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AN ELECTROSTATIC INTERPRETATION OF STRUCTURE-PROPERTY RELATIONSHIPS IN IONIC LIQUIDS Mark N. Kobrak and Nathan Sandalow Department of Chemistry of Brooklyn College and The Graduate Center of the City University of New York 2900 Bedford Ave. Brooklyn, NY 11210

ABSTRACT Room-temperature ionic liquids represent a novel and intriguing class of materials, but little is understood about the relationship between their ionic structure and their macroscopic physical properties. One of the most uniquitous concepts in molecular liquid theory, that of a molecular dipole moment, is in fact ill-defined for ionic materials and useless in the interpretation of their behavior. We derive an alternative electrostatic multipole description of a molecular ion and show how it can be used to interpret the macroscopic dynamic properties of an ionic liquid. Key to this analysis is the derivation of a quantity we call the “charge arm,” which is readily calculable for any ion and permits comparison of ions of both similar and disparate chemical structure. The principles advanced are useful in the interpretation of ionic liquid behavior and in the design of novel ionic liquids with desirable properties.

INTRODUCTION Chemists have long relied on simple physical principles to understand the properties of molecular liquids. These principles are based upon knowledge of the intermolecular forces at work in the liquid, and are invariably introduced to students in the first year of college chemistry.1 When comparing two liquids, variation in properties such as melting point and viscosity are interpreted by comparison of their dispersive interactions, specific interactions (e.g. hydrogen-bonding character), and molecular dipole moment. The emergence of room-temperature ionic liquids (ILs), salts that are molten at room temperature, demands re-assessment of these basic principles. While there is every reason to believe that dispersive and specific interactions will operate in a familiar way in ILs, more careful consideration must be given to the electrostatic interactions that are represented by the dipole moment in molecular liquids. The dipole moment is in fact a second-order term in a power series expansion of the electrostatic potential associated with a distribution of charge, and is mathematically ill-defined for systems including a net charge (i.e. ions). If electrostatic interactions in ILs are to be understood, an alternative framework must be derived.

Such a framework is the goal of the present work. We begin with a review of the derivation of the dipole moment and its failure for charged systems. We then present an alternative formulation in which the ion is treated as a point charge with its location chosen to optimally represent the total electrostatic distribution. We derive a property of an ion that we call the “charge arm,” which is a measure of the rotational response of the ion to an external electric field. We show that this property is can be used to qualitatively understand macroscopic dynamic properties such as viscosity and conductivity in ILs.

MULTIPOLE DESCRIPTION OF MOLECULAR IONS

Multipole Description of Ionic Systems We begin by reviewing the multipole expansion of the electrical potential associated with a distribution of charged sites. We follow the derivation of Wangsness2 and consider the electrostatic potential at a point r arising from an assembly of N charges Figure 1. Coordinate scheme. positioned at coordinates ri N

φ (r ) = ∑ i =1

qi

4πε o Ri

[1]

,

where Ri=|r-ri|=(r2+ri2 –2 r ri cos(θ)). In the case where the sites are arranged such that the distance between them is small relative to their distance from the point r, it is productive to rewrite the 1/Ri term in Eq. [1] as

1 1 = ; Ri r (1 + t )1/ 2 2

[2] 2

⎛r ⎞ ⎛r ⎞ t = −2 ⎜ i ⎟ cos(θ i ) + ⎜ i ⎟ . ⎝r⎠ ⎝r⎠ For ri