A Classical Density Functional Study of Clustering in Ionic Liquids at

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A Classical Density Functional Study of Clustering in Ionic Liquids at Electrified Interfaces Ke Ma, Jan Forsman, and Clifford E. Woodward

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.


J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/ acs.jpcc.6b11491 • Publication Date (Web): 29 Dec 2016 Downloaded from http://pubs.acs.org on January 3, 2017



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A Classical Density Functional Study of Clustering in Ionic Liquids at Electrified Interfaces Ke Ma ,† Jan Forsman,‡ and Clifford E. Woodward

∗,¶

† School of Materials Science and Engineering Tianjin University of Technology Tianjin 300384, P. R. China ‡Theoretical Chemistry, Chemical Centre, Lund University P.O.Box 124, S-221 00 Lund, Sweden ¶ School of Physical, Environmental and Mathematical Sciences University of New South Wales, Canberra at the Australian Defence Force Academy Canberra ACT 2600, Australia E-mail: [email protected]

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Abstract Ion association, leading to the formation of clusters in ionic liquids is investigated within the framework of classical density functional theory. Clusters are incorporated into a generic coarse-grained model for imidazolium-based ionic liquids confined by planar charged surfaces. We find that the short-ranged structure adjacent to surfaces is remarkably unaffected by the degree of ion association. The physical implications of ion clustering only become apparent in equilibrium properties that depend upon the longrange screening of charge, such as the asymptotic behaviour of forces between charged surfaces and the differential capacitance around low surface potentials. Surface forces show a long-range exponential decay, which depends primarily on the concentration of non-associated ions, while the differential capacitance seems to be a sensitive function of the internal structure of clusters. Furthermore, the size of the ion clusters only slightly influences surface forces, but has a significant effect on the differential capacitance. These behaviours would be difficult to observe in simulations, due to the system sizes required.



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Introduction When room-temperature ionic liquids (RTILs) are in confined geometries, the fluid may display properties that suggest an increase in fluid structure, particularly at surfaces. 1,2 Indeed, it is often proposed that RTILs may be considered as a fluid where clusters of ions (Bjerrum clusters) are in equilibrium with free ions. 3 This is usually articulated in terms of ion pair formation (between anions and cations). For example, recent surface force apparatus (SFA) measurements of imidazolium-based RTILs by Gebbie et al. suggest a substantial fraction of ion pairs in equilibrium with free ionic species. 4,5 Using DLVO theory to fit the force-distance curves between gold and mica surfaces immersed in an RTIL, those authors were able to estimate the degree of ion pairing, required to explain the long-range exponential decay of the forces. That is, the strong correlations between anions and cations, which is manifested in ion pair formation, diminishes the ability of the RTIL to exhibit non-electroneutral fluctuations and thus compromises the screening of electrode charges. However, one could reasonably argue that ion pairs are in fact just the smallest example of what should be considered as a hierarchy of ionic clusters of increasing size. Cluster theories are well-known vehicles with which to describe strong correlations in electrolytes, 3,6 but their importance in RTILs remains controversial. 1 For example, contradictory results are found in both experimental and theoretical capacitance studies. 7 Ion clustering has also been used to explain unusual temperature dependence of the differential capacitance (DC) of electrodes immersed in RTILs. Some workers, 8–11 have suggested that a DC which increases with temperature can be explained by a greater number of free charges. That is, the dissociation of clusters, as the temperature increases, leads to more ions being available for screening. On the other hand, Drushchler et al. 12 found a decrease with temperature is observed for rapid capacitive processes at gold electrodes, while Costa et al. observed an increase of the DC with temperature for both imidazolium and pyrrolidinium RTILs at Hg electrodes. 9 Alam et al. 13 found that the DC for 1-buyl-3methyl imidazolium tetrafluroborate increases with temperature at glassy carbon and gold 3 ACS Paragon Plus Environment

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electrodes but decreases at Hg electrodes. Subsequently, those authors found that around the potential of zero charge (PZC), the DC decreased with temperature for 1-ethyl-3-methyl imidazolium tetrafluroborate but increased if the cation was 1-ethyl-3-methyl imidazolium. 2 Theoretical work, relevant to these experimental findings, has also been reported. Recent molecular dynamics (MD) simulations showed increasing DC with temperature at corrugated surfaces 14 and also due to specific adsorption of the imidazolium RTILs at electrode surfaces. 15 Monte Carlo simulations of ion-dipole mixtures also indicate that the presence of ion pairs induces a strongly layered structure at both charged and neutral surfaces. 16 In addition, cluster models have also been used to describe molten inorganic salts 16,17 and, in particular, the effect of temperature on capacitance in these systems. 18 Clustering has also been used to explain the dynamic properties of RTILs. For example, Tokuda et al. 19 introduced the concept of ionicity, which is the ratio of the measured conductivity and that derived from the simple Nernst-Einstein relation. An implied value of around 50%-70% pairing is not unusual for a variety of RTILs. Despite the work done in this area, however, the relationship between ionic clusters and their effect on dynamic and equilibrium properties is not straightforward. For example, ionic clustering may more properly be seen as a manifestation of strong correlations, with slow diffusion over an intermediate time scale, rather than as very long-lived entities in a dynamic setting. On the other hand, from an equilibrium point of view, the average size and number of clusters remains essentially constant. In recent work, we used classical density functional theory (DFT) to investigate the role played by ion pairing on the properties of RTILs at electrified interfaces. 20 There we found that the presence of ion pairs did not have a significant effect on the structure of the liquid at surfaces, but that there was a qualitative difference in both the DC and surface forces between electrodes in the presence and absence of pairs. However, as suggested earlier, it is likely that ionic association in RTILs may go beyond simple cation-anion pairs. For example, Gebbie et al 5 have recently proposed that the temperature-dependence of long-range surface



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forces between electrodes immersed in RTILs can be rationalized in terms of a conceptual model, based on the equilibrium between neutral clusters and dissociated ions. In the current work we have extended our earlier study on ion pairs to include the presence of higher-order clusters. We have carried out calculations in order to investigate the role played by larger ion clusters on the properties of RTILs, using a suitably generalized classical DFT. In our approach, the cluster population in the bulk fluid (with which the non-uniform fluid is in equilibrium) will be set a priori. So, while we do not address the problem of how clusters form (we merely assume that they exist due to strong electrostatic correlations) we do investigate their affect on the properties of RTILs, were they present. This has important implications on, e.g., what experimental signatures should be considered as possible evidence of cluster formation. In the next section we describe the generalization of the classical DFT required to include the presence of clusters. We then report the results of calculations on the RTIL fluid structure, capacitance and surface forces, for different degrees of clustering. We conclude with final remarks about the influence of high-order clusters on the properties of RTILs.

Density Functional Theory with Ion Clusters In this section we describe the generalized classical DFT, which includes the presence of a population of clusters.

Coarse-grained model In this study, we will use a previously developed coarse-grained model for for the class of 1alkyl-3-methyl imidazoliumtetrafluroborate RTILs, with general formula,[Cn M IM + ][BF4− ]. 21 For example, n = 4 corresponds to 1-butyl-3-methyl imidazoliumtetrafluroborate, which is depicted in Figure 1, together with the corresponding cartoon of the coarse-grained model. As is implied by the figure, both cation and anion species are modeled using tangentially


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connected hard spheres with equal diameters so that each sphere can roll over the surface of their bonded neighbors. We chose a sphere diameter of σ = 2.4˚ A, which approximates the molecular volume of the anions and cations. The imidazolium ring on the cation is modelled as a star-like structure of five beads, each carrying a partial charge of +0.2e. The methyl groups are described as neutral spheres. A star of 5 beads (each carrying charge of -0.2e ) is used to mimic the molecular structure of [BF4− ]. The electrostatic interaction between any two partial charges is screened by the relative dielectric permittivity r , due to electronic and intra-molecular polarizability. We use, r = 2.3, which is typical for hydrocarbon groups. In addition to electrostatic and steric forces, all beads are also assumed to interact via the long-range attractive component of the Lennard-Jones potential at separations greater than σ. σ Φatt (r) = −4LJ ( )6 r

(1)

The attractive strength is given by LJ /kB = 100K where kB is Boltzmann’s constant and σ = 2.4˚ A is the same as the hard sphere diameter of the beads. It should be noted, that the model parameters used here have not been fine-tuned to precisely mimic the experimental behavior but merely chosen so that the physical properties of the model fall reasonably within a range expected for this class of RTILs.

Figure 1: Coarse-grained model of cations [C4 M IM + ], anion [BF4− ] and neutral species ion pairs. Coloured spheres are charged whilst the rest are neutral.



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Inclusion of clusters into the free energy functional We allow the possibility of neutral cluster formation in the fluid and, as was the case for our treatment of ion pairs, 20 we will assume that the distribution of of the clusters in the bulk is set a priori. It is important to realize, however, that clusters are entities based on a dissociative equilibrium, thus we expect that their size distribution will be approximately exponential. Given that clusters are presumably held together by strong electrostatic correlations, we shall model them with an internal structure in which the charged groups are in close proximity. This implies that the long hydrocarbon chains in our model will tend to form an outer shell in a type of “inverted micelle” arrangement. The definition of what constitutes a “cluster” has a degree of ambiguity. In this work, we will define a cluster as consisting of an equal number of anions and cations, which are constrained so that that the central charged beads of the molecular ions are restricted to lie inside a “cluster sphere” of given radius. This is accomplished by attaching the central charged beads to a point coincident with the center of the cluster sphere via a flexible “string”. Examples of quadruple and hextuple clusters are depicted in Figure 2. The length of string joining the charge and clus-

(a) Quadruple cluster

(b) Hexagonal cluster

Figure 2: Schematic figure of (a) quadruple cluster (N=4) and (b) hextuple cluster (N=6) formed by linking the central charged beads to a fictitious central point in the cluster sphere, by a flexible string. Black beads are positively charged on the cation while red beads are negatively charged on the anion.

ter centers is equal to the radius of the cluster sphere and it is clear that this length should increase with the cluster size. We generally used the same length for the quadruple cluster (lq = 0.8σ) and hextuple cluster (lh = 0.8σ), while for the pair cluster we used lp = 0.6σ. 7 ACS Paragon Plus Environment

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However, we did examine the effect of varying the length of these bonds. The presence of higher-order clusters requires a generalization of the classical DFT developed for oligomeric fluids and RTILs. 20–25 The total free energy consists of a sum of ideal and excess terms. The excess contributions are generally functionals of appropriate site densities (as we discuss below), and it is reasonable to assume that they will retain the same functional forms even in the presence of clusters. This is consistent with our treatment of non-associated molecules, 21,24,25 as one can view the clusters as just another (albeit more elaborate) molecular species in this context. Thus, the only change to the free energy functional brought about by the introduction of clusters is in the ideal term. In this work, we shall constrain our analysis to clusters containing: two; four, and six ions only, but our methods are easily generalizable to even larger (and even non-neutral) clusters. With this constraint the ideal free energy has the specific form,

id

βF [{N

(α)

α

(R )}] =

XZ

N (α) (Rα ) ln[N (α) (Rα )] − 1 dRα +

α

XZ

(α)

N (α) (Rα )VB (Rα ) dRα

(2)

α

in which {N (α) (Rα ); α = f c, f a, p, q, h} is the set of oligomeric densities describing the free cation, free anion, pair, quadruple and hextuple clusters, respectively. The collective co-ordinate Rα = (r1 , r2 ...), represents the site coordinates of the beads in the particular (α)

molecular(cluster) species of type α. The bonding function VB (Rα ), will contain a combination of rigid bond and flexible string components, relevant to the particular type of cluster. The collective site densities are denoted as {nβ (r); β = c, a, n}, and are the sum over site densities of the same type, e.g., positive (c); negative (a), and neutral (n) beads, respectively. Hence we have, nβ (r) =

XX α

(α)

ni (r)

i(β)


(3)


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where the nomenclature of the sum over i implies that we sum over all sites of type β in the species α. The specific site densities in the species α is given by, (α) ni (r)

Z =

δ(r − ri )N (α) (Rα ) dRα

(4)

where δ(r) is the Dirac delta function. The total site density is given by, ns (r) = nc (r) + na (r) + nn (r). The total Grand free energy functional has the general form,

Ω = F id [{N (α) (Rα )}] + Fhs [¯ ns (r)] + Fdisp [ns (r))] + Fcorr [nc (r), na (r)] Z Z X X α (α) α α (α) (α) + Φ(R )N (R ) dR − (µ + q ΨD ) N (α) (Rα ) dRα α

(5)

α

This free energy functional describes the fluid in the presence of an externally applied potential, in equilibrium with a bulk reservoir. In this case the chemical potentials, µ(α) of species α are fixed, which also determines the molecular weight distribution of clusters in the bulk. The quantity, ΨD , is the Donnan potential, which maintains electroneutrality in the presence of external charges and q (α) is the total charge of species α. The various excess free energy terms have been presented in previous descriptions of the classical DFT 20,21 and will only be described briefly here.

Excess free energy contributions The hard sphere contribution to the free energy, Fhs [n¯s (r)], accounts for the entropy arising from the excluded volume interaction between all beads. 26–28 For this term we shall use a simplified version of the generalized Flory-dimer (GF-D) equation of state, which is assumed to be a functional of the total weighted bead density n ¯ s (r), 29 3 n ¯ s (r) = 4πσ 3

Z

ns (r0 ) dr0

|r−r0 |

A Classical Density Functional Study of Clustering in Ionic Liquids at

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