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Publicly Available Published by De Gruyter January 21, 2016

“Thought experiments” as dry-runs for “tough experiments”: novel approaches to the hydration behavior of oxyanions

  • Ariel A. Chialvo ORCID logo EMAIL logo and Lukas Vlcek ORCID logo

Abstract

We explore the deconvolution of correlations for the interpretation of the microstructural behavior of aqueous electrolytes according to the neutron diffraction with isotopic substitution (NDIS) approach toward the experimental determination of ion coordination numbers of systems involving oxyanions, in particular, sulfate anions. We discuss the alluded interplay in the title of this presentation, emphasized the expectations, and highlight the significance of tackling the challenging NDIS experiments. Specifically, we focus on the potential occurrence of Ni2+SO42 pair formation, identify its signature, suggest novel ways either for the direct probe of the contact ion pair (CIP) strength and the subsequent correction of its effects on the measured coordination numbers, or for the determination of anion coordination numbers free of CIP contributions through the implementation of null-cation environments. For that purpose we perform simulations of NiSO4 aqueous solutions at ambient conditions to generate the distribution functions required in the analysis (a) to identify the individual partial contributions to the total neutron-weighted distribution function, (b) to isolate and assess the contribution of Ni2+SO42 pair formation, (c) to test the accuracy of the neutron diffraction with isotope substitution based coordination calculations and X-ray diffraction based assumptions, and (d) to describe the water coordination around both the sulfur and oxygen sites of the sulfate anion. We finally discuss the strength of this interplay on the basis of the inherent molecular simulation ability to provide all pair correlation functions that fully characterize the system microstructure and allows us to “reconstruct” the eventual NDIS output, i.e., to take an atomistic “peek” (e.g., see Figure 1) at the local environment around the isotopically-labeled species before any experiment is ever attempted, and ultimately, to test the accuracy of the “measured” NDIS-based coordination numbers against the actual values by the “direct” counting.

Introduction

Oxyanions such as nitrates, carbonates, and sulfates are ubiquitous species in geochemical and industrial aqueous phase environments. In particular, the aqueous chemistry in the troposphere usually involves hydrated nitrates resulting from the fast interaction between nitric acid and aerosol particles [1], while their interactions with sulfates are crucial in the homogeneous nucleation of ice particles [2]. Moreover, sulfate ions trigger changes of electrode activity in fuel cells [3], and are usually the signature of water pollution in fresh water environments [4]. Yet, despite its environmental and industrial relevance, little is understood about the hydration structure of the sulfate ion. As in the case of the hydration of nitrates [5, 6], sulfates exhibit weak – though not to the extent of the monovalent oxyanions – interactions with water whose pair correlation functions typically overlap those of the cation–water interactions [7–10]. Its weak hydration behavior, added to the ionic strength effect on the water dielectric permittivity, provides the opportunity to participate in cation-oxyanion (Mν+SO42) pairing as suggested by the wide range of experimental data of stability constants [11], and probes from electric conductivity [12–18], dielectric relaxation spectroscopy (DRS) [19–23], NMR [24], Raman [25–32], spectrophotometry [33–35], IR [36, 37], potentiometry [38–40], and osmometry [41, 42], even when the available XRD evidence [43–52] is still a matter of heated debate.

The current state of affairs is likely the result of a common factor behind the hydration of oxyanions manifested as the overlapping of several inter- and intra-atomic correlation peaks within the radial distance, 2.0<r(Å)<4.5, including in this case the H2OSO42 and the Mν+SO42 pair correlations. Considering that our ultimate goal is an improved understanding of the hydration behavior of the sulfate group, it is essential for us to be able to deconvolute the overlapping contributions prior to any interpretation of the coordinating environment. To achieve that goal we invoke a recently proposed NDIS approach based on the dual substitution on the oxyanions involving heavy- and null-aqueous environments [5], i.e., natOS/18OS and natS/33S for the current case of SO42, which provides a powerful means to settle the matter by detecting, isolating, and assessing directly the microstructural evidence regarding the presence of the Mν+SO42 pairing. This is made possible by the natS/33S scattering length difference, 1.89 fm, i.e., several times larger than the natOS/18OS corresponding difference, 0.204 fm; however, because the oxygen atomic concentration in the substituted SO42 is obviously four times higher than that of the sulfur, the natOS/18OS contribution to the neutron weighted distribution functions will be comparable to (just about half) that of the natS/33S substitution.

Surprisingly, we are not aware of any NDIS data available in the literature for aqueous sulfates involving sulfur isotopic substitutions, despite the significant natS/33S neutron coherent scattering-length contrast [53]. The reason for this shortage might be traced back to the fact that the conventional combination of natS/33S and H/D substitutions provides complementary information for the solvation structure around sulfate anion, yet, neither addresses directly the near-neighbor coordination between the anion’s oxygen and water. In fact, the few NDIS available experiments involving metal sulfates of which we are aware actually targeted the metal hydration behavior rather than the anion [54–57].

In this context, the main goal of the current work is to illustrate how a judicious interplay between statistical mechanics theory and molecular simulation can be used to test the accuracy and internal consistency of the NDIS approach for the determination of the aqueous coordination of the sulfate anion, i.e., as a dry-run for the actual experiments. For that purpose, we explore the eventual deconvolution of the contributions from H2OSO42 correlations using neutron diffraction involving successive heavy- and null-aqueous solutions of NiSO4 under natS/33S and natOS/18OS isotopic substitutions, i.e., Ow···OS, H···OS, Ow···S, and H···S, that would result in a full characterization of the first water coordination around SO42. In Section 2 we discuss the fundamentals underlying the novel first-order difference approach involving natS/33S and natOS/18OS substitutions in 1.72 m NiSO4 aqueous solutions, where we manipulate the isotopic composition of the aqueous environment to allow the detection and isolation of the contact Mν+SO42 pair, as well as the definitive assessment of its strength and effect on the ‘measured’ anion-coordination numbers. After a brief description of the intermolecular potential models and the simulation methodology in Section 3, we discuss the microstructural behavior of the aqueous NiSO4 and its characterization of the coordinating water environments according to the proposed interplay. Finally, we close the manuscript with a discussion and highlight the gist embedded in the title of this contribution.

Fundamentals

Neutron diffraction with isotopic substitution involving monoatomic ionic species has become a mature approach for the microstructural studies of aqueous electrolyte solutions [58–60] as long as the information from the NDIS raw data is corrected for the presence of ion-pairing [6, 61–64]. In contrast, NDIS experiments on aqueous electrolytes for the determination of the hydration microstructure of oxyanions involving dual αY/βY and natO/18O substitution are currently inexistent obviously due to the short time elapsed since Fischer et al.’s findings [65] regarding the more accurate assessment of the natO/18O difference of coherent scattering length, and the challenging nature of the NDIS experiments.

As we have recently highlighted it [5, 6], and due to the considerable experimental challenges behind the NDIS experiments involving null (i.e., neutron transparency to specific atoms) environment, it becomes extremely valuable to have available a molecular-based tool to conduct a priori analyses of the sought NDIS experiments. In this context, the interplay between statistical mechanics theory and molecular-based simulation affords a rigorous way to test unambiguously the accuracy of the approximations underlying the NDIS methodology. This is achievable because molecular simulation provides the full characterization of the system microstructure, i.e., not only the s(s+1)/2 pair correlation functions for a system involving s interacting sites (vide infra), but also the resulting NDIS output for any molecular model representing the system of interest. Precisely for this reason, the test of accuracy of the NDIS formalism becomes independent of the choice of the interaction potential models used in the simulation, yet, the reader might be understandably interested in knowing how realistic the used models are for the description of the aqueous electrolytes under investigation, an issue we have already addressed in great detail elsewhere [6].

The access to the s(s+1)/2 pair correlation functions by molecular simulation provides the unmatched opportunity to locate the relevant diffraction peaks, the presence of peak overlapping, and ultimately to facilitate the interpretation of the diffraction data. This is especially relevant for the study of oxyanion hydration, where it becomes practically impossible to extract any coordination information associated with the oxygen site of the oxyanion, unless we are able to discriminate it from the corresponding water–oxygen correlations. Therefore, in what follows we illustrate what we could expect to observe in a simulated NDIS experiment involving natO/18O and natS/33S in aqueous sulfate solutions, and discuss the implications on the accurate determination of water–sulfate coordination, where all microstructural details are known simultaneously with the corresponding first-order differences of the neutron weighted distribution functions. This feature allows the unambiguous identification of peak overlapping between pair correlation functions, and consequently, the isolation of specific pair correlation peaks for the assessment of meaningful species coordination as well as the opportunity to test conjectured behaviors underlying the potential occurrence of ion-pairing and its effect in the interpretation of the NDIS raw data [62, 66].

Neutron diffraction with dual natS/33S and natOS/18OS isotopic substitution

Here we examine the link between the measured quantities and the targeted microstructures (e.g., ion coordination environment), and identify the (frequently dismissed) problems underlying the meaningful interpretation of the experimental evidence, such as the unavoidable peak overlapping as the signature of ion-pair formation [62, 66]. The starting point is the portion of the neutron scattering differential cross section of the aqueous sample, /dΩ, that comprises the desired information on the microstructure, i.e., the total structure factor F(k) defined as [67],

(1)F(k)=ij(2δij)cicjbibj(Sij(k)1) (1)

where k=(4π/λ)sin(θ/2), λ of the incident neutron wavelength, while θ, ci and bi denote the scattering angle, the atomic fraction and the coherent neutron scattering length of the atomic species i, respectively. Obviously F(k) is a linear combination of the partial structure factor Sij(k) describing the correlation between atoms of types i and j, i.e.,

(2)Sij(k)=1+(4πρ/k)0[gij(r)1]rsin(kr)dr (2)

where ρ and gij(r) are the atomic number density of the solution and the corresponding radial pair distribution function for ij-pair interactions. For most practical purposes we prefer dealing with the total (real space neutron-weighted) pair correlation function G(r) rather than F(k), i.e., its Fourier transform,

(3)G(r)=ij(2δij)cicjbibj(gij(r)1) (3)

Considering that a simple Mν+XOxν aqueous solution of the oxyanion XOxν and the cation Mν+ comprises s scattering species then, in principle, there are s(s+1)/2 independent radial distribution functions gij(r) defining the system microstructure. Consequently, any attempt to determine the ion coordination would in principle require the same number of experiments (i.e., involving different isotopic compositions) to extract the full set of gij(r) [68]. Obviously, this approach would be infeasible because inter alia the overwhelming contribution from water-water correlation functions [either gOH(r) and gHH(r) or gOD(r) and gDD(r)] to G(r) will hamper any attempt to extract relevant information about the local aqueous environment around the ions and the analysis of ion pair formation, and the fact that we need only the profile of a few correlation functions, over a limited [rL, rU ] radial interval, associated with the targeted ion environment.

The reasonable way to circumvent this shortcoming is through the cancelation of the undesired correlations via the implementation of the first-order difference scheme [69], i.e., comprising two diffraction experiments involving a pair of identical solutions except for the isotopic states of the species (labeled α or β) under study. Then, the ion coordination can be determined by the integration of the resulting first-order difference within the radial interval [rL, rU ] where the sought ion-water correlation occurs and exhibits no peak overlapping (vide infra). Specifically, the cancelation of contributions from the solvent–solvent and solvent–(non-substituted) solute species interactions in the first-order difference scheme for the study of the coordination around the SO42 anion might result in the following expressions,

(4)ΔGSsolv,norm(r)=[ASgOWS(r)+BSgDS(r)+CSgHS(r)+DSgMS(r)+ESgSS(r)+FSgOSS(r)]/ΣSsolv (4)

with ΣSsolv=AS+BS+CS+DS+ES+FS under the natS/33S substitution, and

(5)ΔGOsolv,norm(r)=[AOgOWOS(r)+BOgDOS(r)+COgHOS(r)+DOgMOS(r)+EOgOSOS(r)+FOgSOS(r)]/ΣOsolv (5)

with ΣOsolv=AO+BO+CO+DO+EO+FO under the natOS/18OS substitution, where we have explicitly identified the two distinct isotopic variants of the oxygen species, i.e., OW for the natural oxygen isotope in water and OS as the sulfate oxygen undergoing the isotopic substitution. Note that in eqs. (4)–(5) we have also differentiated explicitly the hydrogen isotopes of the water as H≡1H and D≡2H so that the solvent (solv) environment can be represented by light-, heavy-, or null-aqueous environments, i.e., for which BS=BO=0, CS=CO=0, or BS +CS=B0 +CO=0, respectively. Moreover, the prefactors of the six contributions in eqs. (4)–(5) are given by cicjbj[biα(2δjiα)biβ(2δjiβ)], where we emphasize the isotopic dependence of the corresponding coherent neutron scattering lengths (bil with l=α, β) for the isotopic species, and invoke Kronecker delta δjil.

A further manipulation of the system environment could be achieved by making the M-cation transparent to the neutrons by means of the so-called null M-cation (nM) environment, involving the αM/βM isotopic substitution at an isotopic composition cαM/cβM=–(bβM/bαM ) that makes DS = DO = 0, so that eqs. (4)–(5) reduce to the following working expressions,

(6)ΔGS,nMhw,norm(r)=[ASgOWS(r)+BSgDS(r)+ESgSS(r)+FSgOSS(r)]/ΣS,nMhw (6)

with ΣS,nMhw=AS+BS+ES+FS and,

(7)ΔGO,nMhw,norm(r)=[AOgOWOS(r)+BOgDOS(r)+EOgOSOS(r)+FOgSOS(r)]/ΣO,nMhw (7)

with ΣO,nMhw=AO+BO+EO+FO. The goal behind the use of the null-cation environment is to erase any contribution form the Mν+SO42 pair interactions, i.e., M···OS and M···S whose correlation peaks usually overlap with the hydration relevant anion interactions such as H···OS and H···S (vide infra, Figs. 2 and 3). In other words, by eliminating the ion-pair contributions from the onset, we should be able to avoid the CIP correction of the corresponding ‘measured’ coordination numbers, i.e., n¯SD(rs,measured) and n¯OSD(rs,measured) as already discussed and illustrated elsewhere for the alkali and alkaline-earth halides counterparts [6, 62, 66].

Fig. 1: Atomistic peek of a representative contact Ni2+⋯SO42−${\rm{N}}{{\rm{i}}^{{\rm{2}} + }} \cdots {\rm{SO}}_{\rm{4}}^{{\rm{2}} - }$ pair in the 1.72 m NiSO4 aqueous solution at ambient conditions.
Fig. 1:

Atomistic peek of a representative contact Ni2+SO42 pair in the 1.72 m NiSO4 aqueous solution at ambient conditions.

Fig. 2: Normalized first-order difference of neutron weighted distribution functions for the 1.72 m NiSO4 in heavy- and null-water solution at ambient conditions under natS/33S substitution.
Fig. 2:

Normalized first-order difference of neutron weighted distribution functions for the 1.72 m NiSO4 in heavy- and null-water solution at ambient conditions under natS/33S substitution.

Fig. 3: Normalized first-order difference of neutron weighted distribution functions for the 1.72 m NiSO4 in heavy- and null-water solution at ambient conditions under natO/18O substitution.
Fig. 3:

Normalized first-order difference of neutron weighted distribution functions for the 1.72 m NiSO4 in heavy- and null-water solution at ambient conditions under natO/18O substitution.

Typically, assuming that there are additional contributions (other than gDS(r)) to ΔGSsolv,norm(r) from other correlations in the [rL, rU ] radial interval, after invoking the statistical mechanical definition of coordination number, i.e.,

(8)n¯Iβ(rs)=4πρβ0rsgβI(r)r2dr (8)

where β=H, O and rs typically locates the first valley of the radial distribution function gβI(r), we have that the coordination of water around the sulfate’s sulfur site becomes,

(9)n¯SD(rs)=(4πρDΣShw/BShw)rLrUΔGShw,norm(r)r2dr (9)

and,

(10)n¯SOW(rs)=(4πρOWΣShw/AShw)rLrUΔGShw,norm(r)r2dr (10)

while the result corresponding to the sulfate’s oxygen site reads,

(11)n¯OSD(rs)=(4πρDΣOhw/BOhw)rLrUΔGOhw,norm(r)r2dr (11)

and,

(12)n¯OSOW(rs)=(4πρOWΣOhw/AOhw)rLrUΔGOhw,norm(r)r2dr (12)

Interaction potential models and molecular simulation methodology

For the analysis of the hydration behavior of the aqueous NiSO4 and to illustrate the interplay as well as issues discussed above, we performed isochoric–isothermal molecular dynamics simulations at ambient conditions. For that purpose we have chosen simple but reliable intermolecular potential models including the rigid SPC/E water [70], and the Wallen et al. [71], and the five-sites rigid tetrahedral Cannon et al. [72] for the nickel and sulfate ions, respectively. All these potential models involve Lennard–Jones models, with the unlike pair interactions described by the Lorentz–Berthelot combining rules, and site-site electrostatic interactions according to the force field collected in Table 1.

Table 1

Potential parameters for the aqueous NiSO4 model solutions.c

ii-Interactionεii/k(K)aσii(Å)bqi(e)Refs.
Ni···Ni50.342.0502.0[71]
OS···OS30.093.150–1.100[72]
S···S125.73.5502.400[72]
OW···OW125.73.166–0.8476[70]
H···H0.4238[70]

aεij=εiiεjj;bσij=0.5(σii+σjj);cbond-length SO=1.49 Å [72].

All simulations involved NW = 1986 water molecules and Nions=2NNi=124 ions and were carried out according to our own implementation of a Nosé–Poincare algorithm [73, 74] for the integration of the Newton–Euler equations of motion within a cubic simulation box of dimension L under 3D periodic boundary conditions. The initial configuration was generated by the Packmol utility [75] and equilibrated by at least 1.0 ns, followed by production runs of 4.0 ns, using an integration time-step of 2.0 fs, during which the microstructural information was collected. All Lennard–Jones interactions were truncated at a cut-off radius min(rc ≈3.5σSPCE, rc=L/2), while the electrostatic interactions were handled by an Ewald summation whose convergence parameters were chosen to assure an error smaller than 5×10–5εSPCE for both the real and reciprocal spaces [76].

Microstructural results

From the ten simulated pair correlation functions we determined the first-order differences of neutron-weighted radial distribution functions, for the heavy-water –ΔGShw,norm(r) and ΔGOhw,norm(r) – and the null-water – ΔGSnw,norm(r) and ΔGOnw,norm(r) – as well as the null-nickel – ΔGS,nNihw,norm(r) and ΔGO,nNihw,norm(r) – for the two isotopic substitutions in SO42 that will be later used for the estimation of the corresponding coordination numbers as well as the analysis of ion-pair formation.

In Table 2 we collect the values of the coherent scattering lengths used in the study, which were taken from the available literature. The first-order differences ΔGSsolv,norm(r) and ΔGOsolv,norm(r) for heavy- and null-water, as well as the contributions from each pair interactions are plotted in Figs. 2ab–3ab. The coefficients of the first-order differences, eqs. (4)–(5) are collected in Table 3 and 4, where the contrasting magnitudes of these coefficients, i.e., A, B >>D, E, F and AS, BS >AO, BO highlight the facts that, as in the case of aqueous nitrates, the two most relevant contributions to ΔGsolv,norm(r) come from the nearest pair interactions between the – substituted species and water, and the fact that coherent scattering contrast for the natS/33S substitution is larger than that for the natOS/18OS substitution.

Table 2

Coherent neutron scattering lengths for aqueous NiSO4.

Speciesbcoh(fm)Refs.
natS2.847[53]
33S4.74[53]
1H–3.74[53]
2H6.67[53]
natO5.805[53]
18O6.009[65]
natNi10.30[53]
62Ni–8.7[53]
Table 3

Coefficients of the first-order difference of weighted distributions for the natS/33S substitution in 1.72 m NiSO4 heavy- and null-aqueous solutions.

CoefficientaHeavy-waterNull-water
AS6.945079E-026.945079E-02
BS0.15959925.732689E-02
CS0.0–5.732689E-02
DS3.970990E-033.970990E-03
ES1.462520E-031.462520E-03
FS8.952077E-038.952077E-03
ΣNsolv0.24343568.383637E-02

aIn fm2 units.

Table 4

Coefficients of the first-order difference of weighted distributions for the natOS/18OS substitution in 1.72 m NiSO4 heavy- and null-aqueous solutions.

CoefficientaHeavy-waterNull-water
AO2.993758E-022.993758E-02
BO6.879714E-022.472620E-02
CO0.0–2.472620E-02
DO1.711742E-031.711742E-03
EO3.926700E-033.926700E-03
FO4.731388E-044.731388E-04
ΣOsolv0.10484633.604916E-02

aIn fm2 units.

The main features of ΔGShw,norm(r),Fig. 2a, are rather similar to those of the nitrate counterparts, i.e., two prominent peaks at ~2.85 Å and ~3.80 Å corresponding to the nearest D···S interactions, and the combination of contributions from the second nearest D···S, the Ow···S, as well as the contribution from the contact Ni···S pair interactions. Moreover, the occurrence of Ow···S, Ni···S, and D···S correlation peaks within 3.2≤r(Å)≤5.0, hinders the determination of the second D···S coordination by integration of the second peak of ΔGShw,norm(r), a common feature shared by most aqueous oxyanions (vide supra).

However, the first peak of ΔGShw,norm(r),Fig. 2a, comprises almost exclusively the contribution from the first peak of the corresponding gDS(r), with two small contributions from the Ow···S and the contact Ni···S pair interactions. Consequently, the outcome for the first coordination number nDS(r) based on the integration of ΔGShw,norm(r), eq. (9), could be slightly smaller than the true one, eq. (8) and Table 5, and highlights the effect of additional contributions [other than gDS(r)] to ΔGShw,norm(r) from the overlapping with other correlations in the [rL, rU ] radial interval.

Table 5

First water coordination of the sulfate ion in 1.72 m NiSO4 heavy- and null-aqueous solutions according to the direct and the NDIS-based expressions.

αβ interactionsDirect integral (ru)cNDIS integral (ru)d
OW···S14.1(4.51 Å)14.7(4.43 Å)b
D···S11.6(3.40 Å)11.4(3.32 Å)a
OW···OS3.2(3.16 Å)3.6(3.24 Å)b
D···OS2.8(2.37 Å)3.1(2.37 Å)a
Ni···S0.4(3.37 Å)
OS···Ni0.4(2.20 Å)0.4(2.20 Å)b

aHeavy-water; bnull-water; cn¯βα(rs)=4πρα0rsgαβ(r)r2dr;dn¯βα(rs)=(4πραΣβsolv/Iαβsolv)0rsΔGβsolv,norm(r)r2dr with solv=(hw, nw) and Iαβsolv=(Aβsolvifα=Ow;Bβsolvifα=D).

The first peak of ΔGSnw,norm(r),Fig. 2b, is centered at r≈3.8 Å and comprises a major contribution from Ow···S interactions that fully overlap with the contact Ni···S pair configurations. Consequently, the conventional determination of the water-oxygen coordination around the sulfur site of the sulfate group, i.e.,

(13)n¯SOw(rs)=(4πρOwΣSnw/ASnw)rLrUΔGSnw,norm(r)r2dr (13)

will experience the same problem as that for n¯SD(rs) from ΔGShw,norm(r), eq. 9, due to the presence of two smaller overlapping contributions from the contact Ni···S and the OS···S pair correlation peaks (Table 5).

Note that n¯OsNi(CIP)n¯SNi(CIP) rather than what has been usually assumed in the interpretation of XRD based on the nearest intra-molecular coordination [46], i.e., n¯OsNi(CIP)n¯SNi(CIP)/4, resulting from the different sizes of the coordination shells, ru =2.2 Å for n¯OsNi(CIP) and ru =3.37 Å for n¯SNi(CIP).

In Fig. 3a–b we display the corresponding first-order differences ΔGOhw,norm(r) and ΔGOnw,norm(r) counterparts to Fig. 2a–b, where we immediately note that ΔGOhw,norm(r) exhibits a well defined first peak associated with the D···OS correlations centered at r≈1.75 Å similar to the D···OW peak for pure water [77], yet, it overlaps completely the contribution from the contact Ni···OS pair peak. Beyond the first peak, ΔGOhw,norm(r) presents a less defined multi-peak region resulting from the overlapping of the second D···OS coordination, the solvent-shared Ni···OS pair, and the two OW···OS correlation peaks. As a first approximation we can ignore the small contact Ni···OS pair contribution to the first peak of ΔGOhw,norm(r), then its integral will provide an adequate estimate for the first coordination n¯OsD(rs) as clearly depicted in Table 5. This number can then be corrected as soon as we determine the first coordination for the Ni···OS interactions. For that purpose, we turn our attention to the behavior of ΔGOnw,norm(r),Fig. 3b, where the first peak describes the nearest Ni···OS interactions, which by integration of ΔGOnw,norm(r) up to a distance r≈2.2 Å, provides an adequate estimation of n¯OsNi(rs).

(14)n¯OsD(rs,measured)=(4πρD/BO)rLrU(BOgDOs(r)+DOgOsNi(r))r2dr=n¯OsD(rs)+(4πρDDO/BO)rLrUgOsNi(r)r2drCIPcontribution (14)

Then the actual n¯OsD(rs) can be assessed by subtracting the CIP– contribution written explicitly as follows,

(15)(4πρNibNi/bD)rLrUgOsNi(r)r2dr=n¯OsNi(rs)(bNi/bD) (15)

from eq. (12) to attain,

(16)n¯OsD(rs)=n¯OsD(rs,measured)n¯OsNi(CIP)(bNi/bD) (16)

Obviously, according to Figs. 2 and 3, there are no chances of assessing the n¯SNi(CIP) and its contribution to the n¯SOw(rs,measured) for the subsequent correction according to eq. (16) counterpart, even if we could derive the corresponding expression, i.e.,

(17)n¯SOw(rs,measured)=(4πρOw/AS)rLrU(ASgOwS(r)+DSgNiS(r))r2dr=n¯SOw(rs)+(4πρOwDS/AS)rLrUgNiS(r)r2drCIPcontribution (17)

so that,

(18)n¯SOw(rs)=n¯SOw(rs,measured)n¯SOw(CIP)(bNi/bOw) (18)

However, we still have another source of information to characterize further the water coordination of the sulfate anion, i.e., the null-nickel venue through the manipulation of the isotopic composition of the cation. Under this null-nickel environment the relevant distribution functions ΔGS,nNihw,norm(r) and ΔGO,nNihw,norm(r), given by eqs. (6)–(7) and illustrated in Fig. 4a–b, indicate that the absence of any type of Ni···OS or Ni···S pairing. Consequently, we can extract the water-hydrogen coordination of the sulfur and oxygen sites of the sulfate anion with no interference from the potential CIP configurations, i.e.,

Fig. 4: Normalized first-order difference of neutron weighted distribution functions for the 1.72 m NiSO4 in heavy-water null-Ni solution at ambient conditions under either natO/18O or natS/33S substitution.
Fig. 4:

Normalized first-order difference of neutron weighted distribution functions for the 1.72 m NiSO4 in heavy-water null-Ni solution at ambient conditions under either natO/18O or natS/33S substitution.

(19)n¯SD(rs)=(4πρDΣS,nNihw/BS)0rsΔGS,nNihw,norm(r)r2dr (19)

and,

(20)n¯OsD(rs)=(4πρDΣO,nNihw/BO)0rsΔGO,nNihw,norm(r)r2dr (20)

Note that due to the absence of any contact (anion site-cation) pair corrections, the coordination number determined by eq. (19) will provide a test of consistency for the calculations involving eq. (16). As an illustration of the issue, in Table 6 we made the comparison between the reference coordination numbers given by their statistical mechanical definition, i.e., eq. (8).

Table 6

First water coordination of the sulfate ion in 1.72 m NiSO4 aqueous solutions from integrals over ΔGSsolv,norm(r),ΔGS,nNihw,norm(r),ΔGOsolv,norm(r), and ΔGO,nNihw,norm(r).

αβ InteractionsDirect integral(ru)bNDIS integral (ru)aNDIS integral (ru)
D···S11.6(3.40 Å)11.4(3.32 Å)11.4(3.32 Å)c
D···OS2.8(2.37 Å)3.1(2.37 Å)2.8(3.7 Å)d

an¯XD(rs)=n¯XD(measured)n¯XNi(rs)(bNi/bD) where X=[OS, S], n¯OsNi(CIP)0.4 (Table 5), n¯SNi(CIP)0.4 (Table 5); bfrom eq. (8); ceq. (19); deq. (20).

Up to this point we have presented the tools and illustrated their use not only to detect the presence of (CIP) Mν+SO42 but also to isolate it, and finally to estimate a representative coordination number. Yet, we have not addressed the extent of such an ion-pair association, i.e., the degree of Mν+SO42 pair and its link to an association constant, rather than a CIP coordination number n¯XM(rL,rU) with X=[OS, S], as well as the participation of CIP, SShIP, and SSIP configurations according to the Eigen-Tamm classification [78]. For that purpose we invoke a somewhat less-known rigorous formalism to make such a connection [6, 79, 80].

The degree of either Mν+S[O42] or Mν+O[SO32] pair association α–+ can be expressed in terms of the ion-pair distribution function G–+(r) related to the corresponding pair correlation function g–+(r)) as an integral equation, i.e.,

(21)G+(r)=4πρ+g+(r)r2P(r)P+(r) (21)

where P(r) (P+(r)) denotes the probability that either S[O42] or O[SO32] separated by a distance r from an Mν+ (either S[O42]or O[SO32]) does not engage in an ion pair with any other ion of the opposite charge, i.e., P(r)=10rG+(s)ds and P+(r)=10rG+(s)ds. Typically, the solution of this integral equation is analytical and involves the obvious conservation of species and electroneutrality boundary conditions (for more details on the resulting expressions see Refs. [6, 79, 80]). Then, the degree of either Mν+S[O42] or Mν+O[SO32] pair association reads,

(22)α+(d+)=0d+G+(r)dr (22)

under the condition that, in the thermodynamic limit, limd+α+(d+)=1.0, where d–+ typically denotes the largest distance within which the pairs are counted, such as the location of the first (for contact ion pairs) or second valley (for contact plus solvent-shared ion pairs) of g–+(r).

In Fig. 5a–b we display the Ni+2S[O42] radial distribution functions g–+(r), as well as their corresponding ion-pair radial distribution functions G–+(r) and degree of ion-pair association α–+(r) for the 1.72 m NiSO4 aqueous solution at ambient conditions, where we should highlight the significant difference in the magnitude between G–+(r) and g–+(r) functions, and the obvious correspondence between their two main peaks associated with the CIP and SShIP configurations. Moreover, note that even at this low sulfate concentration the system exhibits a significant Ni+2S[O42] pair association characterized by α(CIP)≅0.28, with α(CIP+SShIP)≥0.75. The last value is in remarkable agreement with the reported value of α(1.6≤m≤1.8)≈0.69 for ambient NiSO4 aqueous solutions in Table 5 of Ref. [81].

Fig. 5: Radial pair distribution function gNiS(r), the corresponding Pourier-deLap Ni+2⋯S[O42−]$N{i^{ + 2}} \cdots {\rm{S[O}}_4^{2 - }]$ pair distribution function and the resulting degree of pair association for the 1.72 m NiSO4 aqueous solution at ambient conditions.
Fig. 5:

Radial pair distribution function gNiS(r), the corresponding Pourier-deLap Ni+2S[O42] pair distribution function and the resulting degree of pair association for the 1.72 m NiSO4 aqueous solution at ambient conditions.

Discussion and final remarks

We have described the fundamentals underlying the interpretation of microstructural behavior of aqueous electrolytes according to the NDIS approach toward the experimental determination of ion coordination numbers of systems involving sulfate anions. We discussed the ‘philosophy’ motivating the interplay alluded to in the title of this presentation, and emphasized the expectations as well as significance of tackling the challenging NDIS experiments. Specifically, we highlighted the potential occurrence of Mν+SO42 pair formation with both anion interaction sites, a frequently overlooked hydration phenomenon, identified its signature in the neutron-weighted distribution functions, suggested novel ways for either the direct probe of the CIP strength for the subsequent correction of its effects on the experimentally measured coordination numbers, or the determination of anion coordination numbers free of CIP contributions through the implementation of null-cation environments.

We must emphasize that this interplay becomes a formidable asset after recognizing the inherent molecular simulation ability to provide all pair correlation functions that fully characterize the system microstructure and allows us to “reconstruct” the eventual NDIS output, i.e., to take an atomistic “peek” at the local environment around the isotopically-labeled species before any experiment is ever attempted, and ultimately, to test the accuracy of the “measured” NDIS-based coordination numbers against the actual values by the “direct” counting. In addition, the isotopic differentiation between OW and OS in eqs. (4)–(7) provides a handy way to check the consistency of the experimental raw data sets according to the natural constraint given by the intra-molecular coordination nSOS(SO)=4, where SO is the intra-molecular bond-length. For that purpose we introduce the definition of intra-coordination and invoke un-normalized eq. (4), i.e.,

(23)nSOS(SO)=4πρOSSOδSO+δgSOS(r)r2dr=(4πρOS/FS)SOδSO+δ[ΔGSsolv(r)+ΣSsolv]r2dr (23)

where ΣSsolv=AS+BS+CS+DS+ES+FS.. Likewise, from eq. (5) we have that,

(24)nOSS(SO)=4πρSSOδSO+δgOSS(r)r2dr=(4πρS/FO)SOδSO+δ[ΔGOsolv(r)+ΣOsolv]r2dr (24)

with ΣOsolv=AO+BO+CO+DO+EO+FO, where the parameter δ describes the spread of the distribution of the intra-molecular bond length SO in the real system. Obviously, for a model system with rigid geometry for the sulfate anion, as for Cannon et al. [72], the bond-length distributions in eqs. (23)–(24) become delta functions, i.e., the constraint is satisfied by construction, i.e.,

(25)nOSS(SO)=4πρSgSOS(r)r2δ(rSO)dr=πρOSgOSS(r)r2δ(rSO)dr=nSOS(SO)/4 (25)

where gSOS(r)=gOSS(r). Note that the equality of the two lines in eqs. (23)–(25) is guaranteed as long as there is no overlapping of peaks within the range of intra-molecular interactions, i.e., the self-consistency of ΔGSsolv(r) and ΔGOsolv(r) will mean that,

(26)cOSFF1SOδSO+δ[ΔGSsolv(r)+ΣSsolv]r2dr=4cSFO1SOδSO+δ[ΔGOsolv(r)+ΣOsolv]r2dr (26)

where the identity (26) becomes an indicator of properly normalized experimental first-order differences ΔGOsolv,norm(r) and ΔGSsolv,norm(r).

In summary, in developing the current interplay we have exposed some frequently overlooked issues and highlighted that the pressing challenge at this juncture is to confront the significant difficulties associated with these null-environment NDIS experiments and translate the proposed novel schemes into versatile tools for the accurate full characterization of oxyanion hydration.

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).


Article note:

A collection of invited papers based on presentations at the 34th International Conference on Solution Chemistry (ICSC-34), Prague, Czech Republic, 30th August – 3rd September 2015.



Corresponding author: Ariel A. Chialvo, Chemical Sciences Division, Geochemistry and Interfacial Sciences Group Oak Ridge National Laboratory, Oak Ridge, TN 37831-6110, USA, e-mail: .

Acknowledgments

This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division.

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Published Online: 2016-1-21
Published in Print: 2016-3-1

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