Polymeric Ion Pumps: Using an Oscillating Stimulus To Drive Solute

Mar 13, 2018 - To aid in the design of the chemically active composite membrane, a mathematical model that describes the ability of the membrane to en...
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Polymeric Ion Pumps: Using an Oscillating Stimulus to Drive Solute Transport in Reactive Membranes Sherwood Benavides, Siyi Qu, Feng Gao, and William A. Phillip Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00193 • Publication Date (Web): 13 Mar 2018 Downloaded from http://pubs.acs.org on March 26, 2018

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Langmuir

Polymeric Ion Pumps: Using an Oscillating Stimulus to Drive Solute Transport in Reactive Membranes Sherwood Benavides, Siyi Qu, Feng Gao, William A. Phillip* Department of Chemical and Biomolecular Engineering, University of Notre Dame, 205 McCourtney Hall, Notre Dame, IN 46556, USA. E-mail: [email protected] * To whom correspondence should be addressed: [email protected] KEYWORDS (Word Style “BG_Keywords”). If you are submitting your paper to a journal that requires keywords, provide significant keywords to aid the reader in literature retrieval. Abstract The development of membranes that separate molecules based on chemical factors, rather than physical factors, is one promising approach to meeting the demand for membranes that are more selective. In this study, the design of multifunctional, pH-responsive membranes that selectively pump a target solute is detailed. The membranes consist of two functional components: a gate layer made from an amine-functionalized copolymer and a reactive matrix lined by iminodiacetic acid groups that bind divalent cations reversibly. These two chemistries exhibit concurrent changes in the cation binding affinity and gate permeability in response to the pH value of the surrounding solution such that when the membranes are exposed to an oscillating pH, the combination drives a facilitated transport mechanism that pumps ions. In mixed solute systems, calcium permeated through the membrane four times faster than sucrose in the presence of an oscillating pH even though the solutes possess similar hydrodynamic sizes and permeated through the membrane at the same rate when the pH value was constant. The development of the polymeric ion pumps was 1 ACS Paragon Plus Environment

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guided by a model that provided several critical insights. First, the solute binding capacity and thickness of the membrane define the asymptotic limit for enhanced selectivity. Second, the maximum enhancement in selectivity is realized in the limit of infinitely rapid oscillations. The multifunctional membranes discussed here provide a platform for the development of processes that can fractionate molecules of similar size but varying chemistry. 1. Introduction The science of naturally-occurring membranes was studied well before synthetic membranes became practical technologies. As such, the design of synthetic membranes has often taken inspiration from natural system. For example, this inspiration coupled with advances in materials science led to the successful development of seawater desalination,1-3 wastewater treatment,4-5 dialysis and gas separations6-8 processes based on semi-permeable membranes similar to the animal bladders used to identify the phenomenon of osmosis.9 The state-of-the-art membranes used in these processes are primarily passive sieves that fractionate solutes using sizeselective transport mechanisms.10 Catalyzed by advances in nanotechnology these size-selective membranes are approaching practical limits of performance,11-13 which has driven increased interest into membrane materials that exploit other transport mechanisms. For example, the development of membranes that augment the transport of target solutes through chemical means is one potential route toward exceeding the performance of size-selective sieves.14-15 This class of chemically-active membrane is said to function through the use of facilitated transport mechanisms16-17 and are of interest due to their ability to isolate dilute solutes from solutions containing many species of comparable size.18 Similar to the development of size-selective membranes, natural systems can provide inspiration to guide the design of new chemicallyselective membranes. 2 ACS Paragon Plus Environment

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Naturally-occurring systems realize chemically-selective transport mechanisms using specialized membrane proteins that are embedded within the cell wall. These proteins, which interact selectively with target solutes, are vital for the transport of metal ions such as Na+, K+, Mg2+, and Ca2+ through hydrophobic cell membranes.19-21 In general, the structures formed by these proteins can be classified as ion channels, where the proteins form a hydrophilic passage that traverses the cell wall, or ion pumps, where the proteins exhibit reversible conformational changes that direct the transport of target molecules through the cell wall.22-24 Taking inspiration from these biological systems, prior efforts have attempted to develop synthetic membranes that function by analogous mechanisms.25 Despite the latent possibility to enhance mass transfer and improve selectivity by chemically targeting solutes, realizing the potential of facilitated transport mechanisms has proven difficult.26-27 Supported liquid membranes, where mobile reactive carriers reversibly capture solute at the membrane-solution interface and carry it across the membrane, allow for highly selective processes to be developed.28-32 However, these membranes are mechanically unstable because they consist of a thin layer of organic solvent suspended between aqueous solutions.33 Polymeric membranes, wherein reactive sites are covalently bound to a crosslinked matrix, are a mechanically-stable alternative;34-35 however, the immobile reactive groups are not able to carry captured solutes across the length of the membrane, which limits the ability of these membranes to enhance solute flux.36-38 The prior efforts to mimic the facilitated transport mechanisms39 of biological membranes have focused almost exclusively on emulating the transport mechanisms of ion channels. Few studies have attempted to mimic the reversible conformational changes that ion pumps use to direct solute transport.15,

40

Therefore, due to the efficacy of ion pumps in biological systems and

promising results in the few studies aimed at mimicking their function,40 the present study develops 3 ACS Paragon Plus Environment

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a fundamental understanding of how multifunctional, responsive membranes can enhance the selective throughput of a target solute. In order to develop this understanding, the design of a composite membrane capable of exhibiting coordinated conformational changes is detailed (Figure 1). The composite membrane consists of two functional components (Figure S1). A pH-responsive gate layer, which is based on an amine-functionalized poly[acrylonitrile-co-oligo(ethylene glycol) methyl ether methacrylate-co-glycidyl methacrylate] [P(AN-OEGMA-GMA)] copolymer,41 and a matrix rich with iminodiacetic acid groups that can reversibly capture and release divalent cations. These chemistries were selected due to their ability to exhibit concurrent changes in the solute binding affinity and gate permeability in response to changes in the pH value of the feed solution. At high pH, the deprotonated amines of the gate layer allow solute to diffuse from the feed solution into the membrane where it is readily captured by the deprotonated iminodiacetic acid groups. When the pH is reduced, both functional groups protonate. The protonated iminodiacetic acid groups release the bound cations while the positively-charged ammonium moieties make the gate layer impermeable, which forces the released solute to permeate selectively towards the receiving solution. When exposed to an oscillating stimulus that induces periodic changes in the membrane conformation, the combination of these two elements exhibit a facilitated transport mechanism capable of emulating ion pumps.

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Figure 1. Schematic of the ion pumping mechanism driving transport. The mechanism relies on a cyclic process that consists of loading and unloading stages where the conformation of the membrane directs the flux of solute in a prescribed direction. During the loading cycle, the feed side of the membrane, which is coated with neutral amine functional groups, allows solute to flow into the membrane where it is captured by iminodiacetic acid groups that are dispersed throughout the matrix of the membrane. Subsequently, an external stimulus causes the amine groups to protonate and form positively-charged ammonium groups, which makes the feed side of the membrane impermeable. Concurrently, the stimulus causes the solute trapped within the membrane to be released and directed toward the receiving solution. In this study, solution pH was the stimulus used to periodically cycle between the two states.

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In order to aid in the design of the chemically-active composite membrane, a mathematical model that describes the ability of the membrane to enhance the flux of target solutes is developed. The model describes the performance of the membrane, e.g., the solute flux and selectivity, as a function of its properties, such as thickness, density of reactive sites, and solute diffusion coefficient, and operating conditions such as the concentration of solute in the feed solution, the length of the loading time, and the length of the unloading time. Subsequently, bench-scale experiments validate the model and demonstrate the ability of the polymeric ion pumps to preferentially transport calcium over sucrose even though they possess comparable hydrodynamic diameters.42 Finally, predictions made by the model are used to inform the design of future membranes and systems that exhibit this potentially-transformative transport mechanism. 2. Theory Consider a periodic system wherein a reactive membrane is forced between distinct loading and unloading states using an external stimulus. One side of the membrane is exposed to a donating solution (i.e., a feed stream) that contains a known concentration of target solute; the other side of the membrane is exposed to a receiving solution (i.e., a permeate stream) devoid of solute. By changing the magnitude of the stimulus, the interactions between the membrane and the target solute are altered (Figure 1). During the loading stage, the reactive sites within the membrane are active and trap solute molecules as they diffuse through the membrane. Conversely, during the unloading stage, the feed side of the membrane becomes impermeable and the reactive sites within the membrane release the captured solute. By periodically modulating the stimulus to switch between the two states, these effects can cause the solute to permeate preferentially into the receiving solution. 2.1. An Unsteady-State Mass Balance on the Unloading Stage 6 ACS Paragon Plus Environment

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The analysis begins directly after a change in the stimulus causes the membrane to transition from the loading stage to the unloading stage. At the start of the unloading stage, the solute that was captured during the preceding loading stage is instantly released. In addition, the interface of the membrane in contact with the feed solution becomes impermeable to the solute. Because the transport of the freed solute in the matrix is governed by Fickian diffusion, this physical picture leads to the following governing equation, boundary conditions, and initial condition: !" !#

!" !(

!'"

= 𝐷 !( '

(1)

(𝑡, 0) = 0

(2)

𝐶(𝑡, 𝐿) = 0 𝐶(0, 𝑧) = 𝐶(𝑡1 , 𝑧) where D is the effective diffusion coefficient of the solute, L is the membrane thickness, C is the concentration of the target solute, z is a spatial dimension, 𝑡1 is the loading time, and t is time during the unloading step. A single diffusion coefficient is used to describe diffusion of the ionic solutes through the membrane. This assumption could be relaxed at the cost of solving the more complex Poisson-Nernst-Planck equations for the composite membrane. However, since the matrix is neutral during the unloading stage and no solute transport is assumed through the positively-charged gate layer (i.e., the boundary condition at z = 0), the use of a single diffusion coefficient is a reasonable approximation. The initial condition, 𝐶(0, 𝑧), corresponds to the distribution of captured solute during the prior loading step. This system of equations is made dimensionless using the following constants: "

(

7

𝜍 = " , 𝜁 = 5 , 𝜏 = 5' 𝑡

(3)

3

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where 𝐶8 is the concentration of solute in the feed solution. Subsequently, Equation (1) and Equation (2) become: !9

!'9

!9

(𝜏, 0) = 0

= !; ' !:

!(

(4)

(5)

𝜍(𝜏, 1) = 0 𝜍(0, 𝜁) = 𝜍(𝜏1 , 𝜁) The partial differential equation and the boundary conditions are linear and homogenous. Thus, the concentration of solute within the membrane is given by: 43 R

'

ABC : 𝜍(𝜏, 𝜁) = ∑H 𝑐𝑜𝑠(𝛼? 𝜁) where 𝛼? = O𝑖 + ST 𝜋 ?IJ 𝛾? 𝑒

(6)

where γW are constants defined by the initial condition. R

𝛾? = ∫J 𝜍(𝜏1 , 𝜁) 𝑐𝑜𝑠(𝛼? 𝜁) 𝑑𝜁

(7)

At this stage of the analysis, the 𝛾? are left undefined because the functional form of the initial condition ς(τ\ , ζ), which depends on the nature of the preceding loading step, is not yet known. At the end of the unloading phase, the external stimulus is switched to initiate the loading stage. 2.2. An Unsteady-State Mass Balance on the Loading Stage During the loading phase, the gate layer is permeable and the open binding sites within the membrane matrix sequester solute. Assuming that the binding reaction between the diffusing solute and immobile binding sites is instantaneous, a solution for the concentration of captured solute can be obtained by noting that the interaction between the solute and active sites is confined to a reactive front 𝑙(𝑡), which slowly traverses the membrane as the active sites saturate.36 8 ACS Paragon Plus Environment

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Consequently, during the loading phase, the instantaneous solute flux to the reactive front at 𝑙 is given by: 7

𝑗(𝑡) = 1(#) `𝐶8 − 0b

(8)

Since the solute-active site interaction is much faster than the solute diffusion within the membrane, a time-dependent mass balance can be written for the saturated active sites: c c#

(𝐴 𝑙 𝐶ef# ) = 𝐴 𝑣 𝑗

(9)

where A is the area of the membrane in contact with the feed solution, 𝑣 is the stoichiometric coefficient for the reaction between the solute and active sites, and 𝐶ef# is the concentration profile of available active sites in the membrane at the start of the loading phase. 𝐶ef# may not necessarily be equal to the total membrane capacity, 𝐶h , because residual solute remaining in the membrane at the end of the precedent unloading stage, 𝐶(𝑡i , 𝑧), would be recaptured thereby reducing the number of sites available during the loading cycle, i.e., 𝐶ef# = 𝐶h − 𝑣𝐶(𝑡i , 𝑧)

(10)

The problem is normalized using the following constants: "

"

1

(

7

𝜍 = " , 𝜍j = l"k , ℓ = 5 , 𝜁 = 5 , 𝜏 = 5' 𝑡 3

3

(11)

and Equation (8) and Equation (10) are substituted in Equation (9) to obtain O𝜍j −

9(:n ,;) cℓ l

R

T c: = ℓ

(12)

Rearranging and integrating this expression results in Equation (13), which relates the penetration depth of the reactive front, ℓ, to the loading time, τ\ . 9 ACS Paragon Plus Environment

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ℓ

𝜏1 = ∫J 𝜁𝜍j − 𝜁𝜍(𝜏i , 𝜁) 𝑑𝜁

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(13)

Having written the distance the reactive front penetrates into the membrane in terms of the loading time allows the initial condition for the unloading stage [Equation (5)] to be modeled as a linear combination of the saturated membrane capacity and the residual solute from the preceding unloading step. A step function, 𝐻(𝜁), that transitions at ζ = ℓ was used to express this condition. 𝜍(𝜏1 , 𝜁) = 𝜍j 𝐻(ℓ − 𝜁) + 𝐻(𝜁 − ℓ)𝜍(𝜏i , 𝜁)

(14)

Whereas 𝜍j is a constant membrane property, ℓ and ς(τs , ζ) depend on the operating parameters of the process. At the end of the loading stage, a change in the stimulus forces the membrane to transition to the unloading stage starting a new cycle of the process. 2.3. A Pseudo-Steady State Emerges in Cyclic Systems The prior results that detailed the behavior during individual loading and unloading stages are combined to describe the transport of solute during a full period, i.e., 𝜏 t = 𝜏1 + 𝜏i

(15)

During the loading stage, the membrane acts as a barrier until all of the active sites have been saturated. Thus, no solute flows out of the membrane into the receiving solution while 𝜏1 < 𝜏ve# , ℱxi# (𝜏1 ) = 0 for 0 < 𝜏1 < 𝜏ve#

(16)

where

ℱ=

∫ {(#)c# "3 5

(17)

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and 𝜏ve# is the saturation time of the membrane. 𝜏ve# is identical to the lag time that is commonly defined in the barrier membrane literature.36, 38 The total amount of solute that diffuses into the membrane during the loading stage is given by integrating the concentration of available active sites over the distance the reactive front ℓ advanced. ℓ

ℓ

ℱ?| (𝜏1 ) = ∫J 𝜍ef# dz = ∫J 𝜍j − 𝜍(𝜏i , 𝑧) dz

(18)

During the unloading stage, the upstream interface of the membrane (i.e., the gate layer) is impermeable, hence, ℱ?| (𝜏i ) = 0

(19)

Concurrently, the previously captured solute is released and flows out of the membrane into the downstream receiving solution. The magnitude of this flow can be obtained by integrating the diffusive flux at the downstream interface over the length of the unloading period : c9

ℱxi# (𝜏i ) = ∫J n c; (𝜏, 1) 𝑑𝜏

(20)

where Equation (6) is used to find the local gradient at 𝜁 = 1. ~

R

'

C ABC :n ℱxi# = ∑H T sin(𝛼? ) where 𝛼? = O𝑖 + ST 𝜋 ?IJ B O1 − 𝑒 C

(21)

Over the course a single period, the amount of solute that has been loaded into the membrane is given by: ℓ

ℱ?| = ℱ?| (𝜏1 ) + ℱ?| (𝜏i ) = ∫J 𝜍j − 𝜍(𝜏i , ζ) dζ

(22)

And the amount of solute released from the membrane is provided by:

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~

'

C ABC :n ℱxi# = ℱxi# (𝜏1 ) + ℱxi# (𝜏i ) = ∑H T sin(𝛼? ) ?IJ B O1 − 𝑒 C

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(23)

For a continuously oscillating system where the operating parameters, 𝜏1 and 𝜏i , are kept constant, the system will eventually reach a pseudo-steady state. For this pseudo-steady state to be reached, it follows that the amount of solute loaded into the membrane must be equal to the amount of solute unloaded. ℓ

~

'

C ABC :n T sin(𝛼? ) ∫J 𝜍j − 𝜍(𝜏i , ζ) dζ = ∑H ?IJ B O1 − 𝑒 C

(24)

Additionally, under pseudo-steady state operation, the solute concentration profile becomes periodic. 𝜍(𝜏, ζ) = 𝜍(𝜏 + 𝜏 t , ζ) = 𝜍(𝜏 + 2τ t , ζ) = ⋯ = 𝜍(𝜏 + 𝑛𝜏 t , ζ) where 𝑛 ∈ ℕ

(25)

This constraint allows the distribution of solute at the end of the unloading step to be related to the unloading time. '

ABC :n 𝜍(𝜏i , ζ) = ∑H cos(𝛼? 𝜁) ?IJ 𝛾? 𝑒

(26)

Within the pseudo-steady state regime, the process has now been completely defined in terms of membrane properties and operating conditions. However, the complexity of the equations requires a numerical algorithm to evaluate, which obscures simple relationships between the performance and process parameters. An analytical solution can be produced by linearizing the concentration profile of the solute in the membrane during the unloading stage. This approximation relies on the exponential decay of the solute concentration profile during the stage. Due to this dependence, the non-linear concentration profile at the start of the unloading step decays rapidly to a quasi-linear profile. Consequently, it is assumed that 𝜍(𝜏i , 𝜁)~𝜖(1 − 𝜁)

(27) 12 ACS Paragon Plus Environment

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where 𝜖 is a time-dependent variable used to conserve the mass of solute within the membrane during the unloading stage. R

R

'

ABC :n cos(𝛼? 𝜁) dζ ∫J 𝜖(1 − ζ)dζ = ∫J ∑H ?IJ 𝛾? 𝑒

(28)

The system of Equations (13), (24), and (28) is further simplified by taking the leading eigenvalue 𝛼R =

‰ S

and substituting Equation (27) into each equation such that the following

system results. ℓ

𝜏1 = ∫J 𝜁𝜍j − 𝜁𝜖(1 − ζ) dζ ℓ

∫J 𝜍j − 𝜖(1 − 𝜁) dζ =

R ∫J 𝜖(1

S ~ ‰

(29) ‹'

Š1 − 𝑒 A:n Œ •

'

− 𝜁) dζ =

‹ R A: ∫J 𝛾 𝑒 n Œ

‰

cos O S 𝜁T dζ

(30)

(31)

Equations (29)-(31) are combined to obtain a direct relationship between the operating parameters 𝜏1 , 𝜏i and the membrane property 𝜍j . 𝜏1 = 𝜍j

ℓ' S

‹' ' ℓ •‘n Œ Š AR• ’ ‹' •‘ RA• n Œ (ℓAR )'

RŽ•

(32)

This then allows flow of solute out of the membrane into the receiving solution to be written as

ℓ“RA•

ℱxi# = 𝜍j RA•

‹' •‘n Œ ”

(33)

‹' •‘n Œ (ℓAR)'

where, for simplicity’s sake, the operating parameter 𝜏1 has been replaced by ℓ; Equation (29) can be used to determine the value of ℓ for different loading times. 13 ACS Paragon Plus Environment

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It is reassuring to examine the limiting case where the membrane is fully loaded, ℓ = 1, and fully unloaded at the end of each period, 𝜏i → ∞. In this case, the equations governing the dimensionless saturation time and solute flux simplify to 𝜏1 =

9— S

"

= Sl"k

(34)

3

"

ℱxi# = 𝜍j = l"k

(35)

3

respectively. In this limit, the saturation time, Equation (34), is consistent with the lag time derived for reactive barriers.36 Additionally, the solute flow per period, Equation (35), is equal to the membrane capacity, which is expected for a membrane that is fully saturated and then fully unloaded. 2.4. System Performance is Defined Relative to Diffusive Transport The performance of the system is defined as the solute flow normalized by the steady state diffusive flow of solute across a non-reactive membrane with similar physical properties. ™

𝜓 = ™ šn›

(36)

œC33

where ℱc?88 = 𝜏 t = 𝜏1 + 𝜏i

(37)

Therefore, by substituting Equation (33) and (37) into (36) the performance of the system is fully described in terms of membrane properties and system operating conditions.

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S 9— ℓ “RA•

𝜓=

‹' •‘n Œ ”

(38)

‹' ‹' ' ℓ •‘ •‘ S:n •RA• n Œ (ℓAR)' žŽ 9— ℓ' ŸRŽ• n Œ Š AR• ’

In addition to predicting the performance of the time-varying process described above, Equation (38) can be compared to the results of experimental measurements. 3. Experimental 3.1. Materials All chemicals were purchased from Sigma Aldrich unless noted otherwise. Calcium chloride and sucrose were used as solutes in transport experiments. Hydrochloric acid and sodium hydroxide were used to vary the pH of the feed solution. A Millipore water purification system (Milli Q Advantage A10, Milli Q, MA) was the source for the deionized water (DI water) used in the preparation for all aqueous solutions. Polyacrylonitrile (PAN), nonwoven polyester (PET) sheets (CU 414, Cranemat, MA), macroporous beads of chelating resin (S930, Purolite, PA), and dimethyl sulfoxide (DMSO) were used to fabricate the reactive layer of the composite membrane. A

poly[acrylonitrile-co-oligo(ethylene

glycol)

methyl

ether

methacrylate-co-glycidyl

methacrylate] [P(AN-OEGMA-GMA)] copolymer, which is described in detail in a prior study,40 was used to generate the gate layer. 3.2. Composite Membrane Fabrication The composite membranes developed in this study consisted of a gate layer situated on top of a reactive matrix. Wet-laid, thermally-bonded nonwoven PET were used as a support substrate for the membranes. Macroporous beads of a chelating resin that had been ground using a mortar 15 ACS Paragon Plus Environment

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and pestle and passed through a mesh sieve to remove any particles greater than 75 µm in diameter were used to introduce iminodiacetic acid groups into the reactive matrix. The reactive matrix was fabricated using a standard non-solvent-induced phase separation (NIPS) method. A casting solution consisting of 8% (by weight) polyacrylonitrile (PAN) and 4% or 16% (by weight) ground resin in dimethyl sulfoxide (DMSO) was stirred at 50 °C until the PAN was dissolved and the solution appeared homogenous. Directly after removing the solution from the stir plate, it was cast on the nonwoven support using a doctor blade set at a gate height of 100 or 300 µm. The cast film was immediately plunged into a water bath in order to precipitate the polymer and vitrify its structure. The surface of the newly formed film was dried in a ventilation hood for ~1 h at room temperature. However, the film was not completely dried in order to preserve its mechanical integrity. Subsequently, to form the bilayer composite, the P(AN-OEGMA-GMA) gate layer was cast on top of the reactive matrix. The synthesis of the P(AN-OEGMA-GMA) copolymer as well as the protocol used to introduce the responsive amine functionality that was capable of modulating the membrane permeability have been previously documented.40 Briefly, a 20% (by weight) P(ANOEGMA-GMA) in DMSO layer was cast on the dried surface of the reactive matrix using a doctor blade set at a gate height of 60-100 µm; then the solvent was allowed to evaporate for 180 - 600 s before plunging the composite membrane into a water bath. The introduction of the amine functionality within the copolymer was realized by placing the composite membrane in an aqueous solution of 1 M 1,6-hexane diamine at room temperature for 2 hrs.

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Table 1. Fabrication parameters for membranes used in this study.

Sample Name Membrane 1 Membrane 2 Membrane 3

Reactive Matrix PAN Resin Conc. Conc. [wt %] [wt %] 8 4 8 16 8 16

Evap. Timea [s] 3600

Blade height [µM] 300

3600 3600

100 100

Gate Layer Copolymer Conc. [wt %] 20 20 20

Evap. timeb [s] 600 180 180

Blade Height [µM] 100 60 60

DMSO was used as the solvent to prepare the casting solutions for the reactive matrix layer and the gate layer. aFor the reactive matrix, the evaporation time refers to the length of time that water was allowed to evaporate from the matrix before the gate layer was cast on top. bFor the gate layer, the evaporation time refers to the length of time that solvent was allowed to evaporate before the film was plunged into the non-solvent bath. Three of the composite membranes were fabricated using the parameters summarized in Table 1. For membranes 2 and 3, the mass of ground resin in the casting solution of the reactive matrix was increased in order to increase the concentration of reactive binding sites distributed throughout the matrix. Additionally, the overall thickness, which is directly related to the characteristic diffusion time of the membrane, was reduced by using smaller blade heights for both the gate layer and the reactive matrix. Because less solution was deposited using the lower blade height, the evaporation time for the gate layer was reduced to prevent complete solvent loss prior to plunging the composite film into the non-solvent water bath. The membranes were then rinsed and stored in DI water until future use. The thickness of the membranes was measured using a micrometer. 3.3 Design of the pH-swing Membrane Test Cell A diagram of the test unit developed for this study is shown in the Supporting Information (Figure S2); it is based on a design used in previous studies44-45 with the addition of a module to enable the oscillation of the donating solution (i.e. feed solution) pH. The crossflow membrane unit consisted of two flow channels that were separated by the membrane. Two gear pumps (Cole– Parmer) circulated the feed and permeates solutions from their reservoirs through the channels at a constant flow rate of 1 L min-1. A pH meter (Hanna Instruments), two Digital Remote-Control 17 ACS Paragon Plus Environment

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Solenoid-Diaphragm Metering Pumps (Cole Parmer), current output module (NI 9265), and analog input module (NI 9205) were part of the unit used to control the pH of the feed solution. The pH of the feed reservoir was continuously monitored and regulated by a PID controller program implemented in LabVIEW (National Instruments). In order to match the desired pH set-point, two independent control schemes regulated the flow rates of an acidic solution (0.66 M HCl) or a basic solution (0.5 M NaOH) into the feed reservoir. In this study, the feed reservoir pH was set to follow a constant frequency square wave function that oscillates the feed solution pH between 1.7 and 10. An aggressive controller was used to change the pH from pH 10 to pH 1.7. Specifically, the proportional constant dominated the controller response. In contrast, a controller dominated by the derivative constant was used to change the pH from pH 1.7 to pH 10. As a result, the system was able to reach the acidic set-point within 5 seconds while it took 20 seconds to reach the basic setpoint. This was done to minimize pH overshoots that could damage the membrane integrity at pH values that exceeded 10. 3.4. Execution of Transport Experiments All transport tests were performed in the diffusion test cell described above. Samples of the composite membranes that were 8 cm by 3 cm in size were placed between the two flow channels. After the membrane was secured, 0.5 L of a feed solution and 0.5 L of the permeate solution were placed in their respective containers under constant stirring; the feed solution consisted of 10 mM of the permeating solute and DI water was used as the receiving solution. The diffusion coefficient of the permeating solute was determined by measuring the solute flux at a constant pH of 10. The binding capacity of the membrane was determined by measuring the solute released into the permeate solution after the membrane was fully loaded (pH 10) and fully unloaded (pH 1.7). The flux was determined using the cell geometry and by evaluating the time 18 ACS Paragon Plus Environment

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rate of change of solute concentration in the permeate solution. The rate of change was calculated by sampling the permeate solution every 10 mins over 1 hour. The concentration of the drawn samples was determined using inductively-coupled plasma optically emission spectroscopy (ICPOES Perkin Elmer Optima 8000) for calcium and total organic carbon analysis for sucrose. Experiments where the effects of oscillating pH are studied are automated using the described controller system. A constant period is maintained over the length of the experiment and the solute flux into the permeate solution is measured. Prior to the start of each experiment, the membranes were exposed to an acidic solution at pH 1.7, and 10 full cycles were performed before measuring the concentration of solute in the permeate solution. This process helped to ensure that all membranes were operated with the same initial condition, i.e. fully unloaded, and that the measurements were performed once the system had reached pseudo-steady state. The time needed to reach the pseudo-steady state reflects the asymptotic approach to the pseudo-steady once the oscillations have been initiated. The analytical solution does not indicate how many cycles it takes for a system to reach the pseudo-steady state; hence we used a numerical approach to make this estimate. We looked at the residual solute within the membrane at the end of each cycle and numerical simulations suggest that membranes with the properties shown in Table 1 require 5-10 cycles to be within a 10% difference of the pseudo-steady state residual value (e.g., see Supporting Information Figure S3). And while the number of oscillations needed to reach the pseudo-steady state was a function of membrane properties, process variables, and the initial state of the membrane, we did not attempt to elucidate the exact functional form of this relationship. As such, the 10 cycles and 10% error parameter used in this study should not be taken as intrinsic constants of the transport model but as empirical variables that allowed for an accurate comparison between

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repeated experiments. Subsequently, 20 cycles were performed and the increasing concentration of solute in the receiving solution was used to determine the time-averaged flux. 4. Results and Discussion 4.1. Numerical validation of the analytical model When exposed to a time-varying stimulus, the composite membranes detailed in this work were designed to mimic ion pumps found in cell walls. The extent of the flux enhancement provided by this facilitated transport mechanism is quantified using the performance parameter, 𝜓. Due to the crucial role that Equation (38) served in the design of the polymeric ion pump system, the validity of the simplifying assumptions used in its derivation, in particular the pseudo-steadystate approximation, were tested by comparing the model results with numerical solutions for the full set of governing equations (i.e., Equation (4) and Equation (13)).

Figure 2. A comparison of the analytical and numerical solutions for the governing equations. The performance obtained by numerically solving the governing equations is plotted on the y-axis, and the value predicted by Equation (38) is plotted on the x-axis. Each marker corresponds to a membrane system operated with a unique set of conditions. The membrane 20 ACS Paragon Plus Environment

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capacity, 𝜍j , unloading time, 𝜏i , and loading time, 𝜏1 , were varied. The black solid line indicates perfect agreement between the analytical solution and the numerical solution The performance of membranes with capacities, 𝜍j , ranging from 0.1 to 1000, that were operated at a series of loading/unloading time pairs is displayed in Figure 2. The time pairs were chosen at random subject to the constraint that the saturation time for the membrane was not exceeded (i.e., 0 < ℓ < 1). The long-time values of 𝜓 obtained by solving the system of equations numerically are plotted on the y-axis and the predictions made by Equation (38) are plotted on the x-axis. The results all fall on or near the 45° line, which suggests excellent agreement between the two solution methods and offers support for the simplifying assumptions made during the development of the model. In addition to validating the model, the results in Figure 2 demonstrate that membranes and operating conditions that exhibit enhanced solute transport (i.e., 𝜓 > 1) are feasible. 4.2. Experimental Realization of Ion Pumps Multifunctional composite membranes, which consisted of a gate layer that modulated the solute permeability coated on a matrix lined by reactive solute binding ligands, were fabricated to demonstrate the ability of the time-varying process to enhance solute transport. The bilayer structure shown in Figure 3 was achieved by sequentially coating two distinct polymer solutions onto a non-woven support. The reactive matrix of the membrane, which consisted of S930 cation exchange resin ground to below 75 µm in diameter and dispersed in microporous polyacrylonitrile (PAN), was coated onto the non-woven first [micrographs of the ground resin are provided in the Supporting Information as Figure S4]. The S930 resin is functionalized with iminodiacetic groups that bind divalent cations at high pH and release them at low pH. Next, the gate layer was formed by coating a P(AN-OEGMA-GMA) copolymer onto the preformed matrix. This particular 21 ACS Paragon Plus Environment

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copolymer microphase separates to form a nanoporous structure whose pore walls can be modified with amine functional groups.40,46 At high pH, the neutral amine groups allow solute to permeate freely into the membrane. At low pH, the amine groups protonate to form positively-charged ammonium moieties that impede cation permeation back into the feed solution during the unloading stage. Three of these membranes were fabricated and their properties are summarized in Table 2.

Figure 3. Cross-sectional scanning electron microscopy (SEM) micrograph of the composite membrane. a) A dense P(AN-OEGMA-GMA) copolymer comprises the gate layer, which sits on 22 ACS Paragon Plus Environment

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top of the reactive layer containing S930 chelating resin dispersed in phase inverted PAN. b) The porous polymeric matrix of the reactive matrix surrounds a grain of ground resin. The resin was ground and sieved through a 75 µm mesh; hence, the resin grain within the membrane are not larger than 75µm. The resin in this figure is ~4 µm in diameter. Table 2. Characteristic properties for the membranes used in this study.

Membrane 1 Membrane 2 Membrane 3

𝐶h [mol L-1] 3.0E-3 2.0E-2 2.6E-2

𝐶h 𝑣𝐶8 0.3 2.0 2.6

𝑡ve# [s] 2361 2812

𝐷 [mS s AR ] 8.57E-11 7.58E-12 1.04E-11

𝐿 [µm] 374 150 240

The capacity was determined by performing full-loading and full unloading cycles. It was corroborated by measuring the saturation time. The calcium diffusion coefficient was measured at pH=10. In all cases, the feed solution was 10mM CaCl2. The iminodiacetic acid groups covering the matrix of the membranes reversibly capture a variety of divalent cations as a function of pH.47 However, many of these cations form insoluble hydroxide complexes in the alkaline conditions needed to deprotonate the amine groups of the gate layer, which have a pKa of ~10.48 Therefore, calcium was chosen as the target solute due to the high solubility of its hydroxide salt.49 An aqueous solution containing 10 mM CaCl2 was used as the feed solution for all experiments and DI water was used as the receiving solution. The number of calcium binding moieties dispersed throughout the membrane matrix, i.e., the membrane capacity, was determined by measuring the flow of solute into the receiving solution at the end of "

full loading-full unloading cycles. The values for 𝜍j = l"k were corroborated by measuring the 3

saturation time and using Equation (13) to estimate the capacity for the membranes. In all cases, the two values were on the same order of magnitude. As anticipated, the capacity of the membranes increased as the mass of resin dispersed in the casting solution formulation increased.

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Figure 4. The effect of oscillating pH on solute permeation. a) The concentration of calcium permeating into the receiving solution is plotted as a function of time. The black dots indicate measured values of calcium concentration. The background color specifies the pH value of the feed solution. A blue background represents a pH value of 10; a red background represents a pH value of 1.7. The oscillations were initiated at t = 0.5 h. b) The concentrations of calcium and sucrose in the receiving solution are plotted as a function of time. The feed solution contained both solutes at a concentration of 10 mM. For the first hour, the pH value of the feed solution was held constant. Subsequently, the pH value of the feed solution oscillated between pH 1.7 and pH 10. A loading time of 525 s and an unloading time of 281 s were used. Note: the plot is truncated to compare solute permeation at a constant pH value and at while operating at pseudo-steady-state; an unabridged version is provided in the Supporting Information (Figure S5). 24 ACS Paragon Plus Environment

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Figure 4a displays a characteristic data set showing how the experimentally-measured concentration of calcium in the receiving solution varied as a function of time under both constant pH, i.e., free diffusion, and oscillating pH conditions. In both cases, the time-averaged slope of the data was proportional to the solute flux. Prior to the start of the experiment reported in Figure 4a, the membrane was soaked in a pH 10 CaCl2 solution to saturate all of the available binding sites. As such, the initial three data points increase linearly with time, which is consistent with steadystate diffusion. Because the slope of the line under these conditions is proportional to the flux of calcium, it was used to determine the effective diffusion coefficient for CaCl2 in the membrane. The variations of calcium concentration in the oscillating regime are evidently not linear and can be divided into two well-defined stages. During the loading stage, which is highlighted by the blue background, the calcium concentration in the receiving solution does not change significantly. Instead, the calcium that permeates into the membrane is sequestered by the available binding sites before it reaches the receiving solution. During the unloading stage, which is highlighted by the red background, a rapid increase in the calcium concentration is evident. This is a result of the trapped solute being released and diffusing into the receiving solution. Relative to the slope of the diffusion data collected at the outset of the experiment, the slope of the experimental data measured during the unloading stage is steeper, which is indicative that the instantaneous flux is greater. The increased driving force, i.e., concentration gradient, that developed due to the rapid release of bound solute at the outset of the unloading phase is what generated this increased flux. And as highlighted by the data presented in Figure 4b, this augmented concentration gradient drives enhanced fluxes that allow the composite membranes to selectively transport a target solute from a feed solution containing multiple solutes.

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The composite membranes were designed to target divalent cations; and the ability of the polymeric ion pumps to selectively transport calcium over sucrose, a neutral solute with a comparable hydrodynamic diameter to that of calcium,41 is demonstrated in Figure 4b. In this experiment, membrane 3 was challenged with a feed solution containing 10 mM sucrose and 10 mM calcium chloride. The concentrations of calcium chloride, indicated using black markers, and sucrose, highlighted using white markers, in the receiving solution are plotted as a function of time in Figure 4b. For the first hour, the system was operated at a constant pH value of 1.7. Then, oscillations in the feed solution pH were initiated (𝑡1 = 525 s, 𝑡i = 281 s) and the system was allowed to reach pseudo-steady-state. This experiment was repeated three times and data from the repeated experiments are aggregated in Figure 4b. The data demonstrates that whereas both solutes permeate at a similar rate when the pH value was held constant, calcium permeates four times faster than sucrose when the pH value was varied periodically. Since the gate layer is impermeable during unloading stage, the increased flux must be directly related to the solute released from the binding sites within the reactive matrix. Moreover, this observation highlights the ability of the polymeric ion pumps to selectively transport molecules based on their chemistry not their physical size. The predictive ability of the analytical model, which is displayed in Figure 5, was instrumental to realizing selective calcium fluxes that exceeded those observed in diffusive systems. The experimentally-determined values of 𝜓 for a series of operating conditions (Table 3) are compared to predictions made by the model for the same conditions in Figure 5. As detailed in the Experimental Section, care was taken to ensure that the experimental values were determined once the system had approached pseudo-steady-state. Each operating condition was repeated at least twice and the average value with its respective standard deviation is plotted. For most of the 26 ACS Paragon Plus Environment

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conditions examined, the measured performance agrees quite well with the analytical prediction. Two pair of operating conditions, membrane 2 (tl = 150 s, tu = 300 s) and membrane 2 (tl = 300 s, tu = 600 s), do show relatively large standard deviations, which highlights potential sources of error in the current experimental design. Namely, the pH control loop is effective in correcting departures from the set-point, and these deviations are corrected quickly. However, for high frequency experiments these delays are significant and disturb the pseudo-steady-state.

Figure 5. Comparison of experimental performance to model predictions. The experimentally-determined performance is plotted versus the performance predicted by Equation (35). The experimental values were obtained by subjecting three different membranes to a series of loading/unloading time pairs. Each marker corresponds to a unique pair of operating parameters. The black solid line indicates a perfect agreement between experimental measurements and analytical predictions. In addition to this operational consideration, the analytical solution relies on the assumption of a perfect gate layer. However, the gate layer is not a perfect barrier and some leakage of solute into the feed solution during the unloading stage is anticipated. Despite these issues regarding the current system, the membranes were capable of rapidly capturing solute and then directing its permeation selectively toward the receiving solution and the model predicted the magnitude of this unique response. 27 ACS Paragon Plus Environment

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Table 3. Operating parameters, loading time and unloading time, used in the experimental trials plotted in Figure 5. Membrane 1 𝒕𝒍 [s] 15 60 375

[s] 30 30 30

𝒕𝒖

Membrane 2 𝒕𝒍 [s] 32 150 300 300 869

[s] 225 300 300 600 1170

𝒕𝒖

Membrane 3 𝒕𝒍 [s] 196 525

[s] 39 281

𝒕𝒖

4.3. High-Frequency Oscillations Maximize Performance The relationship between membrane properties and performance described by Equation (38) informed the fabrication of membranes 2 and 3. In particular, relative to membrane 1, membranes 2 and 3 were three times thinner and included four times the amount of ground resin. More critically, Equation (38) aided in the identification of the high frequencies oscillations needed to generate 𝜓 values that exceeded one. Optimum values for the loading and unloading times were sought by searching for the critical points of Equation (38) !ª

O !ℓ T

:n

=0

(39)

The only non-trivial solution for Equation (39) is

𝜏i =

ℓ' S

1' "

𝜍j = S5'l"k

(40)

3

which identifies the value of the unloading time that maximizes performance for a given penetration depth of the reactive front. Equation (40) can be substituted into Equation (32) to directly relate the loading time to the unloading time under this constraint.

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𝜏1 = 𝜏i

‹' '‘ •‘n Œ O n ART ’«— ' ‹' '‘ •‘ RA• n Œ Š¬ n AR • «—

RŽ•

(41)

Furthermore, Equation (40) can be substituted into Equation (38) to identify the maximum value of 𝜓 that can be realized for a system, given 𝜍j and 𝜏i .

𝜓-ej =

® ‹' • n Œ ‘n ‹' '‘ '‘ '‘ • Œ Š¬ n A n AR•° ¬ « n ¯RŽ• «— ’«— —

RA•

(42)

Equation (42) is asymptotically bound by the following constraints.

lim [𝜓, 𝜓-ej ] = 𝜍j

#n →J

‰' ²

"

~ l"k

(43)

3

lim [𝜓, 𝜓-ej ] = 0

#n →H

This behavior is highlighted in Figure 6, where a heat map of the performance is plotted as a function of the unloading time and reactive front location for a membrane with 𝜍j = 10. The pairs of 𝜏i and ℓ that maximize performance, Equation (40), are identified by the diagonal white line on the color map. This line defines the operating parameters that produce local maxima in 𝜓. There is not a universal critical point that maximizes performance within the phase space. Instead, as indicated by Equation (43), the maximum performance is found in the asymptotic limit of 𝜏i → 0 (i.e., high frequency oscillations).

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Figure 6. The performance 𝜓 for a membrane with 𝜍j = 10 is plotted as a color gradient for the phase space of unloading times 10AS < 𝜏i < 10R and reactive front locations 10AS < ℓ < 1. The pair of 𝜏i and ℓ that maximize performance, Equation (40), is identified by the diagonal white line. The need to operate using high frequency oscillations can be rationalized by considering the time-averaged driving force for solute permeation. Namely, performance is maximized when the time-averaged driving force (i.e., concentration gradient across the membrane) is large. As such, the increased fluxes that were observed during the unloading phase in Figure 4a are desirable. These increased fluxes were driven by the steep concentration gradient that developed when solute was released from the binding moieties within the reactive matrix. Then, as the unloading phase progresses and the released solute permeates into the receiving solution, the magnitude of the gradient becomes smaller. Consequently, the instantaneous flux is greatest at the outset of the unloading phase and decays as the unloading phase progresses. Oscillating between the loading and unloading stages quickly prevents a significant mass of solute from permeating out of the membrane during any single unloading stage. Operating in this manner maintains a steep concentration gradient and reduces the amount of time needed to reload the membrane with solute thereby maximizing the time-averaged driving force and performance. 30 ACS Paragon Plus Environment

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Using simple scaling arguments (Supporting Information), the maximum receiving solution concentration, Cp, that solute can be pumped against can be identified in the limit of rapid oscillations. Specifically, a polymeric ion pump with a binding site concentration, Cb, that is exposed to a feed solution with target solute at a concentration, Cf, can move solute from the donating feed solution to the receiving permeate solution as long as the following expression is true. 𝐶³