Solute hindrance in non-porous membranes: An ATR-FTIR study

DES-12516; No of Pages 9 Desalination xxx (2015) xxx–xxx

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Solute hindrance in non-porous membranes: An ATR-FTIR study Derrick S. Dlamini a,b,c, Stanislav Levchenko b, Maria Bass b, Bhekie B. Mamba a, Eric M.V. Hoek d,e, Justice M. Thwala c, Viatcheslav Freger b,⁎ a

University of South Africa, Nanotechnology and Water Sustainability Research Unit, College of Engineering, Science and Technology, Florida Campus, Johannesburg, South Africa Department of Chemical Engineering, Technion—Israel Institute of Technology, Haifa, Israel Department of Chemistry, University of Swaziland, Private Bag 4, Kwaluseni, M201, Swaziland d Department of Civil & Environmental Engineering, University of California, Los Angeles, Los Angeles, CA, United States e California NanoSystems Institute, University of California, Los Angeles, Los Angeles, CA, United States b c

H I G H L I G H T S • Diffusion of dye molecules in poly(vinyl alcohol) films measured by ATR-FTIR • Measured diffusivity values show very large discrepancies with hindered transport theory. • Results suggest that HTT may not be suitable for computing solute mobility in RO and NF.

a r t i c l e

i n f o

Article history: Received 16 November 2014 Received in revised form 4 March 2015 Accepted 6 March 2015 Available online xxxx Keywords: Thin films Diffusion Solute transport Partitioning coefficient ATR-FTIR

a b s t r a c t The hindered transport theory (HTT) is widely used to analyze the size-dependence of transport in membranes such as reverse osmosis and nanofiltration, however, serious concerns have been raised recently about its suitability for molecular solutes. To address this concern, the diffusivities of dye molecules (189 to 961 Da) in a crosslinked poly(vinyl alcohol) film were measured using ATR-FTIR spectroscopy. Possible errors due to liquid layers on both sides of the film were analyzed and appeared to have a minor effect on the results. The values of diffusion coefficient were found to be in the range 3.3 × 10−14 to 4.4 × 10−13 m2/s. Based on the pore size of 2.24 nm estimated from hydraulic permeability, these values were about 3 orders of magnitude lower than the predictions by HTT. This result strongly suggests that HTT may not be a suitable model for the transport of small molecules in membranes such as RO and NF, pointing to the need for more adequate models. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The transport and separation of penetrants by non-porous polymeric membranes in reverse osmosis (RO) and nanofiltration (NF), as well as pervaporation, vapor permeation, and gas separation, involve preferential sorption and diffusion of a small molecule in a polymeric medium. Mass transport models are essential for understanding the mechanisms governing this process and for engineering calculations. The transport of trace organic rejection by NF and RO membranes was one challenging field where elucidation of transport mechanism and modeling attracted a significant research effort [1,2]. Numerous mechanistic and mathematical models have been proposed to describe membrane transport [3] One aspect of permeate transport across membranes is the solute mobility. In porous or fibrous materials where the pore diameter or inter-fiber spacing is comparable to the dimensions of the solute, its mobility ⁎ Corresponding author. E-mail address: [email protected] (V. Freger).

becomes lower than in bulk solution. This phenomenon is termed hindered transport [4]. The classical hindered transport theory (HTT) considers hydrodynamic hindrance of a spherical solute moving in a cylindrical pore filled with a uniformly viscous bulk-like continuum and the membrane is viewed as an array of such pores [5]. The diffusive (Kd) and convective (Kc) hindrance factors express reduction of, respectively, the mobility (diffusivity) and drag velocity relative to the bulk values. For a purely hydrodynamic hindrance these hindrance factors appear to depend exclusively on the ratio of the solute and pore sizes (λ = rs / rp). This version of the hindered transport theory has been widely applied in the modeling of RO [5–8] for predicting hindrance of molecules within the swollen active layer. Kosto and Deen [9] pointed out that the attractiveness of HTT-based estimates is diminished in the absence of a clear connection between pore parameters and the sizes and/or volume fractions of the crosslinked gel polymers. Originally, HTT was developed and tested for relatively large solutes, e.g., colloidal particles or macromolecules, and well-defined pores down only to about 3 nm [5]. While it has been

http://dx.doi.org/10.1016/j.desal.2015.03.009 0011-9164/© 2015 Elsevier B.V. All rights reserved.

Please cite this article as: D.S. Dlamini, et al., Solute hindrance in non-porous membranes: An ATR-FTIR study, Desalination (2015), http:// dx.doi.org/10.1016/j.desal.2015.03.009

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widely used in the modeling of NF and RO for much smaller pores and solutes [10], it is surprising that the extension of HTT to small solutes and dense or gel-like polymeric matrices has not been critically examined, though substantial discrepancies were reported for some macromolecular diffusants in gels [11]. The measurements of diffusion coefficient (D) appear to be a convenient way to test the validity of HTT relations, since it can be measured in static conditions without flow thus diffusivity can be decoupled from convection. Several methods have been used to study the diffusivity of penetrants, mainly, macromolecules, in polymer solutions or gels matrices, in the general context of understanding the reduced mobility in such media [12,13]. Recently, in the specific context of suitability of HTT to RO modeling, Dražević et al. reported first measurements of diffusion of organic solutes in polyamide films of genuine RO membranes using the attenuated total reflectance (ATR)-Fourier transform infrared spectroscopy (FTIR) [14]. This non-destructive approach, first proposed by Fieldson and Barbari [15], is highly suitable for polymer films in equilibrium with a solution. The results pointed to very significant deviations from HTT predictions, by as much as 2–3 orders of magnitude. A similar conclusion was made by Johnston et al. in a study of hindered convection of somewhat larger nacomolecular solutes (proteins) in agarose gels [11]. It was argued that the results were not represented well by the available theories either for parallel arrays of fibers or for straight pores. The dependence of diffusivity measured by Dražević et al. [14] on permeant size also showed no sharp cut-off, a fingerprint of the purely hydrodynamic version of HTT. In the present study we extend the measurements of Dražević et al. [14] to cross-linked poly(vinyl alcohol) (PVA), another polymer used for making NF membranes [16–19] as well for low-fouling coatings in RO and NF [20]. Apart from being relevant to NF and RO, the preparation of PVA as a film or bulk material does not impose any limitation on thickness, in contrast to what interfacial polymerization does for polyamide films. Another advantage is that the swelling (“porosity”) of PVA can be tuned through cross-linking. PVA was then a popular matrix for measuring diffusion in gels. For instance, Wang et al. [21] used pulsed-gradient spin-echo Nuclear Magnetic Resonance (NMR) spectroscopy to study the diffusion behavior of the star polymers, ranging from 1000 to 10,000 g/mol, in aqueous solutions and gels of poly(vinyl alcohol) (PVA). In another NMR-based method, Shapiro and Shapiro [22] investigated the multicomponent self-diffusion of the PVA cryogels prepared by a freezing–thawing treatment of the water/oligooxyethylene glycol solutions of PVA. The ATR-FTIR methods have been used for over three decades, albeit sparingly, in measuring swelling and diffusional mobility in polymers [11,15,23]. It is crucial for accurate measurements of diffusion coefficient that a polymer film of high uniformity and well-defined geometry be used. In particular, the film thickness needs to be chosen to ensure the easily measurable yet not excessively long diffusion times and minimal background IR absorbance by the solution, to which the film is exposed. In the study of polyamide films by Dražević et al. [14] the thickness was increased by overlaying 5–10 polyamide films in a rather tedious preparation, still yielding only a film well under a micron thick. Herein, the thickness of a PVA film was chosen to be much larger, of the order of 10 μm, allowing substantially longer diffusion times and a negligible background. Yet another motivation to examine diffusion in PVA alongside polyamide was the morphology of polyamide that is inherently irregular and the results are harder to interpret. Herein, we used homogeneous PVA films cross-linked with trans-aconitic acid that could be reproducibly prepared using a protocol developed earlier [24]. Finally, the linearity and additivity of IR bands of the solvent, PVA and solutes allow monitoring of appropriate bands of the penetrant along with the bands PVA and solvent to ensure that the swollen film does not undergo any major change during diffusion measurements. The main purpose of this article is to analyze the validity of the hindered transport theory for small solutes in NF membranes. However,

we believe that the results might also be useful in the more general context of transport in cross-linked polymeric gels. Gels and hydrogels are widely used in chromatographic separations, therapeutic devices, drug delivery etc. Predicting rates of molecular transport in these materials and elucidation of mechanism involved are then of significant relevance to both industrial and physiological processes and applications. 2. Theoretical relations The set-up used in this work is schematically shown in Fig. 1a. The film of a known thickness L is attached to the ATR crystal on one side and exposed to a solution on the other side. The solution initially contains no solute. After equilibration with the film, the solution is instantly replaced with one of concentration C0 and the IR band of the solute is monitored over time. The model relations given below are required for deducing diffusion coefficient D of the solute in the film from the variation of band intensity over time. Fitting the result to a full model is preferred here over approximate and often inconsistent calculations based on the time lag or the slope of the pseudo-steady-state portion of the band intensity versus time [25,26]. It is important to consider possible artifacts related to imperfections of the set-up, the most important of which comes from the unstirred solution layer adjacent to the film and the solution-filled gap between the film and the crystal. These interferences are analogous to those faced in electrochemical impedance spectroscopy (EIS) experiments in a similar set-up with the crystal replaced by a solid electrode. For instance, in the present experiments the gap could occur as a result of minor changes in the swelling of the film, which could not be entirely ruled out during the measurements, even though the film was always pre-swollen prior to mounting on top of the crystal. The contribution of the unstirred layer is relatively easy to quantify by performing an experiment without a film (cf. the experiment with a bare electrode in EIS [27–29]). The results (presented in Section 4.1) indicate that the diffusion resistance of the unstirred layer and corresponding diffusion time were small and could thus be ignored. On the other hand, the role of the gap is more difficult to quantify experimentally therefore it needs to be elucidated using a model. For this reason it is useful to consider a more general situation whereby a gap of thickness d is formed between the crystal and film, as schematically shown in Fig. 1b. Note that the set-up ensured that the gap was small (see Materials and Methods, Section 3.3) therefore its diffusion resistance was fairly negligible. The dominant interference by the gap is due to its non-zero solute accumulation capacity, i.e., it mainly acts as a sink rather than an impermeable interface and thus delays reaching the equilibrium profile within the film (Compared to the gap capacitance in EIS [28,29].) In this situation, the model may only include the solute accumulation capacity of the gap and ignore the diffusion resistance across the gap, which is tantamount to assuming a uniform concentration in the gap that only changes with time. The diffusion within the film is then described by a one dimensional Fick's second equation 2

∂C ∂ C ¼D 2 ∂t ∂x

ð1Þ

with the following boundary conditions at the membrane-solution and membrane–gap interfaces: x ¼ L; C ¼ KC 0 ; ∂C d ∂C þ ¼ 0: x ¼ 0; D ∂x K ∂t

ð2Þ

Here C(x, t) is the concentration profile at time t in the films, i.e., for 0 b x b L, and K is the partitioning coefficient between the film and solution, thereby C(x = 0, t) / K is the concentration in the gap, i.e., for

Please cite this article as: D.S. Dlamini, et al., Solute hindrance in non-porous membranes: An ATR-FTIR study, Desalination (2015), http:// dx.doi.org/10.1016/j.desal.2015.03.009

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Fig. 1. Schematic of the ATR-FTIR set-up: (a) without a gap and (b) with a gap present.

−d b x b 0, at time t. The general solution of Eq. (1) with b.c. given by Eq. (2) was given by Carslaw and Jaeger [30]. This yields C(x, t) that may be integrated across the PVA film and the gap to yield the variation of the intensity of the solute band with time in the ATR setup with a gap. The integration involves a convolution of the profile with the exponential decay of the evanescent wave intensity, as follows: Z∞ εC ðx; t Þ exp½−2γðx þ dÞdx;

Aðt Þ ¼

ð3Þ

−d

where ε is the parameter depending on the extinction coefficient of the band and the refractive indices of the crystal, solution (in the gap and above the film) and the film, and γ is the reciprocal attenuation length (penetration depth) of the evanescent wave. We further note that the refractive indices of the film and solution do not differ much, in which case ε and γ can be assumed to be constant and identical for film and solution. Finally, we note that the penetration depth 1/γ is of the order of 1 μm, while L used in this study was an order of magnitude larger, therefore γL ≫ 1. This means that only a fraction of the film closest to the crystal contributes to the IR signal and contribution of the solution above the film may be neglected. This yields the following expression ! ∞ X Aðt Þ 1− expð−2dγ Þ ¼ 1− An sinðα n LÞ Að∞Þ 1 þ expð−2dγÞðK−1Þ n¼1 ! ð4Þ ∞ X K expð−2dγ Þ 2γ sinðα n LÞ−α n cosðα n LÞ 1−2γ ; þ An 2 2 1 þ expð−2dγÞðK−1Þ α n þ 4γ n¼1

swollen (wet) film, D the average diffusivity of the solute in the film in the normal direction, and γ is the reciprocal penetration depth of the specific band in the film. The two unknowns in Eq. (6), D and A(∞), are estimated by fitting the model to the experimental data. 3. Materials and methods 3.1. Preparation of cross-linked PVA films The PVA film used in the diffusion cell was prepared by casting in a polypropylene rectangular plate. Mowiol® PVA 6-98 with a molecular weight of 47 kDa, 98.0–98.8% hydrolyzed (Sigma-Aldrich), was dissolved in deionized water at 90 °C to obtain a 25 w/v % PVA solution. This concentration (25%) was chosen after a series of experiments to select a solution that has a viscosity that allows casting. Trans-aconitic acid (Sigma-Aldrich) was used as a cross-linking agent.

The cross-linking agent was added along with an acid as catalyst under continuous stirring for 2 h to produce the PVA casting solution. The cross-linking agent was added to produce a theoretical crosslinking degree χ of 70% defined as follows:

where

    2 α n 2 þ h2 K 2   exp −Dα n t ; h ¼ ; and parameters αn An ¼   2 d α n L α n þ h2 þ h for n = 1 to ∞ are solutions of the equation tanðα n LÞ ¼

h d or; equivalently; ðα LÞ sinðα n LÞ ¼ cosðα n LÞ: αn LK n

ð5Þ

The latter form of Eq. (5) was found to converge for all n and robustly place all α n within the correct intervals (n − 1)π b α nL b (n − ½)π. Without the gap, i.e., for d = 0, α n = (n − ½)π / L and Eq. (4) is reduced to the well-known formula of Fieldson and Barbari simplified for γ L ≫ 1 as follows:   ∞ X expðgn Þ f n expð−2γLÞ þ ð−1Þn ð2γ Þ Aðt Þ 8γ  2  ¼ 1−  Að∞Þ π ½1 þ expð−2γLÞ n¼0 ð2n þ 1Þ f n þ 4γ 2 ≈ 1−

∞ 4X ð−1Þn expðg n Þ ; π n¼0 2n þ 1

Dð2n þ 1Þ2 π2 t ; 4L2

fn ¼

ð2n þ 1Þπ : 2L

ð6Þ

In Eq. (6), A(t) is the intensity at time t, A(∞) the maximum intensity at infinite t, i.e., in equilibrium with solution, L the thickness of the

W CL  MW PVA;unit  2  100; W PVA  MW CL

ð7Þ

where WPVA and WCL are the weight of PVA and cross-linker, respectively, MWPVA, unit and MWCL represent the molecular weight of the PVA monomer (–CHOH–CH2–) and molecular weight of the cross-linking agent, respectively. A casting knife was used to get a film of defined thickness. Finally, the film was dried in an oven programed to 80 °C for 5 h. The following factors were considered when choosing film thickness range: First, the time required for a small molecule to diffuse through a film of thickness L is of the order L2 / 2D. Times of the order 100–1000 s were considered reasonable. For expected diffusivity of the order 10−13 m2/s this sets the upper limit of film thickness at about 10–20 μm. Second, it was preferable to choose thickness such that γ L ≫ 1 (see Section 2) or L ≫ dp. The depth of penetration dp = 1/γ of the infrared beam into the sample depends on ns, nIRE, θ, and λ as follows: dp ¼

where gn ¼ −

χ ½%  ¼

λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2π nIRE sin2 θ−ns 2

ð8Þ

where λ is the wavelength of IR radiation, θ = 45° the angle of incidence of an IR beam, and ns and nIRE = 2.4 are the refractive indices of the sample (rarer medium) and the diamond ATR crystal, respectively. The refractive index of hydrated PVA film was between that of dry

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Fig. 2. ATR-FTIR diffusion setup: ATR crystal with a stack of PVA films on top exposed to a solution. Fig. 3. Measured water sorption kinetics for a 25 μm thick PVA film.

PVA n ≈ 1.53 [31,32] and that of water n ≈ 1.33. This confines the penetration depth to (0.18 ± 0.03)λ. For the relevant range of wavenumbers 1800 ≥ 1/λ ≥ 1000 cm−1, i.e., wavelengths 5.5 ≤ λ ≤ 10 μm, the penetration depth does not exceed 2 μm. The film thickness L was then selected to be significantly larger that dp to ensure that exp(− 2 L/dp) ≪ 1 (cf. Eqs. (3) to (6)) and below the upper limit set by the diffusion time (see below). The intermediate value of 8 μm in the dry state was chosen to meet both criteria. 3.2. Film characterization

oven at 60 °C overnight to ensure a stable initial weight. Thereafter, single component liquid sorption was measured by immersing the film pieces in deionized water. At time intervals of 10 s, the specimens were removed from the water bath, and the penetrant that adhered onto the surfaces was removed by carefully pressing the samples between soft filter papers with minimum pressure and immediately weighed using an analytical balance. The immersion and weighing of the samples were repeated until the specimen attained equilibrium. The effective pore radius (rp) was determined from pure water permeability (Lp) using the following relation:

3.2.1. Water permeability The separation performance of the PVA films was tested in a stainless steel bench-scale stirred cell with a membrane area of 12.5 cm 2 at 600 rpm. Prior to each test, the membranes were compacted with deionized water at 350 psi for 2 h. After compaction, the pure water flux of the film was determined at 25 °C and applied pressures of 100, 200 and 300 psi by weighing the permeate over time. The pure water permeability Lp was determined from a linear fit of the measured water flux versus the applied pressure data. The reported flux data represent the averages of three separate tests of films prepared on different days using independently prepared PVA casting solutions.

3.3. Diffusion measurements by ATR-FTIR

3.2.2. Swelling measurements Water sorption was measured by the ‘blot and weigh’ immersion protocol. The samples were cut into rectangular sheets and dried in an

Infrared spectroscopy was performed on a Thermo-Nicolet FTIR Spectrometer equipped with a MIRacle single-reflection attachment with a diamond ATR element and Omnic 7 plus software. The set-up

Lp ¼

br2p ϕ μLα

ð9Þ

where b is a geometric parameter that may vary relative to the pore geometry, ϕ is the porosity, and L is the thickness of the film. The value b = 1/8 applies to a wide range of geometries and the value of tortuosity α ≈ 1.8 was estimated for ϕ = 0.26 from the correlation between α and ϕ reported by Torquato [33] for a range of random porous systems (see Dražević et al. [14] and Bason et al. [34]).

Table 1 Characteristics of the dye molecules. Stokes radius (nm)

D0 × 1010 (m2/s)

189.19

0.284

2.84

Naphthol yellow S

358.19

0.354

3.54

Allura red

496.42

0.554

3.94

Trypan blue

960.81

0.769

4.88

Solute AHBSA

a

MW (Da) a

Chemical formula

3-Amino-4-hydroxybenzenesulfonic acid.

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In order to monitor the penetration of the analyte through the film following injection into the chamber a series of spectra were acquired continuously throughout the entire experiment in the spectra series mode. 1 scan per spectrum at a resolution of 4 cm− 1 was used. The experiments started with the flow-through chamber filled with water for about 400 s, at which point an analyte solution was injected. The analyte diffusion was monitored by tracing changes in the area of a characteristic peak in a series mode as a function of time. 3.4. Data fitting ATR-FTIR diffusion results were fitted to Eq. (6) with D and A∞ as fitting parameters using a nonlinear least square routine in MATLAB 2010b. The fitted diffusivity in the membrane D was then used to calculate the experimental diffusion hindrance factor Kd = D / D0, where D0 is the solute diffusion coefficient in bulk solution, as shown in Table 1. The bulk diffusivities D0 of the analytes were unavailable and were estimated using the Polson correlation: Fig. 4. FTIR spectra of Allura red and wet PVA film. It indicates the criteria used when selecting a solute peak to monitor. The area of the 1040 cm−1 band of Allura red was subsequently used for monitoring its concentration over time.

shown schematically in Fig. 2 consisted of a PVA film placed on the diamond crystal and (not shown) overlaid with a fine stainless steel mesh topped with a flow-through attachment. The film was, on one hand, firmly and uniformly pressed with the mesh against the diamond surface with a minimal gap and, on the other hand, exposed to an aqueous solution. The solution could be rapidly replaced by injecting into the flow-through chamber from a syringe. Prior to mounting, the PVA film was immersed in water for 24 h to ensure maximum swelling. The water on the surface was removed by drying the film in open air at room temperature for 15 min and the film was mounted in the setup and stabilized by injecting water into the setup. A series of spectra in the range 650 to 4000 cm− 1 were acquired using 16 scans per spectrum at a resolution of 4 cm−1. The diffusion of organic molecules in the trans-Aconitic–PVA binary system was investigated using four organic dyes shown in Table 1 as model analytes. The compounds were chosen to have a high solubility in water and cover a relatively wide range of molecular weights (189.19 to 960.81 Da). The characteristics of the dyes are summarized in Table 1.

h i 9:40  10−15 T ½K  2 D0 m =s ¼ μ ½Pa  sM 1=3

ð10Þ

where T is the absolute temperature, M the solute molecular weight in Da, and μ is water viscosity. The Stokes radius of the solute rs was then calculated using the Stokes–Einstein equation

rs ¼

kB T : 6πμD0

ð11Þ

The experimental Kd was compared with theoretical estimate of hindered transport theory in order to test their validity. The theoretical Kd was calculated using the following relations [8]: 2

K d ¼ 1−2:30λ þ 1:154λ þ 0:224λ

where λ ¼

3

ð12Þ

rs . rp

Fig. 5. 2D representation of the variation of the spectra prior to and following the injection of Allura red solution. A brighter color corresponds to a larger intensity. The time of injection at 400 s is indicated with an arrow. The integrated area of the 1040 cm−1 band of Allura red was used for plotting the diffusion transient in Figs. 6 and 7.

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standard deviation of 0.11 × 10− 11 m s− 1 Pa− 1. The wet thickness was calculated assuming anisotropic swelling. Using Eq. (9) the effective pore radius was then estimated to be rp = 2.24 nm. 4.2. FTIR results 4.2.1. Solute Band Selection Fig. 4 shows the spectra of a water-swollen PVA film and 10% Allura red solution. It illustrates how the bands used to monitor analyte diffusion were selected to have a minimal overlap with the bands of PVA. Since FTIR spectra are additive, the accuracy was enhanced and interferences minimized by subtracting contributions from undesired peaks, i.e., using the spectrum of PVA prior to analyte injection as a background. The specific bands of the analytes were usually relatively weak, along with the low concentrations (strong exclusion) of the charge analytes in the PVA film that was the main reason for using fairly concentrated solutions Fig. 6. Variation of the characteristic band intensity for Allura red after solute injection for a bare sensor (no film) showing the delay due to the unstirred layer.

4. Results and discussion 4.1. Water sorption, permeability and effective pore radius of PVA films

0.35

0.35

0.3

0.3 Absorbance, a.u.*cm-1

Absorbance, a.u.*cm-1

A typical curve showing the swelling kinetics of the prepared PVA films is shown in Fig. 3. The results show that, for a film with a thickness of 25 μm, equilibrium was reached within 150 s. The swelling of the film was about 33%, corresponding to water fraction in the film ϕ = 0.25. The average swelling of 7.5 μm thick films was slightly larger, about 35%, yielding ϕ = 0.26. These results indicate that swelling of the films was insignificantly affected by the thickness and was fairly robustly determined by the film composition and casting and cross-linking procedure. This also suggests reasonable uniformity of the films. Average pure water permeability obtained for 3 different films of dry thickness 7.5 μm at 3 pressures was 1.09 × 10−11 m s−1 Pa−1 with a

4.2.2. Diffusion experiments Fig. 5 shows the 2D representation (spectrum evolution over time) of a typical result obtained for Allura red. Only the region with bands of interest is shown. The solution was injected at 400 s. In some cases a slight decline (not shown in the presented results) in the intensity of PVA bands was observed. This could indicate that sorption of the organic analytes by the films could cause some variation in polymer swelling and corresponding decrease of the density of polymer segments in the region probed by the evanescent wave [35]. Nevertheless these did not interfere with monitoring the changes occurring in the bands identified as diffusant bands. As explained previously, the observed diffusion kinetics may be affected by artifacts related to the unstirred boundary layer at the solution and the gap between the film and crystal. The effect of the unstirred layer is most easily clarified by an experiment without a film. Fig. 6 shows the result of such an experiment without a film for Allura red. The time delay of about 30 s caused by the set-up is to be compared with the result in Fig. 7 and appears to be significantly shorter than

0.25 AHBSA

0.2 0.15

experiment fit, no delay fit, with dealy

0.1 0.05 0 0

500

1000

1500

0.25 Naphtol yellow 0.2 0.15 experiment fit, no delay fit, with delay

0.1 0.05 0 0

2000

500

Time, s

1000

1500

2000

2500

Time, s 0.25

Absorbance, a.u.*cm-1

Absorbance, a.u.*cm-1

0.25 0.2 Allura red 0.15 0.1 experiment fit, no delay fit, with delay

0.05 0 0

500

1000 Time, s

1500

2000

0.2 Tryptan blue

0.15

0.1 experiment fit, no delay fit, with delay

0.05

0 0

1000

2000 Time, s

3000

4000

Fig. 7. Variation of the intensity of the relevant IR band over time after injection for the 4 solutes diffusing across the PVA film and best fits to Eq. (8) with and without time delay.

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D x 10 12, m 2/s

1

0.1 with delay

no delay

0.01 0

0.5 Solute Radius, nm

1

Fig. 8. Solute diffusion in PVA film versus estimated Stokes radius of the four solutes, experimental data fitted with and without delay.

4.2.3. Fitting of diffusion data: analysis of possible artifacts As explained previously, the observed diffusion kinetics may be affected by the unstirred boundary layer at the solution–film interface and the solution gap between the film and crystal. The former was approximately taken account by adding a time delay, which did not produce a drastic change in the fitted value of D. To clarify the effect of the gap, it was modeled using Eq. (4). Fig. 9 shows the simulated A(t) for representative values of L, D and K and different values of gap thickness d. Several observations and conclusions can be made. On one hand, even a gap as thin as 1 μm, i.e., 8 times thinner than the film, may indeed significantly increase the transient time and thus substantially underestimate the diffusivity in the film. On the other hand, the gap more significantly affects the slope of the rising part of the curve and not as much the time lag. The curve for the largest gap d = 1 μm then assumes a nearly exponential shape with a barely noticeable lag. Since in the ideal situation (d = 0) the slope (or rise time) and the time lag are rigidly related [15], the time lag versus the slope may indicate the presence or absence of the gap artifact. A comparison with Fig. 7 shows that the relation between the lag and rise times was fairly consistent for all solutes, which suggests that the gap effect was not significant. It is reasonable to conclude that the results in Fig. 7 have not been significantly affected by the above artifacts. Therefore, the use of Eq. (8) for fitting and calculating the diffusion coefficient was reasonable and the corresponding errors in the values of D are fairly inessential for the subsequent comparison with HTT predictions (see next). 4.2.4. Comparison with hindered transport theory Fig. 10 compares the HTT estimates using Eq. (12) and experimental values, presented as the diffusion hindrance factor Kd. The difference between blue and red points in Figs. 8 and 10 indicate the magnitude of the modeling and fitting error, which was mainly the uncertainty associated with the use of different models. Another possible source of error was the film thickness L, however, this should be far less than 1 μm, i.e. well under 10%. Given that the fitted diffusivity is ultimately deduced from the diffusion time τD, related to diffusivity as D ~ L2 / τD, this could produce a relative error of at most 20%. These errors are relatively small, compared to the much larger discrepancy between HTT prediction (green circles) and experimental values. The HTT estimates in Fig. 10 are of the order 10− 1 to 1, i.e., the theoretical hindrance is weak, which could be expected, given that the estimated pore size of 2.24 nm was much larger than the solute size. In contrast, experimental values of Kd are in the much lower range 10−4 to 10−3. Clearly, HTT greatly underestimates the actual hindrance and thus overestimates the diffusion coefficients, The failure of HTT to sensibly predict the diffusion of dyes in PVA is consistent with our previous report that showed that HTT largely overestimates the diffusion of organic solutes in the polyamide layer of RO membranes [14]. The reason should probably be traced back to the two main relations forming the basis of HTT estimates, namely,

0.35

0.07

0.3

0.06

0.25

Absorbance, a.u.

Absorbance, a.u.

the transient time of diffusion (time lag) when the film is present. Some effect of the boundary layer cannot then be ruled out, but it was apparently acceptably small. Nevertheless, the possible effect of the boundary layer was considered, namely, it was approximately taken into account by performing an alternative fitting, in which time t in Eq. (6) was replaced with a (t − tdelay), where tdelay was yet another fitting parameter (Note that Eq. (6) modified in such a way computes A(t) only for t N tdelay while A(t) = 0 is assumed for 0 b t b tdelay.). Representative results showing evolution of the characteristic bands over time following solute injection are displayed in Fig. 7. They were fitted with Eq. (6) (zero gap) with and without extra time delay tdelay due to the unstirred layer, which ultimately yields D of the solutes within the film. Noteworthy, the maximal time of experiment was limited to about 2000 s because of the possible drift of the instrument's optical alignment. Therefore, experiments for most slowly diffusing solute were terminated before the intensity saturated and absorbance at saturation A∞ was made a fitting parameter. Nevertheless, this should not compromise the accuracy of the fitted diffusivity, as in the cases where the intensity saturated, the difference between the fitted A∞ and actual A∞ was insignificant. The fitted diffusivity values are plotted in Fig. 8 versus the estimated solute radius. As seen in Fig. 7, fits to Eq. (6) that included a time delay were better, but the fitted values of D were not drastically different as seen in Fig. 8. As expected, the diffusivity clearly decreases with the solute size. The reduced diffusion coefficients that are characteristically observed with membrane separations originate from the combined effect of more tortuous diffusion paths and, mainly, an increase in the hydrodynamic hindrance experienced by the solute in the pore relative to that experienced in an unbounded fluid. However, the observed values of diffusivity in Fig. 8 are significantly lower than expected. This is further discussed later in the paper.

0.2 d=0 d=0.01 d=0.1 d=1

0.15 0.1 0.05 0 0

500

1000 Time, s

1500

7

d=0 d=0.01 d=0.1 d=1

0.05 0.04 0.03 0.02 0.01

2000

0 0

20

40

60

80

100

Time, s

Fig. 9. Simulated effect of gap thickness d (in microns) on the diffusion transients at long and short times. Parameters used in simulations: L = 8 μm, A∞ = 0.35 a.u., and D = 0.1 μm2/s, K = 0.1.

Please cite this article as: D.S. Dlamini, et al., Solute hindrance in non-porous membranes: An ATR-FTIR study, Desalination (2015), http:// dx.doi.org/10.1016/j.desal.2015.03.009

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D.S. Dlamini et al. / Desalination xxx (2015) xxx–xxx

1E+0

References

1E-1 theory 1E-2

fit with delay

d

fit - no delay

K 1E-3 1E-4 1E-5 0.4

0.5 0.6 0.7 Solute Radius, nm

0.8

Fig. 10. Measured (fits to diffusion transient) and calculated hindrance factor Kd for the four solutes vs. the solute radius.

(a) Eq. (9) estimating the effective pore radius based on water permeability and water fraction (swelling) and (b) Eq. (12) that assumes cylindrical pores and a purely hydrodynamic hindrance. Very likely, at least one of these assumptions may not hold in a dense polymeric network. For instance, the effective pore radius may differ for water and solutes, especially, if the polymer is dense and the pore network is random and irregular [14,36]. Similarly, hindrance may involve other mechanisms, not accounted for by continuum hydrodynamics. This may be especially true for charged dyes used in this study, which may encounter various kinetic barriers due to charge or other interactions with the organic PVA matrix. The inherent properties of the polymer matrix, rather than just solute characteristics, may also play an important role. For instance, Lee et al. [23] reported that the diffusion of small diffusants in dense polymer systems may be nearly independent of molecular weight and suggested that solutesize-independent factors, such as segmental mobility of the host polymer, may play the dominant role. In either case, the present and previous data indicate that the use of HTT requires much care and may not apply to molecular solutes in dense polymeric materials and membranes with sub-nanometer pores.

5. Conclusions The presented results demonstrate that the diffusion of charged dye in a cross-linked PVA film can be successfully measured using ATR-FTIR spectroscopy. The accuracy of the method was improved by designing a set-up that suits the use of pre-cast films of virtually any thickness. The diffusion transients measured by this method were fitted to a model to yield the solute diffusivity within the film, totally uncoupled from partitioning. The model was also extended to consider and rule out possible artifacts, such as a solution gap between the films and ATR crystal, which is likely to occur for pre-cast hydrophilic films and could lead to an underestimated diffusivity. Ultimately, the diffusivity values were used to test the relation of the hindered transport theory, often used in NF modeling. The results indicate that HTT relations may not hold for the transport of small molecules in such polymeric systems with a random mesh-like structure and a maximal caution is advised while applying HTT to NF and RO membranes.

Acknowledgments This work was supported by the DST/Mintek Nanotechnology Innovation Centre (NIC) (fellowship to DD) and Technion — Israel Institute of Technology. Partial support by the Israeli Science Foundation grant 1152/11 is gratefully acknowledged.

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Please cite this article as: D.S. Dlamini, et al., Solute hindrance in non-porous membranes: An ATR-FTIR study, Desalination (2015), http:// dx.doi.org/10.1016/j.desal.2015.03.009