## 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 _{4} 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

## 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

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 ^{nat}O_{S}/^{18}O_{S} and ^{nat}S/^{33}S for the current case of ^{nat}S/^{33}S scattering length difference, 1.89 fm, i.e., several times larger than the ^{nat}O_{S}/^{18}O_{S} corresponding difference, 0.204 fm; however, because the oxygen atomic concentration in the substituted ^{nat}O_{S}/^{18}O_{S} contribution to the neutron weighted distribution functions will be comparable to (just about half) that of the ^{nat}S/^{33}S substitution.

Surprisingly, we are not aware of any NDIS data available in the literature for aqueous sulfates involving sulfur isotopic substitutions, despite the significant ^{nat}S/^{33}S neutron coherent scattering-length contrast [53]. The reason for this shortage might be traced back to the fact that the conventional combination of ^{nat}S/^{33}S 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 _{4} under ^{nat}S/^{33}S and ^{nat}O_{S}/^{18}O_{S} isotopic substitutions, i.e., O_{w}···O_{S}, H···O_{S}, O_{w}···S, and H···S, that would result in a full characterization of the first water coordination around ^{nat}S/^{33}S and ^{nat}O_{S}/^{18}O_{S} substitutions in 1.72 m NiSO_{4} aqueous solutions, where we manipulate the isotopic composition of the aqueous environment to allow the detection and isolation of the contact _{4} 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 ^{α}Y/^{β}Y and ^{nat}O/^{18}O 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 ^{nat}O/^{18}O 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 ^{nat}O/^{18}O and ^{nat}S/^{33}S 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 ^{nat}S/^{33}S and ^{nat}O_{S}/^{18}O_{S} 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, *dσ*/*d*Ω, that comprises the desired information on the microstructure, i.e., the total structure factor *F*(*k*) defined as [67],

where *k*=(4*π*/*λ*)sin(*θ*/2), *λ* of the incident neutron wavelength, while *θ*, *c*_{i} and *b*_{i} 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 *S*_{ij}(*k*) describing the correlation between atoms of types *i* and *j*, i.e.,

where *ρ* and *g*_{ij}(*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,

Considering that a simple ^{ν+} comprises *s* scattering species then, in principle, there are *s*(*s*+1)/2 independent radial distribution functions *g*_{ij}(*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 *g*_{ij}(*r*) [68]. Obviously, this approach would be infeasible because *inter alia* the overwhelming contribution from water-water correlation functions [either *g*_{OH}(*r*) and *g*_{HH}(*r*) or *g*_{OD}(*r*) and *g*_{DD}(*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 [*r*_{L}, *r*_{U} ] 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 [*r*_{L}, *r*_{U} ] 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

with ^{nat}S/^{33}S substitution, and

with ^{nat}O_{S}/^{18}O_{S} substitution, where we have explicitly identified the two distinct isotopic variants of the oxygen species, i.e., O_{W} for the natural oxygen isotope in water and O_{S} 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≡^{1}H and D≡^{2}H so that the solvent (solv) environment can be represented by light-, heavy-, or null-aqueous environments, i.e., for which *B*_{S}=*B*_{O}=0, *C*_{S}=*C*_{O}=0, or *B*_{S} +*C*_{S}=*B*_{0} +*C*_{O}=0, respectively. Moreover, the prefactors of the six contributions in eqs. (4)–(5) are given by *l*=*α*, *β*) for the isotopic species, and invoke Kronecker delta *δ*_{jil}.

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 ^{α}M/^{β}M isotopic substitution at an isotopic composition *c*_{αM}/*c*_{βM}=–(*b*_{βM}/*b*_{αM} ) that makes *D*_{S} = *D*_{O} = 0, so that eqs. (4)–(5) reduce to the following working expressions,

with

with _{S} and M···S whose correlation peaks usually overlap with the hydration relevant anion interactions such as H···O_{S} 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.,

Typically, assuming that there are additional contributions (other than *g*_{DS}(*r*)) to *r*_{L}, *r*_{U} ] radial interval, after invoking the statistical mechanical definition of coordination number, i.e.,

where *β*=H, O and *r*_{s} typically locates the first valley of the radial distribution function *g*_{βI}(*r*), we have that the coordination of water around the sulfate’s sulfur site becomes,

and,

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

and,

### Interaction potential models and molecular simulation methodology

For the analysis of the hydration behavior of the aqueous NiSO_{4} 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.

ii-Interaction | ε_{ii}/k(K)^{a} | σ_{ii}(Å)^{b} | q_{i}(e) | Refs. |
---|---|---|---|---|

Ni···Ni | 50.34 | 2.050 | 2.0 | [71] |

O_{S}···O_{S} | 30.09 | 3.150 | –1.100 | [72] |

S···S | 125.7 | 3.550 | 2.400 | [72] |

O_{W}···O_{W} | 125.7 | 3.166 | –0.8476 | [70] |

H···H | – | – | 0.4238 | [70] |

^{a}^{b}^{c}bond-length *ℓ*_{SO}=1.49 Å [72].

All simulations involved *N*_{W} = 1986 water molecules and *N*_{ions}=2*N*_{Ni}=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(*r*_{c} ≈3.5*σ*_{SPCE}, *r*_{c}=*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^{–5}*ε*_{SPCE} 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 –

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 *A*_{ℓ}, *B*_{ℓ} >>*D*_{ℓ}, *E*_{ℓ}, *F*_{ℓ} and *A*_{S}, *B*_{S} >*A*_{O}, *B*_{O} highlight the facts that, as in the case of aqueous nitrates, the two most relevant contributions to *ℓ*– substituted species and water, and the fact that coherent scattering contrast for the ^{nat}S/^{33}S substitution is larger than that for the ^{nat}O_{S}/^{18}O_{S} substitution.

Species | b_{coh}(fm) | Refs. |
---|---|---|

^{nat}S | 2.847 | [53] |

^{33}S | 4.74 | [53] |

^{1}H | –3.74 | [53] |

^{2}H | 6.67 | [53] |

^{nat}O | 5.805 | [53] |

^{18}O | 6.009 | [65] |

^{nat}Ni | 10.30 | [53] |

^{62}Ni | –8.7 | [53] |

Coefficient^{a} | Heavy-water | Null-water |
---|---|---|

A_{S} | 6.945079E-02 | 6.945079E-02 |

B_{S} | 0.1595992 | 5.732689E-02 |

C_{S} | 0.0 | –5.732689E-02 |

D_{S} | 3.970990E-03 | 3.970990E-03 |

E_{S} | 1.462520E-03 | 1.462520E-03 |

F_{S} | 8.952077E-03 | 8.952077E-03 |

0.2434356 | 8.383637E-02 |

^{a}In fm^{2} units.

Coefficient^{a} | Heavy-water | Null-water |
---|---|---|

A_{O} | 2.993758E-02 | 2.993758E-02 |

B_{O} | 6.879714E-02 | 2.472620E-02 |

C_{O} | 0.0 | –2.472620E-02 |

D_{O} | 1.711742E-03 | 1.711742E-03 |

E_{O} | 3.926700E-03 | 3.926700E-03 |

F_{O} | 4.731388E-04 | 4.731388E-04 |

0.1048463 | 3.604916E-02 |

^{a}In fm^{2} units.

The main features of _{w}···S, as well as the contribution from the contact Ni···S pair interactions. Moreover, the occurrence of O_{w}···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 *vide supra*).

However, the first peak of *g*_{DS}(*r*), with two small contributions from the O_{w}···S and the contact Ni···S pair interactions. Consequently, the outcome for the first coordination number *g*_{DS}(*r*)] to *r*_{L}, *r*_{U} ] radial interval.

α–β interactions | Direct integral (r_{u})^{c} | NDIS integral (r_{u})^{d} |
---|---|---|

O_{W}···S | 14.1(4.51 Å) | 14.7(4.43 Å)^{b} |

D···S | 11.6(3.40 Å) | 11.4(3.32 Å)^{a} |

O_{W}···O_{S} | 3.2(3.16 Å) | 3.6(3.24 Å)^{b} |

D···O_{S} | 2.8(2.37 Å) | 3.1(2.37 Å)^{a} |

Ni···S | 0.4(3.37 Å) | – |

O_{S}···Ni | 0.4(2.20 Å) | 0.4(2.20 Å)^{b} |

^{a}Heavy-water; ^{b}null-water; ^{c}^{d}*hw*, *nw*) and

The first peak of *r*≈3.8 Å and comprises a major contribution from O_{w}···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.,

will experience the same problem as that for _{S}···S pair correlation peaks (Table 5).

Note that *r*_{u} =2.2 Å for *r*_{u} =3.37 Å for

In Fig. 3a–b we display the corresponding first-order differences _{S} correlations centered at *r*≈1.75 Å similar to the D···O_{W} peak for pure water [77], yet, it overlaps completely the contribution from the contact Ni···O_{S} pair peak. Beyond the first peak, _{S} coordination, the solvent-shared Ni···O_{S} pair, and the two O_{W}···O_{S} correlation peaks. As a first approximation we can ignore the small contact Ni···O_{S} pair contribution to the first peak of _{S} interactions. For that purpose, we turn our attention to the behavior of _{S} interactions, which by integration of *r*≈2.2 Å, provides an adequate estimation of

Then the actual

from eq. (12) to attain,

Obviously, according to Figs. 2 and 3, there are no chances of assessing the

so that,

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 _{S} 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.,

and,

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).

α–β Interactions | Direct integral(r_{u})^{b} | NDIS integral (r_{u})^{a} | NDIS integral (r_{u}) |
---|---|---|---|

D···S | 11.6(3.40 Å) | 11.4(3.32 Å) | 11.4(3.32 Å)^{c} |

D···O_{S} | 2.8(2.37 Å) | 3.1(2.37 Å) | 2.8(3.7 Å)^{d} |

Up to this point we have presented the tools and illustrated their use not only to detect the presence of (CIP) *X*=[*O*_{S}, *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 *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.,

where *P*_{–}(*r*) (*P*_{+}(*r*)) denotes the probability that either *r* from an M^{ν+} (either

under the condition that, in the thermodynamic limit, *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 *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 NiSO_{4} 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 *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 NiSO_{4} aqueous solutions in Table 5 of Ref. [81].

## 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

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 O_{W} and O_{S} 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 *ℓ*_{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.,

where

with *δ* describes the spread of the distribution of the intra-molecular bond length *ℓ*_{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.,

where *g*_{SOS}(*r*)=*g*_{OSS(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

where the identity (26) becomes an indicator of properly normalized experimental first-order differences

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 34^{th} International Conference on Solution Chemistry (ICSC-34), Prague, Czech Republic, 30^{th} August – 3^{rd} September 2015.

## 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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