Development of Nanostructured Water Treatment Membranes Based on Thermotropic Liquid Crystals: Molecular Design of Sub‐Nanoporous Materials

Abstract Supply of safe fresh water is currently one of the most important global issues. Membranes technologies are essential to treat water efficiently with low costs and energy consumption. Here, the development of self‐organized nanostructured water treatment membranes based on ionic liquid crystals composed of ammonium, imidazolium, and pyridinium moieties is reported. Membranes with preserved 1D or 3D self‐organized sub‐nanopores are obtained by photopolymerization of ionic columnar or bicontinuous cubic liquid crystals. These membranes show salt rejection ability, ion selectivity, and excellent water permeability. The relationships between the structures and the transport properties of water molecules and ionic solutes in the sub‐nanopores in the membranes are examined by molecular dynamics simulations. The results suggest that the volume of vacant space in the nanochannel greatly affects the water and ion permeability.


1-[3,5-Bis(9,11-dodecadienyl)oxy)-4-dodecylbenzyl]pyridinium chloride (5(12)-Cl)
To a solution of 9(12) (0.95 g, 1.41 mmol) in dry toluene (5 mL), pyridine (4 mL) was added and heated at 70 °C for 40 h. The reaction mixture was poured into water and extracted three times with CHCl 3 . The organic phase was washed with 5% HCl aq. and sat. NaCl aq., dried over MgSO 4 , filtered, and the solvent was removed in vacuo. The crude product was purified by flash column chromatography (silica gel, eluent: CH 2 Cl 2 /methanol = 10/1) and recrystallization in acetone to give 5(12)-Cl (0.34 g, 45%) as white solids.  Compound 5(12) was synthesized analogously to 1(12) except for using 5(12)-Cl (0.25 g, 0.33 mmol) as the start compound and purified through flash column chromatography (silica gel, eluent: CH 2 Cl 2 /methanol = 15/1) to give the product as white solid (0.24 g, 0.29 mmol, 88%          The lattice size and number of the molecules and channels (= sub-nanopores) in a unit area and unit volume for the LC structures are summarized in Table S1. a calculation assuming number of the LC channels per the Cub bi lattice to be 4 b calculation assuming total length of the LC channels with in a Cub bi lattice to be 24×a/√8. [6] Numbers of molecules per 0.45 nm thick slice were calculated with following equeation: for the cubic structures
The interaction acting on the LC monomer was estimated as the sum of intermolecular potentials (the Coulomb and the Lennard-Jones (LJ) potentials) for all atoms of the monomer plus the sum of intramolecular potentials (bond, valent angle, and dihedral angle potentials).
Parameters for the LJ on each atom, bond, valent angle, and dihedral angle potentials were determined with a general Amber force field (GAFF). [8] The charge (q) on each atom was generated with ANTECHAMBER 1.4. [9] The potential parameters in the GAFF are optimized with the TIP3P model. [10] Therefore, the TIP3P model was used for estimation of the intermolecular interaction between a pair of water molecules. The interaction acting on each of Na + and Cl − ions was estimated with a model proposed by Joung et al. [11] The LJ parameters, ε and , for the interactions between the LC monomer and water molecule, and between the LC monomer and each of the ions (the LC-water and LC-ion interactions, respectively) were determined by the Lorentz-Berthelot rules. We checked that for more than 20 different LC-water distances, the potential energy, U, for the LC-water interaction in the present potential models was satisfactorily reproduced by the first-principles calculation using the MP2 method with the basis function of 6-311+G**. However, U for the LC-ion interaction in the present potential models was deviated from that calculated by the first-principles calculation. Thus, to reduce the deviation of U,  for the LC-ion interaction, which was determined by the Lorentz-Berthelot rules, was multiplied by 0.9. The reason for use of the MP2 method to evaluate U was that the method has the advantage of naturally and properly accounting for medium-and long-range correlation effects, compared to other methods, such as the density functional theory method. [12 ] This study focused on the structure of an ionic channel enclosed by the arrangement of ammonium moieties of the LC monomers, and the mobility of water molecules and ions in the channel. Thus, for simplicity, MD simulations in this study were performed with a model of the LC monomer from which alkyl chains were eliminated (hereafter, LC monomer model).
The potential parameters for each atomic site of the benzene ring and ammonium moiety of the LC monomer model are listed in Table S2, and the definitions of atomic sites are given in Figure S12. q, and  and  of the Lennard-Jones (LJ) potential, U LJ =4{(/r) 12 −(/r) 6 } (r is the distance between a pair of sites), on the atomic sites of each LC monomer model. q on each atomic site was determined from the first-principles calculation for all atoms of the LC monomer using ANTECHAMBER. The whole LC monomer has a charge of +1. However, the sum of q over the atoms of the LC monomer model was not equal to +1. In this study, the charge of the LC monomer model was adjusted to +1 with q on C3. Figure S12. The definitions of the atomic sites in the LC monomer models. We note that it would be better to use more accurate potential parameters, which were determined so that they matched solution free energies, especially for the LJ interactions, if the purpose of the simulation was quantitative reproduction of the structural and dynamic properties of ions in channels created by real self-assembled LC monomers. However, the purpose of the simulation was qualitative understanding of the difference in the mobility of water molecules and ions in the channel between 1(n), 2(n), and 3(n). Moreover, we checked that for both the LC-water and LC-ion interactions during the simulation, U for the Coulomb interaction was much greater than that for the LJ interaction, suggesting that the LJ interactions did not significantly influence the mobility of water molecules and the ions in the channel. Thus, we believe that the present potential parameters were sufficient for this study.

3-2. Simulation Systems
Two simulation systems were prepared: one was the system used for an MD simulation to investigate the stable structure of the channel created by the LC monomers (system A), and the other was the system used for an MD simulation to investigate the mobility of water molecules, Na + and Cl − ions in the channel (system B).
System A was created in the following way (see Figure S13). Firstly, an LC tetramer layer was formed with four LC monomer models ( Figure S13a). In the layer, the LC monomer models were arranged radially on positions rotated by 90° so that their ammonium moieties were oriented toward the center of the layer. Secondly, an assembly of four LC tetramer layers was constructed to create an ionic channel at the center of it by piling up the layers in the z-axis direction ( Figure S13b). In the assembly, the second and forth layers were rotated by 45° around the center to form a close-packed structure. The distance between the layers was fixed at 0.45 nm. To make the assembly electrically neutral, 16 Cl − ions were inserted into the channel of the assembly. Notably, the experimental study used BF 4 − ions but not Cl − ions. However, we checked that the minimum energy structure of the assembly with Cl − ions did not significantly change even if SO 4 2− ions were used instead. Therefore, we assumed that the minimum energy structure with Cl − ions was the same as that with BF 4 − ions. Again, the purpose of the simulation was qualitative understanding of the difference in the mobility of water molecules and ions in the channel between 1(n), 2(n), and 3(n), but not quantitative reproduction of the structural and dynamic properties of ions in real channels created by the self-organized LC monomers. Then, system A was constructed by putting the assembly into a rectangular-parallelepiped ( Figure S13c). The dimension of system A in the z direction, 1.8 nm, was set to be equal to the length of the assembly in the z direction. The dimensions in the x and y directions were 33 nm 2 . Periodic boundary conditions were imposed in all x, y and z directions.
In this study, several different sizes of the channel in the assembly were examined. The size of the channel was changed by changing the distance between the centers of the LC tetramer layer and benzene ring, R. Figure S13. Schematic of the procedure to create system A for 3(n).
System B was created in the following way. Firstly, a grand canonical MC (GCMC) simulation was performed for system A to fill the system with water molecules. In the GCMC simulation, temperature was set to 298 K and the chemical potential was set to the value for the bulk water at 298 K and 10 atm in the TIP3P model (−26 kJ/mol). The total number of trials (translation, rotation, insertion and removal of water molecules) in the GCMC simulation was 1.410 10 . Then, using the final configuration generated by the GCMC simulation, system B was created by putting a copy of the system filled with water molecules onto the original one in the z-axis direction.
Because we focused on the mobility of water molecules and ions in the channel, water molecules and ions located at positions far from the channel were eliminated from the system.
A water molecule was judged to be located at a position far from the channel if it was not included in a circular column with a radius of R nm from the center of the channel. Two water molecules existing in the channel were replaced with single Na + and Cl − ions. Thus, system B consisted of eight LC tetramer layers, a Na + ion, 17 Cl − ions (16 counter anions of the assembly and the anion dissolved in water), and water molecules. The dimensions of system B were 553.6 nm 3 . Periodic boundary conditions were imposed in all x, y and z directions.

3-3. MD Simulation and Analysis
Computation was carried out using a leap-frog algorithm with a time step of 0.5 fs.
Temperature was maintained at a constant value by means of a Nosé-Hoover thermostat with a coupling parameter of 0.1 ps. [13] The O-H distances in the LC monomer were kept constant at their equilibrium values by means of the SHAKE algorithm. The long-range Coulomb interactions were estimated using the Ewald summation method. The real space cut-off distance was set to 1 nm. The Ewald convergence parameter was set to 4.658 nm −1 for system A and 3.208 nm −1 for system B. The maximum indices of the reciprocal lattice vector in the x, y, and z-directions were 14, 14, and 8 for system A and 16, 16, and 12 for system B. The LJ interactions were cut-off at an interatomic distance of 1 nm. The simulations were performed with DL_POLY 2.20. [14] The distance restraints and angler restraints implemented in DL_POLY 2.20 were used to maintain the bond lengths and angles of the LC monomer the around their equilibrium values.
For system A, MD simulations were performed to obtain the energetically stable structure of the channel in the following way. First, an MD simulation was performed at 298 K for 1.5 ns. Then, using the final configuration of the MD simulation at 298 K, an MD simulation was performed at 0 K for 0.5 ns. These heating and subsequent quenching simulations were repeated three times, starting with different initial configurations. The energetically stable structure of the channel was determined by comparing the potential energy of the final configuration for three quenching simulations.
The stable structure of the channel for each R was analyzed with the final configuration of the simulation. For system B, three MD simulations starting with various initial positions of Na + and Cl − ions were performed at 298 K. The run for each simulation was 15 ns. The mobility of water molecules and ions in the channel was investigated by analyzing the root mean square displacement, dr 2 . For each of the three simulations, the analysis was done using the time-sequence of the coordinates of water molecules and ions at eight different periods. Thus, for each of water molecules, Na + and Cl − ions, the dr 2  as a function of t was created by averaging 24 independent dr 2  functions. Notably, during the simulation, 16 Cl − ions as the counter anions of the assembly and a Cl − ion as the anion dissolved in water were indistinguishable. Thus, the dr 2  function for Cl − ion was created with the dr 2  data for all the 17 Cl − ions.
During the simulation for both systems, the benzene rings of the LC monomer models were fixed at their initial positions. During the simulation for system B, an external potential, U ext = A(R−r xy ) −n (r xy is the distance between the center of the system and species in the x-y plane) was applied to the system in order to prevent water molecules and ions from moving out of the channel. A and n used in this study were 0.001 kJ/mol and 4, respectively.

3-4. MD Simulations for Different Pore Sizes
To check the effect of the space volume size on the mobility of water molecules and the ions in the nanopore, an MD simulation to investigate the mobility of them was also performed for the nanopore of 3(n) with two different sizes, R = 0.8 and 0.875 nm. The simulations results of dr 2  as a function of t for all R are shown in Figure S14. The selfdiffusion coefficient of water molecules increased with increasing R, whereas the coefficients of the ions did not significantly change with R. This result suggests that the lower mobility of water molecules in the nanopore for 3(n) than for 1(n) and 2(n) originated from the smaller size of the space volume.