Nonelectrostatic interactions between ions with anisotropic ab initio dynamic polarisabilities

Colloids and Surfaces A: Physicochem. Eng. Aspects 343 (2009) 57–63

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Nonelectrostatic interactions between ions with anisotropic ab initio dynamic polarisabilities Drew F. Parsons ∗ , Vivianne Deniz, Barry W. Ninham Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia

a r t i c l e

i n f o

Article history: Received 30 September 2008 Received in revised form 5 January 2009 Accepted 30 January 2009 Available online 5 April 2009 Keywords: Dispersion forces Electrolyte Anisotropy ab initio dynamic polarisability Nonspherical ion PACS: 31.15.ap 34.20.Gj 33.15.Bh

a b s t r a c t Ion specific effects are common in colloidal and biological systems. Bubble lifetime before coalescence depends on the electrolyte. The strength of different salts at precipating proteins leads to Hofmeister series. Theories of ion and colloid interactions based on electrostatics alone, including the Debye–Hückel theory of electrolytes or the DLVO theory of colloids, are unable to predict these ion specific effects. Rather, the theories need to be modified to account for quantum mechanically derived nonelectrostatic dispersion interactions [1]. The dynamic polarisability (i) of an ion lies at the heart of its dispersion interactions. In general it may be written as a sum over many quantum modes, each with frequency n. Approximations in the past have reduced it to a single mode with modal frequency derived from the ionisation potential (IP) of the ion [2]. We have calculated the exact dynamic polarisabilities of a wide range of ions [3] using ab initio quantum mechanics and present here comparisons against the single-mode IP approximation. The error in calculated dispersion energies due to the single-mode IP approximation averages around 40%, and reaches as high as 86% error for halide ions. Ionic self-energies, and ion–ion and ion–surface interaction energies are calculated. Applications to activity coefficients in the bulk electrolyte are presented. Hofmeister series in the activity coefficients of alkali halides consistent with experiment are found. We also discuss further development of the theory of dispersion interactions with the aim of obtaining a quantitatively, not merely qualitatively, predictive theory. The steps include (i) taking into account the nonspherical anisotropic features of the ions and (ii) using a nonlocal description of the solvent to include solvent spatial structure. Step (i) should be crucial in resolving the adsorption of anisotropic ions such OH to interfaces, which is currently subject of debate between theory and experiment [4,5]. © 2009 Elsevier B.V. All rights reserved.

Ion specific effects have been well characterised experimentally. Electrolytes exhibit a range of interactions with proteins, ranging from salting it to salting out (the original Hofmeister series). Bubble coalescence is suppressed by certain electrolytes but not others [6,7]. Certain ions are preferentially adsorbed at an interface over other ions. Cations or anions follow Hofmeister series in their activity coefficients, but cations, for example, may follow different series when matched with different anions, and vice versa [8,9]. Developments in the theoretical understanding of ion specific effects have led to an appreciation of the role of nonelectrostatic dispersion interactions between ions [1,2,10–12]. Gaps still remain however. The correct Hofmeister series have not always been predicted by theory. Only simple models of the dynamic polarisabilities of ions, the chief quantity responsible for dispersion interactions,

∗ Corresponding author. Tel.: +61 2 6125 2847; fax: +61 2 6125 0732. E-mail address: [email protected] (D.F. Parsons). 0927-7757/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2009.01.029

have been available. The crucial role of the hydration layer around ions has started to become apparent [13–16]. The theory of nonspherical ion shape has been introduced [17], but a characterisation of the consequences of nonsphericity have not been pursued. In this paper we present developments in the theoretical description of ionic dispersion forces. Firstly we introduce new calculations of ionic dynamic polarisabilities performed using ab initio quantum chemistry. We compare the effect of these polarisabilities on self-energies, ion–ion and ion–surface interaction energies, compared with the simpler model used previously. Next, we present the impact of our ion–ion dispersion interactions, together with current notions of ionic hydration shells, on bulk osmotic and activity coefficients. We explore the success of these calculations at reproducing experimental Hofmeister series in activity coefficients. Lastly, we present a model of a nonspherical ion with anisotropic polarisability and apply the model to the solvation (self-) energy of an ion.

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D.F. Parsons et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 343 (2009) 57–63 Table 1 Selected ionic radii. ag is the Gaussian radius used in nonelectrostatic dispersion energies. as is the hard sphere radius used to define the excluded volume region in calculation of bulk properties. Li+ , Na+ , F− and acetate are treated as strongly hydrated cosmotropic ions and include a water radius of 1.14 Å.

Fig. 1. Dynamic polarisability ˛(iω) fora perchlorate ion along the imaginary axis. The exact ab initio curve is compared with the single-mode curve based on the ionisation potential of perchlorate.

1. Ionic parameters 1.1. Dynamic polarisabilities Ab initio dynamic polarisabilities ˛(iω) for bare ions (gas phase) were calculated [3] at imaginary frequencies using the effective fragment potential method [18]. In principle the ab initio calculation includes contributions from all modes (all electronic transitions). A simple single-mode model has been used previously in studies of nonelectrostatic dispersion interactions [2,10–12], ˛(iω) =

˛0 1 + (ω2 /ω02 )

.

(1)

In the single-mode IP model the single characteristic frequency ω0 is estimated by the ionisation potential of the ion. In contrast the ab initio polarisability may be described at the sum of many such terms, each with a different modal frequency corresponding to a different electronic transition. The consequence of the full ab initio description is that the polarisability gains a stronger high frequency tail than seen in the single-mode model. For instance, the ab initio and single mode polarisabilities for perchlorate are shown in Fig. 1. Single-mode parameters were taken as ˛0 = 5.453 Å3 , ω0 = 1.313 × 1015 Hz [10]. Even though the static ab initio polarisability at iω = 0 is slightly smaller than the single-mode parameter, the single-mode polarisability falls to 0 at frequencies an order of magnitude sooner than in the ab initio case. The appearance of the stronger high frequency tail is consistent in all ions. Since dispersion interactions are calculated by summing over the polarisabilities at all frequencies, the magnitude of the interactions from ab initio polarisabilities are consequently much stronger than from the single-mode model. 1.2. Ionic radii Correct identification of ionic radii in solution is crucial to predicting electrolyte properties such as activity coefficients and their Hofmeister series, as discussed below in Section 3.3. Ionic radii were estimated by calculating the volume of the ions’ electron cloud using ab initio quantum chemistry [19], ascribing a Gaussian spatial spread (cf. Eqs. (3) and (17)) to the charge distribution. All nonelectrostatic dispersion energies were calculated using Gaussian radii. Electrolyte properties (osmotic and activity coefficients) were calculated (Section 3) with hard sphere excluded volumes. The hard sphere radii as were derived from the Gaussian radii ag by conserv√ 1/3 ing the volume of the ion (i.e. as = ag (3 /4) ). Metal cations are treated specially, with the hard sphere radius as taken to be the

Ion

ag (Å)

as (Å)

Li+ Na+ K+ Cs+ F− Cl− Br− I− Acetate

0.38 0.61 0.96 1.47 1.12 1.69 1.97 2.12 2.20

1.56 1.81 1.77 2.14 2.26 1.86 2.16 2.33 3.55

radius corresponding to the volume enclosing the electron cloud, without fitting to the Gaussian radius. This corresponds to a cavity radius, with the cavity determined by the empty outer electron shell [19,20]. Lastly, we follow Collins’ concept of the Law of Matching Water affinities [13,21] to distinguish between cosmotropic and chaotropic ions. Cosmotropic ions are taken to have a tightly bound hydration shell. Therefore a hard sphere water radius of 1.14 Å is added to the hard sphere radii of cosmotropic ions. Of the ions used in Section 3.3, Collins identifies cosmotropic ions to be those with a positive Jones-Dole viscosity B-coefficient. Examples of cosmotropes include Li+ , Na+ , F− and acetate (although Na+ is borderline). Since the question of ion size is important in reproducing Hofmeister series of activity coefficients, we give the radii of the ions used in the Hofmeister calculations of Section 3.3 in Table 1. 2. Nonelectrostatic dispersion energies We calculate dispersion energies using the Mahanty–Ninham formulation of Lifshitz theory [22,23]. In general, the dispersion energy at temperature T is given by U disp = kT



Tr G(n ),

(2)

n=0

summing over imaginary frequencies n = iωn = 2inkT/¯h. The prime against 0 indicates the term at n = 0 is taken with a factor of 1/2. Matrix G derives from the Greens function describing the interaction of the electromagnetic field around the ion with its polarisation current. We assume the ionic polarisability is isotropic, following a spherical Gaussian spatial distribution with Gaussian radius ag , −r 2 /a2

g e ˛(, r) = ˛() √ 3 .  ag

(3)

The case of a nonspherical ion with anisotropic polarisability is considered below in Section 4. 2.1. Self-energy For a spherical ion in a medium with dielectric response ε(iω), the dispersion self-energy (that is, the interaction of the ion with its own electromagnetic field), is determined by Eq. (2), with the trace of the Green’s matrix being ˛() F(). Tr G() = 4 √ 3  ag ε() F() = 1 + 4x2 [1 − x exp(x2 )erfc(x)], with x =

(4)



ε()ag /2c.

D.F. Parsons et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 343 (2009) 57–63

Fig. 2. Dispersion self-energy for ions in water, as a function of ion molar volume. Energy calculated using ab initio polarisabilities (black circles) and using singlemode IP polarisabilities (blue squares). % relative difference between the two models is given (red diamonds). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

The dispersion self-energy typically comprises around 30% of disp the total self-energy Us = Usel + Us , which also includes an electrostatic component. In order to highlight the effect of ab initio polarisabilities, we display only the dispersion component for a range of ions as a function of ionic volume. Dagastine and White’s dielectric function ε(iω) for water is used [24]. The results, shown in Fig. 2, show that ab initio polarisabilities provide a correction of as much as 50% or more over the over the single-mode IP polarisabilities. 2.2. Ion–ion interaction energy The nonelectrostatic dispersion interaction between two ions in a medium with dielectric spectrum ε(iω) with isotropic polarisabilities ˛i () and Gaussian radii ai separated by distance R is again calculated by Eq. (2). The trace of the Green’s matrix describing the twobody interaction is Tr G() = ˛1 ()˛2 ()

F(R, ) 4R6

(5)

where taking x = /c (c is the speed of light), F(R, ) =

2



e

ε()a2 x2 /4



2 2

1+R x +

i





i=1,2



− 1 + R2 x 2 −



+

e





ε()Rx ϕ2i

ε()a2 x2 /4

+

8Re √ ai

i

−

4Re √ ai



(ε() − 1)R2 x2 +2

(1 − ε())R2 x2 −R2 /a2

−

−R2 /a2



i

i=1,2





 1+



+2



ε()Rx − 2 ϕ2i



ε()Rx + 2 ϕ1i

 

ϕ1i = exp −

ϕ2i = exp



 ε()xR

.

ε()xR

2



 erfc



ε()xai

erfc

−

R ai



ε()xai 2

R + ai

The nonelectrostatic interaction between an ion and the interface between two media with dielectric functions ε1 (iω) and ε2 (iω) is given approximately by [10,25]: Usurf (x) =

B x3

(8)

where the ion is at distance x from the interface. The coefficient B is determined from the ion polarisability and the difference between the two dielectric media, with the ion located in medium 1, kT  ˛(iωn )(iωn ) . 2 ε1 (iωn )

(9)

(iω) =

ε2 (iω) − ε1 (iω) . ε2 (iω) + ε1 (iω)

(10)

The B coefficients for a range of ions at the air-water interface are given in Table 2, comparing results drawn from ab initio polarisabilities and single-mode IP polarisabilities. The ab initio polarisabilities



a2i



2.3. Ion–surface interaction energy

As with Eq. (2), ωn = 2nkT/¯h and the prime against 0 indicates the term at n = 0 is taken with a factor of 1/2. (iω) is the dielectric difference

i

(6) with

By way of illustration we present the bromide–bromide dispersion interaction in Fig. 3, comparing the interaction using ab initio polarisabilities with the one using the single-mode IP model. The ab initio interaction in this case is around 40% stronger than the single-mode model. The impact of the increased ion interactions on bulk electrolyte properties (osmotic and activity coefficients) is discussed in Section 3.

n=0



R2

Fig. 3. Dispersion interaction energy between two bromide anions in water, as a function of separation between the ions. Interaction energy calculated using ab initio polarisabilities (black solid line) and using single-mode IP polarisabilities (blue dotted line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

B=−

ε()Rx ϕ1i

59

(7)

Table 2 B coefficients for the ion–surface dispersion interaction for a range of ions at an air-water interface. The first column was calculated using ab initio dynamic polarisabilities ˛(iω), the second used a single-mode ionisation potential model for ˛(iω). Ion

B (10−50 J m3 ) (ab initio pol.)

B (10−50 J m3 ) (single-mode pol.)

Babin. /B1−mode

Li+ Na+ K+ Cs+ Cl− Br− I−

0.21 0.91 3.83 8.93 11.27 12.63 19.69

0.19 0.78 3.14 6.81 5.70 7.07 9.07

1.1 1.2 1.2 1.3 2.0 1.8 2.2

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generate ion–surface interactions which are much stronger than those predicted by the single-mode model. For instance, although there is only a slight correction to the alkali cations, the ab initio interactions are around twice as strong for halides 3. Bulk electrolyte properties The osmotic and activity coefficients of an electrolyte are calculated with the help of the radial distribution function, described as a Boltzmann distribution derived from the ion–ion interaction energy gij (r) = exp[−uij (r)/kT ]. This describes the probability of finding charge j at a distance r from a central charge i. The interaction energy uij (r) between i and j contains both an electrostatic disp

(uel ) and nonelectrostatic (uij ij disp uij (r) + uxij (r).

) component, with uij (r) = uel (r) + ij

uxij (r)

The term is an excluded volume term corresponding to inner-core repulsion of the two ions at very close separations where their electron clouds overlap. In the absence of a detailed quantum-derived description of this inner-core repulsion, we employ a hard sphere potential with radius ah , for which the corresponding radial distribution function is a step function, preventing ions j getting closer than ah from ion i. The hard-core radius ah for central ion i is taken to be the largest ion–ion distance afforded by the hard sphere radii asj of each ion, that is ah = maxj {asi + asj }

Fig. 4. Calculated osmotic coefficients for CsI over a range of concentrations. Calculations performed with both electrostatic and dispersion interactions between ions (solid black curve), and electrostatic only (dotted red curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

to a surface contact term. Hence, 1 

=1− nT kT ij



∞

4r 2 dr r ah

lated from the nonlinear Poisson–Boltzmann equation ∇ 2 i (r) =  −4i (r)/ε0 , where i (r) = qi ı(r) + nj qj gij (r). nj is the number density of ion j, that is nj = j n where j is the stoichiometric factor for ion j in the electrolyte and n is the bulk concentration (number density) of the electrolyte. Modelling the radial distribution as a Boltzmann distribution is strictly speaking only valid at lower concentrations. At very high electrolyte concentrations many-body effects, that is correlations between different ions surrounding the central charge, become significant. These effects may be handled via integral equation theories, such as the hypernetted chain (HNC) approximation [26,27]. Nevertheless, the Poisson–Boltzmann approach provides good predictions up to relatively high concentrations of 1 M [28]. Calculations shown in the figures in this section are presented up to concentrations of 2 M. The deviation of Poisson–Boltzmann calculations away from experimental values between 1 M and 2 M can be seen, for instance, in the activity coefficients for potassium acetate in Fig. 6. disp The nonelectrostatic dispersion interaction uij (r) was calculated in the previous section.

4a3h  nT

ni nj gij (a+ ) h

ij

In Fig. 4 we present the osmotic coefficients for several electrolytes calculated with and without nonelectrostatic dispersion interactions. Osmotic coefficients with nonelectrostatic interactions are calculated using both ab initio dynamic polarisabilities and single-mode IP polarisabilities. We see that the nonelectrostatic correction is not large until the concentration exceeds 0.06 mol L−1 . 3.2. Activity coefficients Activity coefficients are defined as i = ∂A/∂Ni , where Ni is the number of particles of ion i. A is the Helmholtz free energy of the electrolyte, determined by integrating the excess internal energy E*, where E ∗ = ∂(A/T )/∂(1/T )|V [29,30]. E* is determined from the average interaction energy between two ions, that is V ni nj E = 2 ∗



d3 r u˜ ij (r)gij (r)

(14)

ij

In the linear approximation neglecting nonelectrostatic interactions, the excess internal energy at a fixed electrolyte concentration may be written  as a function E ∗ ( ) of the inverse Debye length

2 = (q2e /εv ε0 kT ) i ni q2i (εv is the permittivity of a vacuum). This indirectly provides the temperature dependency required to extract the Helmholtz free energy and hence the activity coefficient. In this  way the mean activity coefficient ln ± = ln i / reduces to i i ln ± =

3.1. Osmotic coefficients

∂r

gij (r) +

(13)

(11)

Radius as is the hard sphere radius of the ion, see Section 1.2. The electrostatic interaction energy uel (r) is calculated by treatij ing charge j to be a point charge, interacting with the electrostatic potential i (r) of charge i, that is uel (r) = qj i (r). i (r) is calcuij

∂u˜ ij

 E ∗ ( ) ni Vi . + nVkT

(15)

i

The osmotic coefficient of an electrolyte with number density (concentration) n is given by [29]:

=1−

1  ni nj nT kT ij



∞

4r 2 dr r 0

∂u˜ ij ∂r

gij (r).

(12)



nT is the total  number density of all ions, that is nT = nj = n, where = j . In this formula u˜ ij is the primitive interaction disp

energy between two lone ions, u˜ ij = u˜ el + uij ij

+ uxij with u˜ el (r) = ij

qi qj /ε0 r. When the excluded volume term is presented by a hard sphere potential, its derivative is ∂uxij /∂r = −kTı(r − ah ), giving rise

The volume V is cancelled by the excess internal energy in Eq. (14). The second term on the right hand side accounts for the compressibility of the electrolyte, with Vi being the ionic volume of ion i [30]. We apply this formula to obtain an estimate for the activity coefficient of an electrolyte. Results are shown in Fig. 5 for [KCl], comparing the electrolyte with and without nonelectrostatic interactions, and comparing the dispersion interactions using both ab initio and single-mode IP models of the polarisability. We see that the nonelectrostatic correction becomes significant at concentrations greater than 0.1 mol L−1 .

D.F. Parsons et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 343 (2009) 57–63

61

Table 3 Calculated activity coefficients for series of potassium and bromide salts at 1 M concentration. Calculated using nonelectrostatic dispersion forces. Ac indicates acetate. Values in brackets are experimental [9]. Li[3w] and Na[2w] refer to cosmotropic Li+ and Na+ with a triple and double hydration layer respectively. Activity coefficients ln Potassium salts KF KCl KBr KI KAc

Fig. 5. Calculated activity coefficients ln for CsI over a range of concentrations. Calculations performed with both electrostatic and dispersion interactions between ions (solid black curve), and electrostatic only (dotted red curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

3.3. Hofmeister series Osmotic and activity coefficients follow a well defined Hofmeister series. For instance, the experimental activity coefficients of potassium halides (and acetate, denoted by Ac− ) and the alkali bromides [9], shown in Fig. 6, exhibit the series Cl− < Br− < I− < F− < Ac− Cs+ < K+ < Na+ < Li+

(16)

The series is generally consistent over all concentrations (the relationship of KF to KI being the exception), we therefore test the ability to predict the Hofmeister series by calculating at a concentration of 1 mol L−1 since activity coefficients of different electrolytes are well spread at this high concentration, showing clear ion specificity. Results are shown in Table 3. The Hofmeister series for the potassium salts is reproduced perfectly, including the unusually large values for KF (falling outside of the normal halide order) and KAc. Moreover quantitative agreement is very reasonable.

Bromide salts −0.449 (−0.439) −0.521 (−0.504) −0.471 (−0.483) −0.458 (−0.439) −0.270 (−0.245)

LiBr Li[3w]Br NaBr Na[2w]Br KBr CsBr

−0.473 (−0.219) −0.200 −0.460 (−0.375) −0.351 −0.471 (−0.483) −0.461 (−0.620)

We have less success in reproducing the correct Hofmeister series for the alkali bromides. We have the correct relationship of Na+ to K+ , but the activity coefficients of Li+ and Cs+ are overestimated and underestimated, respectively. We attribute this discrepancy to the special model used to estimate the hard sphere radii of metal cations. Li+ was treated as a hydrated cosmotrope by adding one water radius aw = 1.14 Å to the intrinsic hard sphere radius of the ion, forming the hard sphere cavity radius used for the ion. Since Li+ is extremely small, its electric field near to the ion is large and therefore its hydration shell is bound tightly with a larger solvation number than other ions [31]. It is reasonable then to expect that the hydration layer for Li+ should be larger than for other cosmotropes, as implied by the larger hydrodynamic radius of Li+ [32]. If, for instance, we double the hydration width to 2aw , then the activity coefficient for 1 M CsBr rises to −0.365, which then reproduces the correct Hofmeister series relative to Na+ and K+ . Tripling the hydration width to 3aw shifts ln ± to −0.200, which quantitatively matches the experimental value. Interestingly, the radius with triple hydration width does indeed correspond to the Nightingale hydrated radius for Li+ , 3.8 Å. The hard sphere radius of the metal chaotropic cations was determined by matching the outer volume of the electron cloud, such that there could be no overlap between ions. However, Cs+ lies on a row where its outer electron levels involves f-orbitals, whereas only d-orbitals are involved in the electron cloud of K+ . It is reasonable therefore to suppose that the ionic radius of Cs+ , enclosing the ion’s electron cloud, should be treated differently to K+ . For instance,

Fig. 6. Experimental [9] activity coefficients ln (single points) for potassium halides (and acetate, denoted by Ac− ) and the alkali bromides over a range of concentrations. Continuous lines indicate theoretical calculations (Eq. (15)). Cosmotropes in theoretical calculations are F− and Ac− (single water radius), Na+ (double water) and Li+ (triple water), all other ions are chaotropic (no fixed hydration layer). The theoretical hard sphere radius for Cs+ was matched to its Gaussian radius.

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D.F. Parsons et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 343 (2009) 57–63

we might treat Cs+ in the same way that the anions were treated, allowing some overlap between electron clouds (achieved by set√ 1/3 ting as = ag (3 /4) , i.e. taking the hard sphere radius which that is equivalent to the Gaussian radius), such that as reduces from 1.86 Å to 1.62 Å. The activity coefficient for 1 M CsBr then drops to –0.502, which now reproduces the correct Hofmeister series relative to the other alkali metal cations. Theoretical estimates over concentrations 0–2 M (using the corrected hard sphere radius for Li+ , Na+ and Cs+ ) are shown in Fig. 6. As discussed above, the Poisson–Boltzmann distribution is valid up to around 1 M. The simplified expression for the activity coefficients, Eq. (15) also breaks down at higher concentrations. Its limit of validity may be determined via the Gibbs–Dunhem equation, relating osmotic and activity coefficients to one another [29,30]. Activity coefficients derived from osmotic coefficients via the Gibbs–Duhem relation agree with those derived from Eq. (15) up to concentrations of around 0.3 M, which is sufficiently high, for instance, to describe physiological systems (at 0.1 M). It is interesting to note that we nevertheless see good agreement in Fig. 6 between experiment activity coefficients and our calculated (Eq. (15)) values at high concentrations up to 2 M. 4. Nonspherical anisotropic ions In general polyatomic ions are nonspherical. Nitrate and chlorate, for instance, are flat, diatomics are cylindrical. Rather than imposing a spherical Gaussian model on such ions, we suggest a nonspherical model with normalised spatial spread given by

 

1 f (r) = √ exp −   ax ay az

x2 a2x

+

y2 a2y

+

z2 a2z



(17)

where ax , ay , az are anisotropic Gaussian radii. The ionic polarisability with spatial spread is ˛(iω, r) = ˛(iω)f(r), where ˛(iω) is the anisotropic polarisability tensor. The electrostatic charge distribution may be similarly modelled by (r) = Qf (r). The same ab initio quantum calculations of the mean isotropic polarisabilities described in Section 1.1 also generate anisotropic polarisability tensors. The axes of the ion may be aligned such that the tensor is diagonal. As an illustration of the effect of anisotropy, the three polarisability components for formamide, the most anisotropic molecule are shown in Fig. 7. The anisotropic electrostatic self-energy (Born energy) calculated in reciprocal space is Usel =

1 2(2)3



 k)  (k)  d3 k(

 = Q exp[−(a2 k2 + a2 k2 + a2 k2 )/4] is the charge distriwhere (k) x 1 y 2 z 3  = bution in reciprocal space, and the electrostatic potential is (k)  4(k)/ε k2 . The three-dimensional integral is not tractable ana0

lytically but is readily calculated numerically. The anisotropic dispersion self-energy is calculated with Eq. (2), where matrix M now derived from a nonspherical anisotropic polar By aligning the axes of the nonspherical ion isability tensor ˛(, k). with the k-axes, the polarisability tensor becomes diagonal and the trace of M reduces to



TrM() = −4

d3 k





3ε0  2  + s · ˛(,  G(, k).  ˛(, ˜ k) k) c2

(19)

 is the polarisability vector ˛(,  = s is the vector [k12 , k22 , k32 ], ˛(, k)  k)  and ˛(,  is the mean isotropic polaris[˛1 (), ˛2 (), ˛3 ()]f (k) ˜ k)  = (˛ () + ˛ () + ability with nonspherical spatial spread, ˛(, ˜ k) 1 2  ˛ ())f (k)/3. G(, k) is a Green’s function, 3

G(, k) =

−1 (ε() 2 /c 2 ) + k2 .

(20)

As for the nonspherical electrostatic self-energy, the triple integral for the trace in Eq. (19) cannot be solved analytically but may be handled numerically. disp The total self-energy is Us = Usel + Us . We present results in terms of the solvation energy Us = Us [water] − Us [vac], the difference in self-energy of the ion in solution (ε0 = 78.36) and in vacuum (ε0 = 1). We present the relative difference in solvation energy for a range of ions and neutral molecules in Fig. 8 as a function of mean eccentricity. The relative difference is the difference between the anisotropic nonspherical model and the isotropic spherical model, calculated as a percentage relative to the nonspherical solvation energy. The mean eccentricity is the average of the eccentricities derived from the three anisotropic radii,  a21 − a22 /a1 (with a1 > a2 ). Naturally taken two at a time, e = the anisotropic correction increases as the eccentricity, that is the degree of nonsphericity, grows, with the correction becoming as high as 5–8% in the least spherical ions and molecules. The relative differences in Fig. 8 appear to split into two series, with one series (following thiocyanate , N2 , nitrate, chlorate, O2 and CO2 ) having a greater nonspherical correction than the other (following tartrate, ethanolamine-H+ , bicarbonate, formate and formamide) for a given mean eccentricity. The first series is distinguished from the latter in that its members are axisymmetric (linear or planar), that is two of their three anisotropic radii are equal. In the second series all three nonspherical radii are different. This observation likely indi-

(18)

Fig. 7. Anisotropic dynamic polarisability ˛(iω) for formamide. The polarisability tensor has been diagonalised to the three principle components shown in this figure.

Fig. 8. % relative difference in total solvation energy between nonspherical anisotropic model and spherical isotropic model. The two curves suggest series of axisymmetric (dotted curve) and asymmetric (dashed curve) ions. EtON denotes ethanolamine-H+ .

D.F. Parsons et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 343 (2009) 57–63

cates that the scalar mean eccentricity is not complete measure of three-dimensional nonsphericity. If a different measure assigning a greater nonsphericity value to planar or linear ions could be determined, it may bring the two series back into one unified curve. However reducing the three-dimensional shape of an ion down to a single number in this fashion is not necessarily possible. Since nonsphericity is expressed in the dispersion self-energy via the polarisability as well as the spatial shape of the ion, the nonspherical correction is greater for the dispersion component of the solvation energy than for the electrostatic component. The nonspherical correction to the electrostatic component ranges from 0.5 to 3% for ions with mean eccentricity above 0.5, compared to 1.2–8% for the dispersion component. That is, the anisotropic effect on the dispersion component is around three times stronger than on the electrostatic component. 5. Other developments We have demonstrated that experimental activity coefficients may be successfully calculated for spherical ions once appropriate ion sizes have been established. We also showed that the nonspherical anisotropic correction to the solvation energy of linear and planar molecules is quite significant. It seems reasonable to expect that the nonspherical anisotropic correction will be far greater in ion–ion interactions than seen in the self- or solvation energy, since either or both of the two ions may be oriented at angles shifting the strength interaction far from that predicted by a spherical model. We therefore conclude that including nonspherical anisotropic effects to ion–ion interactions will enable predictive, even quantitative, calculations of activity and osmotic coefficients for nonspherical salts. Likewise the nonspherical correction to ion–surface interactions should be stronger than the correction seen for ion self-energies since there will be a certain preferred orientation for a nonspherical ion interacting with the surface. We have seen that the single-mode ionisation potential model for the dynamic polarisability of the ion is largely inadequate, with enormous corrections observed when the dynamic polarisability is instead calculated from ab initio quantum chemistry. The difference lies chiefly in a stronger tail at high frequencies. The ab initio dynamic polarisabilities used in this paper were those for isolated ions. An important project for future development is to determine how these ab initio polarisabilities change in solution, due to adjacent hydration layers. It may prove necessary to treat the hydrated ion as a cluster, with explicit water molecules bound to the ion forming an hydration shell. The presence of fixed hydration shell will have implications for the ion radius used in calculations. The previous point addresses the notion of ion–solvent interactions. Ion–ion interactions, of course, define bulk electrolyte properties, osmotic and activity coefficients. A final component in electrolyte theory which will complete the picture is the question of solvent–solvent interactions. At the level of theory used in this paper, solvent interactions were modelled solely through the dielectric response ε(iω), which completely neglects the spatial structure of the solvent. Nonlocal models describing solvent–solvent spatial correlations (via ε(iω, k)) [33–35] may provide an appropriate means of including the molecular or spatial structure of the solvent, while remaining within a continuum model of the solvent without the need for an atomistic description of each individual solvent molecule. Acknowledgements We would like to thank Prof. Dr. W. Kunz for useful discussions. We recognise the support of the Australian Research Council’s Discovery Projects funding scheme.

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