1, we show Coulombic interactions from both polarizable (black curve) and nonpolarizable (red curve) simulations; we also plot interactions computed with scaled charges (blue curve), which will be discussed later. Depicted in dashed lines are the SL screening limits of the respective force fields. For the nonpolarizable simulation, the Coulomb interactions approach unity in the small-wave-vector limit; however, for the polarizable simulation, these interactions are reduced to ∼0.5, corresponding to a predicted ϵ∞ ≈ 2, in good agreement with similar theoretical predictions11 and within experimental uncertainty.26 The SL screening condition influences the structure of neighboring shells of like ions. The peak centered at ∼1 Å−1 in Figure 1 corresponds to repulsive interactions in the IL arising from such neighboring ion shells. The nonpolarizable simulation exhibits a peak of greater height and narrower width than the peak of the polarizable simulation; this indicates greater like-ion repulsion and enhanced ion structuring due to omission of polarization. This enhanced ion structuring at ∼6− 7 Å distances is an effect that propagates from the different asymptotic screening limits of the polarizable and nonpolarizable force fields. Note that such interpretation is distinctly different than what is commonly presented in the literature. Structuring of neighboring ion shells is often interpreted based on the strength of nearest-neighbor counterion interactions. In fact, however, this structuring is insensitive to changes in the ion interaction strength (as demonstrated by scaled charge results; see below) but rather is mediated by the asymptotic ion structure and screening. Polarization alters this intermediate-range ∼6−7 Å ion correlation by modulating the asymptotic structure through the shift in the SL screening condition. For greater insight, we decompose the reciprocal space Coulomb interactions into contributions from anion−anion, cation−cation, and cation−anion interactions. This decomposition, along with the respective ion−ion correlation functions (RDFs) are shown in Figures 2−4. We focus our discussion on the 0.8−1.2 Å−1 wave vector range of Coulomb interactions, corresponding to the peak in Figure 1; at longer
these previous works, our conclusions may be interpretted as independent of subtle force field parametrization details and rather a general consequence of the fundamental SL screening condition. MD simulation details are similar to those from previous work, and we employ the SAPT-FF force field for the IL,16,23 which is a fully polarizable model developed on the basis of ab initio linear response calculations. We also analyze the corresponding model in which polarization is omitted; note that omitting polarization does not significantly alter the density or cohesive energy of the liquid.17 In this work, we utilize a relatively large system of 1600 ion pairs to minimize finite-size effects as the focus is on the long-range (small wave vector) correlations. After equilibration, production runs of 50 ns are used for statistical sampling, as enabled by the OpenMM simulation software24 running on GTX-1080-Ti GPU cards (nonpolarizable simulations are 200 ns to compensate for reduced sampling due to slow dynamics). In Figure 1, we evaluate the SL screening condition for [BMIM+][BF−4 ] and its dependence on electronic polarization.
Figure 1. Reciprocal space Coulomb interactions in [BMIM+][BF−4 ] computed with (black curve) and without (red curve) polarization and with scaled charges (blue curve); the top axis denotes the corresponding real space distance, r = 2π/k. Note that the SL screening condition corresponds to the k = 0 limit, as depicted by the respective dashed lines.
Plotted on the y-axis are the averaged reciprocal space Coulomb interactions S(k)/k2, in units of kBT/4π; these Coulomb interactions are computed from eq 3 utilizing a
Figure 2. (a) Anion−anion RDFs compared with (b) anion−anion reciprocal space Coulomb interactions, computed with (black curve) and without (red curve) polarization and with scaled charges (blue curve). The top axis in (b) denotes corresponding real space distance, r = 2π/k. 4767
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Figure 3. (a) Cation−cation RDFs compared with (b) cation−cation reciprocal space Coulomb interactions, computed with (black curve) and without (red curve) polarization and with scaled charges (blue curve). The top axis in (b) denotes corresponding real space distance, r = 2π/k.
Figure 4. (a) Cation−anion RDFs compared with (b) cation−anion reciprocal space Coulomb interactions, computed with (black curve) and without (red curve) polarization and with scaled charges (blue curve). The top axis in (b) denotes corresponding real space distance, r = 2π/k.
range (k < 0.6 Å−1), there is large cancellation between repulsive like-ion and attractive counter-ion interactions, while at shorter range (k > 1.2 Å−1), important real space contributions have not been considered. The nonpolarizable simulation shows sharper peaks in the Coulomb interactions in this region. For example, the anion−anion contribution to this repulsive Coulomb peak is clearly seen in Figure 2b, which is sharpened in the absence of polarization. Because of their bigger size, the cation−cation repulsion is more diffuse; however, these interactions also become more localized in the 0.8−1.2 Å−1 regime in the absence of polarization, as seen in Figure 3b. The cation−anion interactions (Figure 4b) also show a repulsive peak at this length scale due to a counterion depletion effect. As seen in Figure 4b, this counterion depletion is enhanced in the absence of polarization due to the greater structuring of like ions. Such alteration of ion−ion interactions due to polarization is a direct result of the different SL screening limits depicted in Figure 1. These trends in Coulomb interactions directly correspond to changes in the liquid structure. The enhanced ion structuring in the nonpolarizable simulations dictated by the SL screening condition is evident in the RDFs shown in Figures 2a−4a. Most notable is the enhanced anion−anion structuring both at intermediate (∼6−7 Å) and long (∼15−40 Å) ranges, shown in Figure 2a. Also notable is the enhanced long-range ∼15−40 Å structure of cation−anion interactions shown in Figure 4a. The polarization effect for short-range ∼3.5−6.5 Å cation−
anion RDF peaks is seemingly more significant than was observed in the previous work of Bedrov et al.;18 however, this is due to our choice of ring carbon atoms for computing pairwise correlation functions, rather than the cation center of mass. Alteration of such cation−anion RDF peaks thus reflects changes in nearest-neighbor spatial distribution functions, which is consistent with previous findings;17,18 note that integration of the RDFs out to the minimum at ∼7.3 Å results in an identical ion coordination number predicted by both force fields. We finally comment on the use of reduced or scaled charges in simulations of ILs. This is motivated by the extensive use of such force fields in the literature27 and additionally serves to illustrate that the discussed effects are not a consequence of the ion interaction strength. From the equilibrated nonpolarizable simulation, we scale cation and anion charges by 0.8,27,28 reequlibrate the system, and compute the average Coulomb interactions. We utilize the NVT (number of molecules, volume, and temperature fixed) ensemble to avoid focus on density effects (such charge scaling would decrease the density by ∼5−6% if equilibrated at constant pressure17). The Coulomb interactions from this scaled charge simulation are shown as the blue curve in Figure 1. Note that the Coulomb interactions of the scaled charge simulation satisfy the same asymptotic limit as the nonpolarizable simulation, as dictated for all pair potentials (ϵ∞ = 1) by the SL screening condition. The consequence is that for intermediate and long distances (k 4768
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The Journal of Physical Chemistry Letters < 1 Å−1), the net Coulomb interactions of the scaled charge model appear very similar to the nonpolarizable model, and the like-ion repulsion peak at ∼1 Å−1 is also enhanced relative to the corresponding peak of the polarizable simulation. The primary modulation in Coulomb interactions due to charge scaling actually appears at short range, k > 1 Å−1, for which the cohesive energy is dramatically reduced due to the much weaker attraction of nearest-neighbor counterions (this is implied by the curves shown in Figure 1, but the energetics at these distances would include real space and self-interaction corrections, which are not considered here). Analogous to the nonpolarizable model, the scaled charge model predicts enhanced ion structuring corresponding to narrower Coulomb interaction peaks at around ∼1 Å−1. In Figures 2−4, the corresponding ion−ion correlation functions predicted by the scaled charge force field are plotted as blue curves. It is evident that the modulated anion−anion and cation−anion structuring predicted by the scaled charge force field is very similar to that observed for the nonpolarizable simulation, which is a consequence of the same asymptotic limit of the screening condition (eq 2). This verifies that the structural changes are due to fundamental omission of dynamic response (ϵ∞ = 1) and not to specific ion−ion interaction strength and that scaled charge models do not reproduce asymptotic screening of polarizable models, in contrast to previous suggestions.27 In summary, we have shown that the enhanced ion structuring predicted by nonpolarizable simulations of ILs is a fundamental consequence of the SL screening condition. Polarization enables dynamic electronic response of the electrolyte, ϵ∞ > 1, which reduces the asymptotic limit of the charge correlation structure factor. The extent of ion structuring should therefore not be discussed in terms of the magnitude of nearest-neighbor counterion interactions but rather is largely dictated by long-range Coulomb interactions that must satisfy the asymptotic screening limit. Any computer simulation employing pairwise-additive potentials will thus contain some artifact in the predicted liquid structure due to the modulated asymptotic limit imposed on the structure factor. This effect seems to be more significant for room-temperature ILs composed of asymmetric ions, as indicated by comparison of results of this and previous work18−20 to similar studies of molten salts.12,13,15,29 While thorough analysis of the effect of polarization on the dynamic properties is beyond the scope of this work, it is natural to assume that such effects are at least partially explained by the liquid structure modulation. Our analysis suggests that the enhanced ion dynamics predicted by polarizable force fields is due to the modulated ion structuring imposed by long-range screening conditions; in contrast, the enhanced dynamics predicted by scaled charge IL models27 is a local effect due to modulated short-range ion−ion interactions, which does not reflect the correct physical behavior.
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Letter
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ACKNOWLEDGMENTS
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REFERENCES
This research was supported in part through research cyberinfrastructure resources and services provided by the Partnership for an Advanced Computing Environment (PACE) at the Georgia Institute of Technology, Atlanta, Georgia, U.S.A.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Jesse G. McDaniel: 0000-0002-9211-1108 Arun Yethiraj: 0000-0002-8579-449X Notes
The authors declare no competing financial interest. 4769
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