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Phys Rev E Stat Nonlin Soft Matter Phys. Author manuscript; available in PMC 2008 July 15.

Published in final edited form as:

Phys Rev E Stat Nonlin Soft Matter Phys. 2008 March; 77(3 Pt 1): 031501.

Published online 2008 March 4. doi: 10.1103/PhysRevE.77.031501PMCID: PMC2467436

NIHMSID: NIHMS51041

Sergio A. Hassan, Center for Molecular Modeling, DCB/CIT National Institutes of Health, Bethesda, Maryland 20892, USA;

See other articles in PMC that cite the published article.

Formation of ion clusters in aqueous NaCl solutions at 25 °C is investigated with dynamics simulations in the 0.1–3M concentration range. The medium is found to be highly structured even at moderate concentrations, and clusters of over 20 ions are observed above ~2M. The medium can be viewed as a multicomponent fluid, composed of reacting particles with well-defined average populations, shapes, sizes, and charge distributions. Clusters show substantial morphological variations that are here characterized by their hydrodynamic radii and internal charge distributions. Clusters can be described as prolate spheroids in the subnanometer length scale. The hydrodynamic radius and the radius of gyration follow simple power laws with the number of ions in the cluster. Dipole moment distributions show characteristic peaks in the ~10–60 debye range that reflect conformational preferences modulated by electrostatic and liquid-structure forces.

Electrolytes are commonly used in technological and industrial application, and are ubiquitous in biological systems. They are found in body fluids and tissues, and play a role in the highly regulated electrochemical balance in cells. Electrolytes affect physical and chemical properties of proteins and nucleic acids, the motility of individual cells (chemotaxis), the migration of simple multicellular organisms, and the survival of bacteria. These processes occur at a broad range of concentrations, from highly diluted to nearly saturated conditions.

Thermodynamic and transport properties of electrolyte solutions are well characterized [1]. However, their microscopic nature at ambient and physiological conditions is still poorly understood. Raman scattering experiments have suggested the presence of clusters in unsaturated, saturated, and supersaturated solutions [2,3]. More direct evidence of cluster formation has been obtained in recent years with dynamic light scattering techniques [4]. Assuming colloidal clusters of spherical shape, hydrodynamic radii were obtained for aqueous NaCl solutions at ~20 °C. Two well-defined size scales were identified: below ~1 nm and above ~100 nm [4].

The most direct evidence of cluster formation, however, is provided by computer simulations. They allow a detailed study of properties that cannot be scrutinized experimentally. Among these are the atomistic nature of cluster speciation, the properties of the liquid structure, and cluster morphology. Molecular dynamics (MD) simulations have been used to study structural and dynamic properties of concentrated solutions [5], the formation of small clusters in unsaturated aqueous electrolytes [6,7], early steps of salt nucleation in oversaturated solutions [8], and cluster formation in supercritical conditions [9].

Because of limitations in computer power earlier studies have been restricted to small systems and short simulation times. Recently, longer (10 ns) MD simulations provided direct evidence of clusters of up to six ions in 1*m* NaCl aqueous solutions at 25 °C [7] and permitted reliable calculation of thermodynamic data such as pair dissociation constants. In this paper, results from a set of 40-ns MD simulations of NaCl in water at 25 °C and 1 bar are reported. Salt concentrations are in the 0.1–3M range, which is typical in biological systems and related experimental studies. These relatively long simulations allow the observation of larger clusters, with 10–30 ions above ~2M, which start forming after ~10–20 ns. The simulations provide data for statistically reliable calculations, which permits identification of simple behaviors not yet reported in these systems.

The simulations were performed with classical potentials in the canonical ensemble. A cubic cell of volume ~(46 Å)^{3} containing ~3500 nonpolarizable water molecules preequilibrated at 25 °C was used. The density was ≈0.993 g/cm^{3}, consistent with a pressure of 0.1 MPa. Concentrations of 0.1M, 0.5M, 1M, 2M, and 3M were generated by randomly replacing water molecules with Na^{+} and Cl^{−} ions. Periodic boundary conditions were used (see additional details in [10]). The system was equilibrated for ~2 ns to allow ions to diffuse across the unit cell [11].

A cluster is defined as any array of ions such that (i) every ion is connected to at least one ion of opposite charge; two ions are said to be connected if they are separated by a distance smaller than 3.5 Å [12]; and (ii) every ion can be reached from any other ion through a path of consecutive connections. The following criterion applies to time evolution: If a cluster breaks into two fragments at time *t* but reforms before *t*+*t*^{}, then the cluster experienced just a structural fluctuation, and did not break at *t*. In this study *t*^{} is 2 ps [13]. Similarly, if two clusters coalesce at time *t* but separate again before *t*+*t*^{}, then the clusters just collided, and did not react to form a single, larger cluster at *t*.

Statistical properties are derived from the cluster density *ρ*(*n*,*q*,*t*,*c*), defined as the number of clusters of size *n* (number of ions) and charge *q* per unit volume, at time *t* and concentration *c*. The clusters degree of formation is given by *α*(*n*,*c*) = *c _{n}*(

Degree of formation *α*_{n} as a function of cluster size *n* (number of ions in the cluster) at different concentrations *c* (in moles/liter). The inset shows *α*_{n} as a function of *c* for dissociated ions (*n*=1) and for ion pairs (*n*=2).

The electrolyte undergoes significant structural changes in the course of the simulation. It fluctuates between highly structured and highly dissociated, with characteristic periods of ~0.5–1 ns. These changes can be quantified by the probability *P _{s}* that a fraction

Cluster morphology shows substantial variation within classes (defined by *n* and *q*). Clusters are far from spherical, and present no crystal-like substructures. A measure of compactness can be defined as Ψ = *γ*/*γ _{c}*, where

Water accessible surface area *γ* as a function of cluster size *n*. Bars are standard deviations; statistical errors are small and omitted. The inset (a) shows the compactness parameter *ψ* (solid thick: simulation data; thin: linear arrange; **...**

Hydrodynamic radii *R _{H}* of colloidal particles are accessible experimentally using dynamic light scattering. Calculation of

$${R}_{H}^{(k)}={\left[\sum _{i=1}^{n}{\rho}_{i}^{2}\right]}^{2}{\left[\sum _{i=1}^{n}{\rho}_{i}^{3}+\sum _{i=1}^{n}\sum _{j\ne i}^{n}{\rho}_{i}^{2}{\rho}_{j}^{2}{\langle {r}_{ij}^{-1}\rangle}_{k}\right]}^{-1},$$

where *ρ _{i}* is the hydrodynamic radius of ion (subunit)

Hydrodynamic radius *R*_{H} as a function of cluster size. The inset (a) shows the linear dependence between *R*_{H0} and *R*_{γ}. The inset (b) shows the linear dependence between *R*_{H} and the radius of gyration *R*_{G}. Bars are standard deviations; statistical errors **...**

Also accessible experimentally is the radius of gyration *R _{G}*, which can be measured using static light scattering. Inset (b) of Fig. 3 shows

Cluster charge varies between −4*e* and +4*e*, depending on salt concentration, and will be studied elsewhere. The interest here is in higher moments of the cluster charge distribution, which displays substantial variations as well and provides information on the cluster morphology. Dipole moments are calculated here with respect to the clusters centers of mass, which can be far removed from the corresponding centers of charge (from which point *μ*=0). Thus, a cluster can be viewed as a liquid-excluding spheroidal body with charge *q* and dipole *μ* located at its center of mass. This interpretation may be useful to quantify clusters electrostatic effects on a distant solute (e.g., a protein or DNA molecule) through a multipole expansion of the potential. Bulk and nonbulk effects of concentrated electrolytes on biomolecular interactions have been studied in [21].

A distribution function *p*(*μ*|*n*,*q*) can be defined such that *dP _{n}*

Conditional probability distributions *p* as a function of clusters dipole moments *μ* (in debye) for small clusters. Distributions are normalized as ∫*p*(*μ*|*n*,*q*)*dμ*=1, and are shown to scale for each *n*. Charges are in units of **...**

This study utilized the high-performance computer capabilities of the Biowulf PC/Linux cluster at the NIH. This work was supported by the NIH Intramural Research Program through the Center for Information Technology. The author thanks Peter Steinbach for reading the manuscript.

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11. Concentration-dependent self-diffusion coefficients *D* were calculated from the Einstein-Smoluchowski equation *D*=|**r**(*t*)−**r**(*t*)|^{2}/6*t* at long *t*, where denotes the average over all ions and over all time origins *t*_{0} (0< *t*_{0} <*τ*). The equilibration time was estimated from the diffusion of the slowest ions (Na^{+} at 3*M*, in which case *D*≈1.5 × 10^{−5} cm^{2}/sec).

12. A rather restrictive definition of ion cluster is adopted here where *l*_{0} ≈ 3.5 Å is the position (independent of concentration) of the minimum in the Na^{+}-Cl^{−} pair correlation functions *g*(*r*) calculated from these simulations. A softer definition would use *l*_{0} ≈ 6.0 Å, the position of the second minimum, thus allowing solvent-separated pairs within clusters.

13. The calculation of self-diffusion coefficients [11] implies that the slowest ions in bulk require ~2 ps to travel a distance equal to the radius of a water molecule (~1.4 Å). Thus, ~2 ps is taken here as the maximum time that an ion is allowed to move away from a cluster without exchanging position with a water molecule.

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