Monte Carlo Simulations of Framework Defects in Layered Two

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Environmental Processes

Monte Carlo Simulations of Framework Defects in Layered Two-Dimensional Nanomaterial Desalination Membranes: Implications for Permeability and Selectivity Cody Ritt, Jay Ryan Werber, Akshay Deshmukh, and Menachem Elimelech Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.8b06880 • Publication Date (Web): 08 May 2019 Downloaded from http://pubs.acs.org on May 8, 2019

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Environmental Science & Technology

Monte Carlo Simulations of Framework Defects in Layered Two-Dimensional Nanomaterial Desalination Membranes: Implications for Permeability and Selectivity

Cody L. Ritt1, Jay R. Werber1, Akshay Deshmukh, and Menachem Elimelech*

Department of Chemical and Environmental Engineering, Yale University, New Haven, Connecticut 06520-8286

*Corresponding author: Menachem Elimelech, Email: [email protected], Phone: (203) 4322789 1These

authors contributed equally to this work.

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ABSTRACT

2

Two-dimensional nanomaterial (2-D NM) frameworks, especially those comprising graphene

3

oxide, have received extensive research interest for membrane-based separation processes and

4

desalination. However, the impact of horizontal defects in 2-D NM frameworks, which stem from

5

nonuniform deposition of 2-D NM flakes during layer build-up, has been almost entirely

6

overlooked. In this work, we apply Monte Carlo simulations, under idealized conditions wherein

7

the vertical interlayer spacing allows for water permeation while perfectly excluding salt, on both

8

the formation of the laminate structure and molecular transport through the laminate. Our

9

simulations show that 2-D NM frameworks are extremely tortuous (tortuosity ≈ 103), with water

10

permeability decreasing from 20 to < 1 L m-2 h-1 bar-1 as thickness increased from 8 to 167 nm.

11

Additionally, we find that framework defects allow salt to percolate through the framework,

12

hindering water-salt selectivity. 2-D NM frameworks with a packing density of 75%,

13

representative of most 2-D NM membranes, are projected to achieve < 92% NaCl rejection at a

14

water permeability of < 1 L m-2 h-1 bar-1, even with ideal interlayer spacing. A high packing density

15

of 90%, which to our knowledge has yet to be achieved, could yield comparable performance to

16

current desalination membranes. Maximizing packing density is therefore a critical technical

17

challenge, in addition to the already daunting challenge of optimizing interlayer spacing, for the

18

development of 2-D NM membranes.

19

TOC Art

20 21

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INTRODUCTION

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Increasing water demand, corresponding with an ever-growing global population, often

24

necessitates the desalination of seawater and saline groundwater to augment water supplies.1–3

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Pressure-driven reverse osmosis (RO) has become the benchmark technology for desalination,

26

largely due to its low energy consumption when compared to other methods, such as thermal

27

desalination.1 In addition to energy efficiency, membrane-based separations offer modularity and

28

enhanced space-efficiency.1,4

29

Thin-film composite (TFC) polyamide membranes are currently the gold standard for

30

nanofiltration (NF) and RO applications, due to their relatively high water-salt selectivity, water

31

permeability, and chemical stability during operation and chemical cleaning.5,6 Despite their

32

widespread use, polyamide membranes are limited by their fouling susceptibility, poor resistance

33

to oxidants such as chlorine, and inadequate water-salt selectivity for certain applications.5 Further

34

optimization of membrane-based desalination processes demands the development of membranes

35

which maintain high levels of permeate water flux and salt rejection over a range of operating

36

conditions. However, due to inherent material limitations of polyamide, recent improvements in

37

performance have been marginal.5

38

To address the limitations of polyamide membranes, there has been substantial research

39

interest in novel materials, such as two-dimensional nanomaterials (2-D NMs), as the selective

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layer in desalination membranes.7,8 Stacking of 2-D NMs to form laminate membranes has been

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proposed as a scalable method to induce size-selective sieving of ions based on the vertical spacing

42

between stacked sheets.9,10 Graphene oxide (GO),11–13 molybdenum disulfide,14–16 and zeolite

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nanosheets17 have been considered for laminate membranes, with GO and its chemical derivatives

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being the most prominently studied materials.8

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Interest in GO stems primarily from the possibility of ultra-fast water transport along

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atomically smooth graphitic planes,11 paired with its inexpensive production, monoatomic

47

thickness, and likely oxidative resistance.5,8,9 However, GO membranes have not exhibited the

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sought-after ultra-fast water transport during desalination applications,20 which is likely due to

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extended friction from oxygen-containing groups present on GO nanosheets.19,21 Molecular

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dynamics simulations have found that the presence of only ~5% of oxygen-containing functional

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groups on a graphene sheet (O:C ratio of 0.05) reduces the slip length, a measure of flow 2 ACS Paragon Plus Environment


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enhancement, by 97% compared to the ~48-nm slip length of pristine graphene.19 The marked

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impact of an O:C ratio of only ~0.05 on slip length is further accentuated by experimental O:C

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ratios of GO membranes typically ranging from 0.2 – 0.5.11,22–27

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Water-salt selectivities have been poor for most GO membranes, resulting in NaCl rejections

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of only 20–50% under low NaCl feed concentrations.11,12,28,29 Low ionic strength greatly enhances

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the effects of electrostatic interactions;11,30,31 therefore, it is likely that Donnan exclusion by

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ionized carboxyl groups on the GO nanosheets played a greater role than steric factors in the

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reported salt rejection values. Several research efforts have focused on developing methods to

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decrease the interlayer spacing (i.e., center-to-center vertical spacing of neighboring sheets) in

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order to achieve high levels of water-NaCl selectivity through steric (size) exclusion.11,24,27,32,33 In

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the dry state, GO membranes have interlayer free spacings (i.e., unoccupied space between

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neighboring sheets) of 0.30–0.64 nm that vary with humidity, with spacings greater than 0.40 nm

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allowing for water transport.33 The lower spacings in this range may also exclude salt based on

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simulation studies of transport through carbon nanotubes (CNTs), which found that nanotubes with

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0.9 nm.36,37 Methods such as covalent

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crosslinking,11,27,38 cation association,24 chemical reduction,39 and physical confinement33,40 of the

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GO nanosheets have been attempted to tune the interlayer spacing. Physical confinement in

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particular has produced interlayer free spacings small enough to largely exclude salt (0.40–0.45

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nm),32 with reported NaCl rejections in RO in one study of up to 96%.40 Although the physical-

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confinement method is likely not scalable for desalination processes, it is currently believed that

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competitive desalination membranes can emerge from 2-D NMs with comparably small interlayer

75

spacings.10,33,40,41

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Molecular transport through 2-D NM laminates can occur via several pathways, including

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interlayer channels, intrinsic defects (or “holes”) within NM sheets, and “framework defects”.

78

Framework defects refer to horizontal spacings between sheets that occur due to nonuniform

79

deposition of 2-D flakes during layer build-up. While such spacings are necessary for vertical

80

molecular transport, overlap of these framework defects can lead to a continuous pore network

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through the laminate. Recently called “pinholes” in a study of ultra-thin GO membranes,

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continuous pores formed from vertically overlapping framework defects were shown to allow 3 ACS Paragon Plus Environment

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relatively rapid permeation while negating the steric selectivity of the interlayer spacing.12 While

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it has been superficially shown that increasing the framework thickness can cover up pinholes,12

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the effect of interconnected framework defects on membrane performance has not been thoroughly

86

studied. Furthermore, there have been numerous studies on the transport of molecules via

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interlayer channels and intrinsic NM defects (i.e., nanoporous graphene),13,14,30–40 but these studies

88

fail to capture the effects of geometry and framework defects by focusing on flow in very localized

89

sections of the network. With the effect of framework defects largely being overlooked, there is a

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significant gap in the fundamental understanding of molecular transport through 2-D NM laminate

91

membranes.

92

In this study, we assess the effects of framework defects on the performance of 2-D NM

93

laminate membranes under idealized conditions. Specifically, we assume that steric effects are

94

solely responsible for transport, that an interlayer spacing of 0.84 nm (free spacing of 0.5 nm) is

95

consistently attained, and that water can freely move in interlayer channels of this spacing while

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salt is completely excluded. Under these conditions, we apply Monte Carlo simulations on both

97

the formation of the laminate structure and molecular transport through the laminate, particularly

98

addressing to what extent the transport properties of the overall framework truly rely on flow

99

through the interlayer channels, as opposed to flow through the framework defects ( ≫ 0.5 nm

100

diameter). Our results provide important insight into how framework properties, such as NM areal

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packing density and overall membrane thickness, critically impact separation behavior and

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membrane desalination performance. Through this assessment, we also estimate the achievable

103

selectivities and permeabilities of 2-D NM frameworks with ideal interlayer spacings. We

104

conclude with a discussion on the effect of framework defects on the overall potential of 2-D NM

105

frameworks for desalination.

106 107

METHODS

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Development of In Silico 2-D Nanomaterial Frameworks. 2-D NM frameworks were

109

generated in a layer-by-layer fashion through randomized sequential deposition of squares (or

110

“flakes”) on a 5 µm × 5 µm grid under the constraints described below. Although 2-D NM flakes

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may have a range of irregular polygonal shapes in reality, they were modeled as squares in this

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work for simplicity. Flakes were plotted from a distribution of square sizes (3.5–0.05 µm) in 4 ACS Paragon Plus Environment


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descending order. GO sheets can vary from several micrometers in length to less than 0.1 µm,

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depending on the synthesis and dispersion procedures.52 Prior to plotting, the target packing

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density was specified (0–100 %). Once the layer reached the target packing density, within an

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acceptable error of ± 1%, the plotting finished and the layer was added to the framework. Each

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flake was randomly assigned a location within the plane, an angle of rotation from 0–90°, and

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assessed for overlap with previously plotted flakes. Disallowing flake overlap within the same

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layer resulted in a planar framework of rigid single-layered 2-D NM sheets. During framework

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construction, each flake size in the distribution range was allotted 105 attempts at plotting before

121

proceeding to the next size in the distribution. Periodic boundary conditions were used to

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approximate an infinite planar system. After the layers finished plotting, they were combined into

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a single framework (Figure 1). This model mimics a layer-by-layer assembly as each layer must

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be prepared before proceeding to subsequent layers in the framework. Pressure-assisted assembly

125

methods may have a slight bias towards larger flakes settling out of suspension before smaller

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flakes.53,54

127 128

Figure 1. Construction of in silico 2-D nanomaterial frameworks. Randomized deposition of squares is

129

repeated in a layer-by-layer fashion. Salt (as hydrated sodium ions) and water molecules are probed as hard

130

spheres against the framework. Interlayer free spacings were set to 0.5 nm, which is assumed to allow for

131

water permeation while completely excluding salt. For clarity, probes and interlayer nanochannels are

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enlarged in the figure. 2-D NM flake dimensions (sides of 3.5–0.05 µm) greatly exceed the size of the probe

133

molecules.

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Pinhole Density. Continuous vertical pores or “pinholes,” which are formed from

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overlapping framework defects, are defined as pores that traverse the entire membrane framework 5 ACS Paragon Plus Environment

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and allow unhindered vertical passage of sodium through the 2-D NM laminate framework. To

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assess the depth-persistence of pinholes, the surface of the membrane framework was probed

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randomly using 106 spheres (radius of 0.36 nm)55 to represent hydrated sodium ions. If a probe

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landed completely outside each of the flakes on the layer, then it proceeded to the layer below.

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This check was completed for each subsequent layer until the probe either reached the other side

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of the 2-D NM framework or came into contact with a flake. The cases in which the probe travelled

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vertically all the way through the membrane framework (i.e., tortuosity of 1) were counted towards

143

the probability of sodium passing through a pinhole.

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Framework Permeability. To quantitatively assess the impact of framework defects and

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packing geometry on permeability and selectivity, the distribution of path lengths for the transport

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of water and sodium was assessed. Electrostatic effects were neglected; rather, our model considers

147

water molecules and sodium ions as hard spheres. Although electrostatic effects could influence

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the partitioning and diffusion of ions through the framework (e.g., through carboxyl groups on GO

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nanosheets), their effects would be small when desalinating feed solutions with high ionic strength

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such as seawater (~32000 ppm NaCl)31. For example, it is widely accepted that selectivity in

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conventional polymeric membranes is predominantly achieved by size-based diffusion selectivity,

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stemming from steric resistance to solute (e.g., hydrated ion) diffusion between molecular voids

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in non-porous polymers such as polyamide-RO thin films.56,57 With much of the interest in 2-D

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NM membranes focused on exploiting the interlayer spacing for steric exclusion,10 our model

155

focuses on whether such steric exclusion would be sufficient to achieve adequate performance.

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Furthermore, the focus on size-exclusion effects makes our results more broadly applicable to 2-

157

D NM membranes of different chemical nature.

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First, the top of the framework was probed in 105 random locations with a spherical probe,

159

which was assessed as an adequate sample size for simulation accuracy (Figure S1). Probe radii of

160

0.14 nm and 0.36 nm were used for water and hydrated sodium, respectively. For each layer, the

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probe was checked for overlap with every flake on that layer. If the probe landed on a flake, it was

162

randomly assigned a direction of travel (𝜃). From its landing point, the probe traveled straight in

163

the direction defined by 𝜃 until completely passing over the edge of the flake. The distance traveled

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in this fashion added to the horizontal path length (𝐿H) of the probe. It was assumed that probes

165

representing water molecules can pass through all interlayer channels. Probes representing 6 ACS Paragon Plus Environment


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hydrated sodium ions were assessed for blockage by 2-D NM flakes on the layer above. In our

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model, sodium with a hydrated diameter of 0.72 nm was too large to enter horizontal nanochannels

168

with an interlayer free spacing of 0.50 nm; however, sodium could move freely in spaces of

169

incomplete packing, where the resulting channel height was 1.34 nm (i.e., interlayer spacing,

170

0.84 nm, plus free spacing, 0.5 nm). If a salt probe intersected with the edge of a flake, it was

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assigned a new direction to travel, which was defined by a deflection angle. The deflection angle

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was randomized over a 180° range away from the intersecting flake. Salt probes were allocated

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104 such deflections (more details in Supporting Information; Figure S2) to travel over one of the

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edges of its resident flake. Salt probes were allowed to travel unidirectionally (i.e., laterally or

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forward). Salt probes unable to find their way over an edge under these constraints were removed

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from the framework. Such non-permeating salt probes were factored into a percolation ratio (𝜁),

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representing the ratio of the number of probes permeating through the framework (𝑛p) and the total

178

number of probes tested (𝑛T):

 

179

np nT

(1)

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Inherently, this value is always 1 for water and ≤ 1 for salt probes. Salt probes that managed to

181

completely pass over the edge of the flake under the given constraints advanced to the next layer

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at that location. Layer advancement increased the measured vertical path length (𝐿V) by the

183

specified interlayer spacing (0.84 nm). The same procedure was applied at each subsequent level

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until the probe traveled through the entire framework. For permeating probes, the total path length

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for the probe (L) was determined by summing all of the measured vertical and horizontal path

186

lengths.

187

Permeability (𝑃𝑖) for a permeating probe was then defined as the inverse of its transport

188

resistance (𝑅𝑖):58

189

Pi 

1 Ri

(2)

190

where transport resistance is due to the path length of the probe, making path length a critical

191

measurement parameter. We explicitly define transport resistance as


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Ri  rV LV,i  rH LH,i

192

(3)

193

where the horizontal resistance per unit length (𝑟H) and unit vertical resistance (𝑟V) are dependent

194

on the interactions between the probe molecule and 2-D NM. While much of the analysis in this

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study applies broadly to 2-D NMs of differing chemistries, for estimation of permeabilities we use

196

data for GO. The horizontal resistance can be determined experimentally or through molecular

197

dynamics simulations of flow between GO nanosheets. For this study, a horizontal resistance was

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calculated using a slip length of 1 nm and viscosity of 1.5 × 10 ―3 Pa s. These values were

199

estimated from published molecular dynamics studies for GO to represent our flow conditions with

200

a uniform interlayer free spacing of 0.5 nm.19,43,59–63 Although lacking experimental data,

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simulations of ion transport in GO, CNT, and graphene nanochannels suggest that selectivity

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occurs at the entrance of the channel, whereas the resistances of salt and water inside the channel

203

are similar.44,64,65 Therefore, we assume that the unit horizontal resistances are equivalent for salt

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and water.

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We defined a weighting factor (𝛼) as the ratio of the vertical and horizontal unit resistances:

206

𝛼 = 𝑟V 𝑟H. A molecule traveling vertically between layers (typically through relatively large gaps

207

between sheets) is expected to have a significantly lower transport resistance than a molecule

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traveling horizontally and interacting with the 2-D NM surface. We assumed a value of 10-4 for 

209

as a rough approximation of flow through >100-nm diameter framework defects (see Supporting

210

Information for more details). The permeability of each independent path was then determined as

211

a function of its total vertical and horizontal path lengths using

212

Pi 

1

(4)

rH ( LV,i  LH,i )

213

The average permeability 〈𝑃〉 is then defined as the summation of the permeabilities of all

214

independent, parallel paths traveled by probes through the framework divided by the total number

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of paths traveled: n

216

n

1 T  p P   Pi   Pi nT i 0 np i 0

(5)



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217

Because the permeability of non-permeating probes is zero, the average permeability is thus the

218

percolation ratio (eq 1) multiplied by the average permeability of permeating probes. We refer to

219

the average permeability as Pw and Ps when considering water and salt, respectively. Pw can be

220

related to the commonly used water permeability coefficient, A:5,57

221

PWVˆW RgT

A

(6)

222

The specific volume of water (𝑉W), ideal gas constant (𝑅g), and absolute temperature (𝑇) are set

223

to 18 cm3 mol ―1, 83.145 cm3 bar K ―1 mol ―1, and 298 K, respectively. Ps is identical to the

224

commonly used salt permeability coefficient, B (i.e., B = Ps).5,57

225 226

227

Since permeability is inversely proportional to the path length, we define the effective path length (𝐿e) as the harmonic mean of all independent, parallel path lengths:

Le 

nT

i0  LV  LH  np

1

 1   



  np 1   LV  LH   i 0

np

(7)

228

The harmonic mean is used here to allow for direct relation of the effective path length with

229

permeability:

230

P 

1 rH Le

(8)

231

The probability of permeation is not equal for independent paths, as a molecule would

232

preferentially take a path with less resistance. As permeability scales with 1/L, the harmonic mean

233

captures the relative contribution of each path to overall transport. Non-permeating probes were

234

not included in the summation, as their path lengths were set to infinity, yielding 𝐿𝑖―1 = 0.

235

Water/Salt Permselectivity. To relate water and salt permeabilities of the 2-D NM

236

framework, we calculate permselectivity (𝑃W 𝑃S) as the ratio of average water permeability to

237

average salt permeability. The equivalent unit resistances cancel out, resulting in the water/salt

238

permselectivity essentially being the ratio of the effective path lengths of salt and water. The


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permselectivity captures the contribution of the membrane to molecular separation and can be

240

related to the real (intrinsic) salt rejection (𝑅):57

241

PW VˆW A  p    p     Cp P R T g S R  1  B  Cm 1  A p   PW VˆW   1   p    B P RT S

(9)

g

242

where Cm and Cp are the concentrations of salt at the membrane/feed interface and in the permeate,

243

respectively. The applied pressure (∆𝑝) and osmotic pressure difference (∆𝜋) are taken here as

244

standard seawater reverse osmosis (SWRO) test conditions (55.15 bar applied pressure and 32000

245

ppm or 0.55 M NaCl feed solution).1,66

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RESULTS AND DISCUSSION

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Framework Parameters of Interest. Framework thickness is critical for assessing membrane

248

performance as it can significantly affect solute and solvent fluxes.57,67 In most membrane

249

processes, flux is inversely proportional to the thickness of the active layer.58 Increasing membrane

250

thickness can result in better salt separation by mitigating the effects of defects; however, thick

251

active layers can be extremely detrimental to water flux. Therefore, it is imperative to find a

252

balance between the necessary thicknesses to minimize the influence of defects while still

253

maintaining adequate water flux. The active-layer thickness of semi- or fully-aromatic polyamide

254

membranes ranges from ~20 – 200 nm.68 In this study, framework thickness was varied from 1 to

255

200 layers (or 0.84 to 167 nm).

256

Areal packing densities represent the dense packing state of each layer of the framework and

257

were achieved with polydisperse squares (side lengths of 3.5–0.05 µm) when creating the layers,

258

as discussed in the Methods section. For comparison between our model and experimental data,

259

we approximated the areal packing densities of frameworks from previously published work by

260

using reported GO mass loadings, membrane thicknesses, and nanosheet vertical spacings under

261

vacuum. Referencing the spacing and density of graphene in graphite and including mass

262

contribution from oxide functionalities informed the calculation of areal packing densities for GO

263

membranes. For each published report, mass contributions from oxygen functionalities were

264

determined from measured O:C ratios. Our survey of published works indicated that GO 10 ACS Paragon Plus Environment


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membranes tend to have O:C ratios in the range of 0.2-0.5. Reports failing to disclose O:C ratios

266

for their GO were assumed a value of 0.3. Much of the packing density data collected from

267

previously published works on 2-D NM laminates was incomplete. Some reports lacked essential

268

information for approximating packing densities such as membrane area, thickness, or applied 2-

269

D NM mass.69–71 Meanwhile, data from other studies result in unattainable packing densities (𝜑 >

270

100% or 𝜑 ≈ 0%).15,38,72 Regardless of the substantial variability in literature, several reports were

271

found to produce reasonable packing densities, ranging from 45 – 80% (more details in Supporting

272

Information; Figure S4 and Table S2).11,22–27,73 Random packing of unit squares oriented in parallel

273

has a maximum packing density of 56%,74 indicating that incorporating polydispersity of squares

274

is necessary to achieve greater packing densities. Accordingly, we incorporated polydispersity into

275

the frameworks in order to study packing densities of 50, 75, and 90%, which represent low, high,

276

and optimistically high values.

277 278

Pinholes in Membrane Frameworks. The first assessment of framework defects was the

279

impact of vertically continuous pores, or “pinholes.” Pinholes allow for direct, unhindered

280

transport of salt and water from top to bottom of the 2-D NM framework with the shortest path

281

length possible; therefore, pinholes have minimal hydraulic resistance and were predicted to

282

dominate water permeability if present.

283

The persistence of pinholes was qualitatively evaluated by visualization of the framework

284

(Figure 2). The prevalence of pinholes, while intuitively dependent on the thickness of the 2-D

285

NM framework, was also dependent on the areal packing density (𝜑) of the framework. Figure 2

286

illustrates that pinholes persist much further through frameworks with lower packing densities. A

287

framework with a packing density of 50% often contained pinholes past 10 layers (8.35 nm), while

288

a packing density of 90% resulted in the coverage of pinholes around 5 layers (4.17 nm).


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Figure 2. Framework build-up as a function of packing density (𝜑) and thickness. (A) 1 layer, 𝜑 = 50%.

291

(B) 10 layers, 𝜑 = 50% (C) 1 layer, 𝜑 = 90%. (D) 10 layers, 𝜑 = 90%. A framework defect, representing

292

the horizontal spacing between sheets on the same layer, has been highlighted in panel C. The build-up of

293

frameworks with a lower packing density (e.g., B) resulted in continuous vertical pores (or “pinholes”)

294

which greatly reduce the selectivity of the membrane.

295

To assess the persistence of pinholes, 100 unique 2-D NM frameworks were probed with

296

spheres representing hydrated sodium. The pore size distribution of the pinholes was highly

297

dependent upon the framework packing density and thickness, with higher packing densities

298

reducing the distribution and areal coverage of the pinholes (Figure S5). The probability of sodium

299

passage through pinholes as framework thickness increased was calculated and averaged among

300

all frameworks (Figure 3A). Figure 3A corroborates our observations from Figure 2, showing that

301

pinholes persist past 10 layers in a framework with a packing density of 50%. The data in Figure

302

3A were fit with a binomial curve that stems from the independent geometry of each layer:

303



1   

c

(10)

304

where 𝛿 is the membrane thickness and 𝑐 is the probability of sodium passage through a pinhole,

305

which is, in essence, the areal density of pinholes. We then define the critical thickness (𝛿𝑐𝑟𝑖𝑡) as

306

the membrane thickness of each simulated 2-D NM framework that was required to reduce the 12 ACS Paragon Plus Environment


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maximum solvent (water) permeability through pinholes to ≤ 0.01 L m-2 h-1 bar-1, thus minimizing

308

the impact of pinholes on unhindered salt transport. The maximum pinhole permeability was

309

approximated by combining the areal coverage of pinholes, c, into a single pore (see the Supporting

310

Information for more details). The critical pinhole area, 𝑐crit, similarly represents the maximum

311

areal coverage of pinholes that maintains a permeability ≤ 0.01 L m-2 h-1 bar-1. We calculated 𝑐crit

312

for all thicknesses (Figure S6), allowing us to predict (eq 10) the thickness expected to meet the

313

permeability constraint for packing densities ranging from 0.1–99.9%. The average critical

314

thicknesses, determined from 100 unique frameworks for each packing density, generally fell

315

slightly below the predicted critical thicknesses, but nevertheless followed the same trend (Figure

316

3B). The discrepancy is due to the non-zero size of the sodium probe, which led to underestimation

317

of the pinhole area as 𝑐crit (total area of ~10 nm2 per 25 m2; Figure S6) was approached.

318

Figure 3B displays the importance of high packing densities toward the minimization of critical

319

thickness. Furthermore, critical thicknesses in the range of interest for packing densities (50-80%)

320

compare well with previously published observations.12 Due to the substantial contrast in vertical

321

to horizontal transport resistance values influencing the effective path length (Figure 3C), pinholes

322

governed framework permeation when they were present. This phenomenon was demonstrated by

323

the drop in the effective path length (𝐿e) of water under weighting conditions, with 𝛼 = 10 ―4

324

(Figure 3D). The observed drop in 𝐿e only persisted to ~10 layers, which was the average critical

325

thickness for membranes with the given packing density of 50%. Permeation past this critical

326

thickness did not exhibit large dissimilarities between 𝐿e and the non-weighted path length (𝐿), as

327

permeation became dictated by flow through interlayer spacings. Similar trends were observed

328

when using other weighting factors (Figure S3).

329

While understanding the extent of pinholes and their impact on water permeability is an

330

essential preliminary assessment of framework defects, percolating (not necessarily vertical)

331

networks of framework defects are likely much more important. Because salt cannot travel through

332

interlayer spacings in our model, it must travel through percolating defect paths. Such paths require

333

a framework defect in one layer to overlap, at any point, with a defect in the next layer, whereas

334

pinholes require continuous overlap of framework defects at the same planar location. Therefore,

335

pinholes generally persist up to just ~10 layers and are mathematically simple to describe (eq 10).

336

In contrast, percolating paths are much more persistent, and formulating an analytical description 13 ACS Paragon Plus Environment

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337

for these paths is extremely difficult, if not impossible. Further assessment of percolating paths is

338

critical and will be discussed later.

339 340

Figure 3. Prevalence of pinholes through membrane frameworks and implications on effective path length.

341

Pinholes allow for unhindered vertical passage of hydrated sodium through overlapped defects in the

342

membrane framework. (A) Unhindered access for sodium ions through 2-D NM framework pinholes as a

343

function of thickness and packing density. The data were fit with solid lines representing binomial

344

probability curves (eq 10). Error bars represent the relative standard deviation of 100 sampled 2-D NM

345

frameworks. (B) Critical thickness (𝛿crit) needed to decrease pinhole-driven water permeability to < 0.01 L

346

m-2 h-1 bar-1 as a function of packing density. Curve is the expected critical thickness from eq 10, with more

347

details in the Supporting Information. Error bars represent the standard deviation of 100 sampled 2-D NM

348

frameworks. (C) Schematic representing higher rates of water permeation through areas of lower resistance

349

in the vertical direction, which was incorporated as a weighting factor (𝛼) to calculate the effective path

350

length. (D) Influence of pinholes on the effective path length of 2-D NM frameworks with a packing density



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351

of 50%. With a weighting factor used, the effective path length drops considerably when pinholes are

352

present. Error bars represent the relative standard deviation of 20 sampled 2-D NM frameworks.

353 354

Water Permeability. For the transport of water, we now focus on frameworks with 10 to 200

355

layers (8.35-167 nm) in order to minimize the effect of pinholes. After pinholes were eliminated,

356

water flow through the framework was dominated by flow through interlayer spacings. Because

357

each layer of the 2-D NM framework has roughly the same flake areal coverage, the effective path

358

lengths increased linearly with respect to thickness (Figure 4A). Frameworks with higher packing

359

densities consistently demonstrated higher effective path lengths, which was expected as higher

360

packing densities represent greater areal coverage by 2-D NM flakes on each layer and thus require

361

more horizontal transport before the probe encounters a flake edge. Specifically, a 1% increase in

362

packing density increased the effective path length of water by ~10 nm per layer, leading to slopes

363

of roughly 800, 1075, and 1200 nm/layer for packing densities of 50, 75, and 90%, respectively.

364

Additionally, each framework can be described by its respective tortuosity (𝜏), defined as the ratio

365

of effective path length to thickness (𝜏 = 𝐿𝑒 𝛿), with  ranging from 970 to 1400 (Figure S7).

366

Tortuosity characterizes the packing structure of a porous media and is quite striking in this case,

367

as tortuosities of typical porous media range from 1–2.75 The three order of magnitude increase in

368

tortuosity from typical porous media to our 2-D NM frameworks is attributed to the high aspect

369

ratio of the 2-D sheets.

370

As discussed earlier in the paper, we assumed a slip length and viscosity of 1 nm and 1.5 ×

371

10 ―3 Pa s, respectively. From these values, we obtained a unit horizontal resistance of ~3.9 × 103

372

h m-2 (see Supporting Information for more details), which corresponds to a hydraulic water

373

permeability (per unit length) of 190 L m2 h1 m bar1. The unit resistance allows us to convert

374

from effective path lengths to diffusive water permeability (eq 8). Permeability, defined as

375

inversely proportional to resistance (and the effective path length), was similarly inversely

376

proportional to thickness (Figure 4B). As framework thickness increased, water permeability was

377

found to decrease from ~20 L m-2 h-1 bar-1 to < 1 L m-2 h-1 bar-1. These values appear to be

378

reasonable as GO membranes with similar packing densities have reported water permeabilities

379

around 10–20 L m-2 h-1 bar-1 up to 50 layers and < 1 L m-2 h-1 bar-1 around 150 layers.11,27,76

380

Reported experimental water permeabilities also trended higher because of larger interlayer 15 ACS Paragon Plus Environment

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381

spacings, which allow for more flow, than the 0.5-nm interlayer free spacing simulated in this

382

study. Although flow enhancement from extreme nanoconfinement through smaller interlayer

383

spacings could increase flow rates, low slip lengths for GO membranes suggest flow enhancement

384

would be small.19 However, the impact of our slip length and viscosity assumptions on water

385

permeability was assessed with values representative of lower and upper bounds for possible

386

outcomes for GO-based membranes (Figures S8 and S9).

387

A framework thickness of 200 layers was the maximum that was computationally accessible.

388

As the frameworks approached this maximum, their water permeabilities decreased below the

389

lower limit for what could be considered relevant for desalination applications, as current SWRO

390

membranes have water permeabilities of 1–3 L m-2 h-1 bar-1.66 Therefore, studying permeation

391

through membrane frameworks up to 200 layers was sufficient to critically assess the performance

392

of the simulated 2-D NM frameworks.



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393 394

Figure 4. Effect of packing density, , and thickness, , on water permeation. (A) Effective path length for

395

water as a function of thickness and packing density, scaling as 𝐿e ∝ . (B) Water permeability coefficient

396

(A) as a function of thickness and packing density, scaling as 𝐴 ∝ 1 . Error bars are presented as the

397

standard deviation of 20 sampled membrane frameworks.

398 399

Selectivity. Enhanced water-salt selectivity (also referred to as permselectivity) is a critically

400

important research target for next-generation desalination membranes.66 However, the

401

experimentally demonstrated selectivity of 2-D NM membranes remains far below that of current

402

TFC membranes.11,12,28,29 To gain insight into the achievable selectivity of 2-D NM frameworks,


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403

we assess the water-salt selectivity as the ratio of the water and salt permeabilities (𝑃W 𝑃S).

404

Estimation of selectivity relies mainly on the assumption of equal horizontal resistances for water

405

and salt, whereas estimation of water permeability depends greatly upon assumptions built into the

406

unit horizontal resistance value (e.g., slip length). Therefore, our estimated selectivities are

407

inherently less susceptible to error than the estimated water permeabilities.

408

Water-salt selectivities increased exponentially with thickness, indicating that salt permeability

409

decreased far more sharply with respect to thickness than did water permeability (Figure 5A). The

410

difference between salt and water permeation stemmed from the inability of salt to flow through

411

the ideal interlayer spacings in our model. Instead, salt probe molecules could only travel through

412

percolating paths of connected framework defects. Permselectivity also increased with packing

413

density. Increased thickness and packing density both decrease the likelihood of a path of

414

connected framework defects to fully permeate the network. While these factors affect water

415

transport, as discussed in the previous section, the effect is relatively small since water is never

416

blocked from permeating.

417

To further elucidate the selective transport of salt through the framework, we individually

418

assessed the two components that comprise salt permeability, namely the percolation ratio (eq 1)

419

(Figure 5B) and the effective path length of permeating probes (Figure S10). The percolation ratio

420

is the percentage of salt probes that fully permeate the network, or the percentage of fully

421

permeating paths of connected framework defects. The percolation ratio was clearly the critical

422

factor: the water-salt permselectivity increased at nearly the same rate in which the percolation

423

ratio decreased with respect to membrane thickness and packing density. For the most selective

424

membrane assessed (90% packing density and 200 layers), the percolation ratio reached as low as

425

10-4. In contrast to the percolation ratio, the effective path length of salt did not differ dramatically

426

from water with packing density or thickness (Figure S10). Interestingly, the effective path length

427

for permeating salt probes was always slightly less (2–10-fold) than that of water. This result can

428

be explained by the forced path of salt molecules through framework defects, which are biased

429

towards shorter distances.



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430 431

Figure 5. Impact of packing density and thickness on selectivity. (A) Water/salt permselectivity as a

432

function of thickness and packing density, with the weighting factor (i.e., ratio of the vertical and horizontal

433

unit resistances) applied (𝛼 = 10 ―4). (B) Percolation ratio, representing the ratio of percolating salt paths

434

to the total number of paths (eq 1), as a function of thickness and packing density. (C) Permeability–

435

selectivity trade-off. Each curve represents a different packing density, with each data point representing a

436

different number of layers. Increased thickness, , results in decreased water permeability but increased

437

selectivity. (D) Permselectivity as a function of packing density at relevant operating water permeabilities.

438

A packing density of 90%, an optimistically high value, was the maximum assessed due to computational

439

limits. Rejections in (C) and (D) refer to real rejection and are calculated (eq 9) from the permselectivity

440

under standard SWRO test conditions (55.15 bar applied pressure and 32000 ppm NaCl feed solution).1,66

441

Due to the positive skewness of the data, permselectivity data were reported as median values with

442

interquartile range error bars of 20 sampled membrane frameworks.


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443 444

Permeability-Selectivity Tradeoff. Water permeability and water-salt permselectivity are

445

compared in Figure 5C, showing a clear trade-off relationship. Permeability–selectivity trade-offs

446

occur for a variety of materials, including polymeric desalination membranes.66,77 Membranes

447

falling in the top-right corner of the permeability-selectivity diagram would be ideal, while

448

attaining enhanced selectivity at current water permeability levels is a primary goal for desalination

449

membrane research.66 Although increasing membrane thickness enhanced the water-salt

450

permselectivity of the frameworks, it was also detrimental to their water permeabilities. In contrast,

451

when the packing density increased from 50 to 90% for framework thicknesses ≥ 160 layers, the

452

permselectivity increased over 1000-fold (Figure 5A) with less than a 2-fold drop in water

453

permeability (Figure 4B). In other words, packing density had a much greater effect on selectivity

454

than on water permeability. Permselectivity was found to be proportional to an exponential

455

function (𝑃w 𝑃s ∝ 𝑒𝛽𝛿), where 𝛽 is a constant dependent on packing density that dictates the rate

456

of selectivity increase with respect to membrane thickness (Figure S11). 𝛽 increased sharply with

457

respect to the framework packing density (𝛽 ∝ 𝜑4).

458

The trade-off analysis shows that while thickness is an important parameter, increasing packing

459

density is essential. In fact, reaching very high packing densities (𝜑 > 99%) would begin to

460

resemble nanoporous graphene, which requires only a single layer for water-salt separation and

461

theoretically should allow for simultaneously high water permeability and water-salt selectivity.78

462

In practice, however, it would be unrealistic to create such a tightly packed membrane, and

463

nanoporous graphene membranes themselves have had substandard selectivities due to pore-size

464

distributions and grain boundaries.78,79

465

Model Assumptions. It is worthwhile to review the assumptions in this model, not just to

466

qualify our findings but also to provide more insight into material design requirements. Our model

467

assessed purely geometric effects, essentially considering the movement of hard spheres through

468

assemblies of vertically stacked sheets with uniform interlayer spacings. This model is necessarily

469

simple, but resembles the core mechanism of transport in well-assembled 2-D NM membranes of

470

any chemistry. The geometries in our systems are optimistic as they neglect the distribution of

471

void sizes and interlayer spacings that can occur experimentally due to non-uniform 2-D NM

472

deposition.80 The hard sphere assumption allows for complete exclusion of hydrated ions when the 20 ACS Paragon Plus Environment


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473

interlayer spacing is less than the hydrated diameter. However, hydrated diameters of ions are not

474

absolute quantities. In reality, ions may be able to enter interlayer channels by shedding hydrated

475

water at the expense of ion-water interaction energy.56 Electrostatic interactions were also

476

neglected. Increased water/ion selectivity stemming from electrostatic effects would occur in

477

certain situations, such as the retention of multivalent ions in low-salinity feed solutions, but are

478

likely minimal in the case of monovalent ion retention in high-salinity feed solutions.31

479

Additionally, the assumption of equal unit resistance for water and salt, based in part on molecular

480

dynamics simulations,44,64,65 directly impacts the calculated selectivity, as permeabilities were

481

calculated from the modeled path lengths. For example, a two-fold higher salt unit resistance

482

would increase the water/salt selectivity two-fold. It may be possible to design 2-D NMs that

483

interact more strongly with ions, increasing the resistance to ion flow. For context, in order to have

484

comparable selectivity to polyamide TFC membranes, a 2-D NM membrane with 75% packing

485

density, which is typical of current 2-D NM membranes, would need ~5000-fold higher selectivity.

486

In other words, resistance for flow of hydrated ions within nanochannels would need to be ~5000-

487

fold lower than for water, likely a difficult task.

488

The slip lengths that we used to calculate the unit resistance and permeabilities are another

489

source of potential error. Slip lengths were based on extrapolated values from molecular dynamics

490

simulations for GO with interlayer spacings larger than 0.84 nm, as GO- and other 2-D NM-based

491

membranes have not been well-studied at relevant interlayer spacings for desalination. What we

492

deemed to be realistic values for GO nanochannels are presented in the main manuscript and a

493

range of possible values are shown in the Supporting Information (Figures S8, S9). For GO, the

494

slip length is highly dependent on the content of oxygen functionalities.19 In fact, recent modeling

495

work, where interlayer spacings were not fixed and were studied down to ~0.7 nm, found that the

496

diffusion of water in GO nanochannels was slower than in the bulk for all cases.80 This finding

497

suggests a potential lack of meaningful slip flow in GO nanochannels due to hydrophilic surface

498

interactions. The most optimistic, and possibly unrealistic, conditions (slip length of 2.5 nm and

499

viscosity of 0.89 × 10 ―3 Pa-s) would increase water permeability ~6-fold over the values in

500

Figures 4 and 5.

501


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502 503

Implications for 2-D NM Frameworks. In order to be industrially viable, the performance

504

of 2-D NM membranes must meet or exceed that of polyamide seawater desalination membranes,

505

which currently achieve real salt rejections ranging from 99.85% to 99.91% along with water

506

permeabilities of 1–3 L m-2 h-1 bar-1.66,81,82 Several relevant water permeabilities are highlighted in

507

Figure 5D for the modeled 2-D NM frameworks. At the higher end of achievable water

508

permeabilities for SWRO membranes (3 L m-2 h-1 bar-1), none of the studied packing densities

509

produced comparable salt rejections. However, at the lower end of water permeabilities (1 L m-2

510

h-1 bar-1), 2-D NM frameworks with a packing density of 90% are projected to have nearly

511

comparable performance with a real salt rejection of ~99.2%.

512

This work is another demonstration of how the performance of next-generation membranes

513

with highly selective channels/pores (e.g., channel-based membranes, nanoporous graphene)5,83,84

514

can be limited by defects formed during fabrication. In particular, our results show that percolating

515

framework defects will limit the achievable permselectivities of essentially all 2-D NM

516

frameworks. Furthermore, it is clear that extremely high packing densities must be achieved, in

517

addition to the exceptionally difficult and unrealized goal of scalably producing salt-excluding

518

interlayer spacings with high water permeability (as was assumed in our model). To our

519

knowledge, 2-D NM frameworks with high packing densities (roughly 90% or greater) have not

520

been realized experimentally (Figure S4). Constructing 2-D NM frameworks with such high

521

packing densities requires a highly ordered system which, albeit possible for simulation studies, is

522

difficult to achieve for real systems.20,46

523

Our modeling results suggest that the impact of framework defects could be mitigated by

524

designing 2-D NM frameworks with increased intrinsic water permeability. This scenario would

525

shift the permeability/selectivity curves in Figure 5C towards higher permeability. If a 2-D NM

526

framework can be designed with 5–10-fold greater permeability than modeled in this study—and

527

assembled at 90% packing density with ideal interlayer spacings—the performance could be

528

extremely competitive. Increasing the water permeability (per unit length) may allow these

529

frameworks to be designed to thicknesses great enough to limit the effect of flow through

530

framework defects, thereby achieving enhanced selectivities with competitive water

531

permeabilities. 22 ACS Paragon Plus Environment


Page 24 of 36

532

The results of this study are applicable to any 2-D NM, including unidentified materials with

533

promising properties (e.g., high slip length). Further simulations of water and salt transport within

534

2-D NM nanochannels of defined chemistry are needed at relevant interlayer spacings to determine

535

slip lengths and the ability to exclude salt. Results from these simulations can then be combined

536

with our model to estimate the performance at various packing densities and thicknesses. This

537

combination of different modeling approaches could yield promising design targets for material

538

synthesis.

539

However, the end goal remains highly challenging. In 2-D NM frameworks, the properties of

540

the framework are intricately linked to the chemical nature of the 2-D NM itself. Therefore,

541

chemical changes to optimize one property (e.g., slip length) could negatively impact other

542

important properties (e.g., interlayer spacing and packing density). In contrast, next-generation

543

membrane materials using molecular-design approaches often allow for independent tuning of

544

properties such as channel permeability and defect density.5 The design of 2-D NM frameworks

545

with all of the required characteristics—high packing density, ideal interlayer spacing, enough

546

thickness to minimize the impact of defects, and permeabilities greater than that projected for

547

GO—will likely require major breakthroughs in materials science to be industrially relevant for

548

desalination.

549 550

ASSOCIATED CONTENT

551

Supporting Information

552

The Supporting Information is available free of charge on the ACS Publications website at DOI:

553

XXX.

554

Required number of probes for accurate measurement (Figure S1 and Table S1); justification for

555

the maximum number of deflections before removal of probe from simulation (Figure S2);

556

determination of appropriate weighting factor (Figure S3); characteristics of graphene oxide

557

membranes in literature (Figure S4 and Table S2); framework pore size distribution (Figure S5);

558

permeability through pinholes and prediction of membrane critical thickness (Figure S6);

559

derivation of unit horizontal resistance (used in eq. 8) for slip flow between two sheets; framework 23 ACS Paragon Plus Environment

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560

tortuosity (Figure S7); influence of slip length and viscosity assumptions on unit horizontal

561

resistance and membrane permeability (Figures S8 and S9); effective path length of permeating

562

probes (Figure S10); relation between selectivity and packing density (Figure S11) (PDF)

563

AUTHOR INFORMATION

564

Corresponding Author

565

*E-mail: [email protected]. Phone: +1 (203) 432-2789.

566

Notes

567

The authors declare no competing financial interest.

568

ACKNOWLEDGEMENTS

569

This work was supported as part of the Center for Enhanced Nanofluidic Transport (CENT), an

570

Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science,

571

Basic Energy Sciences under Award #DE-SC0019112. We also acknowledge the National Science

572

Foundation Graduate Research Fellowship awarded to C.L.R.



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Figure 1. Construction of in silico 2-D nanomaterial frameworks. Randomized deposition of squares is repeated in a layer-by-layer fashion. Salt (as hydrated sodium ions) and water molecules are probed as hard spheres against the framework. Interlayer free spacings were set to 0.5 nm, which is assumed to allow for water permeation while completely excluding salt. For clarity, probes and interlayer nanochannels are enlarged in the figure. 2-D NM flake dimensions (sides of 3.5–0.05 µm) greatly exceed the size of the probe molecules. 315x156mm (150 x 150 DPI)


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Figure 2. Framework build-up as a function of packing density (φ) and thickness. (A) 1 layer, φ= 50%. (B) 10 layers, φ= 50% (C) 1 layer, φ= 90%. (D) 10 layers, φ= 90%. A framework defect, representing the horizontal spacing between sheets on the same layer, has been highlighted in panel C. The build-up of frameworks with a lower packing density (e.g., B) resulted in continuous vertical pores (or “pinholes”) which greatly reduce the selectivity of the membrane. 199x194mm (150 x 150 DPI)




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Figure 4. Effect of packing density, φ, and thickness, δ, on water permeation. (A) Effective path length for water as a function of thickness and packing density, scaling as Le∝δ. (B) Water permeability coefficient (A) as a function of thickness and packing density, scaling as A∝1⁄δ. Error bars are presented as the standard deviation of 20 sampled membrane frameworks. 82x137mm (600 x 600 DPI)



Membrane Thickness,

A

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99.5 A (L m-2 h-1 bar-1) 1 3 10

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Water Permeability Coefficient, A (L m-2 h-1 bar-1)


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Monte Carlo Simulations of Framework Defects in Layered Two

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