Dynamical properties across different coarse-grained models for ionic liquids
Joseph F. Rudzinski, Sebastian Kloth, Svenja Wörner, Tamisra Pal, Kurt Kremer, Tristan Bereau, Michael Vogel
DDynamical properties across different coarse-grainedmodels for ionic liquids
Joseph F. Rudzinski , Sebastian Kloth, Svenja W¨orner, Tamisra Pal, Kurt Kremer, Tristan Bereau, , Michael Vogel Max Planck Institute for Polymer Research, 55128 Mainz, Germany Institute of Condensed Matter Physics, Technische Universit¨at Darmstadt,Hochschulstr. 6, 64289 Darmstadt, Germany Van ’t Hoff Institute for Molecular Sciences and Informatics Institute, University ofAmsterdam, Amsterdam 1098 XH, The NetherlandsE-mail: [email protected]
Abstract.
Room-temperature ionic liquids (RTILs) stand out among molecularliquids for their rich physicochemical characteristics, including structural and dynamicheterogeneity. The significance of electrostatic interactions in RTILs results in longcharacteristic length- and timescales, and has motivated the development of a numberof coarse-grained (CG) simulation models. In this study, we aim to better understandthe connection between certain CG parametrization strategies and the dynamicalproperties and transferability of the resulting models. We systematically comparefive CG models: a model largely parametrized from experimental thermodynamicobservables; a refinement of this model to increase its structural accuracy; and threemodels that reproduce a given set of structural distribution functions by construction,with varying intramolecular parametrizations and reference temperatures. All five CGmodels display limited structural transferability over temperature, and also result invarious effective dynamical speedup factors, relative to a reference atomistic model.On the other hand, the structure-based CG models tend to result in more consistentcation-anion relative diffusion than the thermodynamic-based models, for a singlethermodynamic state point. By linking short- and long-timescale dynamical behaviors,we demonstrate that the varying dynamical properties of the different coarse-grainedmodels can be largely collapsed onto a single curve, which provides evidence for a routeto constructing dynamically-consistent CG models of RTILs.
Submitted to:
J. Phys.: Condens. Matter a r X i v : . [ c ond - m a t . s o f t ] F e b oarse-grained ionic liquids
1. Introduction
Of the broad variety of molecular liquids, ionic liquids (ILs) stand out for their richphysicochemical characteristics [1, 2]. ILs are salts, with a melting point or glass-transition temperature that can reach low temperatures—notably, “room-temperature”ILs (RTILs) are in the liquid state at ambient conditions. RTILs are commonlycomposed of an organic cation and an inorganic anion. ILs play an important roleas a solvent in sustainable chemistry, with applications including biomaterials andcatalysis [3, 4, 5, 6]. Being conductive, ILs are also strong candidates in electrochemicalapplications.The organic cations in ILs often consist of a polar ring group along with nonpolarside chains. The amphiphilic nature of these cations facilitates the formation ofnanoscale segregation. Nanoheterogeneous structures have been investigated using bothX-ray and neutron scattering [7, 8, 9, 10, 11]. Analysis of the structure factor revealedstructural inhomogeneity, together with complex heterogeneous dynamics, observed viadielectric spectroscopy [12, 13]. Nanoscale segregation increases when the temperatureis decreased toward the glass transition, evidenced by shifts in the static structurefactor [10]. Low temperatures can also yield specific dynamical effects, such as abreakdown of the Stokes-Einstein-Debye relation [14]. Despite a wealth of studiescharacterizing the properties of ILs, a clear link between the structural and dynamicalproperties of these systems, especially close to the glassy regime, remains elusive.Computer simulations have played a significant role in furthering our understandingof ILs [15, 16, 17, 18, 19]. A review by Maginn highlights that interests in ILscoincided with the advent and development of molecular simulations [20]. Simulationshave provided invaluable insight into the structural, thermodynamic, and dynamicalaspects of ILs [21, 22, 23, 24, 25]. The dramatic breadth of relevant length- andtimescales has made excellent use of multiscale modeling—from quantum mechanicsto classical atomistic to coarse-graining—to shed light on various aspects includingviscosity, interfacial behavior, and dynamic heterogeneity [26, 27, 28, 17, 29, 30, 31].Going up the multiscale-modeling ladder facilitates the study of phenomenaoccurring at longer length- and timescales. The need for large systems and the associatedextensive timescales, due in no small part to the strong electrostatic interactions inILs, lend themselves to a coarse-grained (CG) description of the system. By lumpingtogether several atoms into super-particles or beads, CG models not only decrease thenumber of particles to be simulated, but also effectively smooth the underlying free-energy landscape [32, 33, 34].A variety of CG approaches have been previously applied to study ILs. Wang andVoth used the force-matching-based multiscale coarse-graining method to coarse-grain1- n -alkyl-3-methylimidazolium ([C n mim] + ) nitrate, for both n = 2 (ethyl) and n = 4(butyl) [35]. The mapping involved a single bead for the imidazolium ring, one beadper methylene group, as well as a separate bead for the terminal methyl group. ExplicitCoulomb interactions were used, where the partial charge of each bead was determined oarse-grained ionic liquids et al . proposed a CG model for the family of [C n mim] + cations togetherwith the anion hexafluorophosphate ([PF ] − ) [36]. Three CG beads were devoted to themethylimidazolium atoms to ensure the planarity of the ring, while using one to threeCG beads to represent the alkyl chain, depending on its length. Transferability betweenbutyl, heptyl, and decyl side chains was obtained by means of two bead types: terminaland interior beads. A single bead was used to represent the anionic PF . Both intra-and intermolecular interactions were represented by simple functional forms: harmonicpotentials for the bonds and bending angles; 9–6-type Lennard-Jones potentials forthe short-ranged nonbonded interactions; and explicit Coulomb electrostatics. Theparameters of the model were optimized to reproduce several properties of [C mim][PF ]under ambient conditions: ( i ) the mean of reference all-atom (AA) distributions alongeach order parameter that governs an intramolecular interaction in the CG potential,( ii ) the density, and ( iii ) the surface tension. The authors’ analysis highlighted therole of the alkyl chains in the morphology of the liquid, leading to nanoscale ordering,in good agreement with X-ray scattering experiments. The study remains, to date, anexcellent landmark for CG models in their capability to reproduce various features ofRTILs.Karimi-Varzaneh et al . constructed a model that focuses on the same chemistries,i.e., [C n mim][PF ] with n = { , , } , and additionally considered two mappings [37].While the first mapping resembled the one used by Bhargava et al ., the second onenotably used a single bead to represent the imidazolium ring. The CG model wasparametrized using iterative Boltzmann inversion, fitting all radial distribution functionswhile employing a single short-ranged pairwise potential per pair type, i.e., withoutemploying explicit electrostatics. While only targeting structural features, excellentagreement was also found for the surface tension, when compared against both the CGmodel from Bhargava et al . and experiments. Aspects of temperature transferabilitywere probed, with a focus on the change in the major peak positions of the X-rayscattering structure factors. The choice of mapping had a direct impact on the qualityof the peak positions, highlighting that the aromatic ring and the alkyl chains impactdifferent regions of the structure factor. Finally, the dynamics of the CG models werecharacterized. Previous studies employing both pulsed field gradient NMR [38] andAA simulations have demonstrated that the diffusion coefficient ( D ) of the cation ishigher than that of the anion for n = 4. Karimi-Varzaneh et al . reported that theratio D cation /D anion depends on the mapping scheme, the value of n , and temperature.Moreover, a better performance of the second mapping in terms of reproducing thenon-Gaussian particle displacement statistics of the AA model highlighted the role of amore detailed representation of the alkyl chains.More recently, Deichmann and van der Vegt revisited the bottom-up parametriza- oarse-grained ionic liquids mim] + with three different anions: [PF ] − , tetrafluoroborate, and chloride [39].The CG model employed a two-bead representation for the cation: one for the imida-zolium ring and one for the alkyl chain. They relied on the conditional reversible workmethod, which determines the CG parameters from (biased) AA simulations by in-voking a thermodynamic cycle. They used Morse-type potentials for the short-rangednonbonded interactions, in addition to explicit Coulomb electrostatics. The resultingthermodynamic properties of the model—both the liquid–vapor surface tension and massdensity—agreed well with AA simulations after an a posteriori correction of the forcefield. Overall, good agreement was found in terms of the radial distribution functions,although a few outliers were observed, in particular those involving both the imida-zolium ring and the [PF ] − anion. Regarding dynamics, it was found that the diffusivespeedup with respect to the AA model differs for the cation and anions. We also notethe development of other CG models for RTILs using alternative methodologies: New-ton Inversion [40] (also known as Inverse Monte Carlo), relative entropy [41], and graphneural networks [42]. While each of these studies features different parametrizationschemes and a variety of resulting properties, they share the common difficulty of rep-resentability —simultaneous reproduction of structural, thermodynamic, and dynamicalproperties.The dynamical properties of CG RTIL models are of particular interest, sincethe link between structural and dynamical heterogeneity in these systems remainsunclear. Unfortunately, the interpretation of CG dynamics is inherently difficult, asthe removal of degrees of freedom from the system results in both a loss of frictionand a “smoothing” of the underlying free-energy landscape [43]. Although generalizedLangevin dynamics can be applied to correct for these effects, this approach remainsextremely challenging for complex soft matter systems, and also (partially) removesthe beneficial speedup provided by the CG model [44]. Instead, many researchershave taken a much simpler approach by determining an effective dynamical speedupfactor, e.g., by calibrating against a long-timescale dynamical observable. For specificsystems, and in particular when a clear timescale separation exists between characteristicprocesses, this time-rescaling approach can be highly effective [45, 46]. In general,though, the presence of multiple coupled kinetic processes, which may be independentlyaccelerated by the coarse-graining, leads to inconsistent dynamics [47, 48]. A similardynamical inconsistency typically arises in CG models of multicomponent systems withdistinct molecular species, e.g., cations and anions in RTILs [37, 39]. For bottom-up models, improving the description of the many-body potential of mean force (i.e.,the theoretically-ideal CG potential) offers a systematic route toward one importantaspect of dynamic (in particular, kinetic) consistency: barrier-crossing dynamics [49].This link further justifies efforts to improve the structural accuracy of CG modelsvia many-body [50, 51, 52, 53, 54, 55, 56, 57, 58, 42, 59, 60] or environmental-dependent [61, 62, 63, 64, 59, 65, 66, 49, 67, 68] interactions.Recently, Pal and Vogel investigated the relationship between various dynamicalmodes and the relevance of spatially heterogeneous dynamics for AA and CG models of oarse-grained ionic liquids mim][PF ] [69]. They found that, despite the nontrivial transformation of dynamicalprocesses upon coarse-graining, several relationships between dynamical modes on verydifferent timescales are preserved for both ionic species. In particular, time scales ofmaximum non-Gaussian and spatially heterogeneous dynamical behaviors collapse ontothe same curve when plotted as a function of the structural relaxation time, independentof the value of the ionic charges and, thus, of the electrostatic interactions [70], andconsistent with results for various types of viscous liquids [71]. Somewhat related,Douglas and coworkers have put forth an energy-renormalization method for coarse-graining that makes use of a relationship between the structural relaxation time of aliquid and the temperature-dependent activation free energy, which is linked to theconfigurational entropy of the system [72]. This approach determines a temperature-dependent adjustment to the cohesive energy of the molecular interactions, whichhas been demonstrated to result in consistent CG dynamics over temperature andfrequency (for small-amplitude oscillatory shear molecular dynamics) for liquid ortho-terphenyl [73] and several polymer melts [72, 74, 75]. These studies demonstratethat by preserving certain short-timescale quantities under coarse-graining of theserepresentative systems, the long-timescale dynamics can be predicted.In this study, we aim to better understand the connection between certain CGparametrization strategies and the dynamical properties and temperature transferabilityof the resulting models. We systematically compare five CG models:(i) “thermo”: the model from Bhargava et al . [36], largely parametrized fromexperimental thermodynamic observables;(ii) “thermo*”: a refinement of the thermo model to increase its structural accuracy;(iii) “struct-anabond”: a structure-based model focused on reproducing intermoleculardistributions;(iv) “struct”: a structure-based model that targets both intra- and intermoleculardistributions;(v) “struct-260K”: a structure-based model that targets both intra- and intermoleculardistributions, parametrized at a lower reference temperature than the struct model.All five CG models display limited structural transferability over temperature, and alsoresult in various effective dynamical speedup factors relative to a reference AA model.For each model, we quantify the speedup, analyze the relative diffusivity of cations andanions, and study the relationship between vibrational and diffusive motions. In thisway, we provide further evidence of a route to constructing dynamically-consistent CGsimulation models, in particular for RTILs.
2. Methods
The present work makes use of AA simulation trajectories of [C mim][PF ] generatedand analyzed in previous works [31, 69, 70]. Briefly, the force field from Bhargava and oarse-grained ionic liquids . e and − . e , respectively, harmonic potentials for bond stretching and bond-angle bending as well as dihedral interactions. The simulations were performed using theGROMACS software package [77] for N = 256 ion pairs and a time step of 1 fs. Periodicboundary conditions were applied and the Particle Mesh Ewald (PME) method [78] wasutilized to calculate the Coulomb interactions. At all set temperatures T , the systemwas first equilibrated at a pressure P = 1 bar to adjust the density, employing the Nos´e-Hoover thermostat [79, 80] and Parrinello-Rahman barostat [81]. The production runswere carried out in the isochoric-isothermal ( N V T ) ensemble. We note that, as can beseen in Fig. 5, the simulations at the lowest temperatures (i.e., below 300 K) may not befully equilibrated, as the particles do not completely reach the diffusive regime. For thisreason, we do not include the data from these temperatures in the dynamical propertyanalysis. Nonetheless, we have used the lowest temperature simulation to construct oneof the CG models (see below), based on the structural properties, which presumably aremuch less sensitive to the full equilibration of these simulations. Additionally, we usethe structural properties from these lower temperatures to compare to the properties ofthe CG models.
We employ the CG mapping proposed by Bhargava et al . [36] and used previously [69],which represents each imidazolium cation with 4 CG sites and each [PF ] − anion witha single CG site. The imidazolium ring is represented by 3 sites, I1, I2 and I3, mappedto the center of mass of a corresponding group of atoms, as illustrated with the largetransparent spheres in Fig. 1. Note that the I1 and I2 sites overlap, sharing contributionsfrom the 2-carbon of the 5-membered ring (i.e., the carbon flanked by two nitrogens).The butyl chain is represented by an additional site, denoted CT, while the anion siteis denoted PF. In the following, this mapping is applied to 5 different CG models, eachemploying the same set of interactions, e.g., nonbonded pairwise interactions betweenall unique pair types, but with variations in the functional forms and parameters ofthese interactions. 4 bonded interactions are employed to retain the imidazoliumconnectivity: I1–I2, I1–I3, I2–I3, I2–CT. Accordingly, each model employs 5 bond-angle interactions: I1–I2–I3, I1–I3–I2, I3–I1–I2, I1–I2–CT, I3–I2–CT. There are nodihedral angle interactions in these models. Finally, 15 pairwise nonbonded interactions,corresponding to all unique pair combinations of the 5 site types, are employed, whileexcluding intramolecular imidazolium pairs. There are two distinct contributions toeach nonbonded interaction: (i) a short-ranged van der Waals-like interaction that isparametrized separately for each model, and (ii) a long-ranged Coulomb interaction thatis kept fixed for all models. The Coulomb forces are calculated by mapping the partialcharges of the AA model to the CG representation: q = +0 . e , +0 . e , +0 . e , 0 e ,and − . e for the I1, I2, I3, CT and PF sites, respectively. oarse-grained ionic liquids CT I1I2 I3 PF
Figure 1.
CG representation. Each molecule is partitioned into a number of atomicgroups, as indicated by the large transparent spheres: 4 for the cation and 1 for theanion. All CG beads were mapped to the center of mass of the corresponding group ofatoms: ( i ) CT – the last 3 carbon groups of the alkyl chain; ( ii ) I2 – the first carbongroup of the alkyl chain, the 1-nitrogen of the ring, and “half” of the 2-carbon groupof the ring; ( iii ) I1 – the 3-nitrogen and associated methyl group of the ring and also“half” of the 2-carbon group of the ring; ( iv ) I3 – the 4- and 5-carbon groups of thering; ( v ) PF – the entire anion. Iterative Boltzmann Inversion (IBI): IBI is a bottom-up method that determines theCG potential that will reproduce a given set of 1-D distribution functions through aniterative refinement that assumes independence of the CG potential parameters [82, 83].Given an initial model and a set of reference distributions, each interaction potential, U , is updated at each iteration according to: U ( i +1) ( ξ ) = U ( i ) ( ξ ) − αk B T ln P ( i ) ( ξ ) P (ref) ( ξ ) , (1)where k B is the Boltzmann constant, ξ is a scalar order parameter that governs theinteraction, and P ( ξ ) is the Jacobian-transformed distribution function along ξ . Forpairwise nonbonded interactions, P ( ξ ) corresponds to the radial distribution function(RDF), g ( r ), where r is the distance between two particles. α ∈ [0 ,
1] is a dampingfactor used to increase the stability of the procedure.Iterative generalized Yvon-Born-Green (iter-gYBG): We also considered an alternativeiterative parametrization scheme, using a generalization of the Yvon-Born-Green (YBG)integral equation framework [84, 85]. This approach relates a set of structural correlationfunctions, b , to the parameters of the CG interaction potentials, φ , via a matrixequation, which quantifies the cross-correlations, G , between each CG degree of freedom: b = Gφ . (2) oarse-grained ionic liquids b is directlyrelated to the corresponding RDF, and G characterizes the average cosine of the anglebetween triplets of particles [86, 87]. b can also be expressed in terms of force correlationfunctions through a connection to the multiscale coarse-graining method [85, 88]. Thismethod employs force and structural correlation functions that are determined froma set of AA reference simulations, b AA and G AA , i.e., calculated from a set of AAsimulation trajectories mapped to the CG representation, to determine an optimal setof CG parameters φ . Note that if the model derived from this method fails to reproducethe target vector of the equations, i.e., b AA , it implies that the cross-correlation matrixgenerated by the higher resolution model does not accurately represent the correlationsthat would be generated by the resulting CG model. This indicates a fundamentallimitation of the model representation and interaction set. Nonetheless, Equation 2 canbe solved self-consistently to determine the interaction parameters φ ∗ that reproducethe target correlations b AA [89]: b AA = G ( φ ∗ ) φ ∗ . (3)This approach has been previously denoted as an iterative generalized YBG (iter-gYBG)method [89, 90, 91]. When the multiscale coarse-graining model is employed as the initialset of parameters, this procedure has been demonstrated to converge very quickly (e.g.,in less than 10 iterations), although it may be less robust than IBI [89, 92]. thermo: As already mentioned in the Introduction, the model proposed by Bhargava et al . [36] was parametrized in a top-down fashion to reproduce experimentalthermodynamic properties, and will thus be denoted as ”thermo” in the following. Thismodel employs analytic functional forms for the CG interaction potentials: harmonicpotentials for intramolecular interactions and 9–6 Lennard-Jones potentials for thenonbonded interactions. Initial equilibrium distances and force constants for theintramolecular potentials were determined by fitting the Boltzmann-inverted potentialsfrom AA simulations to the harmonic functional form. These parameters were thenrefined to reproduce the mean of the 1-D distributions along each order parametergoverning an intramolecular interaction in the CG model. The Lennard-Jones energiesand particle radii were then tuned to reproduce the density and surface tension fromexperimental measurements at 300 K. It is also reported that the site–site RDFs fromAA simulations were taken into account when determining these parameters, althoughit is not clear to what extent or how this was carried out.thermo*: We performed a refinement of the thermo model to improve its structuralaccuracy, while attempting to minimally perturb the thermodynamic properties of themodel. To achieve this, we applied the iter-gYBG approach to each of the short-rangednonbonded interactions, but restricted our calculation to relatively short distancesbetween CG sites. More specifically, we calculated the change in the force coefficients up oarse-grained ionic liquids G (i.e., the initial model was obtained via the multiscalecoarse-graining method). The iter-gYBG model was then calculated following the iter-gYBG framework, by solving Eq. 3 iteratively, using cross correlations determined fromthe CG simulations at the previous step, until a pre-set accuracy threshold was achievedwith respect to the 1-D distribution functions along each order parameter governing aCG interaction (8 iterations in this case). The first 3 iterations scaled the calculatedparameter update by a factor of 0.25, while further iterations applied a factor of 0.5.For these calculations, the fixed long-ranged Coulomb interactions were incorporatedvia the reference potential method [96], which subtracts the contributions to the forcecorrelations, b , from a given set of fixed terms in the force field. To increase numericalstability, we also used the direct Boltzmann inverted forces for each intramolecularinteraction as reference, even though we still calculate the optimal forces for theseinteractions (i.e., we calculate the change in the force parameters, relative to thereference forces). Solution of each set of linear equations and post-processing of thepotentials followed previous work [89, 92].struct-260K: The struct-260K model was parametrized in the same way as the structmodel, but using the reference AA simulations at 260 K. In this case, 7 iterations were oarse-grained ionic liquids All CG simulations were performed using the GROMACS simulation package [97]. Eachsimulation, with the exception of those associated with the thermo model (see furtherdetails below), was performed at the same density as the corresponding AA simulationof the same temperature (see Table S2 for the specific densities and comparisons toexperiments). Starting from an equilibrated AA configuration mapped to the CGrepresentation, each CG model was applied to energy minimize the configuration,followed by a 10 ns simulation in the
N V T ensemble at the respective temperature.All structural analysis was performed using these simulations. For the dynamicalanalysis, some models/temperatures required longer simulations. In particular, for thethermo* model, we performed additional 25 ns simulations at 400, 350, 320, and 300 Kand additional 100 ns simulations at 280, 270, and 260 K. Similarly, for the struct-anabond, struct and struct-260K models, we performed additional 25 ns simulationsat 280, 270, and 260 K. All CG simulations used the stochastic dynamics integratorwith a temperature coupling constant of 2 ps, a 1 fs time step, and periodic boundaryconditions. Electrostatic interactions were employed using the PME method [78] witha Fourier grid spacing of 0.10 nm. A cutoff of 1.5 nm was used for both the short-rangenonbonded interactions and for the real-space contribution to electrostatic interactions.Configurations were sampled every 0.1 ps during the simulations, and then used forsubsequent dynamical analysis. The structural analysis was performed using trajectoriesparsed to yield configurations every 0.4 ps.The dynamical properties of the thermo model were taken from previously publishedsimulations [69]. These simulations were carried out at slightly different densities thanthe AA model (see Table S2), corresponding to the equilibrium density of the modelat 1 bar. To assess the impact of this change in density on the model properties, weran 10 ns simulations of the thermo model at each temperature and corresponding AAdensity, following the protocol described above. All structural analysis presented inthis work was taken from these simulations for consistent comparisons with the othermodels. We note that the slight change in density has negligible impact on the RDFs fornearly all of the simulations. At the lowest temperature (260 K) there are noticeable,albeit very small, deviations in a few of the RDFs. (The full set of RDFs are availablein the manuscript repository [93] for comparison). Similarly, we expect only a relativelysmall impact on the dynamics, although we did not explicitly verify this.Throughout this manuscript we will describe the dynamical timescales of each CGmodel in terms of physical time units, in order to achieve a consistent comparisonwith the AA model. However, as described in the Introduction, the process of coarse-graining results in a lost connection between the CG and AA dynamics. Thus, thereported CG timescales are not meaningful without a proper reference or calibration.However, comparison of relative timescales can assist in assessing if a CG model exhibits oarse-grained ionic liquids
3. Results
In this work, we investigate a spectrum of particle-based CG models for [C mim][PF ],each parametrized to target specific structural or thermodynamic observables of theunderlying system, characterized either with respect to experiments or using AAsimulations. We consider five models—denoted thermo, thermo*, struct-anabond,struct and struct-260K—as described in detail in the Methods section. In brief, thethermo model was previously parametrized to reproduce the experimental density andsurface tension [36]. The thermo* model is a refinement of the thermo model, whichmore accurately reproduces the anion-cation radial distribution functions (RDFs) byadjusting the (very) short-ranged region of the nonbonded interactions, while keepingthe remaining interactions fixed. By focusing on adjusting the force functions at shortdistances, the thermo* model largely retains the thermodynamic accuracy of the original(thermo model) parametrization. In particular, the equilibrium density of the thermo*model at 300 K was determined to be 1.301 g/cm , compared with 1.357 g/cm forthe thermo model, 1.389 g/cm for the AA model, and 1.369 g/cm from experiments.Additionally, the surface tension of the thermo* model at 300 K was determined tobe 38.1 mN/m, compared with 39.0 mN/m for the thermo model and 42.5 mN/mfrom experiments (see Supporting Information for calculation details). (Note that, forconsistent comparisons in the following analysis, the CG models were simulated at thecorresponding AA equilibrium density. However, the dynamical analysis of the thermomodel, taken from previously published work [69], employed slightly different densities.See the Methods section for details.) The struct-anabond model uses the analytic(harmonic) functional forms and parameters for the intramolecular interactions fromthe thermo model, while employing tabulated potentials for the short-ranged nonbondedinteractions to reproduce the RDFs from reference AA simulations at 300 K. The structmodel was also parametrized to reproduce these same RDFs, but additionally employstabulated intramolecular interactions to reproduce the corresponding 1-D distributionsalong each CG (intra)molecular degree of freedom. Finally, the struct-260K model isequivalent to the struct model, but parametrized using the reference distributions at260 K.Fig. 2 compares the force functions for each model, for a representative setof interactions. Row (a) presents the only three intramolecular interactions whosecorresponding 1-D reference distributions (i.e., calculated from the AA simulationsafter mapping each configuration to the CG representation) displayed multimodalbehavior. It is worth noting that the I1–I2–CT and I3–I2–CT interactions aresignificantly coupled to one another. As mentioned above, the thermo, thermo* andstruct-anabond models (red, yellow and green curves, respectively) employ identical oarse-grained ionic liquids Figure 2.
Comparison of representative intra- (a) and inter- (b) molecular forcefunctions for the thermo (red dashed curve), thermo* (yellow dash-dotted curve),struct-anabond (green dotted curve), struct (blue dashed curve), and struct-260K(purple dash-dotted curve) models. The full set of force functions is presented inFigs. S1–S3 of the SI.
Row (b) presents the three short-range nonbonded interactions corresponding to theRDFs with the largest errors in the thermo model (described further in the followingsection). The thermo model (red curve) employs a 9–6 Lennard-Jones functional formwith potential minima of ≈ ≈ oarse-grained ionic liquids N V T simulations that prohibit the characterization of thetypical thermodynamics properties, e.g., density and surface tension, as is commonfor structure-based CG models of liquids [98, 99, 100, 101].
Figure 3 presents the 1-D distributions corresponding to the interactions presented inFig. 2. Panel (ai) shows that the harmonic potential of the thermo model leads to anaccurate description of even the most complicated bond distribution in this molecule,the I2-CT bond, although the fine features (e.g., shoulders) of the distribution are notreproduced. The struct model nearly quantitatively reproduces the distribution, byconstruction. The small remaining discrepancies are due to numerical difficulties ofthe iter-gYBG procedure which can occur at the tails of the distribution. This issuecould likely be resolved by employing a regularization scheme, but we do not pursuethis here, as we don’t expect these discrepancies to affect our results. The struct-260K model matches this distribution for most of its range, but completely neglects thesmall shoulder at short distances, due to this same numerical issue. Panels (aii) and(aiii) demonstrate that the angular distributions involving the alkyl chain of the cationare quite complex, and cannot be accurately described with the harmonic potentialemployed by the thermo model. The struct and struct-260K models reproduce thesedistributions by construction, albeit with some small discrepancies at the tails of thedistributions. Panel (aiv) presents the average mean squared error (mse) for each modelover all intramolecular distributions along the CG degrees of freedom that govern aninteraction in the CG model (see Fig. S4 for explicit comparisons of each distribution).For each intramolecular distribution, the mse was calculated over a range encompassingall sampled instances of the corresponding order parameter, e.g., bond distance.Row (b) presents three sets of RDFs, demonstrating the improvement of thethermo* model, relative to the thermo model, in terms of the description of the anion-cation packing. The remainder of the RDFs show similar behavior between the twomodels, analogous to panel (biii). The struct-anabond, struct, and struct-260K modelsreproduce all RDFs by construction. Panel (biv) presents the average mse for eachmodel over all 15 RDFs (see Figs. S5 and S6 for explicit comparisons of each RDF). Foreach RDF, the mse was calculated from 0 to 1.2 nm. oarse-grained ionic liquids Figure 3.
Comparison of representative intra- (a) and inter- (b) molecular 1-Ddistributions at 300 K for the AA (black solid curve), thermo (red dashed curve),thermo* (yellow dash-dotted curve), struct-anabond (green dotted curve), struct (bluedashed curve), and struct-260K (purple dash-dotted curve) models. The full set ofdistributions is presented in Figs. S4-S6 of the SI.
We also analyzed the structural properties of the models as a function oftemperature. Fig. 4 characterizes the structural accuracy of each model, with respectto the reference AA model, in terms of the average mse of the RDFs (panel a) andthe full width at half maximum (FWHM) of the first peak in the I1-PF RDF (panel b).Although it is somewhat difficult to see on the logarithmic scale, panel (a) demonstratesa decrease in structural accuracy for the thermo and thermo* models upon cooling,despite these models being parametrized from reference data at 300 K. On the otherhand, the structure-based models (struct-anabond, struct, struct-260K), with muchlower errors overall, show divergence of the error away from the reference temperatureof parametrization (300 K for struct-anabond and struct; 260 K for struct-260K). Panel(b) clearly demonstrates an improved accuracy of the thermo* model’s description ofthe I1–PF RDF over the entire temperature range, relative to the original thermomodel. The struct-anabond and struct models very accurately reproduce the AA FWHMfor temperatures near the reference temperature of parametrization, but demonstrateincreasing error for the much higher temperatures. The struct-260K model provides asimilarly accurate description of the I1–PF FWHM, with slightly larger errors at thehighest temperatures, further indicating a limited range of temperature transferabilityfor these structure-based models.
To investigate the molecular dynamics of the AA and CG models, we calculate the mean-square displacement (MSD) of the cations and anions based on the time-dependent bead oarse-grained ionic liquids Figure 4.
Structural accuracy as a function of temperature. (a) Average mean-squared error (mse) of the RDFs relative to the AA model. (b) Full width at halfmaximum (FWHM) of the first peak of the I1-PF RDF for each model. coordinates r i ( t ), (cid:104) r ( t ) (cid:105) = (cid:42) N N (cid:88) i [ r i ( t + t ) − r i ( t )] (cid:43) , (4)where the pointed brackets denote an average over various time origins t . From thelong-time diffusive regime of the MSD, we determine the self-diffusion coefficients D ofthe cations and anions by fits to (cid:104) r ( t ) (cid:105) = 6 Dt . We note that the AA model generatesdiffusion constants in good agreement with experiments (see Table S3 in the SupportingInformation).Figure 5 presents the MSD of the AA model and two representative CG models (thethermo and struct models) at various temperatures. Typical of viscous liquids, a regimeof vibrational motion at short times and a regime of diffusive motion at long times areseparated by a plateau, which becomes longer upon cooling [102]. This plateau regime oarse-grained ionic liquids (a)(b)(c) Figure 5.
Mean squared displacement (MSD), (cid:104) r (cid:105) , as a function of time for the (a)AA, (b) thermo, and (c) struct models. Results for the cations and anions are shownas dashed lines and solid lines, respectively. oarse-grained ionic liquids D .Therefore, we restrict the following quantitative analysis to temperatures T ≥
300 K.To quantitively compare the diffusive behavior of the different models, we calculatedthe diffusion coefficients for the anions and cations, using the MSD curves presented inFig. 5. Fig. 6 presents the temperature-dependent ratio of these coefficients for eachCG model, relative to the AA model: D s = D CG /D AA . These ratios, presented for thecations and anions separately in panels (a) and (b), respectively, represent an effectivespeedup factor for connecting the CG and AA dynamics, at a particular thermodynamicstate point and for a specific molecular species. As one may expect based on thesmoothening of the energy landscapes upon coarse-graining, the speedup factors D s are significantly larger than unity. All CG models also demonstrate an increase in D s upon cooling, consistent with the above observation of the difference in temperaturedependence of the MSDs between the AA and CG models. By refining the structuralaccuracy of the thermo model, the thermo* model results in slower dynamics overall,and displays the smallest speedup factors of all the models. Similarly, by refining theintramolecular structure relative to the struct-anabond model (which displays the largest D s values), the struct model also exhibits slower dynamics. On the other hand, thechange in the reference state of parametrization between the struct and struct-260Kmodels appears to have minimal impact on the effective dynamical speedup. Althoughthe thermo* model yields the best absolute reproduction of the AA dynamics, accordingto this metric, the more relevant quantity for assessing the dynamical consistency ofeach CG model is the relative mobility of the distinct ionic species. Panel (c) of Fig. 6presents the ratio of speedup factors between the anions and cations, D anis /D cats , ateach thermodynamic state point. In this plot, a value of unity indicates consistent CGdynamics at a single temperature, since the anions and cations experience the samespeedup upon coarse-graining. The structure-based models (struct-anabond, struct, oarse-grained ionic liquids Figure 6.
Speed-up factor of the diffusion coefficient, D s = D CG /D AA , for (a)cations and (b) anions. (c) Relative speedup of anions versus cations, D anis /D cats , at agiven thermodynamic state point. oarse-grained ionic liquids D anis /D cats ,over the entire range of temperatures. Interestingly, the struct-anabond model appearsto show the most consistent dynamics, despite having the largest absolute speedupfactors, although the scattering of the speedup ratio is slightly larger than for the otherstructure-based models. The thermo and thermo* models display larger speedup ratios,which increase slightly upon cooling. By refining the structural accuracy relative to thethermo model, the thermo* model does result in slightly more consistent dynamics forall but the highest temperature. r t = 1 ps [nm ] D [ n m p s ] AA catAA anithermo catthermo anithermo* catthermo* anistruct-anabond catstruct-anabond anistruct catstruct anistruct-260K catstruct-260K ani
Figure 7.
Diffusion coefficient, D , versus the magnitude of the MSD in the plateauregion, r ≡ (cid:104) r ( t = 1 ps) (cid:105) . Solid (dashed) gray curves are guides for following thetrend of anions (cations). To ascertain the origin of the observed differences between AA and CG dynamics,we relate long-time and short-time motions for both levels of resolution. We anticipatethat smoother energy landscapes and higher temperatures result in not only fasterdiffusive motion but also larger amplitudes of cage rattling motion in the plateauregime. Elastic [103] and elasticity-based [104] theories of glass-forming liquids proposerelations between the structural relaxation time and the cage rattling amplitude, whichare consistent with simulation results [105, 106, 107]. Fig. 7 presents the diffusioncoefficients, D , versus the amplitude of the cage rattling motion, which we characterizeby the MSD in the plateau regime, r ≡ (cid:104) r ( t = 1 ps) (cid:105) . It is apparent that the verydifferent dynamical behaviors of the variety of models characterized above collapse toa similar D ( r ) dependency, where the curves for the cations and anions are somewhatshifted relative to each other. Overall, the structure-based models (struct-anabond,struct, struct-260K) demonstrate a tighter collapse than the thermo and thermo*models, relative to the AA behavior, although these models also show deviations atsmall r (i.e., lower T ). The collapse of CG dynamics implies that a smootheningof the energy landscape leads to a facilitation of the diffusive motion via a softeningof the local cages. This may provide a promising route for constructing dynamically- oarse-grained ionic liquids r is preserved under coarse-graining. This provides a tractable method forCG parametrizations, since r can be efficiently determined from (relatively) short AAreference simulations.
4. Conclusions
This work employed five distinct structure- and/or thermodynamic-based coarse-graining parametrization strategies for modeling the room temperature ionic liquid[C mim][PF ]. All five models displayed limited transferability in accuratelyrepresenting structural properties over a range of temperatures, although the absoluteerror of the structure-based models was much lower overall, by construction. Thesestructural discrepancies might be systematically addressed via recently proposedapproaches for estimating the entropic contribution to the many-body potential of meanforce (i.e., the theoretically-ideal CG potential) [108, 109, 110, 111].The focus of our study was to better understand the link between the variousparametrization strategies and the dynamical properties of the resulting models. Wequantified the dynamical speedup for each model and for each ionic species via theratio of diffusion constants, relative to the AA model, D s = D CG /D AA . All fivemodels showed a systematic increase in D s , for both cations and anions, with decreasingtemperature, indicating dynamical inconsistency relative to the AA model (i.e., a lack oftransferability). At a given temperature, the structure-based models tended to providea more consistent description of the relative anion-to-cation behavior. Interestingly, theaccuracy with which the model described the intramolecular distributions of the cationand also the reference parametrization temperature for the structure-based modelsappeared to play a limited role in the resulting diffusive properties of the model.Moreover, our results demonstrate that the absolute speedup is independent of thedynamical consistency of the model: in this case the model with the largest speedupprovided the most consistent description of anion-cation relative diffusion.We also investigated the relationship between vibrational and diffusive motions foreach model. Following previous work on theories of glass-forming liquids, we assessedthe relationship between the structural relaxation time and the cage rattling amplitudevia the diffusion constant and the value of the mean-squared displacement in the plateauregime, respectively. Despite their range of parametrization strategies, we found acollapse of CG model behavior in terms of this relationship (Fig. 7). The structure-basedmodels demonstrate a slightly tighter collapse, relative to the AA behavior, althoughclear discrepancies still arise for the lowest temperatures considered. While these modelsreproduce a set of low-dimensional structural distribution functions by construction,the need for iterative refinement relative to the initial (force-matching-based multiscalecoarse-graining) model indicates fundamental limitations of the chosen CG interaction oarse-grained ionic liquids Data Availability
Additional methodological details and results can be found in the SupportingInformation of this work. We have also compiled a repository [93] containing allthe models (in GROMACS format) developed in this work (i.e., the thermo*, struct-anabond, struct, and struct-260K models), some parameter files for the calculationsperformed with the VOTCA and BOCS packages, and also some of the raw dataused to construct the manuscript figures. Additional data can be obtained from thecorresponding author upon request.
Author Contributions
JFR, KK, TB, and MV designed the work. JFR, SK, SW, and TP performedcalculations for the work. JFR, SK, KK, TB, and MV analyzed the results. JFR,SK, KK, TB, and MV wrote the manuscript.
Acknowledgments
This work was supported by the TRR 146 Collaborative Research Center (project A6)of the Deutsche Forschungsgemeinschaft.
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