Electrokinetic transport of monovalent and divalent cations in silica nanochannels

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1 Microfluid Nanofluid (2016) 20:8 DOI /s RESEARCH PAPER Electrokinetic transport of monovalent and divalent cations in silica nanochannels Shaurya Prakash 1 Harvey A. Zambrano 2 Kaushik K. Rangharajan 1 Emily Rosenthal Kim 1 Nicolas Vasquez 2 A. T. Conlisk 1 Received: 29 May 2015 / Accepted: 12 October 2015 Springer-Verlag Berlin Heidelberg 2015 Abstract Electrokinetic transport of aqueous electrolyte solutions in nanochannels and nanopores is considered important toward the understanding of fundamental ion transport in many biological systems, lab-on-chip, and organ-onchip devices. Despite the overall importance of these systems and devices, detailed calculations showing velocity and concentration profiles for multi-component, multi-valent ionic species are limited. In this paper, molecular dynamics simulations were employed to compute velocity and concentration profiles for an electrolyte mixture containing sodium, magnesium, and chloride ions with water as the solvent in a ~7-nmdeep amorphous silica nanochannel. The results indicate that addition of trace quantities of divalent Mg 2+ ions to monovalent (NaCl) electrolyte solutions while preserving overall system electroneutrality increases the maximum electroosmotic velocity of the solution by almost two times. Additionally, analyzing concentration profiles of individual ions revealed that Na + was found to be preferentially attracted to the negatively charged silica wall in comparison with Mg 2+ likely due to the hydrated divalent cation having a larger size compared to the hydrated monovalent cation. Keywords Nanochannel Electrokinetic flow Monovalent ion Divalent ion Silica Nanofluidics * Shaurya Prakash prakash.31@osu.edu * A. T. Conlisk conlisk.1@osu.edu 1 2 Department of Mechanical and Aerospace Engineering, The Ohio State University, 201 W.19th Ave, Columbus, OH 43210, USA Department of Chemical Engineering, Universidad de Concepcion, Concepcion, Chile 1 Introduction Fundamental study of fluid mechanics and the implementation of engineered nanoscale devices in the nm functional length scale are now commonly referred to as nanofluidics (Conlisk 2013; Prakash and Yeom 2014). One reason for interest in nanofluidics is the desire to investigate, and eventually emulate, the cell-level transport structures observed in biological systems which rely inherently on nanoscale transport for many key functions and exhibit exquisite control over ion and molecular species transport (Aguilar and Craighead 2013). Typically, these biological transport processes occur within nanoarchitectures (i.e., channels or pores) at sub-20 nm length scales (Hille 1992, Tybrandt et al. 2012). Over the past decade, significant nanofluidic-driven technological demonstrations (Prakash et al. 2008, 2012) have shown progress toward biosensing (Fan et al. 2005; Vlassiouk et al. 2009; Prakash et al. 2012), fluidic logic components (Kuo et al. 2003a, b; Karnik et al. 2005, 2006, Flachsbart et al. 2006; Karnik et al. 2007; Fuest et al. 2015a, b), energy conversion devices (Siria et al. 2013), and two-phase flow systems (Duan et al. 2012; Lee et al. 2014). Since all nanofluidic devices contain at least one operational dimension at sub-100 nm length scales, experimental determination of quantitative species and velocity distributions through direct measurements is extremely challenging. While some experimental measurements of electric current or optical methods such as fluorescence have provided significant information, determination of exact species concentration and velocity profiles continues to be experimentally elusive. In most technological demonstrations, the working fluid has been an aqueous solution of KCl or other simple electrolytes such as NaCl (Prakash et al. 2008;

2 8 Page 2 of 8 Swaminathan et al. 2012). However, many practical applications (Prakash et al. 2007; Datta et al. 2009; Prakash et al. 2009; Conlisk 2013; Prakash and Yeom 2014) such as separations (Han and Craighead 2000; Kim et al. 2010) and biological system mimics (Siwy et al. 2006) use electrolyte mixtures which include multi-valent ions (Gillespie et al. 2008). Technological demonstrations with complex multi-component electrolyte solutions, i.e., those containing more than one type of ion including multi-valent ions at the nanoscale, are limited (Siwy et al. 2006; Guan et al. 2014; Fuest et al. 2015a, b), while fundamental studies showing dependence of nanoscale transport on ion type have been previously reported (Nishizawa et al. 1995; Mammen et al. 1997; Kemery et al. 1998; Zheng et al. 2003; Gamble et al. 2014). In particular, it has been shown that a charged silica or glass wall preferentially attracts calcium (Ca 2+ ) over potassium (K + ) ions in aqueous solutions in multi-component systems (Zheng et al. 2003). Furthermore, in mixtures containing Na +, Cl, Mg 2+, and Ca 2+, continuum models supported by wall zeta potential measurements for microchannels also showed a preference for Ca 2+ over monovalent ions and lower adsorption of Mg 2+ to the silica wall than Ca 2+ based on electrostatic interactions (Datta et al. 2009). Previous work with multi-valent ions has also led to the development of empirical models to determine effective ion mobilities in capillary electrophoresis systems (Friedl et al. 1995). Additionally, calculations of potentials around cylindrical poly-ions using non-linear Boltzmann equations with excluded volume effects to determine ion distributions have been shown (Gavryushov and Zielenkiewicz 1998) along with an estimation of co-ion concentrations in capillaries for symmetric 3:3 (trivalent) electrolytes (Vlachy and Haymet 1989; Vlachy 2001). However, reliance of continuum models on the Poisson Boltzmann descriptions for the electric double layer (EDL) with electrolytes containing multi-valent ions has been questioned, especially at high electric fields, with the need to account for near-wall steric effects being highlighted. Several investigations evaluate these unanswered questions by use of advanced modeling tools such as molecular dynamics simulations (Qiao and Aluru 2003; Zhu et al. 2005; Qiao et al. 2006; Zambrano et al. 2012; Yoshida et al. 2014). Consequently, the observations of velocity profiles and species concentrations in nanochannels obtained through numerical modeling along with the near-wall effects as function of cation type in multi-component mixtures continue to be investigated. Furthermore, influence of ionic charge state on transport within nanoarchitectures is not yet fully understood, for example, in devices with rectification of ionic current in nanochannel devices (Gamble et al. 2014; Wang et al. 2015). Cation-dependent transport also assumes importance due to recent results showing that Microfluid Nanofluid (2016) 20:8 DNA transport through nanopores is affected by cation type as measured by resistive pulse methods (Kowalczyk et al. 2012). In this paper, long-range and large-scale equilibrium and non-equilibrium molecular dynamics (MD) simulations were conducted for a ~7-nm-deep amorphous silica nanochannel containing an aqueous solution with a mixture of NaCl and MgCl 2 with an electrokinetic flow. The results reported in this paper for the molecular dynamics simulations of electrokinetic flow in aqueous electrolytes explicitly account for the surface charge density of a silica wall forming the nanochannel in contrast to previous work with multi-valent ions (Calero et al. 2011). Therefore, the transport of multi-ionic mixtures is captured more realistically than in previous work by providing a detailed description of the concentration, velocity, and net space charge variation across the nanochannel for confined electrokinetic flows. Therefore, the purpose of this paper was to use molecular dynamics simulations to elucidate the behavior of an aqueous electrolyte containing a monovalent and divalent cation in the presence of the same monovalent anion as a function of applied axial electric field to determine resultant velocity and concentration profiles. 2 Methods The simulation methods used in this paper have been reported and discussed previously (Zambrano et al. 2012; Zambrano and Conlisk 2013; Prakash et al. 2015). In the sections to follow, a brief description of these methods is presented for completeness. Non-equilibrium molecular dynamics simulations were conducted using the MD package FASTTUBE (Walther et al. 2001). 2.1 Silica nanochannels The silica nanochannels form a slit-like architecture with two amorphous silica walls that were generated by annealing two identical crystalline silica slabs, as discussed previously for symmetric, monovalent electrolyte simulations (Zambrano et al. 2012; Prakash et al. 2015). The silica interactions were described using a Coulomb potential and a Buckingham potential augmented with a 16-8 Lennard Jones potential term to avoid fragmentation of the slab at high temperature (Tsuneyuki et al. 1988; Guissani and Guillot 1996). A time step of 1 fs was used to integrate forward using the leapfrog algorithm, and an annealing procedure, similar to that reported previously (Huff et al. 1999; Cruz-Chu et al. 2006; Zambrano et al. 2014). The system was coupled to a Berendsen thermostat (Berendsen et al. 1984), and the cristobalite slab was heated to 3700 K and

3 Microfluid Nanofluid (2016) 20:8 Fig. 1 A schematic of the ~7-nm-deep nm-long amorphous silica nanochannel. Periodic boundary conditions were imposed along X and Y. The silica surface charge was set at σ s = C/ m 2. The direction of the applied axial electric field is indicated by the orange arrow. An aqueous solution comprising Na +, Cl, and Mg 2+ ions was used as electrolyte mixture for all the simulations (color figure online) holding the temperature constant for 10 ps. The system was quenched from 3700 to 300 K by imposing a cooling rate of 70 K/ps until the equilibrium state was reached. In this work, no surface reactions with the silica wall arising from the presence of silanol and siloxane surface groups were considered; therefore, a hydrophilic silica surface was reproduced by following a calibration process of the silica water potential parameters to yield a final water contact angle of 27. These two silica slabs form the top and bottom walls of the nanochannel (Fig. 1). 2.2 Channel species interaction Water molecules were modeled using the rigid extended simple point charge SPC/E model (Berendsen et al. 1987). This model has been used with reliable results in several previous computational nanofluidic studies including simulations of electroosmotic flows (Freund 2002; Qiao and Aluru 2005; Joseph and Aluru 2006; Kim and Darve 2006; Zhang et al. 2011). The potentials used for the water interaction potentials, ion water interactions, silica ion interactions, and silica water interactions were all obtained from past published values with a complete summary of the potentials being reported previously (Zambrano et al. 2012; Zambrano and Conlisk 2013). The charge density on native silica was maintained at σ s = C/m 2 as shown schematically in Fig. 1 to evaluate velocity and ion distributions within the silica nanochannels. The magnitude of the surface charge density, σ s, and streamwise electric field (E ax, V/nm) was chosen to minimize contributions from thermal noise for reliable results similar to previous reports (Karniadakis et al. 2001; Conlisk 2013; Yoshida et al. 2014). The surface charge density of silica walls ( C/m 2 ) also falls within experimentally estimated surface charge for silica nanochannels (Stein et al. 2004; Karnik et al. 2005). Page 3 of 8 8 Overall, 20,500 SPC/E water molecules were confined in the ~7-nm amorphous silica nanochannel to reproduce the bulk water density at the center of the channel. Na +, Mg 2+, and Cl ions were described as charged Lennard Jones spheres with partial charges placed on the center of the spherical particles using values reported previously (Koneshan et al. 1998; Joseph and Aluru 2006; Larentzos and Criscenti 2008; Bonthuis et al. 2010). These ions were added to the water to reproduce the electrolyte mixture with 170 total ions. It is worth noting that the space charge within the volume of the fluid must balance the charge on the silica walls to maintain overall electroneutrality in the system. Therefore, the relative number of ions was manipulated across various simulated cases to maintain system electroneutrality. Furthermore, in order to provide a baseline case to evaluate the effect of adding divalent ions on the electrokinetic flow, simulations were also conducted on a silica channel identical to that shown in Fig. 1 but without Mg 2+ ions in the solution. In this baseline simulation, the ionic strength of the NaCl solution was 0.23 M with a corresponding Debye length of 0.63 nm, suggesting that the nanochannel is in the non-interacting EDL regime for electrokinetic transport (Prakash et al. 2008; Conlisk 2013; Prakash and Yeom 2014). It is worth noting that past research for analyzing transport of biological species such as DNA has evaluated translocation events using electrolyte concentration in the range of 10 5 to 3 M for KCl, and therefore, 0.23 M NaCl concentration considered in this study is within the concentration range used in numerous applications in biological systems and also for other applications including water desalination and for energy conversion (Heng et al. 2005; Gracheva et al. 2006; Qiao et al. 2006; Dekker 2007; Kim et al. 2010). 2.3 Simulation details All-atom MD simulations were conducted for the geometric configuration depicted schematically in Fig. 1. The simulation system comprised the amorphous silica walls that were approximately nm 2.53 nm 3.50 nm with a separation of ~7 nm to form the channel. Therefore, a computational box with dimensions nm 2.53 nm nm was implemented with periodic boundary conditions in X- and Y-directions. Based on previously reported methods, a cut-off radius of 1.0 nm for all van der Waals interactions and a SPME algorithm with slab correction for Coulomb interactions with a real-space cutoff of 1.2 nm were also implemented (Essmann et al. 1995; Yeh and Berkowitz 1999; Zambrano et al. 2012; Zambrano and Conlisk 2013; Prakash et al. 2015). The water temperature was controlled by connecting a Berendsen heat bath at 300 K (Berendsen

4 8 Page 4 of 8 et al. 1984) to the system, and the temperature of the electrolyte was computed to confirm isothermal operation during the entire 75-ns simulation with an integrating time step of 2 fs (Zambrano and Conlisk 2013). MD simulations without an electric field were conducted first to equilibrate the system, and then, NEMD simulations were performed. The data reported here were extracted from the last 50 ns of the 75 ns simulation. Combined with a significantly longer computational box and simulation time, the results reported here capture the electrokinetic transport more accurately than past work (Lorenz et al. 2008; Calero et al. 2011) with choice of parameters representing many experimental conditions (Heng et al. 2005; Gracheva et al. 2006; Dekker 2007) likely providing clearer insight into nanoconfined electrokinetic flow with divalent Mg 2+ cations in solution along with the monovalent Na + cations. 3 Results and discussion The methods section describes the wide range of parameters that were simulated. In this section, the salient results that allow the underlying flow physics to be discussed in a relatively succinct manner are presented. Figure 2a shows the water velocity profile within the silica nanochannel for the baseline case containing only the monovalent ions, i.e., Na + and Cl. The velocity profile follows previously reported trends with an overall plug-like shape (Lorenz et al. 2008; Prakash et al. 2015) for the general flow profile for a total ion concentration at 0.23 M. The electric fields used in this work are high ( V/nm) and generally not realizable in practical devices but are similar to the electric fields used in several past MD reports on electrokinetic flows in nanochannels (Qiao and Aluru 2003; Zhu et al. 2005; Joseph and Aluru 2006; Prakash et al. 2015). The effect of these relatively high electric fields was also seen on the velocity profile with non-linear increase in centerline (~ nm) channel velocity, also in agreement with previous reports (Zhu et al. 2005). The non-linear trends for the velocity shown in Fig. 2a are believed to be a consequence of increased driving force on ions with increasing axial field, which causes the ions near the walls to move more toward the center or near the bulk fluid region of the channel, as shown previously for the relatively high electric fields in MD simulations (Zhu et al. 2005). The velocity profile as a function of channel depth for the three applied axial potentials in the V/nm range is shown in Fig. 2b with Mg 2+ now added to the NaCl solution. It is important to note that in the representative case shown in Fig. 2b, the total charge carried by 30 Mg 2+ Microfluid Nanofluid (2016) 20:8 Fig. 2 Water velocity profiles are shown for two representative cases. a The water velocity profile as function of applied axial electric field is shown for NaCl at 0.23 M. The velocity trends reported here agree with the overall trends reported in the literature previously for monovalent electrolyte solutions (Zhu et al. 2005; Lorenz et al. 2008; Prakash et al. 2015). b The velocity profiles as a function of nanochannel depth are shown for the case of 30 Mg 2+ ions in solution with 60 Na + and 80 Cl ions for various applied electric field. As expected, the velocity was found to increase with increasing electric fields ions is the same as the charge carried by 60 Na + ions. The velocity was found to once again increase non-linearly with increasing electric field along the lines previously reported for electrokinetic flow with model electrolytes (Zhu et al. 2005), and also as discussed for the general trends observed in Fig. 2a. It is also noteworthy to compare that at similar electric fields (E ax = 0.7 V/nm) in Fig. 2a, b, the average velocity in the bulk fluid is nearly twice for the case where Mg 2+ comprises ~18 % of the total ion composition to that with NaCl only. Figure 3 shows the effect of adding Mg 2+ ions while reducing the amount of available Na + for the case of the axial electric field, E ax = 0.7 V/nm. Since the charge density at the silica wall is kept constant, the amount of Cl ions in the aqueous electrolyte solution was also adjusted to maintain overall system electroneutrality. Addition of only 10 Mg 2+ ions to simulate a trace impurity addition to the electrolyte and reduction of Na + ions from 105 to 90 led to an increase in the bulk water velocity from ~0.5 m/s to nearly 1 m/s at the channel centerline. Interestingly, increasing the amount of Mg 2+ by another factor of ~5 (i.e., increasing from trace impurity to a multi-component electrolyte solution) caused a further increase in the centerline velocity to ~1.5 m/s. In order to shed light on the observations of the velocity profiles as a function of channel depth reported in Figs. 2 and 3, concentration profiles for the ions are discussed next

5 Microfluid Nanofluid (2016) 20:8 Page 5 of 8 8 Fig. 3 Velocity profiles at a representative applied axial electric field of E ax = 0.7 V/nm are shown. The velocity was found to increase with the addition of Mg 2+ to the electrolyte solution, which was accompanied with reduction of Na + to maintain system electroneutrality for the representative case of E ax = 0.7 V/nm. Figure 4 shows the ion concentration profile for the case with no added divalent ion, Mg 2+. Figure 4a shows that in the bulk region of the nanochannel (~2.0 to 5.1 nm), nearly equal number of Na + and Cl was found inside the nanochannel. Figure 4b shows that the net charge density within the EDL was positive. The net positive charge density would be expected as the space charge within the volume of the fluid must balance the negative charge on the silica walls in order to preserve system electroneutrality. Considering once again the case where the 30 Mg 2+ ions in solution contain equivalent charge as 60 Na + ions, the concentration profiles and net charge density are shown in Fig. 5a, b, respectively. The Debye length was estimated to be 0.51 nm for the total ionic composition shown in Fig. 5. In agreement with previous reports of electrolyte solutions containing divalent ions (Gillespie 2015), charge inversion was observed (Fig. 5b) with the net charge density at ~2 nm from both the top and bottom walls being negative, i.e., the same polarity as the silica wall. Furthermore, in a contrasting result to previous continuum calculations (Mammen et al. 1997; Datta et al. 2009), it was found that Na + was preferentially attracted to the walls as opposed to Mg 2+. Specifically, in the first ~0.8 nm from either the top or bottom walls, a higher packing density (Zambrano et al. 2012) or concentration of Na + was observed. Mg 2+ also shows a preference for higher accumulation near the walls as opposed to the bulk fluid but in the region ~0.8 to 1.3 nm from either wall. The results therefore are in agreement with the general assertion that the thickness of EDL is about 3 4 times the Debye length (Conlisk 2013). Fig. 4 Concentration profile for NaCl (0.23 M) along the channel depth and implication for net charge density within the ~7-nm-deep silica nanochannel are shown. a The red (Na + ) and green (Cl ) lines show that the bulk fluid (~2.0 to 5.1 nm) is electrically neutral with equal amounts of cation and anion within the nanochannel. b The plot shows the net space charge density along the channel depth. Since the silica wall is negatively charged and the bulk fluid is electrically neutral, it would be expected that the EDL would be positively charged to maintain electroneutrality. In the figure, e denotes the elementary charge (e = C) (color figure online) Fig. 5 a Concentration profiles as a function of nanochannel depth are shown for the representative case of 30 Mg 2+ ions in solution with 60 Na + and 80 Cl ions for E ax = 0.7 V/nm. b The net charge density distribution along the channel depth shows that the charge density ~2 nm from the silica wall was negative, suggesting charge inversion in the EDL Next, a hypothesis is presented to explain the ionic layering observed in Fig. 5 with respect to preferential organization of Na + closer to the silica wall than the Mg 2+. Typically, continuum calculations assume all species as point charges and do not consider hydration of ions during

6 8 Page 6 of 8 electrostatic interactions. As water molecules exhibit an inherently dipolar nature, dissociated ions are enclosed by a shell of water molecules, known as the hydration sphere (Ghiu et al. 2003). Though the ionic radii of Na + (~0.116 nm) are higher than that of Mg 2+ (~0.072 nm) (Nightingale 1959; Tansel et al. 2006), the magnesium ion has a higher net charge due to the native divalency. Therefore, the electrostatic interaction and subsequently the force exerted on water molecules is higher for Mg 2+ in comparison with Na + leading to an overall higher hydration radius for Mg 2+ (Ghiu et al. 2003; Tansel et al. 2006; Tansel 2012). The hydrated ion radius for Na + has previously been estimated to be ~0.36 and ~0.43 nm for Mg 2+ (Nightingale 1959, Barthel and Jaenicke 1982; Tansel 2012). It is known that the hydration energy of an ion is directly proportional to the square of ion valence and varies inversely with the ionic radius (Shannon 1976; Burgess 1999). Therefore, it would be expected that the hydration energy for an Mg 2+ ion should be higher than Na +. Indeed, this observation is borne out by the MD simulations here. The hydration of the Na + and Mg 2+ ions was evaluated as the water ion binding or hydration energies. It was found that Mg 2+ has a significantly higher binding energy (~ 1550 kj/mol) magnitude compared to Na + (~ 300 kj/ mol). This observation for the calculated hydration energies is also in agreement with past reports that show higher hydration energy for Mg 2+ in comparison with Na + (Ghiu et al. 2003; Tansel et al. 2006; Tansel 2012). The hydration effects of ions are important, as it is now generally agreed that water causes layering effects near silica walls in nanochannels (Joseph and Aluru 2006). Another report using density functional theory (DFT) has also discussed the layering of ions in nanochannels (Gillespie et al. 2008) with the larger diameter ions located further away from the wall, and the EDLs for the monovalent ions were found to be expanded in contrast to the divalent ions (Hoffmann and Gillespie 2013; Gillespie 2015), similar to the observation for the EDLs discussed above for Fig. 5. Therefore, the preference of Na + in contrast to Mg 2+ near the silica wall in this work can be attributed to the higher magnitude of the hydration energy of Mg 2+ leading to a larger hydrated magnesium ion. Previous studies also show that in silica nanopores and slit-like silica nanochannels, the net flux and therefore the velocity of water molecules in the presence of a solution containing only Ca 2+ is reduced in contrast to a solution containing only Na + ions in simulations with only one type of cation (i.e., Na + or Ca 2+ ) in the nanoarchitectures due to charge inversion (Lorenz et al. 2008; Haria and Lorenz 2012), with charge inversion having also been confirmed experimentally recently (Li et al. 2015). By contrast, in an MD simulation of KCl, MgCl 2, and a mixture of KCl and Microfluid Nanofluid (2016) 20:8 MgCl 2 at 1 M concentration confined in a cubic computational domain, it was found that the flux of water molecules for the same electric field was highest for a Mg 2+ electrolyte, followed by a mixture of K + and Mg 2+ ions, with K + showing the lowest observed electroosmotic water molecule flux (Calero et al. 2011). Furthermore, the trends observed in water flux molecules were attributed to differences in hydration of K + and Mg 2+ (Calero et al. 2011). Overall, the trends reported for velocity of the Na + and Mg 2+ ions here also agree (cf. Fig. 2) with these general trends that have been reported previously. 4 Summary and conclusions Using MD simulations, ~7-nm-deep amorphous silica nanochannels were investigated for electrokinetic transport of a multi-component mixture containing a monovalent (Na + ) and a divalent (Mg 2+ ) cation with a monovalent anion (Cl ). The velocity profiles were found to be similar to a plug-like flow for NaCl which is in agreement with existing theory. However, the maximum bulk water velocity increased almost two times with the addition of only ~6 % of Mg 2+ (10 ions out of total 170 ions) to the electrolyte solution at the lowest applied electric field. Furthermore, the bulk velocity was found not scale linearly upon further increasing the concentration of Mg 2+ ions to over 30 % of the total ions in solution. In contrast to previously reported continuum calculations that considered purely electrostatic effects which showed preference of divalent cations over monovalent cations at a negatively charged silica wall, the MD simulations here show a preference for monovalent cations near the negatively charged silica wall when the hydration of the ions was considered. Acknowledgments The authors acknowledge partial financial support from Defense Advanced Research Projects Agency (DARPA) through the US Army Research Office (ARO) Grant W911NF09C0079, the US Army Research Office (ARO) for Grant W911NF , and NSF CBET Nicolas Vasquez acknowledges support from CONICYT through Scholarship Number The authors also acknowledge discussions with Jim Giuliani in Mechanical Engineering at OSU and the computational support from the Ohio Supercomputer Center (OSC). References Aguilar CA, Craighead HG (2013) Micro- and nanoscale devices for the investigation of epigenetics and chromatin dynamics. Nat Nanotechnol 8(10): Barthel J, Jaenicke R (1982) B. E. Conway: Ionic Hydration in Chemistry and Biophysics. Vol. 12 aus: Studies in Physical and Theoretical Chemistry. Elsevier Scientific Publishing Company, Amsterdam and New York Seiten, Berichte der Bunsengesellschaft für physikalische Chemie 86(3):

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