A Review of Steric Interactions of Ions: Why Some Theories Succeed and Others Fail to Account for Ion Size

Microfluid Nanofluid DOI 10.1007/s10404-014-1489-5 REVIEW A review of steric interactions of ions: Why some theories succeed and others fail to account for ion size Dirk Gillespie Received: 20 June 2014 / Accepted: 23 September 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract As nanofluidic devices become smaller and 1 Introduction their surface charges become larger, the steric interactions of ions in the electrical double layers at the device walls In the last several decades, new fabrication techniques have will become more important. The ions’ size prevents them transformed nanofluidics by, among other things, mak- from overlapping, and the resulting correlations between ing devices smaller and smaller. Now, devices have length the ions can produce oscillations in the density profiles. scales of 10 nm or even smaller (e.g., Duan and Majumdar Because device properties are determined by the structure 2010). Other fabrication techniques have embedded elec- of these double layers, it is more important than ever that trodes in the walls of nanofluidic devices to vary the elec- theories correctly include steric interactions between ions. tric field interacting with the passing ions (Ai et al. 2011; This review analyzes what features a theory must have Branagan et al. 2012; Contento et al. 2011; Hlushkou et in order to accurately account for steric interactions. It al. 2009; Huang et al. 2010; Jiang and Stein 2011; Jin and also reviews several popular theories and compares them Aluru 2011; Lenzi et al. 2011; Maleki et al. 2009; Nam et against Monte Carlo simulations to gauge their accuracy. al. 2009, 2010; Paik et al. 2012; Piruska et al. 2010; van Successful theories of steric interactions satisfy the contact der Wouden et al. 2008). These two advances are creating density theorem of statistical mechanics and use locally devices of previously unattainable size and surface charge averaged concentrations. Theories that do not satisfy these on the device walls. criteria, especially those that use local concentrations From a modeling point of view, having wall-to-wall (instead of averaged concentrations) to limit local packing distances of only a few nanometers and effective surface fraction, produce qualitatively incorrect double layer struc- charges of several hundred mC/m2 bring some fundamen- ture. For which ion sizes, ion concentrations, and surface tal challenges. One of the most important of these is the charges monovalent ions steric effects are important is also inclusion of ion size in the calculations of the electrical analyzed. double layers near the walls. With such small devices, the ions’ sizes become comparable to the slit height in rectan- Keywords Electrical double layer · Steric interactions · gular nanochannels or pore diameter in cylindrical nanopo- Excluded volume effects · Theory res and so only a finite number of ions can fit between the walls. On the other hand, with such high surface charges, a very large concentration of ions will accumulate at the wall, preventing other ions from getting close the wall. In both cases, this results in ions becoming correlated; the presence of an ion in one location affects the probability of another ion being nearby. For example, at high surface charges, counterions form a dense first layer next to the * D. Gillespie ( ) wall. The physical size of the ions in that layer prevents Department of Molecular Biophysics and Physiology, Rush University Medical Center, Chicago, IL, USA a second layer of ions from approaching too close to the e-mail: [email protected] first because the ions’ van der Waals forces prevent them 1 3 Microfluid Nanofluid from occupying the same space (overlapping). This pro- theories. It is also the simplest generalization of point-ion duces a peak in the counterion concentration near the wall PB theory to include steric interactions. and a second peak one ion diameter behind the first peak. These Monte Carlo simulations are extremely help- If the second peak is large enough, a third one can form ful because, up to statistical noise, they can give the exact and so on, producing oscillations in the concentration pro- answer. This allows one to compare an approximate theory file. While this sort of steric hindrance and resulting density to a known answer under a wide range of ion sizes and con- oscillations occur in any electrical double layer, they take centrations and wall surface charges. Only by doing such a on special importance in a nanochannel because under the comparison can one truly judge the accuracy of an approxi- right circumstances, these ion–ion correlations can extend mate theory. Here, we will compare against two results from all the way across a nanochannel (Gillespie 2012; Nilson the group of Lamperski. In one paper, Lamperski and Kłos and Griffiths 2006). They are also important in nanochan- (2008) simulated a molten salt at very low (0.025 C/m2) nels because the ions’ axial velocity is determined by their and very high (0.5 C/m2) surface charges (Fig. 1) and sev- concentration and electrostatic potential profiles, and peaks eral in between (not shown in this review). In another paper, in concentration far from the wall will significantly affect Lamperski and Outhwaite (2008) simulated a dilute elec- those velocities. trolyte with two different-size counterions at a wall with The purpose of this review is to examine several differ- surface charge 0.2, 0.4, and 0.6 C/m2 (Figs. 2, 3). − − − ent continuum models (i.e., generalizations of the original (All simulation parameters are shown in Table 1.) These Poisson–Boltzmann, PB, theory) that attempt to include ion simulations were chosen because they systematically var- size and to understand why some succeed in reproducing ied surface charges, allowing us to see the accuracy of sev- simulation results while others fail to be even qualitatively eral theories as surface charge increases. correct. To do that, the focus here will be on the primitive The ion density profiles from these simulations can be model of ions where the ions are modeled as hard spheres used to assess the accuracy of approximate theories; only with a charge at their center and water is modeled implicitly theories that correctly treat steric interactions can repro- as a uniform dielectric constant. Also, to make the analysis duce these Monte Carlo results. Figures 1, 2, and 3 com- as simple as possible, only one wall with a single electri- pare these profiles from the simulations (symbols) to four cal double layer is considered. This model is used because commonly used steric PB theories (solid lines). In order it has been studied for many decades, and therefore, there to analyze the models’ accuracies in an unbiased way, a is a substantial amount of understanding gleaned from description of each theory is deferred until later. How- hundreds of Monte Carlo simulations and approximate ever, even without knowing these details, one can observe ABCD 50 50 50 50 σ = 0.025 C/m2 40 40 40 40 30 30 30 30 20 20 20 20 concentration (M) 10 10 10 10 0 0 0 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 x (nm) x (nm) x (nm) x (nm) 200 200 200 200 σ = 0.5 C/m2 150 150 150 150 100 100 100 100 50 50 50 50 concentration (M) 0 0 0 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 x (nm) x (nm) x (nm) x (nm) Fig. 1 (Color online) Comparison of four electrical double layer the counterion concentration profiles are shown. Both the counterions models (Theory A–D in the corresponding columns) to the simula- and co-ions have radius 0.2 nm and a bath concentration of 8.672 M. tion results of Lamperski and Kłos (2008) for two different surface See Table 1 for all the simulation parameters charges (0.025 C/m2 in the top row and 0.5 in the bottom row). Only 1 3 Microfluid Nanofluid ABCD 10 10 10 10 σ = -0.2 C/m2 8 8 8 8 6 6 6 6 4 4 4 4 concentration (M) 2 2 2 2 0 0 0 0 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 x (nm) x (nm) x (nm) x (nm) 30 30 30 30 σ = -0.4 C/m2 25 25 25 25 20 20 20 20 15 15 15 15 10 10 10 10 concentration (M) 5 5 5 5 0 0 0 0 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 x (nm) x (nm) x (nm) x (nm) 60 60 60 60 σ = -0.6 C/m2 50 50 50 50 40 40 40 40 30 30 30 30 20 20 20 20 concentration (M) 10 10 10 10 0 0 0 0 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.6 0.8 1.0 1.2 1.4 x (nm) x (nm) x (nm) x (nm) Fig.

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