7/29/2019 Dielectric Interpretation of Specificity of Ion Pairing in Water

1/4

Published on Web Date: December 02, 2009

r2009 American Chemical Society 300 DOI: 10.1021/jz900151f |J. Phys. Chem. Lett. 2010, 1, 300303

pubs.acs.org/JPCL

Dielectric Interpretation of Specificity of Ion Pairing inWaterMikael Lund,*, Barbara Jagoda-Cwiklik, Clifford E. Woodward,z Robert Vacha, andPavel Jungwirth

Department of Theoretical Chemistry, Lund University, P.O.B 124, SE-22100 Lund, Sweden, The Fritz Haber Research Centerfor Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel, zSchool of Physical,Environmental and Mathematical Sciences, University College, University of New South Wales, Australian Defence ForceAcademy, Canberra ACT 2600, Australia, and Institute of Organic Chemistry and Biochemistry, Academy of Sciences of theCzech Republic and Center for Biomolecules and Complex Molecular Systems, Flemingovo nam. 2, 16610 Prague 6,Czech Republic

ABSTRACT We present a dielectric continuum model that, at a semiquantitativelevel, explains why ion pair formation in water is favored by like-sized ions overunlike-sized pairs. Using both classical and ab initio continuum approaches, we

show that the now well-established empirical rule, the so-called

law of matchingwater affinities, can be rationalized in terms of ion solvation. Namely, pairing ofdifferently sized ions is weakened due to a shadowing effect where the larger ionshields thesmaller ionfrom thesolvent.It is shown that this empirical lawceases tobe valid for less polar solvents where strong ion-ion coulomb interactionsdominate the pairing free energy. The presented model demonstrates that certainion-specific effects, such as those connected with the Hofmeister series, can bequalitatively captured by classical continuum electrostatics, although a fullyquantitative description would require explicit molecular treatment of the solvent.

SECTION Statistical Mechanics, Thermodynamics, Medium Effects

The degree of ion pairing in solution is the result of asubtle balance between ion-solvent interactions andCoulomb forces between cations and anions.1,2 While

the former keep the ions isolated in solution, thelatter enableion pairing in the form of contact as well as solvent-separatedpairs. In solvents with a low dielectric constant, this balancetends to be tilted toward strong ion pairing with the extremecase ofr = 1, where solvent effects disappear. Water as asolvent with r = 80 is approaching the other extreme, that is,infinite dielectric constant, where ion-ion Coulomb interac-tions (but not solvation!) are scaled to zero. Nevertheless, ionpairing in water does take place insofar as is indicated by theoccurrence of distinct peaks in the cation-anion radial dis-tribution functions, hinting at contact and solvent-separated

pairs.3-5

While the number of ion pairs obviously grows withincreasingsalt concentration,the strengthof pairing at a givenconcentration is stronglyion-specific. Thatis,itdependson thechemical nature of cations and anions.1,5,6 Phenomenologi-cally, this ion specificity has been formulated as an empiricalrule, the so-called Law of Matching Water Affinities.7,8 It statesthat monovalent cations and anions of similar hydrationenergies, that is, (roughly speaking) of similar size, pair morestrongly than dissimilar ions. It is the purpose of the presentstudy to provide, using a dielectric solvation model and abinitio calculations, an elementary microscopic interpretationof thisremarkablyoperationalempirical rule, which haslackedtheoretical justification.

The Law of Matching Water Affinities is reflected in ther-modynamic data for bulk electrolyte solutions, as shown inFigure 1; for example, exchanging cesium with sodium incontact with iodide (big-big f big-small) is unfavorable,while the oppositeis true for fluoride (big-small fsmall-small). This behavior is also reflected in explicitsolvent molecular dynamics (MD) simulations, as shown inFigure1, right, for the free energyof interaction between NaF,CsF, NaI, and CsI. From explicit solvent MD simulations aswell as from analysis of ion coordination structures, it followsthat ion specificity is intimately connected to ion-water andwater-water interactions.5,9,10

While explicit solvent models account in molecular detailforsolvation changes when ions form a contact ionpair, thisis

not the case for the widely used primitive model (PM) ofelectrolytes, where ions are described by hard, chargedspheres in a uniform dielectric continuum.12 Here, the inter-action between any two ions is given by a solvent-mediatedCoulomb potential, u q1q2/rr, where r is the relativedielectric constant, q1 and q2 are the charges, and r is thedistance between them. Thus, the critical parameter thatdetermines the ion pairing strength in the PM is the distance

Received Date: October 16, 2009Accepted Date: November 17, 2009

7/29/2019 Dielectric Interpretation of Specificity of Ion Pairing in Water

2/4

r2009 American Chemical Society 301 DOI: 10.1021/jz900151f |J. Phys. Chem. Lett. 2010, 1, 300303

pubs.acs.org/JPCL

of closest approach, that is, the sum of ionic radii, a1 a2. Inconflict with the idea of matching radii, this implies thatbig-small combinations will always lead to stronger pairingthan big-big combinations. The PM is often employedsuccessfully to describe bulk electrolyte properties, and acommon, pragmatic, approach is to fit the ionic radii toexperimental data. Table 1 shows examples of distances ofclosest approach that best match experimental osmotic and

activity coefficients for two halide salts of sodium and ce-sium.13 These lead to inconsistent bare ionic radii, within thePM, as the sums of the two diagonals in Table 1 are not equal.The observed distances shown in Table 1 are a consequenceof the fact that sodium fluoride (small-small) and cesiumiodide (large-large) pairs are stronger than sodium iodide(small-large) and cesium fluoride (large-small) pairs follow-ing, The Law of Matching Water Affinities.

Can one account at least semiquantitatively for theseobservations within a continuum solvent model? Consider aspherical cavity of radius a and dielectric constant i im-mersed in a continuum solvent of dielectric constant r. Theelectrostatic free energy of interaction between a pair ofcharges qi and qj located at positions ri and rj inside of the

sphere is then described by an infinite-order multipole ex-pansion14-16

wij qiqj

40i

1rijXn0

rirjn

a2n1

!rirj< a 1

where 0 is the permittivity of vacuum, = [(i - r)Pn(cos)]/[r- in/(n1)],andPn(cos) is the Legendrepolynomialevaluated for the angle between ri and rj. The solvation freeenergy for each charge is given by

Gj q2j

80i

Xn0

r2nj

a2n1 2

which for rj= 0 reduces to the well-known expression for theBorn energy.

Imagine the process of bringing two isolated, solvatedionsintothe contactdistance r12= a1 a2 sothattheyformanionpair with a common, spherical solvation shell, as shown inFigure 2. The pairing free-energy change of this process isGpair= w12G1 G2-G1

Born- G2

Born, where GjBorn = Gj,rj=0

is the Born energy as defined above. The last two terms forGpair thus describe the solvation energies of the two isolatedions. WhileGpair can be evaluated from eqs 1 and 2, we canfor comparison also approximate the pairing energy bydescribing theformedionpair as a point dipole, with moment = q1r1 q2r2, using Onsager's solvation model.17,18 This isequivalent to neglecting terms in the summations above forn > 1. Following the thermodynamic cycle illustrated inFigure 3, we obtain the pairing free energy

Gdipolepair

140

-2

2a3

q212a1

q22

2a2

!1i-

1r

q1q2jq1 -q2j

2i

8