This journal is© the Owner Societies 2017 Phys. Chem. Chem. Phys., 2017, 19, 4701--4709 | 4701

Cite this:Phys.Chem.Chem.Phys.,

2017, 19, 4701

Towards open boundary molecular dynamicssimulation of ionic liquids

Christian Krekeler and Luigi Delle Site

We extend the use of the adaptive resolution (AdResS) method in its grand canonical-like version

(GC-AdResS) to the molecular dynamics simulation of 1,3-dimethylimidazolium chloride. We show that

the partitioning of the total system in a subsystem of interest with atomistic details and a reservoir of

coarse-grained particles leads to satisfactory results. The challenging aspect of this study, compared to

previous AdResS simulations, is the presence of charged particles and the necessity of addressing the

question about the minimal physical input needed to model the coarse-grained particles in the reservoir.

We propose two different approaches and show that in both cases they are sufficient to capture the

decisive physical characteristics that allow a valid system–reservoir coupling. The technically satisfactory

results pave the way for the multiscale analysis of ionic liquids and truly open boundary molecular

simulations.

I. Introduction

Ionic liquids (ILs) are gaining great popularity due to the rangeof amazing properties and potential groundbreaking develop-ments in chemistry, physics, biology and materials science.1

The possibility of designing such systems at the molecular levelhas led to a large effort by theoreticians; the aim is to under-stand them deeply from the perspective of manipulating andoptimizing their chemical design for obtaining large scaleproperties on demand (see e.g. the themed collection of Phys.Chem. Chem. Phys., 2010, 12, 1629 and Faraday Discuss., 154,2012, 1–484). In this respect, the field of molecular simulationhas been particularly active and has led to important progress(see e.g. ref. 2 and references therein). Molecular simulation,by construction, allows for a molecular-based understanding oflarge scale properties and thus represents the most powerfultool to draw strategies for the development and use of ILs.In previous work,3–5 we have shown that ILs (or at least a largeclass of them), are characterized by a local scale where importantproperties, which intuitively one may expect to be dominatedby large scale correlations, are instead highly localized anddepend only on the immediate neighbouring molecules. Theseresults became the inspiration for the extension of the AdResSmethod6,7 in the grand canonical-like version8–11 to the studyof ILs. In fact, AdResS allows us to obtain full atomistic detailsof a region of interest and couple it with a region with very ageneric, coarse-grained, molecular model, without atomistic details,that acts as a reservoir of particles and energy; this implies that

one could use GC-AdResS as a tool to identify the essentialatomistic degrees of freedom required to have a certain property.This idea was already successfully used in previous AdResSstudies to infer about the locality of solvation properties forhydrophobic molecules in water12 and about the locality of theIR spectrum in water.13 We believe that it would be useful to havea similar tool for analysis also for ionic liquids; in this perspec-tive, the aim of this work is to show the technical feasibility of anAdResS study of ILs. The challenging aspect for the application of(GC-)AdResS to ILs is the presence of explicit ions and thecapability of building a coarse-grained procedure for a physicallyvalid system–reservoir coupling. Previous work has already dealtwith the presence of explicit ions14 but only for dilute solutions inwater, that is, the main process was dominated by the solvatingcharacter of water. For ionic liquids we must make a step forwardand treat, inspired by the previous work of water–salt solution,a dense liquid of positively and negatively charged ions. We willreport about two different approaches to build a sufficientlyaccurate coarse-grained model for our study; one where thecoarse-grained model does not carry charges but reproduces theion–ion radial distribution functions of a full atomistic simu-lation (according to the so-called inverse Boltzmann iteration,IBI, procedure15) and the other one where ions in the coarse-grained model carry proper charge (according to the atomisticmodel) and at the same time reproduce, as for the neutralmodel, ion–ion radial distribution functions of a full atomisticsimulation (IBI with explicit charge–charge interaction). In thiswork, we have chosen 1,3-dimethylimidazolium chloride as thetest system for our simulation because it is complex enough totest the general robustness of our simulation method but at thesame time is simple enough for allowing a large number of

Institute for Mathematics, Freie Universitat Berlin, Germany.

E-mail: ch.krekeler@fu-berlin.de, luigi.dellesite@fu-berlin.de

Received 1st November 2016,Accepted 6th January 2017

DOI: 10.1039/c6cp07489h

rsc.li/pccp

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numerical tests in a straightforward way and without parti-cularly expensive computational efforts. Beside the conceptualuse of AdResS as a tool for analysis that identifies the essentialatomistic degrees of freedom, one may want to consider thepossible gain in terms of computational saving due to the drasticreduction of the number of degrees of freedom. Currently, theconceptually elaborated version of GC-AdResS is implementedin GROMACS16 and its computational architecture is not yetoptimized in such a code thus the computational gain occursonly for very large systems and it is still modest. The currentperformance gain for systems of 20 000–40 000 ion pairs, forthe ionic liquid considered in this work, is a speedup factor of1.5–2.0 compared to the performance a full atomistic simulation;the precise factor depends on the size of the atomistic region andthe transition region. However, independent of the future tech-nical optimization of the code, the success, at the conceptuallevel, of the use of AdResS for ILs paves the way for the simulationof open systems with virtually an infinite size when the imple-mentation of AdResS which is coupled to a continuum17–19 isused; for ILs this may certainly represent an interesting futureperspective.

II. GC-AdResS approach

The original AdResS method for molecular dynamics6,20 wasdeveloped following the intuitive principle that the couplingof different regions where molecules have different molecularresolutions should be done in such a way that the passage fromthe dynamics of one resolution (region) to another must besmooth enough so that the perturbation in each region is negligible.From the computational point of view, this principle was imple-mented via a space-dependent interpolation formula for the forceacting between two molecules, a and b (also see Fig. 1, for a pictorialrepresentation):

Fab = w(Xa)w(Xa)F ATab + [1 � w(Xa)w(Xa)]F CG

ab (1)

where F ATab is the force derived from the atomistic potential and

FCGab is the force derived from the coarse-grained potential. The

interpolating (switching) function is defined as:

wðxÞ ¼

1 xo dAT

cos2p

2 dDð Þx� dATð Þ

� �dAT oxo dAT þ dD

0 dAT þ dD ox

8>>>><>>>>:

where, dAT is the size of the atomistic region and dD is the sizeof the hybrid (transition) region as reported in Fig. 1; x is thex-coordinate of the center of mass of the molecule.

The weighting function smoothly goes from 0 to 1 in thetransition region so that the practical effect is that a coarse-grained molecule transforms into an atomistic molecule andvice versa. The system is coupled to an external thermostat thattakes care of supplying or removing the required energy forthermodynamic stability. This simple approach led to numericallyhighly satisfactory results for various systems12,21–27 and broughtthe need of finding a deeper conceptual justification. In fact inthe next step, the basic thermodynamic relations for a propercoupling were defined and then this led to the derivation of aone-particle force acting on the center of mass of the moleculein the hybrid region, named thermodynamic force, Fthf(x).Such a force assures that the effective chemical potential ofthe system is that of the atomistic resolution.28 Later on, thethermodynamic force was derived in a more rigorous way, con-sidering the atomistic and coarse-grained regions as two openregions interfaced by a filter (hybrid region) which is definedby the balance of the grand potential, as if each of the tworegions is in equilibrium with a finite particle reservoir (theother region): pAT þ r0

ÐDFthfðrÞdr ¼ pCG, with pAT being the

reference pressure of the atomistic system (region), pCG thepressure of the coarse-grained model and r0 the referencemolecular density of the atomistic system. The thermodynamicforce was then written as the gradient of the particle density inan iterative form which is computationally highly convenient:29

F thfkþ1ðxÞ ¼ F thf

k ðxÞ �Ma

rref½ �2krrkðxÞ, where Ma is the mass of the

molecule, k a constant chosen to optimize the calculation,rk(x) is the molecular density at the k-th iteration as a functionof the position in the transition (hybrid) region. The choice ofthe convergence criterion depends on the accuracy needed forthe simulation, however, as a rule of thumb, rfinal � r0 shouldnot be larger than 10% in the transition region. Finally, in thelast few years the method has found mathematical and physicalrigorous formalization either in terms of a global Hamiltonianapproach (H-AdResS)30,31 or in terms of the grand ensembleapproach where the coarse-grained region is composed by aliquid of simple spheres whose minimal requirement is that itacts as a reservoir for the atomistic region (GC-AdResS);8–11 thetwo starting points are of course compatible.32–34 Moreover, theAdResS method in each of its revised current formulations hasbeen successfully applied to a rather large class of liquids,mixtures and solvation processes.13,35–43 In this work, we considerthe GC-AdResS approach and require that the coarse-grained

Fig. 1 (top) Pictorial representation of the GC-AdResS scheme; CG indicatesthe coarse-grained region, HY is the hybrid region where atomistic and coarse-grained forces are interpolated via w(x) (in yellow) and AT is the atomistic region(that is the region of interest). (bottom) The molecular representation ofthe atomistic, hybrid and coarse-grained molecules, according to the regionthey belong to.

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region is characterized by some minimal physical input so thatthe numerical results obtained in the atomistic region aresatisfactory when compared with the results obtained in theequivalent sub-region of a full atomistic simulation. In the nextsection, we propose two coarse-grained models and then showthat in both cases the results are highly satisfactory.

III. Coarse-grained models

In previous work of AdResS, it has been shown that coarse-grained models as simple as liquids of generic spheres whichreproduce only the particle density and the temperature of theatomistic liquid are already sufficient to act as a valid reservoir ofparticles and energy for the atomistic region (thus the definitionof grand canonical-like AdResS).8–10,13,42 Kremer’s group took aneven more drastic step and built a reservoir using a gas of genericspheres.44 The robustness of the AdResS method is in thederivation of the thermodynamic force (from first principles) inthe hybrid region; in fact it has been shown that a necessary (andnumerically sufficient) condition for an accurate grand canonicaldistribution in the atomistic region is that the particle density inthe hybrid region must be (ideally) equal to that of the atomisticregion. Next, it has been shown that such a condition and theaction of the thermostat to maintain the same temperatureeverywhere automatically fix the chemical potential at the valueone would obtain from a full atomistic simulation under thesame thermodynamic conditions. The systems considered insuch case where liquids and mixture of small neutral molecules.The question of how to treat the presence of ions in the systemwas then addressed in ref. 14. In such a case, the study concernsthe solvation of Na+ and Cl� in water at concentrations compa-tible with biological conditions. The study was aimed at extendingthe use of AdResS to the solvation of biological molecules and thepresence of corresponding counter-ions.40 The work of ref. 14represents an important starting point for the extension of theidea to liquids with high concentrations of ions, for example theIL treated here. It must be clarified, and this is actually the mostrelevant point of our work, that we do not intend to develop acoarse-grained model of ILs that can be used for full coarse-grained simulations per se; our aim, as already underlined in theIntroduction, is to develop coarse-grained models which aresufficiently good to act as a reservoir for the atomistic region;such a region is the only region of physical interest in ourstudy, while the hybrid and coarse-grained regions are only oftechnical/computational interest (i.e. to be as efficient as possiblein numerical terms). Compared to the approach of ref. 14 we takea further drastic step, in the very spirit of ref. 8–10 and of ref. 44,and consider as a first approach a coarse-grained model withoutexplicit charges; each ion has only one interaction site, that is itscenter of mass (trivial for Cl�), as pictorially illustrated in Fig. 1.The interaction potentials of such a coarse-grained model isderived using the inverse Boltzmann iteration procedure (IBI)15

so that the coarse-grained potentials obtained reproduce theanion–anion, anion–cation, cation–cation full atomistic radialdistribution functions (see Fig. 2–4).

The coarse-grained particles are not charged and once theyenter into the hybrid region they slowly start to switch on theelectrostatic interactions and thus, effectively, acquire the

Fig. 2 Main figure: anion–anion radial distribution function obtained fromatomistic simulation (black line), compared with the anion–anion distribu-tion function obtained from a coarse-grained simulation (red circles) whichemploys the IBI potential for the uncharged coarse-grained model (inset).

Fig. 3 Equivalent of Fig. 2 for the anion–cation case.

Fig. 4 Equivalent of Fig. 2 for the cation–cation case.

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corresponding charge according to the atomistic model(vice versa ions slowly lose their charges upon transit fromthe atomistic to the coarse-grained region). According to thetheoretical principles of GC-AdResS (in which we also countref. 44), the model used carries more details and thus it shouldalready lead to good results; indeed this is true as it will beshown in Section V. The reason why a generic liquid (or amixture if we consider each ion as a species per se) was not usedis that it is not convenient from the numerical point of view; infact the reproduction of the radial distribution functions allowthe ions to be at a sufficiently large distance when just enteringthe hybrid region, instead a generic liquid does not always assurethat this can happen and thus leads to numerical instabilities.In general, in a very simplified spherical coarse-grained model asthe one we employ here, the methyl groups of cations are forcesin a spherical region and they may represent a problem whencations acquire atomistic resolution. In fact two methyl groupsbelonging to two different cations, due to the spherical rotationalinvariance of the coarse-grained model, may be too closeindependent of the accuracy of the coarse-grained model. Insuch a case, very small capping forces are applied, as done forlarge polymers,43 to avoid this problem. Capping forces couldalso be used for the case of a generic coarse-grained model, butthe use of the IBI procedure is more general and straightforward.The conceptual disadvantage of a neutral coarse-grained modelis that the global system, that is the atomistic region and thereservoir, may instantaneously be charged and thus be charac-terized by possible instantaneous spurious net electrostatic(artificial) interactions. Moreover, when ions just enter intothe atomistic region the ion–ion electrostatic interaction maystill not be optimal and thus the equilibration while crossingthe hybrid region to enter in the atomistic region could be lesssmooth than one may desire. For this purpose, we have tried asecond option, we derive the coarse-grained potential using theIBI procedure but maintaining the corresponding charge in thecoarse-grained model. This implies that the potential refined ateach step of the IBI procedure is such that its sum with theelectrostatic potential (untouched by the IBI procedure) mustlead to the reproduction of the radial distribution functionof the target (see Fig. 5–7). In this way, possible limitations ofa neutral coarse-grained model reported above are removed.However, we will see in Section V that the two models, for theproperties considered, are essentially equivalent (with a differenceof 2% at worst).

IV. Technical details

All reported simulations were performed using the packageGROMACS.16 The force field parameters for 1,3-dimethyl-imidazolium chloride were taken from a study by Dommert et al.2

We set up two systems, the first one (350 ion pairs) was used toderive two coarse grain potentials, which we then transferred tothe larger system (1000 ion pairs). We optimized both systemsusing full atomistic NpT calculations. The simulation tempera-ture was set to 400 K, the time step was set as 2 fs and the

electrostatic interactions were calculated through the particlemesh Ewald (PME) technique. For the first 2 ns, we used theBerendsen barostat,45 after that we switched to the Parrinello–Rahman barostat46 for the following 2 ns. We monitored the boxsize and considered that we have reached convergence, whenthe changes in the box length were of the order of 0.0001 nm.

Fig. 5 Equivalent of Fig. 2 for the charged coarse-grained model.

Fig. 6 Equivalent of Fig. 3 for the charged coarse-grained model.

Fig. 7 Equivalent of Fig. 4 for the charged coarse-grained model.

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The density obtained is 1155 kg m�3 which agrees well with theexperimentally found density of 1165 g cc�1.47 For 350 ion pairswe obtained a cubic box with 4.08525 nm box length, whilefor the 1000 ion pairs we obtained a box with a = 12.09408 nm,b = c = 4.03137 nm. Furthermore, the radial distributionfunctions were obtained after 10 ns full atomistic NVT simulationsat 400 K and with 2 fs time steps. The results are shown in Fig. 9and agree well with those from the literature.48 The 350 ion pairsconfiguration was used to derive the two coarse-grained models.We used an inverse Boltzmann iteration, IBI, procedure15 toreproduce the radial distributions functions of the full atomistictarget system (see Fig. 2–4 and for the charged model see Fig. 5–7).After we derived these tabulated potentials, we used the configu-ration from the 1000 ion pairs to set up the GC-AdResS system. Anatomistic region of a = 3 nm is bordered by hybrid regions of thesame length, while the remaining a = 3.09408 nm is the coarse-graining region. As in standard AdResS simulations, a Langevinthermostat is used with G = 10 ps�1. The electrostatic interactions,as usually done in AdResS, were treated by the generalizedreaction field method with a self-consistent dielectric constantas calculated by GROMACS.49 The thermodynamic force throughthe iterative procedure was considered converged after 8 itera-tions, sufficient to reach an accuracy of 3% for the particle densityand for the ion–ion radial distribution functions (compared to thefull atomistic of the reference). The use of the charged coarse-grained model implied that the reaction field method is alsoapplied to the coarse-graining region with effective ionic strength(dielectric constant) one would have in the reaction field methodapplied to the full atomistic simulation.

V. Results

The simulation set up of AdResS reported in Section IV wasspecifically chosen in order to test the robustness of the methodunder critical technical conditions, that is we use a relatively smallreservoir (hybrid plus coarse-grained region). The method wouldperform at its best when the reservoir is very large compared tothe atomistic region, but if it performs sufficiently well undercritical technical conditions, obviously its performance would besatisfactory under ideal conditions. Our aim in this work is toshow that basic distribution functions calculated in the atomisticregion agree in a satisfactory way with those calculated in theequivalent sub-region of a full atomistic simulation of thereference. In fact, distribution functions such as particledensity and radial distribution functions correspond to thefirst- and second-order terms of the statistical distribution inthe atomistic region, respectively (see ref. 9). This implies that ifthe distribution functions of the atomistic region of GC-AdResSagree in a satisfactory way with the corresponding functionscalculated in the equivalent sub-region of the full atomisticsimulation of the reference, then up to (at least) the secondorder, statistical averages calculated in the atomistic region areequivalent to those calculated in the sub-region of a fullatomistic simulation. Moreover, a sub-region in a full atomisticregion (of a size that is statistically relevant) is a natural open

system embedded in a thermodynamic environment providedby the rest of the system; this implies that if the atomisticregion of GC-AdResS reproduces the statistical distribution (atleast up to a certain order) of the sub-region of the reference fullatomistic system, then the atomistic region is equivalent (up toat least a certain order) to a subsystem in a given thermo-dynamic bath/reservoir as the open subsystem of the full atomis-tic simulation. In this perspective, the first quantity to consider isthe particle density, in fact if the coupling would induce evidentartificial effects, the density will display sizable deviations fromthe results of the full atomistic simulation. Fig. 8 reports thecomparison between the full atomistic simulation and theGC-AdResS results for the two different coarse-grained models.The region of major interest is the atomistic region and thelargest deviation from the density of the reference of the fullatomistic simulation is given by the charged coarse-grainedmodel. The difference is not larger than 3% which, also con-sidering the critical technical conditions, we consider highlysatisfactory. Next, we considered the cation–cation, cation–anionand the anion–anion radial distribution functions; the resultsare reported in Fig. 9. Also in this case, the GC-AdResS resultsobtained using both coarse-grained models satisfactorily agreewith the reference full atomistic simulation and they actuallyoverlap (the error is within the thickness of the lines). However,a deeper check is to show that microscopic radial distributionfunctions are satisfactorily reproduced, since the atomisticregion must be characterized by accuracy at the very atomisticlevel. For this reason, we calculated three representative atom–atom radial distribution functions, gHCl(r), gCC(r), gHH(r), (for thedefinition see Fig. 10). The results are reported in Fig. 11–13,in all cases the agreement is satisfactory.

Fig. 8 Particle density as a function of the position in space along thedirection where the change in resolution occurs. The black line corre-sponds to the full atomistic simulation, the red to the GC-AdResS simula-tion with uncharged coarse-grained model and the red line refers to theGC-AdResS simulation with charged coarse-grained model. The top panelrefers to ion-pairs, the middle panel to cations and the bottom panel toanions. As expected, the largest deviation occurs in the hybrid region withan upper value of about 7%. However in the atomistic region, which is theregion of interest the largest deviation is about 3%.

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With regard to structural properties, one must also checkthe possibility of an artificial orientational alignment of themolecules (in this case of the cations) due to the interfaceintroduced by the hybrid region. In order to perform this study,we define an orientational order parameter, |cos y|, calculated

with respect to the x-direction (direction of change in molecular

representation): j cos yj ¼ v

jvj � x����

����. Here, v/|v| is the unit vector of

the principal axis of symmetry of the cation and x is the unitvector along the x-direction (see the inset of Fig. 14 for a pictorialrepresentation). Fig. 14 shows (a) the orientational order para-meter as a function of the x position in the atomistic region and(b) its probability distribution; as it can be seen there is in generalno preferential orientation of the cations and in particular, evenat the interface with the hybrid region the perturbation w.r.t. thefull atomistic case, due to the change in resolution, is negligible.According to the procedure usually employed for the validation ofthe AdResS approach, it is important now to show that there is aproper exchange of particles among the different regions. In fact,the possibility that artificial barriers in the hybrid region hinderthe exchange of molecules must be excluded. If molecules do notdiffuse from one region to the other, the results obtained forthe atomistic region would correspond to those of an effectivelyartificial closed system. Fig. 15 and 16 show the diffusion ofthe instantaneous particle distribution from each region to the

Fig. 9 Cation–anion (top), cation–cation (middle) and anion–anion (bottom),radial distribution functions calculated in the atomistic region of GC-AdResS forboth coarse-grained models and comparison with the corresponding quantitycalculated in the equivalent region of a full atomistic simulation. The accuracy ishighly satisfactory and the curves actually overlap.

Fig. 10 Pictorial illustration which shows the type of atoms that definegHCl(r), gCC(r), and gHH(r).

Fig. 11 The results for the gHCl(r) function. The calculation and the legendare the same as that of Fig. 9. In this case also, the agreement is satisfactory.

Fig. 12 Equivalent of Fig. 11 for the gCC(r) function. As before, the agree-ment is satisfactory.

Fig. 13 Equivalent of Fig. 11 for the gHH(r) function. As before, the agree-ment is satisfactory.

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others and thus a proves that the method behaves reasonablywell. The instantaneous exchange is slightly asymmetric becausewhile the ions in the atomistic region diffuse as in a full atomisticsimulation, the dynamics in the coarse-grained region is insteadfaster. In fact, a consequence of the reduced number of degreesof freedom is a time scale difference in the dynamics of thecoarse-grained system compared to the atomistic one. Thisproblem is well known for the AdResS simulation and can befixed by slowing down the dynamics in the coarse-grained andhybrid regions with an increase of the friction parameter of theLangevin thermostat. However, it has been shown that statis-tical averages in the atomistic region are not affected by theasymmetry.23 Moreover, a quantity related to the exchange ofparticles and the statistical distributions is the particle prob-ability distribution in the atomistic region, P(N), where N is thenumber of ion pairs. Its shape should follow a Gaussian functionand should be compatible as much as possible, within a certainaccuracy, with the corresponding quantity calculated in theequivalent sub-region of the atomistic region. Fig. 17 shows theP(N) calculated for GC-AdResS for the two models and compar-ison with the results of the full atomistic simulation. There is asystematic shift of the location of the peak (number of ions withthe highest probability), the largest deviation occurs for the casein which the charged coarse-grained model is used. This devia-tion of the most probable number of molecules is about 3% andit is due to the systematic 3% excess of particle density. However,Fig. 18 shows the three curves when they are superimposed(with a systematic shift along the axis of N), the curves show asatisfactory agreement regarding their shapes. The latter isactually the relevant aspect of the problem because a systematicshift of 15 ion pairs over 500 ion pairs is numerically negligible,while a sizable divergence of the shape of the distributionswould imply a non-valid statistical behavior of the GC-AdResSsystems. Once again, it should be underlined that we are treatingcritical technical conditions and thus the results here can beconsidered as a sort of upper bound of the disagreement

Fig. 14 Orientational order parameter: j cos yj ¼ v

jvj � x����

����. Panel (a) shows

the ensemble average of |cosy| as a function of the position in the atomisticregion. Panel (b) shows the probability distribution of |cosy|.

Fig. 16 Equivalent of Fig. 15 for the charged coarse-grained model. As forthe uncharged model; in this case also the flux behaves properly.

Fig. 15 Evolution in time of an instantaneous distribution profile along thetrajectory for the ion pairs that are at time, t = 0, located in the atomistic region(top panel), hybrid region (middle panel) and in the coarse-grained region(bottom panel). Here, we consider a GC-AdResS simulation with the unchargedcoarse-grained model. The results indicate that there is an exchange of ionpairs among different regions and it is consistent with the GC-AdResS set up.

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for studies with ideal technical conditions. Moreover, onemay run further iteration steps for the thermodynamic forceand reach an agreement for the particle density within 1–2% ofaccuracy.

VI. Conclusion

We have shown the applicability of GC-AdResS to the study of1,3-dimethylimidazolium chloride which represents the proto-type of a large class of ILs. We have performed our study undertechnical conditions which can be considered a ‘‘worst casescenario’’ of a GC-AdResS set up and have shown that theresults are satisfactory. This study is relevant for two mainreasons: (a) GC-AdResS can be applied as a tool for analysis inorder to identify the essential atomistic degrees of freedom thatcharacterize a given property of the system; it has been provenvery useful for other systems and thus it is, in our view, animportant tool for ILs as well, (b) AdResS has already allowed

the coupling of particle-based regions with the continuum; thiswas done for generic fluids and liquid water but one may think tomake this extension for ILs as well, so that one can have systemswith virtually infinite size and span a large spectrum of scales in aconcurrent fashion. In this perspective, this work represents theinitial step for an open boundary molecular dynamics simulationof ILs.50 Future applications of the adaptive method to ionicliquids in the short term concern the study of other ionic liquids.In particular those where the extension of the current model canbe done straightforwardly, e.g. those systems where the molecularstructural modification of the cations involves the addition ofalkyl groups of a certain size, a molecular structural aspect thatrecent technical advancements of AdResS can now systematicallytreat.43 Next, GC-AdResS allows us to treat an atomistic sphericalregion embedded in a large reservoir of structureless molecules,thus the dependence of structural and dynamic properties as afunction of the size of the atomistic region can be systematicallystudied. By comparing the results of different simulations, eachcharacterized by an atomistic region of a different size, one candetermine the connection between the time and length scales ofgiven properties. For example, one can calculate the hydrogenbond–hydrogen bond autocorrelation function as a function ofsize of the atomistic region. Such a quantity links the informationconcerning the relevance of the hydrogen bonding network of thebulk for the formation of hydrogen bonds at the local level, i.e. inthe atomistic region (length scale), to the life time of a hydrogenbond (time scale) (also see ref. 10 and 13, where this study wasdone for liquid water). Moreover, the comparison between theresults obtained with different ionic liquids will clarify the effectof the specific molecular chemical structure of the ions on thescale interplay. Such a study would allow for a direct check and adeeper understanding of the hypothesis of rattling ions in long-living ion cages proposed in the literature.51–55

Acknowledgements

This research has been funded by Deutsche Forschungsge-meinschaft (DFG) through grant CRC 1114 (project C01) LDSand by the European Community through project E-CAM for CKand LDS. We thank Ruth Lynden-Bell for a critical reading ofthe manuscript and useful suggestions.

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Fig. 17 The ion pair number probability density in the atomistic regioncompared with the corresponding quantity in the equivalent sub-region ofa full atomistic simulation. Compared to the results of the full atomisticsimulation, there is a systematic shift of the peak which in the worst case(charged coarse-grained model) is of 3%. This is due to the 3% excess ofdensity reported in Fig. 8 which we have chosen as acceptable accuracy.

Fig. 18 The three curves of Fig. 17 shifted in such a way that they arecentered at the same point. The result shows that the shape of the threecurves is very similar and thus indicates an appropriate statistical behaviorof the GC-AdResS simulation.

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