Computer simulations of water flux and salt permeability of the reverse osmosis FT-30 aromatic polyamide membrane

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Journal of Membrane Science 384 (2011) 1– 9

Contents lists available at SciVerse ScienceDirect

Journal of Membrane Science

jo u rn al hom epa ge: www.elsev ier .com/ locate /memsci

omputer simulations of water flux and salt permeability of the reverse osmosisT-30 aromatic polyamide membrane

un Luoa, Edward Hardera,1, Ron S. Faibishb, Benoît Rouxa,b,∗

Department of Biochemistry and Molecular Biology, The University of Chicago, Gordon Center for Integrative Science, 929 East 57th Street, Chicago, IL 60637, USAArgonne National Laboratory, Argonne, IL 60439, USA

r t i c l e i n f o

rticle history:eceived 2 March 2011eceived in revised form 25 August 2011ccepted 27 August 2011vailable online 10 September 2011

eywords:

a b s t r a c t

The relative permeability of salt to water across an atomistic model of the FT-30 reverse osmosis (RO)membrane is studied using molecular dynamics (MD) simulations. The membrane model is built using aheuristic approach and gives a membrane density, water solubility and flux that are in good accord withthe experimental values. The salt permeability is calculated from inhomogeneous solubility–diffusiontheory using ion pathways from non-equilibrium targeted MD simulations, yielding an estimated saltrejection of 99.9% that is similar to the experimental value. The encouraging agreement with experimentaldata of FT-30 membrane suggests that MD simulations based on atomic models offer a useful way to

iffusionick’s lawolvationater

onsodium-chlorideolecular dynamics

support the experimental exploration of RO membrane development.© 2011 Elsevier B.V. All rights reserved.

. Introduction

Reverse osmosis (RO) membranes, based on aromatic polyamidehin-film composites (TFC), are extensively used for desalinationnd purification of sea water and brackish water [1]. One of theost common RO membrane, FT-30, is composed of m-phenylene

iamine (MPD) and trimesoyl chloride (TMC) that react to form cross-linked polymeric material via interfacial polymerizationIP) [2]. The performance of RO membranes is judged both by theolume of the water flux under applied pressure, Jw, and the per-entage of salt rejection R. The latter is defined as the percentage ofalt removed from the feedwater stream, R = (1 − Cp/Cf), where Cp

nd Cf are the salt concentrations on the permeate and the feed side.easurements indicate that a 0.125 �m thick FT-30 membrane can

upport a water volume flux of about 7.7 × 10−6 m/s under a pres-

ure difference of 3 MPa [3], with a salt rejection estimated to beoughly between 99.6% and 99.8% [4,5].

∗ Corresponding author at: Department of Biochemistry and Molecular Biology,he University of Chicago, Gordon Center for Integrative Science, 929 East 57thtreet, Chicago, IL 60637, USA. Tel.: +1 773 834 3557.

E-mail address: [email protected] (B. Roux).1 Present address: Schrödinger Inc., 120 West 45th S., New York, NY 10036-4041,SA.

376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.memsci.2011.08.057

Ideally, one would like to engineer the molecular properties ofa membrane in order to maximize the water flux while retaininga high salt rejection rate. However, progress in the developmentof new polymeric material for improved RO membranes is slow,in large part due to the need to carry laborious experimental testsfor each new design. Furthermore, the limited information aboutthe three-dimensional molecular structure of the polymeric mate-rial [6], and the underlying mechanism of water and ion transportthrough the membrane increases the difficulties in formulatingclear guiding principles for the design of new and improved ROmembranes.

In principle, a combination of theory, modeling and simula-tions (TMS) offer a powerful virtual route to better understandthese complex materials. Molecular dynamics (MD) simulations,in particular, make it possible to simulate the diffusion processof ions and water at the atomic level to quantitatively charac-terize the transport properties of membranes. As examples, MDsimulations have been used to understand the transport mech-anism of gas phase separation [7–9], the mobility of water andions in amorphous polyamide [10,11], and more recently, to studywater transport through carbon nanotube membranes [12,13]. Onecomputational strategy is to directly simulate the non-equilibrium

conditions giving rise to the macroscopic flux. Alternatively, it ispossible to calculate the transport properties of a complex sys-tem from equilibrium simulations via statistical mechanical linearresponse theory [14–16]. With this approach, the water flux and

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Y. Luo et al. / Journal of Me

he salt flux can be calculated from the partition coefficient andiffusion constant determined from equilibrium MD simulations.

similar approach has been used to estimate the water permeabil-ty across lipid bilayers [17–19] and the conductance of biologicalon channels [20].

A prerequisite for MD simulation studies is to have a completehree-dimensional atomic model of the complex polymeric mate-ial forming the RO membrane. This is challenging because theross-linking of the MPD and TMC monomers is random and theesulting polymerized material is highly disordered. To resolve thisroblem, a heuristic MD-based approach was previously developednd used to build complete atomic models of the FT-30 aromaticolyamide membrane [6]. The goal of the present study is to uti-

ize this heuristic approach to construct and simulate a realistictomic model of the FT-30 membrane in order to better understandhe microscopic processes underlying water permeation and saltejection. Comparison of the computational results with availablexperimental data (e.g., material density, water flux and salt rejec-ion) demonstrates the reliability of the current approach and theromising future application of these methods in the developmentf new RO membranes.

In the following sections, the methods, system preparation andhe details about simulations are briefly described. Then, the waterux and salt permeability are calculated and compared with exper-

mental data. Finally, the main results are summarized.

. Methods

.1. Atomic model of FT-30 membrane and simulation details

All the atomic systems were constructed and equilibrated usinghe program CHARMM [21]. The parameters of the Lennard–JonesLJ) 6-12 potential for Na+ and Cl− were taken from Ref. [22,23].he TIP3P water potential [24] as modified for the CHARMM forceeld [25] was used. Models for benzene-1,3,5-tricarboxylic acidhloride (TMC), m-phenylenediamine (MPD) and the amide bondedligomer, 1,3,5-tricarboxylic phenylamide (see Fig. S1 in Support-ng Information) were built using the generalized AMBER forceeld (GAFF) [26]. The topology and force field parameters for theompounds (Tables S1 and S2 in Supporting Information) were gen-rated with ANTECHAMBER 1.27 and with AM1-BCC partial charges27,28]. Atomic charges were averaged for symmetrically relatedtoms in each monomer.

Previously, an atomic model of the reverse osmosis (RO) mem-rane was built using a heuristic simulation approach [6]. Thectual time-course of the polymerization process is artificial, butesults in a fairly realistic configuration for the FT-30 polymer-zed membrane. Briefly, an assembly of 2000 TMC and 2000 MPD

onomers, with random positions and orientations, is arrangedithin a cubic volume according to the experimental hydrated FT-

0 membrane layer density of 1.38 g/cm3 and water content level of3 wt% [29]. Membrane “polymerization” is conducted in vacuumnder constant volume (cubic unit cell length = 96 A), and at an ele-ated temperature of 340 K maintained via a Langevin thermostatNVT) [30]. The criterion used to form a new amide bond duringhe MD simulation is based on an inter-molecule distance determi-ation that the nitrogen of a free amine group approaches within.5 A of the carbonyl carbon of a free acyl chloride group. Sim-lations are conducted with periodic boundary conditions (PBC),hough bond formation restricted to molecule pairs that lie withinhe primary unit cell of the system. After 2 ns of simulation time,

he rate of establishing new random monomer-monomer links fallso nearly zero because the formation of the polymer cluster lim-ts the diffusion of the TMC and MPD monomers. To encourageurther association of unreacted functional groups, an additional

e Science 384 (2011) 1– 9

inter-molecule potential is introduced between the nearest amineand acyl chloride groups. After 800 ps of simulation with the addi-tional potential energy, the rate to establish new links falls to zeroagain. The whole polymerization process used up 3761 monomers,including 1987 MPD and 1774 TMC monomeric units. The longestpolymer chain contains 3013 monomeric units. The remaining freemonomers were deleted from the system, resulting in a final mem-brane structure with a cubic volume of 96 A×96 A×96 A.

To generate the hydrated membrane system, a pre-equilibratedorthogonal box of TIP3P water molecules was used. The water boxwas constructed with dimensions matching the xy Cartesian coor-dinates of the membrane and an extended dimension of 226 Aalong the z coordinate. The equilibrated box was overlaid and watermolecules overlapping with the membrane were removed fromthe system [31]. Equilibrium MD simulations of the solvated mem-brane systems were carried out in the constant temperature andpressure ensemble (NPT) using a temperature of 300 K and a pres-sure of 1 atm in NAMD [32]. The Langevin piston Nose–Hoovermethod [33,34] was used to control pressure and temperature. TheSHAKE algorithm was used to constrain covalent bonds involv-ing hydrogen atoms to their equilibrium value [30]. PBC wereapplied in all directions, and the long-range electrostatic forceswere calculated with the particle-mesh Ewald (PME) method [35].The final configuration of the membrane model was used to buildmembrane–water interfaces. To gain better spatial averaging ofthe polymerized material, three independent systems were built,each using one of the three Cartesian axes (xyz) of the final cubeof polymerized material as the membrane normal. An equilibra-tion period of 1 ns was used before carrying out the production MDsimulations of 5 ns for the three independent hydrated membranesystems. The properties of solvated membrane, such as water den-sity and water diffusion constant, were calculated from averagingthe three systems. A model with 1 M NaCl solution in bulk was builtby randomly replacing the appropriate number of water moleculesby ions according to the molar scale (i.e., salt concentration is setfrom the number of ions per unit of volume of the bulk region). Anequilibrium MD simulation of 25 ns was then carried out using thesame protocol as for the hydrated membrane systems.

2.2. Targeted MD simulations

The targeted MD (TMD) approach is used to explore the path-ways of ions moving inside the membrane [36]. The initial positionsof ions are chosen from the final configuration of 25 ns brute-forceMD simulation Fig. 2(a). Among those ions penetrated into themembrane through the porous surface, 5 Na+ and 5 Cl− were pickedfrom each side of membrane surface, and 20 separate TMD trajecto-ries were carried out to pull each ion toward the opposite side. Thisexplores ten pathways inside the membrane for each ion. The forceis only added on z direction. At each step, the magnitude of force isgiven by Fz

i= −k × (Z ion

i− Zm

i− Ztarget

i), where the force constant k

is set to 0.5 kcal/mol A2, Z ioni

is the current ion position on z-axis,

Zmi

is the current center of mass of membrane on z-axis and Ztargeti

is the target z position for current step, which depends on the totalnumber of steps (1,600,000 steps) and the length of the pulling(∼90 A). The magnitude of the TMD force constant and the rate ofpulling were chosen to insure that the ion makes steady progressalong the z-axis while having sufficient time to explore different xypositions in the complex topology of the polymerized membraneregion without distorting its structure.

2.3. Poisson–Boltzmann calculations

Macroscopic continuum dielectric models offer useful approx-imations to treat the free energy of ions in molecular systems

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37]. The electrostatic free energy to transfer an ion from theulk solution to a point r inside the membrane can be calcu-

ated as the difference between the electrostatic free energy of theon-membrane complex (im) and the free energy of the isolated

embrane (m) and ion (i) in bulk: �Gi(r) = �Gim(r) − �Gm − �Gi.he permeating ion and the polymerized membrane are treatedxplicitly with all atomic details, while the water molecules arereated implicitly as a featureless dielectric continuum. Each elec-rostatic free energy �G is calculated as a sum over all the atomicharges qi in the system:

G = 12

∑i

qi�(ri) (1)

here �(ri) is the electrostatic potential at the position of the ithtomic charge qi. The electrostatic potential �(ri) is calculated byumerically solving the Poisson–Boltzmann (PB) equation,

· [�(r) ∇�(r)] − �2(r) �(r) = −4��(r) (2)

here �(r) =∑

iqiı(r − ri), is the charge distribution from theermeating ion and the membrane atoms, and �(r) is the posi-ion dependent dielectric constant. The position-dependent ioniccreening factor, �(r), was set to zero in the present calcula-ions. The coordinates of the membrane with a Na+ or Cl− atosition r are extracted from four TMD trajectories by 1.0 A incre-ents in z direction through the whole traces. Each ion pathway

xtends from roughly −40 A to 40 A along z-axis, for a total of 80 zositions. Four Na+ and Cl− pathways were analyzed with PB cal-ulations, each covering four different positions in the xy plane ofhe membrane. This yields a total of 80 × 4 = 320 different posi-ions sampled for each ion. The aqueous solution was representeds a uniform continuum media with a dielectric constant of 80.he dielectric constant of the membrane was assumed to be 2.ll calculations were performed according to the finite-differenceB solver included in the PBEQ module [38–40] implemented inhe program CHARMM [21]. The atomic charges are taken fromhe force field (see Supporting Information). The atomic radiised to define the protein-solvent dielectric boundary were takenrom previous free energy simulation studies with explicit water

olecules [38]. A water probe of 1.4 A radius was used to definehe molecular surface corresponding to the dielectric boundary39]. Periodic boundary conditions applied in the membrane xylane. Each PB calculation was performed in two steps with aubic grid of 2403 points, starting with a grid spacing of 1.0 A, fol-owed by a focusing around the main region with a grid spacingf 0.5 A.

.4. Free energy perturbation

The relative free energy for transferring an ion from the bulkhase to some specific position z inside the membrane was cal-ulated by computing the reversible work to substitute a waterolecule by an ion in the bulk and at point r in the membrane using

lchemical free energy perturbation molecular dynamics (FEP/MD)imulations. The relative free energy is

Gi(r) = �Gmw→i − �Gbulk

w→i + �Gbulk→mw , (3)

here �Gbulkw→i

and �Gmw→i

are the free energy of transforming aater molecule into an ion in the bulk phase and in the mem-

rane. The free energy for transferring a water molecule from theulk to the membrane, �Gbulk→m

w , was extracted directly from theater density profile, −kBT ln(�w(z)/�w); see Fig. 1. Because of the

nhomogeneity of the membrane, �Gmw→i

is highly position depen-ent. Therefore different snapshots from TMD trajectories are usedo calculate �Gi(r). The alchemical FEP/MD simulations were car-ied out using the dual-topology method implemented in NAMD

e Science 384 (2011) 1– 9 3

[32]. The dual-topology method has been discussed in detail else-where [41]. Briefly, the water molecule and Na+ ion tagged forFEP/MD simulation were inserted at position r without interac-tions between them. The FEP/MD simulations are then carried outwith a hybrid Hamiltonian function of the thermodynamic cou-pling parameter � introduced to control the interactions of thetagged moieties with their surrounding environment (comprisingall the atoms in the rest of the system). When � = 0, the interactionsinvolving the water molecule are turned off and those involvingthe Na+ ion are active; when � = 1, the interactions involving theNa+ are turned off and those involving the water molecule areactive. Each FEP perturbation was divided into 10 windows byevenly increasing the value of �(0, 0.1, 0.2, . . ., 1). Each FEP/MDsimulation was carried out forward and backward with 50 ps ofequilibration and 200 ps of data collection, producing a total of5 ns of simulation time for each FEP/MD transformation. Harmonicpositional restraints were imposed on the tagged ion and watermolecule to keep them at a stable position during the FEP/MDsimulation.

3. Results and discussion

3.1. Water flux

Experimentally, water permeability is determined from a mea-surement of the total flux Jw through the membrane under apressure gradient �P (Jw is the volume of water passing througha plane per unit of time). In the present treatment, it is assumedthat there is no convection and that the purely diffusive waterflux is within the linear response regime of the system. A com-putational strategy based on non-equilibrium simulations couldbe used by monitoring the water flux under a hydrostatic pres-sure difference across the membrane [42,43]. However, such anapproach is less advantageous for the present system. Under a typ-ical experimental pressure gradient 3 MPa, the anticipated waterflux is roughly on the order of 7 × 10−6 m/s [3]. The actual numberof water molecules expected to cross the model membrane dur-ing MD simulations corresponding to such a flux is prohibitivelysmall, less than one water molecule crossing the membrane every10 ns. Even by scaling up the pressure gradient by a factor of100, obtaining reliable statistics would be challenging with sucha small rate of permeation. Scaling further to unphysical highpressure would increase the flux, but the mechanism underlyingthe water flux may then not be representative of experimentalconditions.

According to linear-response theory, the water volume fluxacross a membrane of width L can be estimated from Fick’s law[15,16], as Jw = −vwDmKm �C/L, where vw = 1/�w is the molar vol-ume of bulk water (�w is the density of bulk water), �C is theconcentration difference across the membrane, Dm is the waterdiffusion coefficient of water inside the membrane, and Km is thepartition coefficient of water in the membrane. The concentrationdifference can be expressed in terms of the pressure differenceas, �C = �wˇT �P, where ˇT is the isothermal compressibility ofbulk water. (It may be noted that this treatment is equivalent withDarcy’s law.) The membrane diffusion constant and water parti-tion coefficient are determined from equilibrium MD simulationsof the hydrated membrane. An illustration of a membrane–waterconfiguration with the corresponding density profile of wateralong the membrane normal is shown in Fig. 1. The water densitydrops smoothly at the membrane–bulk interface, and is distributed

evenly in the porous membrane interior (−38 A ≤z ≤ 38 A). FromFig. 1, the partition coefficient K is determined to be 0.30, in goodagreement with the experimental value of 0.29 [29]. It is worthnoting that the atomic density in the model membrane interior

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4 Y. Luo et al. / Journal of Membrane Science 384 (2011) 1– 9

Fig. 1. (a) Equilibrated configuration of water solvated polymer membrane model. The water molecules are shown in VDW model and the polymer membrane is shown ing embri to the

(simtpdDfittiittmiwtomtoceui

reen licorice model. (b) A close up view of the interior of the hydrated polymeric mnterpretation of the references to color in this figure legend, the reader is referred

1.34 g/cm3) remains near the experimental water hydrated den-ity of 1.38 g/cm3 [29]. The diffusive properties of water moleculesn the interior of the membrane is calculated by employing a

ethod that is based on solving the diffusion equation subjecto absorbing boundary conditions [44]. In the current application,lanar absorbing boundaries are set at z = − 20 A and z = 20 A toefine the diffusion coefficient inside the membrane. This gives am as 0.5 × 10−5 cm2/s, which is in good accord with the results

rom a previous study [6]. The rate of diffusion in the membranes ten times smaller than the estimated diffusion coefficient ofhe TIP3 water model in the bulk phase [45]. Recent experimen-al estimates of the water flux across a 0.125 �m thick membranes 7.7 × 10−6 m/s for a pressure difference of 3 MPa [3]. Using thesothermal compressibility of water (4.5 × 10−4 MPa−1 at T = 300 K)o obtain �C for this pressure difference, along with a membranehickness of 0.125 �m, and the estimated values for K and Dm deter-

ined from the MD simulations, a water flux of 2.1 × 10−6 m/ss obtained. The calculated flux result is in excellent agreement

ith experimental measurements. The agreement is better thanhe value of 1.4 × 10−6 m/s reported in a previous MD study basedn a smaller atomic model of the FT-30 membrane [6]. The atomicodel of the polymeric membrane simulated here is 8 times larger

han in the previous study, which is likely to diminish the influencef undesired finite-size effects. The current model yields a partition

oefficient Km of 0.30, which is in excellent agreement with thexperimental measure of 0.29 [29]. In contrast, the previous studynderestimated Km as 0.21, most likely due to finite-size artifacts

n the smaller system.

ane. (c) Density of the water molecules along the membrane interface normal. (For web version of the article.)

3.2. Salt permeability

According to inhomogeneous solubility–diffusion theory, theion permeability coefficient Pi can be expressed as [46],

Pi =[∫ L

0

dze�Gi(z)/kBT

Di(z)

]−1

(4)

where �Gi(z) is the relative free energy of an ion along z relative tothe bulk phase (set at z equal to 0 and L), and Di(z) is the position-dependent diffusion constant (variations in x and y are averagedout).

The density profile of ions along the membrane normal cal-culated from the last 10 ns of equilibrium trajectory is shown inFig. 2(b). The relative free energy of an ion i, obtained from thelocal density, �Gi(z) = −kBT ln(�i(z)/�i), is shown in Fig. 2(c). Itis observed that the free energy �Gi(z) rises gently as the ionsapproach the membrane surface and no barrier opposing ion entryis present around ±48 A where the membrane begins. However,because the permeation of ions through the membrane region is arare process on the MD time scale, no information is available aboutthe local equilibrium density of the ions deep in the membrane inte-rior. To supplement the missing information, TMD trajectories wereused to generate ion permeation events and discover possible ion

pathways across the membrane within a tractable computer time.The TMD technique is described in Section 2.

Fig. 3 shows the traces of trajectories of 10 Na+ and 10 Cl− inzy plan and xy plan. The twenty pathways from TMD represent a

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Y. Luo et al. / Journal of Membrane Science 384 (2011) 1– 9 5

Fig. 2. (a) The final configuration of 25 ns brute-force MD simulation of membrane in 1 M NaCl aqueous solution. The membrane is shown in grey line; sodium ions arei es arec free e( rred to

saiwsostm

�cewspttattapc

rof(nsmh2tl

n yellow VDW model and chloride ions are in green VDW model. Water moleculalculated from the last 10 ns brute-force MD simulation trajectory. (c) The relativeFor interpretation of the references to color in this figure legend, the reader is refe

mall sample of the possible energetically favorable configurationsble to support the ion permeation process. During the TMD, theons are free to move in the xy plane and are able to find their

ay to cross the membrane. Interestingly, a few of the TMD pathstarted at different positions in the xy plane appear to merge intone another inside the membrane, suggesting the existence of a fewpecific energetically favorable permeation pathways. This meanshat the membrane is not a uniformly porous and homogeneous

aterial allowing the passage of ions at arbitrary xy position.Two methods were used to estimate the relative free energy

Gi(z) along the TMD pathways. First, continuum dielectric cal-ulations based on the PB equation were used to determine thelectrostatic free energy for moving an ion along the TMD path-ays. Such calculations are fairly inexpensive and provide a good

emi-quantitative estimate of the relative free energy along theathways (see Section 2.3). In addition, alchemical free energy per-urbation MD (FEP/MD) simulations with explicit solvent were usedo calculate the free energy for transferring an ion from the bulk to

few specific locations inside the membrane (see Section 2.4). Forhe FEP/MD calculations, it is assumed that the configuration fromhose pathways generated by non-equilibrium TMD can be relaxednd equilibrated appropriately. Because this method is more com-utationally expensive, the results from FEP/MD mainly serve toomplement and validate the information obtained from PB.

Fig. 4 shows the combined time series for the TMD trajecto-ies (top: Na+, bottom: Cl−). From top to bottom are the z positionf the ion (a), the instantaneous pulling force on the ion (b), theree energy (c) and the number of water in the first solvation shelld). In the case of Na+, some abrupt movement by the ion can beoted (a), corresponding to sudden rise in the TMD force (b). Thisuggests that the diffusion process is not smooth and that the ionust “hop” from one pocket to another during permeation. The

ydration shell of Na+ (d) undergoes some rapid changes between and 6 water molecules. A few of the hopping transitions appearo be correlated with changes in the hydration number, and it isikely that the polymeric matrix is at the origin of the obstacles

not shown. (b) The density profile of ions along the membrane interface normal,nergy of the ions along the z-axis, �Gi(z), estimated from the local density profile.

the web version of the article.)

opposing diffusion. The FEP/MD results (stars in c) are generally ingood agreement with the PB values (line in c), though the FEP/MDresults appear to be slightly below the PB curves. This suggest thatthe continuum model offers a reasonable and efficient approxima-tion to explore the properties of the system. Those calculations arebase on the same configurations, which were extracted from TMDtrajectories. The overall picture that emerges is that the free energyof Na+ in the membrane can vary up to ∼20 kcal/mol relative to thebulk phase.

In the case of Cl−, the ion movement is very progressive (a) andthe applied TMD force is relatively small (b), indicating that the per-meation process is smoother than for Na+. The hydration shell of Cl−

(d) also appears to be maintain around 7–8 water molecules most ofthe time. This observation is in accord with indications from exper-iments that permeating ions retain partially their hydration shellwhile they cross the RO membrane [47]. In the free energy profileof Cl−, seven configurations were chosen for alchemical FEP/MDcalculations. Two of the relative free energy alchemical FEP/MDare considerably lower than the PB values at the correspondingpositions.

Analysis shows that PB slightly overestimates the cost for trans-ferring an ion from the bulk to the membrane region by about+3 kcal/mol compared with FEP/MD. The discrepancies might bedue to some slight structural re-arrangements within the polymermatrix during the FEP/MD calculations. In the continuum PB model,the membrane structure is fixed. If there are local conformationalchanges, the continuum model is expected to give a higher freeenergy than FEP/MD. Therefore, the results from PB may be con-sidered as an upper bound of the real transfer free energy. Theoverall picture that emerges is that the free energy of Cl− in themembrane is on the order of 10 to 15 kcal/mol relative to the bulkphase.

To produce a more reliable estimates, the PB free energieswere calculated from 4 Na+ and 4 Cl− pathways. Those pathwayswere chosen because represent typical movements of the ions inthe membrane and they do not overlap. All the free energy

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6 Y. Luo et al. / Journal of Membrane Science 384 (2011) 1– 9

F xy plab color i

pTaewpcpbibioatmwsttEt

ig. 3. The traces of TMD trajectories of 10 Na+ and 10 Cl− in zy plan (a and c) and

lue lines. Water molecules are not shown. (For interpretation of the references to

rofiles are summarized in Fig. 5(a) for Na+ and Fig. 5(b) for Cl−.he solid lines show the electrostatic free energy of different ionslong membrane normal calculated from PB equation. The freenergy profiles from FEP/MD, shown in red points, are comparedith the results from PB for the same ion, shown in red line. Theolymer membrane extends from z = − 48 to +48 A. The dashed lineorresponds to the free energy �Gi(z) extracted from the densityrofile of the ions �i(z) and is directly taken from Fig. 2(c). Thelack squares near the dashed line are calculated from PB for one

on moving from bulk towards the membrane surface. In the mem-rane region, the free energy shift largely, corresponding to the

nhomogeneity of the membrane interior. The maximum heightf the free energy barrier is about 30 kcal/mol in the case of Na+,nd about 25 kcal/mol in the case of Cl−. Again, it is observed thathere is no barrier opposing ion entry around ±48 A, where the

embrane begins. There are a few points inside the membranehere the free energy is close or even lower than the bulk value,

uggesting the possibility that the ion can make favorable interac-

ions with water or polyamide groups in specific location. However,he permeation rate is dominated by the unfavorable energies.ven though the surface of the polymerized material is very rough,he PB free energy of ions near the border of the membrane are

n (b and d). Each pathway is shown in different colors. The membrane is shown inn this figure legend, the reader is referred to the web version of the article.)

comparable with the free energy extracted from the density pro-file. The agreement between the two methods is a good evidencefor the reliability of the computed free energy.

The Di(z) in Eq. (4) is the position-dependent diffusion con-stant of ion i inside the membrane. Because of the low densityof ions inside the membrane and the high inhomogeneity of themembrane, it is difficult to calculate D(z) from the mean-squaredisplacement or the force correlation method. From Eq. (4), it isclear that the free energy �Gi(z) is the dominant factor to deter-mine the permeability coefficient. Quantitatively, the effect of thediffusion coefficient on the ion permeability in Eq. (4) is linear,whereas the effect from the free energy is exponential. Thus, theimpact of D(z) on the estimated permeation coefficient is expectedto be much smaller than any of the uncertainties in the calculated�Gi. For the sake of simplicity, we approximate Di(z) by the dif-fusion coefficient of the ions in the bulk phase, and use 0.24 A2/psfor Na+, and 0.33 A2/ps for Cl− [48–50]. It follows that the constantdiffusion coefficient can be taken out of the integral in Eq. (4) to

approximate Pi, which can then be calculated for each ion usingthe PB profiles (see Table 1). From the table, it is noted that theaverage value of ion permeability coefficient is dominated by thepathway with the lowest free energy barrier. The situation is akin

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Y. Luo et al. / Journal of Membrane Science 384 (2011) 1– 9 7

-40

0

40

Z(t

) (Å

)

0

10

F(kc

al/m

ol)

0

30

ΔG

(kc

al/m

ol)

0 400 800 1200 1600

Time (ps)

4

8

Nw

(t)

aa

b

b

c

d

-40

0

40

Z(t

) (Å

)

0

5

10

F(kc

al/m

ol)

0

15

ΔG

(kc

al/m

ol)

0 400 800 1200 1600

Time (ps)

4

8

Nw

(t)

a

b

c

d

Fig. 4. Results from one TMD trajectory for Na+ (top) and one Cl− (bottom). (a) Theposition of the ion along the z-axis as a function of time. (b) The instantaneous pullingforce on the ion as a function of time. (c) The free energy of transferring the ion frombulk to a position in the membrane as a function of time. The blue line presents theelectrostatic free energy calculated from the PB equation. The red stars correspondto the free energy calculated from the FEP/MD simulations. (d) The number of watermCr

tb

biane

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Fig. 5. The free energy of (a) four Na+ and (b) four Cl− . The solid lines shows theelectrostatic free energy of the ions along membrane normal calculated from PB.Different colors represent different ions. The FEP energy profiles are shown in redpoints, comparing with the red line of the same ion. The polymer membrane extendsfrom z = − 48 to 48 A. The dashed line outside membrane is calculated from the localdensity profile of the ions taken from Fig. 2(c). The black squares near the dashed

olecules in the first solvation shell. Radius cutoffs of 3.5 A for Na+ and 3.7 A forl− are used. (For interpretation of the references to color in this figure legend, theeader is referred to the web version of the article.)

o a system of resistors in parallel, in which transport is dominatedy the lowest resistance.

An important criterion for the performance of the RO mem-rane is the salt rejection, R = (1 − Cp/Cf). To test if the calculated

on permeability coefficients are reasonable, one can compare withvailable experimental data under realistic conditions. Due to theeed to maintain electro-neutrality, the total salt flux is roughlyqual to the average ion flux to a good approximation. The salt

able 1on permeability coefficients.

Pathway Permeability coefficient, P (m/s)

Na+ Cl−

1 5.0 × 10−14 9.0 × 10−17

2 1.3 × 10−17 1.4 × 10−17

3 4.4 × 10−11 2.6 × 10−10

4 1.4 × 10−23 5.0 × 10−13

Avg 1.1 × 10−11 6.5 × 10−11

line are calculated from PB for one ion in bulk or near membrane surface. (For inter-pretation of the references to color in this figure legend, the reader is referred to theweb version of the article.)

flux, Js = Ps(Cf − Cp), depends on the salt concentration across theRO membrane. For the sake of simplicity, let us assume that thesalt concentration on the feed side (Cf) is 1 M, and that the con-centration on the permeate side (Cp) is very small and negligible.Thus, the ion flux Js is approximately equal to PsCf (this is actuallyan upper bound to the ion flux, as Cp is not strictly zero). Underconditions supporting a water volume flux Jw and an ion flux Jsacross the membrane, an aqueous salt solution of concentrationCp will accumulate on the permeate side, with Cp = Js/Jw = PsCf/Jw[51]. This upper bound estimate for Cp lead to a simple lowerbound estimate of the salt rejection as, R ≈ (1 − Ps/Jw). By usingthe permeability coefficients estimated from the computations, anestimated salt rejection of 99.998% is obtained. The calculated saltrejection is overestimated compared with the experimental valueof 99.6–99.8% [4,5]. The inaccuracy follows directly from the saltpermeability of 3.8 × 10−11 m/s, which was estimated on the basisof transfer free energies calculated with the continuum electro-static PB approximation. The PB approximation was used here in anattempt to avoid carrying out computationally expensive FEP/MDcalculations at multiple ion positions. However, while the free ener-gies obtained from the PB calculations are useful, it is clear that they

are somewhat inaccurate. Upon further examination, PB systemat-ically overestimates the cost for transferring an ion from the bulkto the membrane region by roughly +3 kcal/mol when comparedwith the more rigorous FEP/MD results. Such systematic inaccuracy

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rom PB in the complex porous environment of the polymerizedT-30 membrane is not surprising and is actually consistent withrevious studies indicating that a continuum electrostatic approxi-ation tends to overestimate the cost of partial dehydration when

ons are located in narrow pore regions [52]. In principle, oneight re-calculate the transfer free energies at all ion positions

sing FEP/MD simulations, although such an undertaking becomesomputationally prohibitive given the large number of positionshat are required to effectively sample the heterogeneous structuref the FT-30 membrane. For the sake of saving time and efforts,he transfer free energies estimated from PB were empiricallyorrected to account for the systematic shift of ∼3 kcal/mol rela-ive to FEP/MD. Using this simple re-calibration of the PB transferree energies yields a corrected salt permeability of 5.8 × 10−9 m/s,hich is in very good accord with the experimental NaCl salt per-eability of 8.0 × 10−9 m/s for the FT-30 membrane as measured by

direct osmosis method [47]. Using the corrected salt permeabil-ty and the water flux from MD simulation yields a salt rejectionf 99.6%, which is in excellent agreement with the experimentalata. In fact, the experimental permeability coefficient correspondso an effective free energy barrier of about 10.6 kcal/mol, showinghat the membrane strongly oppose ion permeation on energeticrounds. In conclusion, the calculated permeability coefficientsalculated on the basis of the current atomic models yields rea-onable results that are consistent with the experimental datahowing that the FT-30 polymeric membrane has a high saltejection.

. Conclusion

An atomic model of the FT-30 aromatic polyamide membraneidely used in reverse osmosis (RO) was constructed using a MD-

ased heuristic approach, and key properties (hydrated membraneensity, water density, water flux and salt rejection) were cal-ulated from MD simulations. The agreement with experimentalesults is a strong evidence that the present membrane modelrovides a realistic representation of the investigated polymericO membrane. To the authors’ best knowledge, the present studyhows for the first time that computational simulation can be usedo calculate the salt rejection from atomistic model of RO mem-rane. The results are very encouraging and suggests that MDimulation based on atomic models can provide an efficient way toredict membrane properties. Although the computational modelsnd classical force fields are certainly burdened by approxima-ions, they appear to be sufficiently accurate to support and guidehe experimental design of new RO membranes. Membrane struc-ure and chemical modification have been carried out for decades53–58] in order to find a higher performance RO membrane struc-ure. However, the laborious organic synthesis and the difficultiesf characterizing the polymeric material in experiments is a time-onsuming and expensive development process. It is also worthoting that, by design, it might be possible to improve the rejectionf specific ions (e.g., sulfate) compared to other ions contained inhe raw water. Increased use of molecular modeling and MD sim-lation offer a promising route for the discovery of new materialo design improved membranes with higher water flux and higherejection of the desired solutes.

cknowledgements

This work was supported by the U.S. Department of Energy

DOE), Office of Basic Energy Sciences, under Contract No. DE-C02-06CH11357, and by Grant No. 2007-110-R2 from theaboratory-Directed Research and Development (LDRD) programt Argonne National Laboratory.

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Appendix A. Supplementary Data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.memsci.2011.08.057.

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