Putting Together the Puzzle of Ion Transfer in Single-Digit Carbon Nanotubes: Ab Initio Meets Mean-Field

2 Nature employs channel proteins to selectively pass water across cell membranes, 3 which inspires search for bio-mimetic analogues. Carbon nanotube porins (CNTPs) are 4 intriguing mimics of water channels, yet ion transport in CNTPs still poses questions. 5 As alternative to continuum models, here we present a molecular mean-ﬁeld model, 6 computing ab initio all required thermodynamic quantities for KCl salt and H + and 7 OH − ions present in water. Starting from water transfer, the model considers transfer 8 of free ions, along with ion-pair formation to address ion-ion interactions. High aﬃn- 9 ity to hydroxide, suggested by experiments and making it dominant charge carrier in 10 CNTP, is revealed as an exceptionally favorable transfer of KOH pairs. Nevertheless, 11 free ions, coexisting with less mobile ion-pairs, apparently control ion transport. The 12 model explains well the observed eﬀects of salt concentration and pH on conductivity, 13 transport numbers, anion permeation and its activation energies, and current rectiﬁca- tion. The proposed approach is extendable to other sub-nanochannels and help design novel osmotic materials and devices.

transport numbers, anion permeation and its activation energies, and current rectifica-14 tion. The proposed approach is extendable to other sub-nanochannels and help design 15 novel osmotic materials and devices. 16 The world faces a water stress, which is predicted to increase and spread to areas not 17 experiencing the shortage of fresh water today. 1 Production of fresh water via desalination 18 of sea, brackish, and waste water is a viable solution, yet currently used membrane desalina-19 tion technology still leaves room for improvement and selectivity-tailoring. This motivates 20 research that looks into alternative materials for next-generation membranes with improved 21 water-salt and ion-ion selectivity. 2 Natural membrane proteins aquaporins efficiently separate 22 water from ions by forcing it through a short and narrow channel in a single-file arrangement 23 at rates exceeding 10 9 water molecules per second with nearly ideal water-ion selectivity. 3 24 Intriguingly, while the use of degradable aquaporins might be impractical, stable nanomate-25 rials, such as atomically thin nanoporous nanosheets 4-7 or narrow nanotubes, 8-10 that can 26 mimic transport in aquaporins and offer an exciting next-generation alternative to currently 27 used polymeric membranes. 11,12 28 Single-digit carbon nanotube porins (CNTP) share many unique features of aquaporins 29 and demonstrated a water-salt selectivity of 10 5 , on par with benchmark polyamide reverse 30 osmosis membranes. 13-15 Numerous theoretical 16-24 and experimental 25-29 studies indicate 31 that, due to wall roughness smaller than the de Brogli length, water transport in CNTPs 32 narrower than about 1.5 nm and similarly narrow graphene slits, occurs in a scatter-less man-33 ner, at rates greatly exceeding hydrodynamic predictions 9,30,31 and even faster than water 34 permeation in aquaporins. 22,25 However, while there is an overall consensus regarding water 35 transport in narrow CNTs, the physical mechanisms behind ion rejection still pose many 36 questions. For instance, it has been long believed that negative carboxylic charges at CNTP 37 rims control salt rejection, 32-34 yet recent data on pH dependence of anion permeation down-38 played this mechanism. Adsorption of OH − ions was proposed as an alternative charging 39 mechanism in CNTs and a number of continuum-type models, solving Poisson-Boltzmann 40 and Navier-Stokes equation employed this and other ad hoc assumptions to describe trans-taining their full translational freedom, which is shown to agree semi-quantitatively with 68 most experimental results. Subsequently, we add to the picture formation of ion pairs, as a 69 proxy of ion-ion-CNTP interactions, which removes most remaining inconsistencies. The re-70 sulting physical picture rationalizes most results on ion permeation, selectivity, conductance, 71 and current rectification in CNTPs reported so far. 72 Internal arrangement of water and ion hydration: not necessarily 73 a single file 74 The narrowest experimentally studied CNTPs, showing the largest water-ion selectivity, 75 have been the (6,6) nanotubes. Classical MD simulations suggested that water in (6,6) 76 tubes forms a single file, similar to (5,5) nanotubes, believed to be the narrowest ones that 77 allow water and ion transport. 16,40,48,49 Ions in (5,5) tubes are then solvated by only two 78 adjacent water molecules, which is confirmed by computations. 47 The low density of water 79 in a single file and resulting high entropy were suggested 50 to be an important factor in 80 experimentally confirmed spontaneous filling of CNTPs with water. 51 However, ab initio 81 simulations recently indicated a possibility of a significantly distorted arrangement in (6,6) 82 CNTPs, both in presence and absence of ions. For instance, while larger K + ions were still 83 solvated by two water molecules, smaller Na + cations displayed a four-molecule solvation. 52 84 Here, we find that significant distortions in (6,6) tubes may be even more common, even 85 without ions.  Fig. 1a Fig. S3) . This contribution is then about rigidly shifts 143 ∆G ex and ∆H ex of all ions relative to ion-specific base values.

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Compared to its effect on water transfer, ∆S ex seems to plays a lesser role in ion transfer, 145 therefore ∆G ex is mainly controlled by enthalpy. Due to more favorable interaction with 146 Figure 1: Molecular arrangement and transfer quantities for water and ions in (6,6) CNTP. (a) Zigzag (top) and triple-bonded (bottom) arrangement of water in CNTP and (b) computed ∆G ex and −T ∆S ex for water transfer to CNTP for each arrangement at different ǫ. Water arrangement around chloride (c) and hydroxide (d) ions in water-filled CNTP. (e) Schematic illustration of single ion transfer process from bulk water to waterfilled CNTP and (f ) computed transfer quantities, ∆G ex and ∆H, for transfer of H + , K + , OH − , and Cl − as single ions plotted versus 1/ǫ. The sloped line and the value of the slope highlight the effect of dielectric energy. Oxygen, carbon, and hydrogen atoms and chloride ions are depicted in red, grey, white and green, respectively.
CNTP, cations has a significantly lower transfer energy than anions. For instance, for ǫ = 147 2, K + transfer into CNTP is nearly athermal and, for (6,6) tubes, it is even more favorable 148 than transfer of water, while transfer of anions is highly unfavorable. Enhanced interaction 149 of potassium was already noted by Aydin et al. for slightly wider tubes and is reminiscent of 150 the long-known complexation of cations with benzene, "cation-π interaction". 54 Partly but 151 less significantly, the differences between the ions are also related to different arrangement of 152 water molecules and water-water interaction around the ion, different for cations and anions. 153 We also note that proton transfers about as favorably as K + . Practically, that means that, 154 in experiments that involve KCl solutions, K + will outcompete the more dilute protons and 155 must be the dominant cation species within CNTPs. However, uptake of K + is subject to 156 limitations imposed by the requirements of overall electroneutrality. The latter will apply 157 when either CNTP is much longer that the screening length 55 or bonding of ions to CNTP 158 delocalizes their charge and sufficiently smears the potential variations. 47 Electroneutrality 159 dictates that the uptake of a K + cation needs to be counter-balanced by uptake of an anion, 160 either Cl − and OH − , both having a highly unfavorable ∆G ex . As the simplest mean-field 161 approximation, we may assume a uniform mean potential φ within the CNTP relative to 162 bulk thus ion uptake is given by diciously select the ions that need to be considered in eq. 1. Since the results of most 171 measurements typically represent ion permeabilities rather than partitioning, the differences 172 in ion mobilities need to be considered as well. We presume that in highly constrained ar-173 rangements within narrow CNTPs, the mobilities of water and ions may be fairly similar, 174 except for proton and hydroxide that may employ the much faster Grotthuss mechanism. We as follows The average ∆G ex s = 1 2 (∆G ex K + ∆G ex Cl ) essentially plays here the role of excess Gibbs energy 185 for pH-independent salt transfer. However, the non-linear scaling of conductivity observed 186 at pH 7.5 in Fig. 2a indicates that such a pH-independent scenario operates only at low pH.

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Apparently, pH comes into play in neutral conditions as preferential uptake of OH − ions, 188 as reported for wider tubes and observed in ab initio simulations of graphene surfaces in 189 water. 27,57 When OH − is strongly favored over Cl − , eq. 1 has to be replaced with where C OH = 10 pH−14 in M units. the other hand, Cl − will transfer as a trace species and its concentration in CNTP will be 195 given by Since, as the minority species, Cl − controls KCl permeability in this regime, the salt and The observed rates were interpreted as a fast proton transfer, presumably involving the 220 Grotthuss mechanism. We note, however, that proton flux J H is indistinguishable from i.e., OH − permeation, to much slower transfer of K + . In absence of electric current, its rate 229 is limited by K + diffusivity and may no more benefit from the fast Grotthuss mechanism.

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The reported blocking effect of Ca 2+ is also consistent with this picture, as it should bind 231 to CNTP more strongly and have a lower mobility than K + due to double charge and thus 232 further slow down hydroxide permeation. for Cl − permeation should be -using notation analogous to ∆G ex -either ∆H s or ∆H s , 240 respectively. Li et al.'s reported E a = 52 kJ/mol for chloride permeation in vesicles, which 241 they compared with computed chloride transfer energy 63 kJ/mol. It is unclear why the 242 latter value, computed for CNTP in vacuum (ǫ = 1), is so different from the present ∆H Cl ≈ 243 166 kJ/mol for ǫ = 1 and is much closer to the present result for ǫ = 100. We presume this 20 kJ/mol. Nevertheless, the above ∆H s is also fairly close the ∆G ex h = 50 kJ/mol that fits 258 the experimental chloride permeation rates in Fig. 2b   Indeed, we do not anticipate any ion-specific effect in solution outside CNTP therefore far 294 more dilute hydroxide would be unable to outcompete chloride and would have a negligible 295 effect on potassium transfer. Conversely, hydroxide transfer as a lone species within CNTP 296 must be strongly suppressed by its prohibitive transfer energies (Fig. 1f), which disagrees 297 with its high transfer number. It seems that experimental data and present ab initio results 298 rule out the electroneutrality breakdown mechanism.

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A more plausible alternative is that the system may substantially deviate from the simple in CNTP is given by where C w =55.6 M in the denominator comes from the fact that the ideal solution entropy 323 needs to be computed using concentrations expressed in molar fractions. We compare eqs. 324 2 and 5 and, specifically, consider the exponential factor that multiplies in eq. 2 the product 325 (C K C OH ) 1/2 that gauges the activity of the KOH "salt" in solution. This shows that the 326 free-ion transfer energy ∆G ex h ≈ 62.6 kJ/mol (for ǫ = 2) in eq. 2 is to be compared with where ∆ designates differences between the two solutions and t OH and t K are respective ion 409 transport numbers within CNTP. G is the effective CNTP conductivity, having the following 410 dependence on the solution concentrations where the omitted proportionality constant accounts for the partitioning (related to the 412 transfer energies), ion mobilities and CNTP geometry. Eq. 6 shows that G is proportional to Large red arrows indicates current direction. Thin arrows next to CNTP mouths indicate ion diffusion, resulting in concentration polarization and a change in local pH. The star indicates local pH controlling CNTP conductivity.
follows 14 where D OH is the OH − diffusivity in solution, r c is the channel radius, and the sign is positive 434 or negative when OH − ions move away from or towards the CNTP. The conductivity will be 435 controlled by the higher pH faced by CNTP, marked with the star Fig. 4. Thus the higher 436 pH will always rise and the current will flow unobstructed, when CNTP faces two identical 437 solutions (Fig. 4a). Similarly, no blockage will be observed when the current -by definition, 438 opposite to OH − flow -is towards the low-pH solution, since it increases C OH,mouth at high-pH 439 end and hence G, as shown in Fig. 4b. However, as depicted in Fig. 4C, when the current 440 reverses, C OH,mouth at high-pH end drops, sharply reducing G and blocking the current.
Since chloride does not allow as much conductivity as hydroxide (cf . Fig 2a), we ignore the 442 takeover by chloride at low pH and take the maximal (limiting) current I lim as approximately 443 corresponding to C OH,mouth = 0. Using D OH = 6.8 × 10 −9 m 2 /s, r c = 0.4 nm, t OH = 0.9, and 444 C OH,bulk = 10 −6.5 M (pH 7.5), we estimate I lim = (2πF D OH r c C OH,bulk )/t OH ∼ 1 fA, which is 445 is far smaller than pA currents measured in forward direction, thereby backward current will 446 be effectively blocked, i.e., rectification will be observed. More accurate relations, accounting is H + rather than OH − , as indicated in Fig. 4b  CNTP at neutral conditions, which change to K + and Cl − in acidic conditions. We conclude 467 that the ion transport is apparently controlled by free ions, coexisting with more abundant,  The analogous expression for transfer of a cation C + and an anion A − from bulk solution 517 and formation of an ion pair C + A − within CNTP was as follows We emphasize that, since thermodynamic parameters of water in CNTP depended on the The transfer quantities critically depend on the reference values for hydration in water. 527 We then first benchmarked computational procedures versus experimental bulk hydration