Electric Potentials of Metastable Salt Clusters | The Journal of Physical

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C: Physical Processes in Nanomaterials and Nanostructures

Electric Potentials of Metastable Salt Clusters Evgenii O. Fetisov, William C Isley III, Gregg J Lumetta, and Shawn M Kathmann J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 09 May 2019 Downloaded from http://pubs.acs.org on May 9, 2019

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The Journal of Physical Chemistry

Electric Potentials of Metastable Salt Clusters Evgenii O. Fetisov,† William C. Isley III,†,¶ Gregg J. Lumetta,‡ and Shawn M. Kathmann∗,† †Chemical Physics and Analysis, Pacific Northwest National Laboratory, Richland WA ‡Nuclear Chemistry and Engineering, Pacific Northwest National Laboratory, Richland WA ¶Current address: Departments of Chemistry, Biochemistry and Molecular Biology, University of Chicago, Chicago IL E-mail: [email protected]

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Abstract Electric potentials and fields experienced by the ions in small metastable NaCl clusters and cubic crystals in vacuum and aqueous electrolyte solutions are used to characterize features underlying condensed phase nucleation. The range of electrostatic potentials and fields experienced by NaCl nanocrystals and metastable clusters are analyzed using point charges, and quantified as a function of particle size and charge state. We show how the potentials and fields of these crystals can be classified into various subgroups corresponding to corners, edges, faces, and interior sites. The differences between the interior and face potentials are correlated with the interfacial surface energy. This, in turn, influences cluster free energies and their corresponding populations in solution. As a result, the nucleation rate is correlated to these potentials through the formalism of classical nucleation theory. Additionally, we consider the importance of representation of the charge density, comparing point charge and continuous densities obtained from quantum mechanical electronic structure. A key result of this work is the identification of the electric potential as a possible order parameter to understand nucleation pathways.

Introduction Nucleation is the first step in aqueous salt formation where small metastable clusters of ions form as precursors of the new phase. Nucleation and crystallization processes are influenced by many factors including internal and external electric potentials and fields (e.g., fast amorphous-to-crystalline transitions in phase-change memory applications, 1 field-induced crystallization, 2 and the emission of visible, UV and deep-UV light called crystalloluminescence 3,4 ). It has also been shown that the formation of small ionic clusters can generate new interfaces with the solvent, and the resulting interfacial fields are strong enough to break bonds, distort molecules, and induce ionization or radicalization. 5,6 However, to the best knowledge of the authors there are no studies quantifying and classifying the distributions of

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electric potentials and fields experienced by free ions, ions in metastable clusters, and ions in finite crystals. In this study, we use electric potentials to provide insight into ion clustering during nucleation. The nucleation process can be greatly influenced by factors that are difficult to control experimentally, such as local temperature and concentration fluctuations, mechanical, acoustic, and field effects, variations due to contaminants or the influence of heterogeneous interfaces. In classical nucleation theory (CNT), the free energy of cluster formation is proportional to σ 3 , where σ is the interfacial surface energy, for which small variations can 3

profoundly impact the nucleation rate, J, since J ∼ e−σ . Dynamical nucleation theory 7–9 has shown that this sensitivity can be traced back to the underlying cluster interaction potentials since favorable energetics lead to more stable clusters. A further challenge exists as there is evidence that as clusters of the new phase approach their crystalline states, they pass through various distorted or amorphous configurations, including those with trapped water molecules. 10,11 Thus, simply defining the relevant clusters becomes more tenuous. This leads to three critical challenges to overcome when modeling nucleation and crystallization processes: 12–14 1. Homogeneous nucleation rates depend on a critical cluster size that may contain tens to hundreds of atoms making atomic level precision impractical and continuum approaches imprecise (e.g., CNT using bulk interfacial surface energies). 2. Most condensed phase nucleation processes occur via heterogeneous nucleation with trace impurities, secondary phases, or other interfaces influencing nucleation. 3. Experimental detection of the cluster size distribution evolution within aqueous solution, as a benchmark for theory, has been unattainable due to lack of contrast, length and time scales required, and the inverse scattering problem. 15 Together, these provide stringent requirements to have both accurate nucleation measurements and models. Employing ab initio statistical mechanics of condensed phase nucleation 3



rapidly outpaces modern computational resources and new insights are needed. Even though nucleation requires a thermodynamic and kinetic description, thermodynamics of cluster formation is the most accessible path forward to understand homogeneous nucleation. Given that electrostatic contributions in ionic systems dominate the clusters energy and free energy, we will generalize the concept of the electric potential experienced by an ion in a infinite crystal, known as the Madelung potential, 16,17 to analyze stability and crystallinity of NaCl nanoparticles. For ionic systems, it has been shown that chemical trends in surface adsorption energies, basicity, and chemical reactivity can be explained by focusing on electrostatic interactions. 18 The electric potential experienced by an ion is very sensitive to the structure of its surroundings. Any real crystal is finite and will have edges, faces, and corners where the potentials Vi experienced by an ion i will differ significantly from the infinite crystal limiting value (Vi 6= VMadelung ). This allows us to use the deviations of an ion’s potential from the infinite crystalline limit as a feature to characterize cluster structure and stability (further discussion of the electrostatic contribution to the thermodynamic cluster free energy is provided in the Supporting Information). Furthermore, a recent paper by Zimmermann et al. 19 on the nucleation of NaCl also underscored the importance of cluster ion site characterization and concluded that an “optimal nucleus size metric should include bulk, surface, edge, and corner sites.” Previous works by Baker and Baker 20 and Baker et al. 21 have had success in using the Madelung potentials to study convergence of ionic systems in various configurations and found good correlation between lattice energies, as well as correlation of the size dependence with quantum energetics. Also, Harrison 22 and Gaio et al. 23 have both provided methods to calculate the bulk Madelung constants for cubic systems. Using the electric potential as a general measure of ionic arrangement can be connected to physical observables that use the quantum electric potential V (r) to indicate structure or crystallinity. This can be seen most directly with the electron diffraction structure facR tor V (k) = Ω−1 V (r)e−ik·r dr, where k is the electron scattering wavevector, Ω is the unit

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cell volume, and V (r) is the quantum mechanical electric potential determined from the total quantum mechanical charge density ρ(r) (nuclear + electronic) via solution of Poisson’s equation ∇2 V (r) = −4πρ(r) (not coincidentally, the X-ray diffraction structure factor is similarly defined with the electron density replacing the electric potential). The formation of salt clusters from the aqueous phase enhances the local scattering power as indicated by electron holography measurements of NaCl rocksalt having a mean inner potential V (k = 0) = Vo R = Ω−1 V (r)dr = +8 V compared to +4 V for the surrounding aqueous electrolyte. 24 This means a potential difference of ∆Vo ∼ +4 V exists between a salt crystal and the surrounding aqueous electrolyte arising from the quantum electric potentials. Electron holography is able to directly probe the mean inner potential of matter, using an electron beam split into two parts, one part propagates through a sample of known thickness t, interfering with the sample’s quantum electric potential, causing a measured phase shift ∆φ = CE Vo t with respect to the second (reference) part of the beam (CE is a constant). The differentiation between effective potentials and fields inferred via the Madelung potential and the actual “near” electric potentials and fields probed by high-energy electron diffraction and holography 24 is important in this context, but beyond our current scope, and will be the focus of future work. Crystal-field effects on the electric potentials or electronic states experienced by atoms or ions at various locations (surfaces, edges, corners, tips, etc.) have been of interest in many areas of physical science since Madelung’s original work. In X-ray photoelectron spectroscopy (XPS) of alkali halide crystals, the Madelung potential represents the largest solid state influence on the core-level electron binding energy shifts. 25,26 NEXAFS experiments on concentrated aqueous NaCl electrolytes underscores the sensitivity of the spectral fingerprints to hydrated NaCl clusters, specifically their models found the distances d (Na–Cl) and d (Na–OH2 ) were sufficient to describe the evolution of their spectra. 27 Photoabsorption at the Na K-edge of NaCl clusters in vacuo have been interpreted with models employing the electric potentials. 28 Furthermore, recent XPS experiments on NaCl nanocrystals 29 have

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shown variations in the Na+ 2p and Cl – 3p electron binding energies are well described by using the Madelung potentials for cubic nanoparticles. Thus, the connection between electric potentials and electron binding energies provides measurable signatures of the structural transformations realized as these clusters go from ion pairs to ion clusters to solid nanocrystals. These studies further underscore the use of these electric potential signatures as a basis upon which to characterize the salt clusters underlying nucleation from concentrated electrolytes. The proceeding discussion suggests that the effective electric potentials and fields experienced by the ions within the clusters underlying nucleation may provide geometrical information and hence serve as an order parameter. As used here, an order parameter (or cluster definition) provides a low-dimensional means of understanding the structural distortions of the ions within metastable clusters leading to different nucleation pathways. In the present paper we will show: (1) the behavior of the distributions of electric potentials and fields experienced by Na+ and Cl – ions present in perfect nanocrystals of salt in vacuo, (2) how size-dependent interfacial surface energies correlate with electric potential differences, (3) how the condensed phase aqueous environment affects the potentials and fields of a perfect 43 crystal compared to in vacuo, (4) how the metastable cluster ion average potentials can be used as a nucleation order parameters, (5) the charge states of the metastable clusters and what they say about the nucleation pathways, and (6) the performance of several fixed point charge electric potential protocols for reproducing quantum electronic structure potentials.

Methods Theory The classical Madelung potential can be thought of as the electric potential (or voltage) experienced by an ion from all other charged species in an infinite crystal. When discussing 6


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the electric potential ‘experienced’ by an ion an important distinction must be made between “near” and “far” electric potentials and fields. The electric potential and field evaluated at an ion site are going to depend on the nature of the site (i.e., if the ion acts as a source, receiver, or both, of the surrounding electric potentials and fields). In the present paper, we neglect the ion’s “self” or “near” potential and field arising from it’s own charge or, for the quantum case, from the electrons and nucleus of charge Z. Thus, we only consider “far” potentials and fields. The Madelung potential can be expressed as

VMadelung = M3D ke

e , r

(1)

where e is the unit of electronic charge, r is the nearest neighbor distance, M3D is the Madelung constant (which depends on crystal or cluster environment), and ke is the Coulomb’s constant. For the 3D NaCl cubic crystal structure, M3D is computed as

M3D =

∞ X

(−1)i+j+k = −1.74756... , 2 + j 2 + k 2 )1/2 (i i,j,k=−∞

(2)

i=j=k6=0

where i, j, and k are indices of ion locations in the crystal lattice and the summation is converged using the method of expanding cubes. 30 The literature extensively covers the elegant mathematics associated with Madelung potentials. For example, in the case of an infinite 3D NaCl crystal, assuming unit charges for Na+ and Cl – , and rmin = 2.81 ˚ A is the nearest-neighbor ion–ion distance, the Madelung potential 3D is VMadelung = ±8.95 V, where the positive (+) value corresponds to the location of a Cl –

site and the negative (−) value corresponds to the Na+ site. The Na+ and Cl – ions are described by ±1 e δ-functions placed at the lattice sites. To give a better representation of the ideal even- and odd-sized NaCl crystals and their charge states, some of them are shown in Figure 1. There, the even sizes (top) are stoichiometric and hence neutral, whereas the odd sizes (bottom) fall into two classes: Cl – -centered and Na+ -centered. Both odd classes

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Even

23

43

63

183

Clcentered

33Cl+

53Cl-

73Cl+

19 3Cl+

Odd Na+ centered

33Na-

53Na+

73Na-

19 3Na-

Figure 1: Illustration of the even and odd cubic crystallite clusters. Na+ are blue spheres and Cl – are green spheres. The even sizes (top) are stoichiometric and hence neutral, whereas the odd sizes (bottom) fall into two classes: Cl – -centered and Na+ -centered.

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are non-stoichiometric and have charges ± 1 that oscillate as a function of size. It is helpful to note that the identity of the corner ions indicates the overall charge for the odd sized crystals, i.e., if Na+ is at corner, then the charge = +1, if Cl – is at corner, then the charge = −1. In addition, for convenience, we will refer to the crystal “size” as the number of ions on the edge, e.g., size 2 denotes 23 , size 3 denotes 33 , etc. Formulations of the Madelung potential for infinite 1D and 2D systems can be found in the Supporting Information and 1D 2D yield VMadelung = ±7.10 V and VMadelung = ±8.28 V.

In 1918, the Madelung potential was first used to compute the large electrostatic contributions to crystal lattice energies within the Born–Landé formulation. 31 For a crystal, the total lattice energy, Utot (r) = eVMadelung + Urep , can be determined from the Madelung constant using the Born–Landè ionic model 31,32 M k e e2 Utot (r) = r

1 1− , n

(3)

where the first term corresponds to the electric potential energy and the second models the Pauli repulsion energy as Urep = M ke e2 /nr with n being the Born exponent that is experimentally determined from the compressibility of the solid. The Born exponent has been calculated from compressibility data as n = 9 for NaCl. This yields a lattice energy, LE = |Utot (r = rmin )|, of 7.97 eV 33 that when compared to experiment gives an error of ca. 0.2%. If we compare the magnitude of the Madelung potential arising from an infinite crystal to the voltage experienced by the Na+ or Cl – ions in the isolated NaCl dimer (at 2.81 ˚ A apart), which is 5.1 V, then we find that an additional ∼3.8 V is provided by the infinite crystal lattice. The electric potential and field distributions are given by P (Vi ) = δ[Vi − Vi (Γ)] and P (Ei ) = δ[Ei − Ei (Γ)], where Vi and Ei are the potentials and field magnitudes, respectively, experienced at ion i from all other ions in a crystal configuration Γ. The potential (V ) and field (E) distributions for even sizes and both classes of odd sizes were calculated at all ion

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sites. The potentials and fields on ion i in a crystal configuration Γ are defined by

Vi (Γ) =

N X j6=i

Ei (Γ) =

ke qj , |ri − rj |

N X ke qj (ri − rj )

|ri − rj |3

j6=i

(4)

,

(5)

where N is the total number of ions in the crystal, ri and rj are the position vectors for ions i and j, respectively, and qj is the charge on ion j. All potentials and fields (note that Ei = |Ei |)). See Supporting Information for visualization of the electric potentials and fields for some selected crystals. Because of how they are related, E(r) = −∇V (r), the information content in the field distributions complements that in the potential distributions. It is useful to split the distributions of potentials and fields into groups corresponding to Na+ (blue) and Cl – (green).

Computational Details Accurate molecular dynamics simulations of concentrated electrolyte properties present a grand challenge in chemical physics research. Concentrated aqueous NaCl electrolytes and NaCl nucleation have been explored, at length, with theoretical models in the literature, 34–41 ranging in scale from atomistic quantum molecular or classical molecular dynamics models 24,42 to continuum treatments like CNT or phase field models. 43 A review of these studies highlights both the strengths and limitations of fixed point charge descriptions of ion–ion and ion–water interactions to yield a wide range of concentrated aqueous electrolyte properties in agreement with experiment. 10,34,37–40,44,45 To date, it remains an open research question as to what interaction potentials yield the most accurate cluster size distributions. This is due, in part, to the absence of measured cluster size distributions to validate theoretical models. Indirect information about clusters through consistency between simulation and X-ray and neutron experiments on the inhomogeneities (arising from the existence of these clusters)

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in concentrated aqueous electrolyte structure maybe the last resort until direct cluster size distributions are probed. In the absence of measured cluster size distributions, we choose well-studied ion and water interaction potentials that are consistent with existing neutron measurements of concentrated electrolyte solution structure. 35,46–48 The various NaCl aqueous electrolytes were modeled classically using the simulation packages CP2K 49 and LAMMPS. 50 The H2 O interactions were described using the SPC/E model, 51 whereas the Na+ and Cl – interactions were described using the Smith–Dang model. 52 These ion-ion and ion–water interaction potentials use a combination of Lennard–Jones plus Coulomb interactions and the Lorentz–Berthelot combining rules. Five concentrations were considered: 1.0, 1.9, 3.8, 5.2, and 6.7 M with the corresponding cubic box lengths of 49.380 ˚ A, 49.114 ˚ A, 48.606 ˚ A, 48.894 ˚ A, and 48.692 ˚ A, respectively. The simulation volumes were calculated from the experimental densities. 53 N V T simulations were performed at a temperature T = 298 K using the Nosé–Hoover chain thermostat and the simulation time step of 2.0 fs. Pre-equilibration for 5 ns was performed prior to the production runs (lasting at least 20 ns) that were used in the analysis. It should be noted, that the most recent estimated solubility of the Smith–Dang model is 0.67 M, 54 meaning all solutions studied in this work are metastable with respect to crystallization. However, the system sizes and run times were not sufficient to lead to crystal nucleus formation resulting in equilibrium sampling of supersaturated NaCl solutions. This was confirmed by tracking cluster size distributions as discussed later in the text. Additionally, density functional quantum mechanical (QM) calculations were performed on NaCl crystals to compute QM electrostatic potentials. Two different levels of theory were employed. For the smaller crystals, the Gaussian 09 E1 software package 55 was used with the B3LYP density functional and def2-SVP basis sets. For the larger crystals, CP2K 49 was used with the PBE density functional and the DZVP-MOLOPT-SR-GTH basis sets. The QM potentials were computed by removing the central Na+ or Cl – ion, i.e., its nucleus and electrons, and computing the electrostatic potential at the location it occupied. This is

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an important distinction between the fixed point charge and quantum electronic structure descriptions of electric potentials and fields at ion sites. In both cases the self-potential or field is not included in the summation.

Results Before we begin our discussion of how the potentials of all of the ions in both even and oddsized crystals vary with size, we consider how the potential experienced by a single central ion converges to the infinite bulk Madelung potential. Odd-sized crystals, by symmetry, contain a middle/central ion (see Figure 1). Figure 2 shows the potentials experienced by the central ion for odd-sized NaCl crystals ranging in size from 3 to 19 for the Cl− -centered crystal class (the Na+ -centered crystals have equal and opposite potentials by symmetry and thus are not shown). As the crystals increase in size, the potential experienced by the central ion converges to the infinite crystal Madelung potential. One can also see the alternating total charge associated with these odd-sized crystals in Figure 1. In Figure 2, we also show the average potential in addition to the potentials experienced by the ions at the corners for both even and odd-sized crystals (these results will be discussed in more detail below). These variations in the potential of the central ion provide a conceptually simple example to better understand deviations from the bulk Madelung potential. In the next two sections we will analyze the distributions of potentials and fields for all the ions in both even- and odd-sized crystals. We provide a 3D visualization of the crystal electric potentials in the Supporting Information.

Even-Sized Crystals We start by investigating the patterns that emerge in the size-dependent potential and field distributions for even-sized crystals. For convenience, from this point forward, we consider only the absolute values of the potentials and the magnitudes of the fields given the

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11 Odd Cl Middle All Average Even Corners Odd Corners

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9 Potential (V)

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8

7

6

5

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 Size

Figure 2: Fixed point charge electric potentials experienced by middle Cl− ion (green circles #) in odd-sized non-stoichiometric NaCl crystals as a function of size and charge. The infinite bulk limit Madelung potential (green horizontal dashed line) for Cl− centered oddsized crystals. The average potential (black asterisks ∗) for all ions for both even- and oddsized crystals. The potentials experienced by the corner sites for both even- (blue squares 2) and odd-sized (red squares 2) crystals. symmetry between Na+ and Cl – ions. Figure 3 shows how the distributions of potentials experienced by Cl – ions vary in the even-sized crystals, ignoring for the moment the number of ions that have a particular potential. Each cubic crystal size has a unique distribution of electric potentials resulting in a unique electric signature that characterizes its size and shape (some representative 3D plots that include the electric potential and field distributions as functions of size for both some even- and odd-sized crystals are provided in the Supporting Information). Conveniently, the size-dependent potentials and fields (to be discussed later) can be split into subgroups corresponding to their location in the crystal: corners (c), edges (e), faces (f ), and interior (i). Thus, all subsequent data for crystals in vacuo is expressed as those potentials or fields experienced by either Na+ or Cl – , dropping the explicit charge superscript, and instead indicating its subgroup as a superscript, e.g., an edge Cl – = Cle . It is also important to recognize the connection between the large range of potentials and the corresponding energy scale by noting that 1 V acting on a unit charge e is 1 eV = 23 kcal/mol. The range of potentials for these even-sized crystals goes from 6.9 V to 9.2 V, giving a difference of 2.3 V in energies for the ion interactions within the crystal subgroups.

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18

interior faces edges corners

16 14 12 Size

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10 8 6 4 2 9.0

8.5

8.0 Potential (V)

7.5

7.0

Figure 3: Distributions of fixed point charge electric potentials on Cl – (green) ions for even-sized crystals of sizes 2 to 18 (potentials on Na+ ions have the same absolute values). The distributions are divided into subgroups (corners (c = 2), edges (e = 4), faces (f = 3), and interior (i = #)) corresponding to where the ions are located in the crystalline lattice. Note that the potentials are largest in the interior and smallest at the corners. The black symbols show the averaged potentials by subgroup with a vertical dashed line at the Madelung potential. The subgroup potentials vary as: interior > faces > edges > corners. As the crystal size increases, the interior, face, and edge distributions start to broaden into subgroup bands. These subgroup potential bands fall into the approximate ranges (for the sizes considered here): interior (i ) 8.8 to 9.2 V, faces (f ) 8.4 to 8.8 V, edges (e) 7.9 to 8.5 V, and corners (c) 6.9 to 7.5 V. The subgroups remain fairly distinct, however, as the size increased some bands start to overlap. For the smallest even-sized (23 ) crystal there are only four Na+ and four Cl – ions all occupying corner sites (there are no edge, face, or interior sites in this case), and thus they experience the same absolute magnitude of potential 7.4 V. The corner site potentials decrease from 7.4 V down to 6.9 V as the size increases from 2 to 18. Also, recall from the theory section that the potential experienced by the Na+ and Cl – ions in the dimer was 5.1 V. For (NaCl)2 the potentials are 6.6 V. If we compare these to the 23 case, it leads to the progression from the ion pair to the first cubic crystal of: (NaCl) = 5.1 V → (NaCl)2 = 6.6 V. The interior subgroup increasingly dominates the overall distribution of potentials as the crystals increase in size. The interior subgroup forms a band with an average that approaches 14


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0.5 4 6 8 10 12 14 16 18

0.4 Normalized populations

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0.3

0.2

0.1

0.0

9.0

8.5

8.0 Potential (V)

7.5

7.0

Figure 4: Normalized populations of electric potentials for even-sized crystals 4 to 18 (inset size legend) showing the convergence of the potentials for the larger crystals tends toward the 8.95 V Madelung potential (vertical black dashed line). the bulk limit VMadelung = ±8.95 V. In Figure 4, we show the normalized populations for crystal sizes 4 to 18 where the distributions have been divided by the total number of Na+ or Cl – ions. It can be seen that the highest peak for the largest crystal is positioned at the Madelung potential. Insofar as the interior subgroup potentials converging with size to the bulk Madelung potential, we can get a very good estimate even for crystals of size 6 or 8 consistent with that achieved by Gaio and Silvestrelli. 23 As far as the ions in the other subgroups are concerned, their potentials will remain smaller than the Madelung potential no matter how large the crystal simply because of the lower ion coordination. We provide the electric field analysis for the even-sized crystals in the Supporting Information. We can estimate 56–59 the size dependent interfacial surface energy, σ, for the h100i plane of NaCl by computing the difference in interior and face subgroup potential as:

σ ≈ e∆V ns = e(VBulk − VSurf )ns ,

(6)

where VBulk and VSurf are the bulk and surface electric potentials, respectively, and ns is the number density of surface ions. In this way, the interfacial surface energy arises from the difference between the interior and surface potentials of the crystallite. Using size 18 as an

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example, letting VBulk = Vi and VSurf = Vf we have ∆V = (8.95 − 8.61) V = 0.34 V and ˚2 (obtained from the lattice constant) that yields σcalc,18 = 343.4 ns = 63.322 × 10−3 NaCl/A ergs/cm2 , which compares well with the experimental value in vacuum, σexpt,vac = 283 ± 30 ergs/cm2 . 57 Note that the difference in these surface tensions only arises from a difference in the interior and face electric potentials of about 0.06 V. To place this sensitivity in proper context, the experimental surface tension of NaCl in liquid N2 is σexpt,N2 = 317 ± 30 ergs/cm2 , showing that the ∆Vexpt,N2 for the NaCl–N2 interface is only about 0.03 V larger than the ∆Vexpt,vac for the NaCl–vacuum interface. Interestingly, if both the Na+ and Cl – ion charges are scaled down to agree with the experimental surface energy in the vacuum, then we obtain a charge of ∼0.84 e, which is close to the 0.85 e used in the recently parameterized AH/TIP4P-2005 interactions used to model some NaCl electrolyte properties at equilibrium. 37 For size 4, which is the smallest crystal for which we can estimate the surface tension, σcalc,4 = 379.9 ergs/cm2 . Note that these calculated values are for perfect crystals, with the Na+ and Cl – ions taken to be equal and opposite, without zero point energy, and at 0 K. In contrast, Zimmermann et al. used CNT 38 to obtain an interfacial surface energy of ∼47 ergs/cm2 for small NaCl crystals in supersaturated aqueous NaCl. This would give rise to ∆V ≈ 0.046 V and at this point we anticipate that the presence of water up against the crystal faces increases the ion’s electric potentials about ∼0.3 V closer to the interior Madelung potential, yielding a smaller interfacial surface energy compared to the vacuum. Similarly, using a slab geometry of NaCl in water, Bahadur et al. 60 found an interfacial surface energy of 70 ± 23 ergs/cm2 . The following analysis will highlight the utility of quantifying the size-dependent potentials and their connection with the interfacial surface energies underlying nucleation. Moreover, interfacial surface energies for different crystallographic faces for multivalent cations and polyatomic anionic nanocrystals could also be calculated in a similar manner.

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Odd-Sized Crystals We now consider the distributions of potentials and fields for odd-sized crystals of sizes 3 to 19. There are several simplifying consequences of charge symmetry in the odd-sized crystals: 1) the fields at the central ion are exactly zero by symmetry, 2) the potentials and fields experienced by (Na+ /Cl – ) ions in a (Na+ /Cl – )-centered crystals are equal and opposite to those experienced by (Cl – /Na+ ) ions in a (Cl – /Na+ )-centered crystals. Using these symmetries, we need only consider the electric potentials and fields for Cl – -centered crystals. As with even-sized crystals, we will show that the odd-sized crystals also display unique size and structural signatures of potentials and fields. In Figure 5 (left panel), the distributions of potentials experienced by both Na+ and Cl – ions in Cl – -centered odd-sized crystals of sizes 3 to 19 as well as the averaged subgroup potentials (right panel) are shown. From the data in Figure 5 the important role of the middle (m) Cl – ion potentials are highlighted compared to the potentials experienced by the other Cl – ions in the crystals in contrast to the potentials shown in Figure 2. In this way, one can see how single Cl – ion potentials converge to the infinite bulk Madelung potential. There are some important differences between the size dependence of the distributions of potentials for even- versus odd-sized crystals: (1) in general, the odd-sized electric potential distribution widths are much larger (5.4 to 10.9 V, i.e., 5.5 V wide) starting at the smallest crystals than the widths ending at the largest crystals for the even-sized crystals (6.9 to 9.2 V, i.e., 2.3 V wide), (2) the even-sized crystal subgroup potential distributions start narrow and grow wider with size whereas the odd-sized crystal subgroup potential distributions (m, i, f , e, and c) start wide and grow more narrow, (3) the band width of electric potential location subgroups as a function of size are much larger for the odd-sized crystals compared to the even-sized crystals (for example, the interior and middle site widths for the odd-sized crystals varies from about 3 V for size 3 down to 1 V for size 19 compared to the interior sites for the even-sized crystals that vary from ∼0 V for size 4 up to 0.4 V for size 18), (4) the corner potentials for the odd-sized crystals increase from 5.4 V for size 3 up compared 17



19

19

m

Cl i Na i Cl f Na f Cl e Na e Cl c Na c Cl

15 13 11 9

m/i

Na m/i Cl f Na f Cl e Na e Cl c Na c Cl

17 15 13 Size

17

Size

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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11 9

7

7

5

5

3

3

11.0 10.5 10.0

9.5

9.0

8.5

8.0

7.5

7.0

6.5

6.0

5.5

5.0

11.0 10.5 10.0

Potential (V)

9.5

9.0

8.5

8.0

7.5

7.0

6.5

6.0

5.5

5.0

Potential (V)

Figure 5: Distributions of potentials on Na+ (blue) and Cl – (green) ions in Cl – -centered odd-sized crystals 3 to 19. The distributions are divided into subgroups [corners (c = 2), edges (e = 4), faces (f = 3), interior (i = #), and middle (m = ×)]. The dashed lines are provided for comparison with Figure 2 (for the Cl – -centered crystals) to show how the central (m) Cl – ion potentials converge to the infinite bulk Madelung potential (black dashed line at 8.946 V). Note that the potentials are largest in the interior and smallest at the corners. The left panel shows each unique potential observed per crystal, and the right panel shows the average potential by subgroup. to the even-sized crystals where they decrease from about 7.43 V for size 2 down to 6.93 V for size 18 (a difference of 0.5 V). The alternating groups of potentials for Na+ ions can be compared to those for Cl – ions as a function of size, where each ion group with the largest potentials is opposite to the crystal charge. For example, the group of Cl – ions for size 3 has the largest potentials and this crystal carries an overall positive charge, whereas the group of Na+ ions for size 5 has the largest potentials and the crystal is negatively charged, and so on for the larger sizes. It is not until size 15 that these interior and edge Na+ and Cl – ion groups begin to overlap and share similar magnitude electric potentials albeit with different signs (which, as described previously, has been suppressed for ease of presentation of the results). Furthermore, the potential oscillations between Na+ and Cl – , each in their respective subgroups, give rise to alternating high and low potential layers (the h111i planes), cutting diagonally (with respect to the faces) throughout the crystal (see Supporting Information for visualization of the interior and exterior electric potentials). For example, compare and contrast the interior (i),

18


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face (f ) , and edge (e) subgroups for sizes 5 and 7. For size 5, there is a difference in Na+ and Cl – potentials of ∼2.4 V with Na+ having larger potentials than Cl – . For size 7, there is a difference in Na+ and Cl – potentials of ∼1.7 V with Cl – having the larger potentials than Na+ . Thus, these i, f , and e alternating potential layer variations extend diagonally throughout the odd-sized crystals. By size 19, these variations show differences by about ∼1 V. The electric field analysis for the odd-sized crystals is provided in the Supporting Information.

Salt Crystal in Aqueous Solution The first part of this work dealt with the potentials and fields experienced by perfect crystals in vacuum in order to provide quantitative basis for understanding finite crystals. In what follows we will show: (1) how the potentials and field distributions in the vacuum are modified in the condensed aqueous and electrolyte phases, and (2) how the metastable amorphous clusters underlying nucleation differ from perfect crystals.

Figure 6: Snapshot of a 43 NaCl crystal in aqueous solution showing how the interfacial H2 O molecules associate with the surface Na+ and Cl – ions to increase their Madelung potentials and make them more bulk-like.

19



To make connection with in vacuo crystal electric potentials, we investigate a 43 crystal in the aqueous phase, shown in Figure 6, to illustrate how water modifies the interfacial electric potentials and fields. The 43 crystal was chosen for computational accessibility and because the interior ions of the 43 lie very close to the bulk Madelung potential - albeit for the case of a perfect crystal in vacuo. Figure 7 shows the normalized distributions of potentials, using exactly the same trajectory, on the Na+ and Cl – ions of a 43 crystal with the surrounding water molecule charges ON (left panels) and OFF (right panels) - OFF means the charges are set to zero. In the OFF case, one can clearly see the similarity to the interior, face, edge, and corner subgroup potential distributions found in vacuo. But, in the ON case, there is a clear shift of the face, edge, and corner voltage distributions to larger and more uniform values. H2O ON Corner Edge Face Interior + Na total

P(V)

0.02

0.02

0.01

0.01

0.00

0.00

0.02 P(V)

Corner Edge Face Interior − Cl total

0.02

0.01

0.00 10.0

H2O OFF

0.03

P(V)

0.03

P(V)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.01

9.0

8.0 7.0 Potential (V)

6.0

5.0

0.00 10.0

9.0

8.0 7.0 Potential (V)

6.0

5.0

Figure 7: Normalized distributions of electric potentials experienced by ions (Na+ = top row, Cl – = bottom row) in a 43 NaCl crystal with the H2 O charges ON and OFF. Note the difference in the range of potentials between H2 O ON and OFF. 20


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For both Na+ and Cl – cases, the potential ordering is still corners < edges < faces < interior, however, the Cl – ordering is more pronounced. This ordering is consistent with recent work by Lanero and Patey, 61 who found that the exposed surface area of various NaCl crystals is proportional to their dissolution rate with the crystals dissolving from the corners first. Furthermore, Yang et al. 62 found that a 43 crystal dissolves by the Cl – corner leaving first. From our discussion of even- and odd-sized crystals it should be clear that the distributions of potentials and fields are unique signatures of those crystals. Our previous classical molecular dynamics studies of concentrated aqueous NaCl electrolytes 35 resulted in Gaussian distributions of electric potentials for all the ions in the solution, including those ions within salt clusters, and spanned a range that included the bulk Madelung potential about 0.5 V from the upper edge of the distribution (for Na+ : mean ≈ −8.5 V, FWHM ≈ 1 V; for Cl – : mean ≈ +8.25 V, FWHM ≈ 0.75 V). But, in that study we did not separate the distributions of potentials between solvated ions and those ions involved in salt clusters or crystals which we address in the current study. Also, the concentration dependence of the potential distributions showed opposite trends for Na+ (shift to smaller potentials ∼0.3 V) and Cl – (shift to larger potentials ∼0.1 V). Interestingly, the magnitudes of the average potentials may be indicative of intermediate dimensional crystalline structures. The distributions of potentials and fields presented here provide the signatures or basis upon which order parameters of the crystallinity of the aqueous salt clusters can be investigated further. Structurally, as shown in Figure 6, these changes arise from hydrogen bonding and oxygen coordination around the exterior Cl – and Na+ ions, respectively. The peaks of the interior Na+ and Cl – ion site distributions are not shifted by the influence of water providing justification for our use of the 43 crystal as a representative case to illustrate the key concepts. But, in contrast to the perfect crystal, the interior Na+ and Cl – potentials, in general, are different from each other due to their different interaction potentials such that hV (Na+ )i = 8.4 V and hV (Cl− )i = 8.7 V. One should also note the order of the potentials, i.e., hV (Cl− )i

21



> hV (Na+ )i, and in what follows below we will see how the ion’s electric potential ordering and magnitude differ for the metastable clusters in concentrated aqueous NaCl electrolytes. The average of the interior Na+ and Cl – potentials yields ±8.55 V. This is lower than the interior potential of the perfect 43 crystal in the vacuum and results from an expansion of ∼0.1 ˚ A due to the solvating water molecules pulling on the ions in the crystal. This was verified directly from a shift in the first peak of the Na–Cl radial distribution function (not shown) between a thermal average of the 43 crystal in vacuum versus the aqueous phase. The Na+ and Cl – total potential distributions can be modeled as the sum of four Gaussian distributions corresponding to interior, face, edge, and corner sites. From the relevant potential differences we find an aqueous interfacial surface energy of σH2 O−ON = 67.5 ergs/cm2 including water (compare with simulations by Bahadur et al. 60 who found σ = 70 ± 23 ergs/cm2 ), and σH2 O−OFF = 329.3 ergs/cm2 without the electric potentials or fields from water (compare with σexpt,N2 = 317 ± 30 ergs/cm2 ). Here, we can see that water plays a key role in lowering the interfacial surface energy, compared to vacuum, by increasing the potentials on the ions on the crystal faces by ∼0.35 V and thereby reducing the difference between interior and surface potentials. Similar analysis of electric fields is provided in the Supporting Information.

Metastable Clusters in Aqueous Solution There is increasing evidence from simulations that amorphous and/or metastable phases can play important roles in nucleation. 10,11,19,45,63–66 Consider Ostwald’s rule of stages 67 where the mother phase transforms into intermediate phases with nearby free energies before the most stable phase is reached. Here we limit our discussion to thermodynamic pathways noting that kinetics 68 can also play a decisive role in forcing nucleation away from the minimum free energy path (e.g., kinetic trapping). Through the lens of CNT, Stranski and Totomanow 69 showed that Ostwald’s stages can be viewed as the sequential formation of metastable phases with lower interfacial surface energies, and hence lower barriers, evolving 22


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prior to the formation of the most stable phase. In this way, the Na+ and Cl – ions do not need to surmount the large nucleation barrier associated with the direct pathway from free ions in solution to the bulk rocksalt crystal (with a larger surface energy), however, can do so gradually by sampling lower barrier indirect metastable hydrated pathways (with a smaller surface energy). Next, we investigate the ordering and magnitudes of the Na+ and Cl – ion potentials as measures of cluster metastability through their deviations from the free ions and bulk crystals. 1

10

1.0 M 1.9 M 3.8 M 5.2 M 6.7 M

0

10

-1

10 ni (M)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-2

10

-3

10

-4

10

-5

10

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 # of ions

Figure 8: Cluster size distributions of NaCl clusters as a function of electrolyte concentration. In order to investigate how the cluster potentials deviate from their bulk behaviour, we first need to define the clusters. We adopt a nonrestrictive distance-based cluster definition by counting aggregates of ions in the electrolyte whose ions are within a cutoff distance, from at least one other ion, less than 3.75 ˚ A, i.e., the minimum after the first peak in the Na–Cl radial distribution function. Note that this definition does not include any restriction on charge state or imposition of rocksalt coordination. The resulting cluster size distributions are shown in Figure 8, where by size, in this context, we refer to the number of ions within a cluster. In Figure 9, the average electric potentials experienced by the ions as functions of size and concentration are presented. We only show average potentials for clusters whose distributions were adequately sampled – i.e., for cluster concentrations greater than ∼1 mM (except for the 23



1.0 M 1.9 M 3.8 M

8.50

Potential (V)

8.45 8.40 8.35 8.30 8.25 8.20 1

2

3

4

5 6 # of ions

7

8

30

35

9

5.2 M 6.7 M

8.45

Potential (V)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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8.40 8.35 8.30 8.25 0

5

10

15

20 25 # of ions

40

Figure 9: Average electric potentials experienced by Na+ (blue) and Cl – (green) as functions of cluster size for the following concentration: (top) = 1.0, 1.9, and 3.8 M and (bottom) = 5.2 and 6.7 M. 6.7 M case where a longer trajectory was run to sample the largest clusters i.e., ∼0.3 mM). For all concentrations, the free Na+ monomer ions have larger average potentials than the free Cl – monomer ions, i.e., hV (Na+ )ifree−ion > hV (Cl− )ifree−ion . Figure 9 (top) shows that average potentials for Na+ get smaller as the cluster size increases as well as showing a slight concentration dependence toward smaller potentials. Conversely, the average potentials for Cl – get larger as the size increases and also show a larger variation (toward larger potentials) with concentration compared to Na+ . For the 3.8 M case, the average potentials for Na+ and Cl – actually cross just after size 4. That a crossover was possible at all, much less at this or higher concentrations may have been anticipated by noting that the the average potentials of the ions in the 43 crystal in water are reversed with respect to the free ions (as shown

24


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in Figure 7), where hV (Na+ )i43 < hV (Cl− )i43 . It appears that at some intermediate cluster size, the average potentials experienced by the Na+ and Cl – ions crossover each other to reach the crystalline state. Figure 9 (bottom) shows the average potentials experienced by Na+ and Cl – ions within the clusters for concentrations 5.2 and 6.7 M. Interestingly, one can see a crossover in the average ion potentials (at ∼8.32 V) when the # of ions = 4, i.e., the (NaCl)2 cluster. But, once the clusters get beyond that size they do not cross. In both the 5.2 and 6.7 M cases, one can observe some degree of symmetric structure in the average potentials between the Na+ and Cl – ions for the larger clusters. 8.8 +

Na− cluster Cl cluster + Na crystal − Cl crystal − Cl int. crystal + Na int. crystal

8.7

Potential (V)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


8.6

2

8.5

67.5 ergs/cm

8.4 8.3 8.2 0.0

0.1

0.2

0.3

0.4

0.5 i

0.6

0.7

0.8

0.9

1.0

−1/3

Figure 10: Cluster scaling plot showing how average cluster ion potentials in the 6.7 M solution (open circles) for Na+ (blue) and Cl – (green) can be used as order parameters describing the gradual transition toward their average crystal potentials (filled squares). The average interior crystal potentials for Na+ and Cl – are also shown (stars). In order to get a better understanding of how the average cluster potentials converge to the bulk crystal (chosen to be the 43 crystal), and thus serve as useful order parameters, in Figure 10, we show the progression of ion potentials in the 6.7 M solution, plotted as a function of i −1/3 , where i is the number of ions in a cluster. This size scaling is reminiscent of that used in CNT to describe how the cluster free energies scale with size. 7,70 Here we assume that the size-dependent average electric potential of crystal clusters and the average bulk crystal electric potential are surrogates for the size-dependent crystal cluster chemical

25



Page 26 of 45

potentials and bulk crystal chemical potential, respectively, so we can write: 1

2 (36π) 3 σ − 1 hVclusters i ≈ − i 3 + hVbulk i, 3 ρ 23

(7)

lim [hVclusters i] ≈ hVbulk i.

(8)

where the limit i→∞

Thus, the average crystal cluster potentials converge to the average bulk crystal potential. Using the 43 crystal number density, ρ, and the surface energy, σ, we obtain the (red) dashed line in Figure 10. The average potentials of both ions in these metastable clusters tend toward their respective average crystal limits, however, they do so differently. Again, one can see the early crossing of the potentials (∼8.32 V) at the tetramer (NaCl)2 cluster. The free monomer ions are in a very different electric environment than those in metastable clusters or the crystal, indicating a gradual coordination transformation from fully hydrated free monomer ions, to partially hydrated metastable clusters, and then to the fully dehydrated crystal. The Cl – ion average potential increases almost linearly from free ions → clusters → bulk crystal – with a difference in average potential of hV (Cl− )i43 − hV (Cl− )ifree−ion = +240 mV = +5.52 kcal/mol. In contrast, the Na+ ion average potential decreases starting from the free ions to the clusters, however, it must reach a minimum and then increase to reach the bulk crystal limit – with a difference in average potential of hV (Na+ )i43 − hV (Na+ )ifree−ion = −110 mV = –2.53 kcal/mol. For comparison, we also show the average interior ion potentials for the bulk crystal to indicate how much the ions’ potentials change between the free ions, metastable clusters, and when they are within the interior of a bulk crystal. Using these interior potentials, we can see that the Cl – ion undergoes the largest shift in potential compared to the free ions yielding an increase of +440 mV = +10.12 kcal/mol. For the Na+ ion, first there is a decrease by −200 mV followed by an increase of +120 mV, yielding an overall difference of −80 mV = –1.84 kcal/mol between the free Na+ ions and the crystal interior.

26


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Using the results of Figures 7 and 10, we can state only qualitatively that the Cl – ions in the metastable clusters tend to be at locations with potentials similar to those found for the 43 crystal Cl – corner and edge sites, whereas the Na+ ions tend to be at locations similar to corner sites. Taken together, this shows that the average ion potentials can be used as order parameters to characterize how the metastable clusters underlying nucleation differ from both the free ions and the crystalline state. The difference in behaviour between Na+ and Cl – ions is striking and subsequent studies will determine the sensitivity of this behaviour to the ion-ion and ion-water interaction potentials. Since the Cl – average potentials vary nearly linearly with size, it is perhaps a better order parameter than the Na+ average potentials, however, there is interesting charge information correlated to the Na+ ion’s non-linear size behaviour. Further structural characterization of the average metastable cluster potentials into general interior and surface potential subgroups will be the subject of future work, however, we anticipate this analysis will yield even smaller interfacial surface energies consistent with metastable cluster stages purported by Stranski and Totomanow. 69

Charge States of Metastable Clusters in Aqueous Solution Finally, we turn to the composition and charge states of the metastable clusters shown in Figure 11. Here we exclude, by nature of the classical model employed in our simulations, the possibility of these intense fields inducing charge transfer, non-adiabaticity, or radicalization during the clustering underlying nucleation of NaCl even though there is experimental measurements of crystalloluminescence, piezoluminescence, lyoluminescence, and theoretical evidence that this may indeed occur including the role of trace impurities undergoing radicalization. 3,4,35,71,72 Here we explore the commonly invoked assumption of charge neutral pathways by addressing the following questions: are charged pathways relevant to metastable clusters and, if so, is there a sign preference? In general, cluster evolution can also include free ion or odd-sized cluster addition and loss. This opens the question of whether these metastable cluster pathways evolve equally through positive or negative states or is one sign 27



preferred over the other.

Figure 11: Cluster charge state analysis for 6.7 M NaCl. (top) corresponds to the even-sized clusters and (bottom) correspond to the odd-sized clusters. We already know from our cluster distribution functions that both even- and odd-sized metastable clusters occur as shown in Figure 8. If we analyze the charge states of the 6.7 M clusters, we uncover several interesting features. Figure 11 shows the most probable charge states for both even-sized (top) and odd-sized (bottom) clusters. The charge state distribution for each cluster size are independently normalized and thus the probabilities between sizes are not comparable. For the even-sized clusters (top), the smaller clusters display a preference to be charge neutral, however, this bias switches above i = 16 where the the +2 state is most probable. Also, we find that the charge distribution is asymmetric and significantly favors +2 state compared to the –2 state. There are even some +4 charge states that are more probable than the –2 states e.g., for i = 16, 18, and 20. For the special case of the NaCl cluster, it only occurred as a charge neutral population and thus is not included in Figure 11 (top). For the odd-sized clusters shown in Figure 11 (bottom), the free Cl – monomer ions are preferred over the free Na+ ions. But, for clusters of size ≥ 3, this charge 28


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asymmetry is reversed with the, Na+ rich, +1 charge state clusters becoming most probable – by a factor of ∼1.5 to 2 relative to the –1 charge states. By the time we reach clusters i = 15 and 17, even the +3 charge states become more probable than the –1 charge states. From the charge analysis we find that the metastable clusters show a preference for non-stoichiometry using a neutral → Na+ rich → neutral → Na+ rich → etc., nucleation pathway compared to the fully neutral or symmetric charge pathway. At this point it is tempting to speculate that these Na+ rich (Nax Cly )x−y clusters, along with an electric field of sufficient strength and appropriately directed from a Cl – toward Na+ ions could yield the right electronic conditions, using non-adiabatic condensed phase quantum mechanics, to allow for electron transfer from Cl – to Na+ yielding long-lived excited homolytic triplet electronic states, corresponding to Cl · and Na · radicals, consistent with the experimental observations of aqueous NaCl crystalloluminescent lifetimes. This type of condensed phase non-adiabatic multi-reference quantum simulation is currently computationally intractable (the authors are unaware of any existing software capable of treating this problem directly). However, in the future we will explore these mechanisms using high-level unrestricted openshell QM/MM simulations, re-sampling our trajectories, to compute the distributions of ground and excited singlet and triplet states associated with these metastable clusters.

Classical and Quantum Crystal Potentials Given the dominant role played by electrostatics in these ionic systems, as well as the large absolute magnitudes and variations in electric potentials with size, we want to explore the differences between classical (CLS) point charge and quantum mechanical (QM) electric potentials. Recall that Figure 2 (green circles) showed the progression of potentials experienced by a middle/central Cl – ion due to all other charges within the odd-sized crystals. The electric potentials shown in Figure 12 display similar variations with size for both the CLS and QM descriptions (i.e., for sizes 3, 5, 7, and 9). But, there is a large difference between the absolute values of the CLS (V3 = 10.9 V, V5 = 7.8 V, V7 = 9.8 V, and V9 = 8.3 V) and 29



QM potentials (V3 = 7.9 V, V5 = 4.2 V, V7 = 6.2 V, V9 = 4.6 V). In Figure 12 we also provide the potentials resulting from a uniform scaling of the CLS charges. Unfortunately, the QM results cannot be obtained simply by scaling the charges uniformly as shown in Figure 12 for the other CLS charges (e.g., |q|= 0.9, 0.725, and 0.5 e). We also tested the ability of Gaussian representations of the ion charges to reproduce the quantum results. For the Gaussians, we used 2σ = rion , where reasonable values rion = 1.16 ˚ A and 1.67 ˚ A were used for the Na+ and Cl – ionic radii, respectively. This resulted in potentials equivalent to the ±1 point charge results. 12

QM B3LYP/Def2-SVP CLS +/-1.0 CLS +/- 0.9 CLS +/- 0.725 CLS +/- 0.5

11 10 9 Potential (V)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

8 7 6 5 4 3

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 Size

Figure 12: Classical fixed point charge (CLS) and quantum mechanical electronic (QM) potentials experienced by a central Cl – ion in odd-sized NaCl crystals as a function of size and ion charge (see Figure 1 for charge assignments). Infinite crystal (bulk limit) estimates for the Madelung potentials are indicated by the dashed lines. The QM voltages cannot be obtained by uniform scaling of the CLS charges (|q|= 1.0, 0.9, 0.725, and 0.5 e). These results suggest that one cannot predict an experimental lattice energy from only the QM electrostatic properties, and we attribute this behavior to inherently QM many-body interactions (Pauli repulsion, exchange, and correlation). Subsequent work will explore the QM decomposition of the lattice energy. One can see in Figure 12 that small changes in the charges due to scaling give rise to quite large changes in the electric potentials. This is important because the electric potentials comprise the largest contribution to the lattice/cluster energy and hence strongly influence their thermodynamic stability during the nucleation process. 30


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We also extended the size range of the QM calculations using DFT/GPW/PBE including Na+ -centered odd-sized crystals (see Figure 13) for comparison to the Cl – -centered crystals. The exact symmetry found between idealized point charges of opposite sign is not found (for sizes 3, 5, 7, and 9) with the QM results for smaller odd-sized NaCl crystals due to different electronic configurations of Na+ and Cl – , as discussed previously. In particular, the QM ± ion non-equivalence manifests itself in near-field differences and we can compare the odd-sized Na+ -centered 3 and 5 crystal potentials (−8.75 and −4.49 V) to those for Cl – -centered crystal potentials (7.89 and 4.20 V) yielding differences of 0.86 V and 0.29 V, respectively. However, this difference quickly diminishes as the crystal size increases confirming the far-field equivalence of the ions. And even with the large discrepancy in absolute electric potentials, a striking similarity in the convergence is observed between the CLS and QM size dependent potentials. Even though the descriptions of the CLS and QM charge distributions are very different, it appears that an overall shift of the CLS (±1.0 e) electric potentials by ∼3.75 V would bring them into close agreement with the QM electric potentials. QM ±ions are only equivalent in the far-field sense – clearly not the case for QM ions near each other in the crystal lattice. Additionally, one can also provide, to the best of our knowledge, the first estimate of the infinite crystal QM Madelung potential using the QM average of the potentials for sizes 11 and 13, yielding a value of VMadelung = ±5.20 V.

Several quantum-based charge-partitioning schemes were used to obtain point charges to determine their ability to reproduce the electric potential at the center. Both the Mulliken and Hirshfeld population analyses are based on partitioning the electron charge density into atomic populations. 73 CM5 is a population analysis scheme based on Hirshfeld population analysis that was parameterized to produce atomic charges that yield better molecular dipole moments. 74 CHELPG 75 and Mertz–Kollman–Singh 76,77 (MKS) are protocols that derive atomic charges by fitting the charges to reproduce the quantum mechanically computed electrostatic potential in regions of space outside the atoms. Once the charges from each atomic charge assignment model are obtained, the electrostatic potential at the center is

31



12 QM B3LYP QM PBE (Cl-centered) QM PBE (Na-centered) CLS +/- 1.0

11 10 9 Potential (V)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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8 7 6 5 4 3

3

4

5

6

7

8 Size

9

10

11

12

13

Figure 13: QM and CLS potentials experienced by both a central Cl – ion as well as a central Na+ ion in odd-sized NaCl crystals as a function of size and ion charge (see Figure 1 for charge assignments). The red and green dotted lines correspond to the infinite bulk CLS and QM Madelung potentials (V∞CLS = 8.95 V and V∞QM = 5.20 V), respectively. computed for each model (see Figure 14. The quantum-based point charges for the size 3 crystal show fairly reasonable agreement across all charge schemes, however as the crystals increase in size, the potentials predicted by population analysis derived charges diverge from the quantum mechanical potential. The CM5 and Hirshfeld charges produce potentials that are close to the (±0.5 e) point charge models for larger sizes. As expected, the electrostatic potential fitted charge schemes (CHELPG and MKS) reproduce the quantum mechanical electrostatic potential very well across all sizes. However, it is important to realize that for all quantum-based charge schemes the charges representing each of the Na+ and Cl – ions are unique by symmetry and size dependent. This means that one would need a different set of charges for each ion, crystal size, and charge state. For example, in the case of CHELPG we find the following charges (all in units of e): q Na = 0.58 to 0.88, q Cl = −0.71 (for size 3), q Na = 0.62 to 0.76, q Cl = −0.71 to −0.78 (for size 5), q Na = 0.74 to 0.91, q Cl = −0.76 to −0.87 (for size 7), q Na = 0.68 to 0.85, q Cl = −0.74 to −0.84 (for size 9). For this case, the smallest charges are on the corners and the largest charges are in the interior of the crystals. The CHELPG and MKS schemes only use the QM potential outside a set of spheres, centered on each atom, and thus do not reproduce the interior potentials as

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12 11 10 9 Potential (V)

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CLS +/- 1.0 CLS +/- 0.5 QM Mull. QM Hir. QM CM5 QM CHELPG QM MKS QM (exact)

8 7 6 5 4 3 2 1

3

4

5

6

7 Size

8

9

Figure 14: Classical (CLS) potentials from uniform point charges, or charges assigned by QM charge partition schemes (Mulliken, Hirshfeld, CM5, CHELPG, MKS), and QM (B3LYP denoted as ‘exact’) potentials experienced by a central Cl – ion in odd-sized NaCl crystals as a function of size and ion charge (see Figure 1 for charge assignments). The CLS potentials from charge partitioning schemes can differ quite dramatically from the QM potentials. well, particularly as the clusters grow in size – see “near” and “far” potentials and fields discussion in the introduction. Thus, as one might have anticipated, in order to mimic the quantum nature of the electrostatic potential, one needs a description of ionic charge that can naturally adjust to its environment. This has important consequences with regard to ion stabilization at various locations on the crystal as well as quantifying chemical shifts in electron binding energies in XPS experiments.

Conclusions In the current work, we have shown how the potentials and fields of salt crystals can be classified into various subgroups corresponding to corners, edges, faces, and interior sites. The differences in interior and face potentials can be used to obtain interfacial surface energies of the crystals, which can further be used to connect with continuum approaches like CNT. When the average Na+ and Cl – potentials are plotted versus i −1/3 , the Cl – ions undergo a nearly linear increase when going from free ions to metastable clusters to crystals. Whereas the Na+ ions display non-linear behaviour where the average potentials first decrease, then

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level off, and then increase to reach the average crystal potential. The metastability order parameter of an individual cluster can be determined using the average Cl – electric potential, where we have observed a linear increase. As such, the deviations of the cluster’s average ion potential from their average bulk Madelung potentials provides a measure of cluster metastability. Analyzing the average charge states from the cluster distribution functions shows that these (Nax Cly )x−y clusters display a preference for asymmetric stoichiometry by going through a neutral → Na+ rich → neutral → Na+ rich → etc. nucleation pathway compared to the fully neutral or symmetric charge pathway. We find that the point charge potentials differ significantly from the quantum electronic structure potentials and that small changes in the charges lead to large changes in the electric potentials. We obtain a classical Madelung potential for the infinite NaCl crystal to be 8.95 V and that simply scaling the point charges will not reproduce the quantum potentials. From this analysis we estimate the quantum Madelung potential to be 5.2 V and that the classical (± 1.0 e) potentials can be shifted by −3.75 V to be consistent with quantum potentials. We characterized the fluctuations in charge, electric potentials and fields for metastable clusters in aqueous NaCl electrolytes and provide insight into how these properties can affect nucleation. We find that the magnitudes of the electric potentials and fields at established crystal interfaces or on metastable amorphous clusters are relatively large and thus may influence condensed phase electronic excitation during nucleation (e.g., crystalloluminescence) as well as producing other electronic effects like core electron binding energy shifts as measured by X-ray photoelectron spectroscopy. Future studies will use these clusters and their electric potentials to simulate appropriate X-ray photoelectron spectra. With regard to the sensitivity of our results to a particular water and ion force field, we expect that different force fields will yield quantitative differences in the magnitudes of the averaged electric potentials and fields on the ions. For example, in the section on classical and quantum crystal potentials we show how much the Madelung potential varies

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as the charges are uniformly scaled as well as when they are determined from quantum calculations. However, we think using the progression of electric potentials as robust order parameters will still help provide insights into the various clustering pathways resulting from a particular representation of ions and the surrounding solvent. It still remains an outstanding challenge to directly connect simulation and experiment when it comes to nucleation. No experiment to date has directly measured the production of critical clusters to yield a true nucleation rate. The particle size distributions relevant to the nucleation event, i.e., those critical clusters being produced as they come over the top of the nucleation barrier, from simulation still remain out of experimental reach. These freshly nucleated particles still need to undergo growth and coagulation to reach the length and times scales probed by conventional laboratory methods. Simulating these processes atomistically quickly becomes computationally intractable. Instead, it is possible to take the particle size distributions relevant to nucleation and model the combined nucleation, growth and coagulation processes through the general dynamic equation. This allows simulators to reach the length and time scales relevant to Small Angle X-ray Scattering (SAXS). On the experimental SAXS side, getting particle size distributions as a function of time suffers from the inverse scattering problem. Current synchrotron X-ray beams as well as the new free electron laser sources of X-rays may be the only way to achieve consistency between measurements and theory of crystallization in condensed phase chemical physics.

Acknowledgement The authors acknowledge helpful discussions with Marcel Baer, Chris Mundy, and Gregory Schenter. WCI and GJL were supported by the Nuclear Process Science Initiative, a Laboratory Directed Research and Development Initiative at Pacific Northwest National Laboratory (PNNL). SMK and EOF were supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences

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and Biosciences. PNNL is operated by Battelle for the U.S. Department of Energy under Contract No. DE-AC05-76RL01830. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

Supporting Information Available Discussion of the electrostatic contribution to the thermodynamic cluster free energy, formulations of the Madelung potential for infinite 1D and 2D systems, analysis of electric fields, and visualization of the electric potentials and fields for some selected crystals. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Electric Potentials of Metastable Salt Clusters | The Journal of Physical

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