Temporal evolution of the selectivity-permeability relationship during porous membrane filtration of protein solutions

Journal of Membrane Science 514 (2016) 385–397

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Temporal evolution of the selectivity-permeability relationship during porous membrane filtration of protein solutions Jing Ma a, Lanlan Qin a, Xiaolei Zhang a, Haiou Huang a,b,n a State Key Joint Laboratory of Environmental Simulation and Pollution Control, School of Environment, Beijing Normal University, No. 19, Xinjiekouwai Street, Beijing 100875, China b Department of Environmental Health Sciences, Bloomberg School of Public Health, Johns Hopkins University, 615 North Wolfe Street, Baltimore, MD 21205, United States

art ic l e i nf o

a b s t r a c t

Article history: Received 5 February 2016 Received in revised form 30 April 2016 Accepted 11 May 2016 Available online 14 May 2016

Currently, there is no clear understanding on the effects of membrane properties on temporal evolution of membrane selectivity-permeability relationships during microfiltration or ultrafiltration-based water treatment. In this study, porous polyvinylidene fluoride (PVDF) membrane, polyvinyl chloride membrane, and glass-fiber membrane with distinctive physical and chemical properties were employed to filter bovine serum albumin (BSA) at different pH, coupled with model simulation approaches. Both the experimental results and model prediction showed that, due to the existence of different filtration mechanisms, selectivity of the PVDF membrane and the PVC membrane increase with decreasing permeability, whereas an opposite trend was observed for the glass-fiber membrane. Also, pH strongly affected permeability by changing the zeta potentials of the solutes and the membrane surfaces. Moreover, membrane selectivity was less significantly affected by pH for the polymeric membranes than for the glass-fiber membrane. Practically, high selectivity and permeability may be simultaneously achieved during water treatment by: (1) selection of membranes with suitable pore structure (such as control of solute diameter 41/5 pore diameter); (2) control of feed water condition (such as pH) to suppress intermolecular interactions or solute-membrane interactions, further influence the deposition/adsorption of solute on membrane. & 2016 Elsevier B.V. All rights reserved.

Keywords: Microfiltration Ultrafiltration Selectivity-permeability relationship Pore structure Physicochemical interactions

1. Introduction Fundamentally, microfiltration (MF)/ultrafiltration (UF)-based water treatment is a pressure-driven filtration process designed to remove particulate contaminants from water. Since mass transports of the contaminants or solutes are dominated by convection, the rate of mass transport of a solute is proportional to the filtrate flux and the corresponding solute sieving coefficients [1,2]. The selectivity and permeability of a MF/UF membrane are closely related to the sieving coefficients and filtrate flux, respectively. Therefore, studying the relationships between the selectivity and permeability is of importance to the understanding of solute transport and further studying of membrane fouling. Permeability and selectivity are key performance parameters for membrane filtration processes [3]. To simultaneously improve the permeability and selectivity of membrane is a consistent aspiration of membrane scientists and engineers. Permeability and selectivity n Corresponding author at: State Key Joint Laboratory of Environmental Simulation and Pollution Control, School of Environment, Beijing Normal University, No. 19, Xinjiekouwai Street, Beijing 100875, China. E-mail address: [email protected] (H. Huang).

http://dx.doi.org/10.1016/j.memsci.2016.05.022 0376-7388/& 2016 Elsevier B.V. All rights reserved.

are intimately linked because increases in permeability are often achieved as a result of increases in solute diffusion coefficients [4,5]. Membranes having higher water transport coefficients, and therefore, higher permeability, often have lower diffusive selectivity and, in turn, lower solute selectivity. In general, the increase of the permeability comes at the expense of the selectivity [3]. It is well known that the water permeate flux and selectivity of porous membranes depend on their structures and solute geometry, such as pore density, pore size distributions, selective layer thickness, pore and solute geometry, membrane, solute, and solution electrokinetical characteristics, etc. [6–8]. Generally, a thinner selective layer and larger pore density can result in high membrane permeability, whereas a thicker selective layer and smaller pore size can lead to a better selectivity in the sacrifice of permeability [9,10]. However, for a specific membrane, the pore size is fixed after preparation and the membrane exhibits an uncontrollable sieving/selectivity, which limits its application scope [11]. In addition, the electrostatic properties and geometry of different solutes depend upon their isoelectric point (IEP) and ambient pH environment [12]. Therefore, it is increasingly desirable to understand different membranes with respect to the capabilities to adjust their sieving/selectivity performance according to the distinct dimension and electrostatic charge of solutes for separation.

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Fig. 1. Three potential mechanisms for solute filtration by porous membranes. (i) monolayer adsorption, (ii) multilayer adsorption, (iii) sorption to the membrane (depth filtration).

Kanani et al. [13] studied the effect of pore geometry on permeability–selectivity relationship, by using both commercially available UF membranes and novel silicon membranes having slitshaped nanopores of uniform size. The results clearly demonstrated that membranes with slit-shaped pores had greater selectivity at a given permeability, than membranes with cylindrical pores. Theoretical calculations indicated that this improved performance became less pronounced as the breadth of the pore size distribution increases. Fang et al. [14] studied the effect of polyethyleneimine (PEI) additive on the permeability and selectivity of polyethersulfone (PES) UF membrane. The results demonstrated that membrane permeability and selectivity were improved simultaneously by optimizing the PEI content. When the PEI content was 0.3 wt%, the permeability reached a maximum of 359.0 L/m2 h under 0.1 MPa, which was 35.9 times that of the pure PES membrane. Meanwhile, the rejection of bovine serum album (BSA) was as high as 96.1%. The optimized PEI-PES membrane outperforms a commercially available membrane with analogous characteristics in regard to permeability and molecular weight cut-off (MWCO). Mehta and Zydney [1] examined the relationship between permeability and selectivity for different UF membranes using filtration data for BSA. Results for a number of different UF membranes fell par, or below, an “upper bound” that reflects the current state-of-the-art in commercially available UF membranes, analogous to the Robeson Plot [5] used to analyze the performance of gas separation membranes. The shape of this upper bound is consistent with a theoretical analysis of solute and solvent transport through a membrane composed of a parallel array of cylindrical pores having a log–normal pore size distribution. A mathematical model was further developed for the comparison of the performance characteristics of different filtration membranes based on a plot of the membrane selectivity (or separation factor) as a function of the membrane hydraulic permeability (Lp). The appropriate selectivity for conventional filtration processes is equal to the flux of solvent divided by the flux of the macromolecule to be retained, which is simply equal to the reciprocal of the sieving coefficient (Sa). Experimental data for BSA filtration using a wide range of membranes were used to validate the reciprocal relationship between permeability and selectivity. The model calculations results were in good agreement with the experimental data, confirming that hydrodynamic interactions and membrane pore geometry (or membrane types) governed the overall performance of these filtration membranes. The distinctive selectivity-permeability relationships demonstrated by different membranes revealed the presence of various filtration mechanisms for UF/MF membranes. Depending upon the ratio of the membrane pore size to the solute size, the following

filtration mechanisms have been recognized in the literature. When the solute and pore sizes are comparable, the filtration may be caused by size exclusion of solutes at pore entrance, solute adsorption on pore walls, cake/gel formation, as well as the resulting pore blockage, pore constriction, and cake/gel layer filtration [15]. The resulting effects also reduce membrane permeability while increasing solute rejection. According to Polyakov et al. [16,17], the pore constriction and cake filtration are controlled by the value of the adsorption coefficient, which is highly dependent upon the zeta potentials of the solutes and the membranes, as well as solute/pore geometry [18,19]. The size exclusion of solutes is controlled by the values of the actual sieving coefficient, which highly depend on the electrostatic energy of interaction between solute and the pore wall, given certain solute/pore geometry [20–22]. The filtration mechanisms become relatively simple when the pore size to solute size ratio obviates from the unity. Specifically, when the pore size is approximately five times greater than the solute, the dominant filtration mechanism at the early stage is pore constriction, followed at the late stage by size exclusion and then cake/gel filtration [8]. In comparison, when the pore size is much larger than the solute, the filtration is dominated by the adsorption of solutes onto internal membrane surfaces, which is further controlled by solute size and the zeta potentials of the solute and the membrane [23]. The basic concepts of these filtration mechanisms are illustrated in Fig. 1. It is noteworthy that, in addition to promoting pore constriction by sieving, adsorption of solute onto the membrane may also alter membrane electrokinetic properties, resulting in increased solute rejection by electrostatic repulsion. Despite the aforementioned reports in the literature, the effects of membrane properties on the selectivity-permeability of MF/UF membranes for water treatment have not been well understood [1,15,24]. Recently, MF and UF have been widely utilized in wastewater treatment and reclamation because of its relatively low energy consumption, high particle filtration efficiency, and low requirements on chemical pretreatment [25,26]. However, membrane fouling is still a main limitation for broader application of membrane technology [27,28]. Protein-like substances, which are abundant in wastewater, have been identified as one of the major types of membrane foulants [29,30]. Furthermore, effects of membrane properties on the rejection of other contaminants (e.g., viruses) are also to be determined for membranes fouled by proteins in water [31]. Indeed, application of the developed theories to practical wastewater treatment processes is hindered by the absence of experimental studies providing reliable data on electrokinetic characteristics of solutions and membranes, as well as solute/pore geometry, and their effect on the selectivity and permeability during the filtration. Therefore, the

J. Ma et al. / Journal of Membrane Science 514 (2016) 385–397

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Table 1. Characteristics of membrane employed in this study. Membrane characteristics Membrane type Membrane material Membrane geometry Contact angle (degree) Nominal pore size (μm)a Transmembrane pressure (MPa) Clean membrane permeability (m s  1 Pa  1) Number of fibers for mini-modules Effective length of mini-modules (mm) Inner diameters of membrane (mm) Outer diameters of membrane (mm) Membrane thickness (μm) Porosity of selective layer Packing density of membrane Diameter of the filamentous fibers (μm) Length of the filamentous fibers (μm) Operational mode a

UF poly(vinylchloride) hollow fiber 88.87 0.01 0.096 (4.377 0.05)
10  10 3 120 1.2 1.5 150 0.89 – – – inside-out

MF poly(vinylidene fluoride) hollow fiber 53.93 0.1 0.007 (1.497 0.02)
10  9 3 180 0.8 1.0 100 0.86 – – – inside-out

MF glass fiber flat-sheet 0 1.2 0.001 (1.86 7 0.03)
10  7 – – – – 500 0.90 0.16–0.18 0.3–2.0 3.0–2.0
104 –

Provided by the manufacturer.

overarching objective of this research was to fill in the gap by comprehensively using various filtration and characterization techniques to study the filtration of model solute by commercially available MF/ UF membranes with existing or potential use in wastewater treatment. BSA was used in this study as a model protein because it is readily available in highly purified form and has been extensively characterized. For example, the molecule sizes of BSA have been determined by different techniques to be in a wide range: 3.11– 3.66 (stokes radius), 3.52 nm (translational diffusion), 3.48 nm (gel filtration), approximately 6.2–6.8 (fluorescence anisotropy) [32], and prolate ellipsoid 14
4
4 nm3 (transient electric birefringence) [33]. The zeta potential of BSA at different pH was also different, dependent upon the solution chemistry. For example, the zeta potentials of 5% BSA in 0.001 M KCl were approximately 10.5 mV at pH ¼3.8, 4.8 mV at pH ¼4.5, 3.0 mV at pH ¼ 5.0,  9.0 mV at pH ¼6.0,  13 mV at pH ¼7.0,  13.5 mV at pH ¼8.0, and  15.0 mV at pH ¼ 8.0, respectively [34,35]. In this study, these data were used in conjunction with comprehensive membrane characterization and filtration experiments to investigate the selectively-permeability relationship under conditions relevant to wastewater treatment practices.

2. Materials and methods 2.1. Membranes Three commercially available membranes were used in this study: a glass-fiber (GF) flat sheet membrane, a PVDF, MF, hollow fiber membrane and a PVC, UF, hollow fiber membrane. The PVC membrane and the PVDF membrane were purchased from the Litree Purifying Technology Co., Ltd., China and the Tianjin Motimo Membrane Technology Co. Ltd., China, respectively. The asreceived membrane fibers were fabricated into mini-modules with an effective membrane area of approximately 0.0017 m2. The flow velocity at lumen entrance was 4.8
10  4 m/s. These mini-modules were then flushed with ultrapure water prior to use for the removal of wetting agent or other preservatives on the membrane, using 50 L of ultrapure water per m2 of membrane area. The GF, flat-sheet membrane employed was purchased from the Membrane Solutions Industries, Shanghai, China. The circularshaped membrane has an effective diameter of 20 mm for filtration, or an effective area of 3.14
10  4 m2. The properties of the three membranes are summarized in Table 1.

2.2. Protein and chemicals Reagent-grade BSA (99% purity) with an IEP of 4.78 and a molecular weight of 67 kDa, was obtained from Beijing Aoboxing Biotech CO., LTD., and used as a model protein to represent the proteinlike substances in water. A BSA stock solution (concentration of 0.2 g/l) was prepared by dissolving BSA in ultrapure water. The stock solution was stored in sterilized glass bottles at 4 °C until use. A fresh feed solution was prepared for each filtration experiment by diluting the stock solution in an electrolyte solution containing 0.01 M of NaCl. The pH of the feed solution was adjusted with 0.01 M HCl or NaOH to desired values that were measured by a pH meter (pHS-3C, Shanghai REX Instrument Factory, China). All reagents and chemicals were analytical grade with purities 4 99%. 2.3. Membrane filtration experiment In order to control the selectivity conditions for BSA, membrane filtration experiments were performed at pH values of 3.78, 4.78, 5.78, and 10.0 at a fixed ionic strength of 10.0 mM. The membrane filtration systems used in this study are shown schematically in Fig. 2. BSA solution in the sample bottle was pumped through the membrane at a constant flux of 1.4
10  5 m/s using the dead-end, inside-out filtration mode. The filtrate samples collected during hollow fiber membrane filtration were measured by a UV spectrophotometer at λ ¼280 nm (DR 6000, HACH, USA). The UV absorbance results were recorded with a personal computer at a constant time interval during the flat-sheet membrane filtration. 2.4. Structure and size distribution of membrane pores Membrane pore structure was determined using a scanning electron microscope (SEM, Hitachi S-4800, Japan). Specimens of membrane fibers were carefully cut from the membrane modules using a razor blade, and then, immersed in liquid nitrogen for approximately 15 s The embrittled fibers were broken into short pieces with lengths of approximately 5 mm. The prepared fiber specimens were mounted on the sample stage using conductive double-side tapes and coated with a thin layer of platinum (ca. 1– 2 nm) to improve surface conductivity prior to SEM imaging. Membrane pore size distributions (PSD) were measured by using two porometers (PSMA-10 and PSDA-20, GaoQ Functional Materials Co., Ltd., China) under a room temperature of 25 °C. The PSMA-10 porometer has a measurement range of 6–200 nm based on a liquid-liquid displacement mechanism, and therefore, was

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Fig. 2. Schematic diagrams of the bench-scale membrane filtration systems used in the study. a. Hollow fiber membrane filtration set-up: 1 – magnetic stirrer; 2 – feed water bottle; 3 – constant flow pump; 4 – pressure gauge; 5 – hollow fiber membrane module; 6 – automatic fraction collector; 7 – sample for UV measurement. b. Flat membrane filtration set-up: 1 – magnetic stirrer; 2 – feed water bottle; 3 – constant flow pump; 4 – pressure gauge; 5 – glass fiber membrane; 6 – data acquisition computer; 7 – UV detector; 8 – waste liquid bottle.

used for the PVC membrane and the PVDF membrane. The PSDA-20 porometer possesses a measurement range of 0.05–500 mm by utilizing the capillary-flow method and was used for the GF membrane in this study. 2.5. Zeta potential measurement Zeta potentials of the membrane surfaces were measured using an electrokinetic analyzer (EKA, Anton Paar GmbH, Graz, Austria) under solution chemical conditions analogous to those used in membrane filtration experiments. In this study, the transmembrane mode was used during the measurement. Zeta potentials of BSA solution at different pHs were measured by using a Malvern Zeta sizer (Nano ZS90, United Kingdom). All zeta potential measurements were performed at a room temperature of 25 °C. 2.6. Membrane permeability–selectivity analysis Recent studies [1,16] have demonstrated that an appropriate framework for describing the performance characteristics of different membranes is the determination of the relationship between

membrane selectivity and membrane permeability. In the context of the study, membrane selectivity ψ is defined as the reciprocal of the protein's sieving coefficient S or the transmission of protein through the membrane (S¼C/C0, where C and C0 are the solute concentrations in the permeate and bulk feed solution, respectively.): ψ ¼C0 /C. Membrane permeability Lp is defined as: LP = J /ΔP , (where J is the permeate flux through the membrane at a given transmembrane pressure of ΔP) [15]. The permeability–selectivity analysis for membranes has been discussed in detail by Mehta and Zydney [1].

3. Results 3.1. Characteristics of BSA in the feed solutions The zeta potential and hydrodynamic diameter of BSA were both dependent upon solution pH. Fig. 3 shows that, when pHo4.78, the zeta potential of BSA molecules was positive due to completely association of basic residues in BSA with protons. When the pH approached 4.78, the zeta potential of BSA molecules gradually

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decreased to 0 due to proton dissociation from the BSA molecules; this was coincident with the smallest hydrodynamic diameter for the BSA molecule (ca. 6.8 nm). With pH increasing above 4.78, the negative charge of the protein increased due to the deprotonation of amino acid residues on BSA, and thus, the zeta potential of BSA molecule became increasingly negative. As the amino acid residues in the BSA molecule surface were completely dissociated at pH¼10, the zeta potential of BSA reached a stabile level of ca.  15 mV, comparable to the literature-reported value [2]. 3.2. Membrane characteristics 3.2.1. Membrane structures The pore structure of the three membranes The surficial and cross-sectional SEM images membrane employed in this study is made of (Fig. 4a) and possesses a homogeneous

is shown in Fig. 4. show that the GF filamentous fibers porous structure

Fig. 3. Zeta potential and particle size of BSA in different pH. The hydrodynamic diameter was calculated according to the diffusion coefficients for BSA determined by Raj and Flygare [36].

389

throughout the depth (Fig. 4b). The PVC membrane and the PVDF membrane have a tight surface layer (Fig. 4c and e, outer surface) that functions as the selective layer for BSA filtration. The thickness of this layer is approximately 0.3–1.0 mm (Fig. 4d and f). 3.2.2. Pore size distributions The PSD of the studied membranes are shown in Fig. 5. The PVC membrane, the PVDF membrane, and the GF membranes possess pores with sizes ranging from 12.0 to 16.0 nm, 56.0–100.0 nm, and 0.20–1.85 mm, respectively. The PVC membrane and the PVDF membrane have relatively narrow pore-size distributions and main peaks were observed at 12.0 nm (88.23%) and 56.0 nm (45.99%), respectively. However, the GF membrane exhibits two peaks with a relatively wide distribution. The corresponding diameters of the two peaks were 0.55 mm (21.63%) and 1.05 mm (3.47%), respectively. Overall, the measurement results suggest that the PVC membrane, the PVDF membrane, and the GF membrane belong to UF, MF, and MF membranes, respectively, which generally agree with what claimed by the manufacturers. 3.2.3. Zeta potential results Zeta potentials of the studied membranes varied with solution pH and the presence/absence of BSA (Fig. 6). In the absence of BSA, the pHIEP of the PVC, PVDF and GF membrane were 3.39, 2.85 and 2.25, respectively. As the solution pH increased from 2.87 to 10.04, from 2.70 to 10.42 and from 2.18 to 10.32, respectively, the zeta potentials of the PVC, the PVDF and the GF membrane decreased from 3.03 mV to 16.12 mV, from 0.18 mV to 16.08 mV and from 1.46 mV to  100.87 mV, respectively. According to these results, the GF membrane was highly negatively charged while the PVC membrane and PVDF membrane were moderately negatively charged at neutral to alkaline pH. In the presence of BSA, the pHIEP of the three membranes were changed to 4.70, 4.65 and 4.50 (Fig. 6), respectively, which were close to the pHIEP of BSA (Fig. 3). Also, the zeta potentials of the PVC membrane, the PVDF membrane, and the GF membrane were

Fig. 4. Scanning electron micrograms of the glass fiber membrane, with (a) scale bar¼ 500 mm, (b) scale bar¼ 50 mm; the PVC membrane, with (c) scale bar¼ 0.3 mm, (d) scale bar¼ 500 mm; and the PVDF membrane, with (e) scale bar¼ 3.0 mm, (f) scale bar¼500 mm. a, c, e and b, d, f are top views and cross-sectional views of the membranes, respectively.

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differences in membrane properties resulted in noticeable differences in the evolution of the selectivity-permeability relationship during membrane filtration of BSA solutions. 3.3.1. The PVC membrane BSA filtration reduced the permeability of the PVC membrane, coincident with increases in membrane selectivity, regardless of solution pH (Fig. 7). After 120 min filtration, the highest selectivity and lowest permeability occurred at the pH of 4.78, the lowest selectivity and highest permeability appeared at the pH of 10.0, and the selectivity and permeability at the pH of 3.78 and 5.78 had the middle values. Moreover, the permeability at pH of 3.78 and 4.78 was similar, but the selectivity at the pH of 4.78 was considerably higher than that at the pH of 3.78. Overall, the evolution of the selectivity-permeability relationship showed two distinctive stages. At the beginning stage (1.0
10  10 m s  1 Pa  1 oLp o4.25
10  10 m s  1 Pa  1), the selectivity was relatively stable while the permeability decreased rapidly; the selectivity-permeability relationship was not significantly affected by pH. At the second stage (0oLp o 1.0
10  10 m s  1 Pa  1), the selectivity increased at accelerating speed, while the decreases in permeability decelerated. pH exerted more important effect on the selectivity-permeability relationship at the second stage than at the first stage. The permeability at a given selectivity (less than 10) slightly increased except for the pH of 10.0. 3.3.2. The PVDF membrane Similar to the PVC membrane, BSA filtration by the PVDF membrane caused a reduction in membrane permeability with a simultaneous increase in membrane selectivity; the process evolved in two stages, i.e., a beginning stage with fast permeability decrease and slow selectivity change and a second stage with slow permeability decrease but fast selectivity increase. However, solution pH exerted different and more distinctive effects on the temporal evolution of the permeability-selectivity relationship of the PVDF membrane than that of the PVC membrane during both stages. Unlike the PVC membrane, the initial selectivity of the PVDF membrane varied with solution pH and increased in an order of pH ¼10.0 4pH ¼ 5.784 pH ¼3.78 4pH ¼4.78 (Fig. 8). This trend was maintained during the beginning stage of filtration. Afterwards, the second stage occurred at noticeable different permeability levels, also following a decreasing order of pH ¼10.0 4pH ¼5.78 4pH ¼ 3.784 pH ¼4.78. During the second stage, when the membrane reached a certain selectivity, e.g., 2.2, the membrane permeability significantly varied between ca. 0.50
10  9 m s  1 Pa  1 or 66% decrease from the initial at pH ¼4.78 and ca. 0.86
10  9 m s  1 Pa  1 or 42% decrease from the initial at pH ¼10.0. This trend appeared to be different from that observed for the PVC membrane whose permeability decreased slightly faster at pH ¼3.78 than at pH ¼4.78 when the selectivity was below 10 (Fig. 7). Fig. 5. Pore size distributions of (a) the PVC membrane, (b) the PVDF membrane, and (c) the glass fiber membrane.

reduced from 6.96 mV to  14.93 mV, from 5.30 mV to 13.94 mV, and from 36.75 mV to  60.02 mV (Fig. 6), respectively, as pH increased from approximately 2 to 10. After the BSA was adsorbed on the membrane, zeta potentials of the studied membranes became close to those of BSA at similar pH (Fig. 3); this clearly suggests that BSA adsorption changed membrane electrokinetic properties. 3.3. Membrane selectivity-permeability relationships As elaborated above, the three membranes employed in the study had very different physical and chemical properties. The

3.3.3. The glass fiber membrane As the only fibrous membrane employed in this study, BSA filtration by the GF membrane resulted in simultaneous decreases in membrane permeability and selectivity (Fig. 9); this unique pattern was different from the other two membranes (Figs. 7 and 8). Regardless of solution pH, the selectivity gradually approached the unity within 50 min of filtration, indicating complete breakthrough of BSA. Despite the unique trend of continuously decreasing selectivity, the temporal evolution of the selectivity-permeability relationship for the GF membrane also exhibited two stages as the filtration time increased. During the beginning stage of the filtration, the selectivity was slowly reduced while the permeability decreased rapidly from ca. 1.66
10  7 m s  1 Pa  1 to ca. 0.60
10  7 m s  1 Pa  1. pH primarily affected the selectivity at any given permeability. The

J. Ma et al. / Journal of Membrane Science 514 (2016) 385–397

Fig. 6. The zeta potential of (a) the PVC membrane, (b) the PVDF membrane, and (c) the GF membrane as a function of solution pH, in the presence/absence of BSA. Temperature ¼25 °C.

selectivity at different pH followed an order of pH¼10.04 pH¼5.784pH¼3.784pH¼4.78, similar to that of the PVDF membrane (Fig. 8) but more distinctive. During the second stage, membrane permeability slowly decreased, while the selectivity was sharply reduced to the unity. The selectivity-permeability relationship was significantly affected by pH as well. pH exerted stronger effects on permeability than on selectivity as the latter decreased monotonously. The final permeability at different pH varied by more than a half and followed an order of pH¼ 104pH¼ 5.784pH¼3.784pH¼4.78. It is noteworthy that, because the initial permeability of the GF membrane were more than two orders of magnitude greater than that of the PVDF membrane or

391

Fig. 7. The permeability (a) and selectivity (b) versus time (lines represent model simulation results) and the evolution of the selectivity-permeability relationships for the PVC membrane at various solution pH (c, arrows denote the evolution of the selectivity-permeability tradeoff during ultrafiltration).

the PVC membrane (Table 1), the GF membrane was still more permeable to water than other membranes when they were clean.

4. Discussion 4.1. Mechanistic models for BSA filtration by different membranes 4.1.1. Monolayer adsorption in membrane pores The selectivity and permeability versus time plots for the PVC

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Fig. 8. The permeability (a) and selectivity (b) versus time (lines represent model simulation results) and the evolution of the selectivity-permeability relationships for the PVDF membrane at various solution pH (c, arrows denote the evolution of the selectivity-permeability tradeoff during ultrafiltration).

membrane are shown in Fig. 7c. In this case, the overall filtration is governed by the interplay among three processes: the reduction in solute mass flux due to the narrowed pore diameter, the reduction in intrapore area available for solute adsorption/deposition, and the increase in steric rejection due to the reduced pore mouth flow diameter (pore constriction). Fig. 1-(i) shows that the initially high solute permeability caused by solute adsorption/deposition on the pore walls starts declining as soon as a significant area of the pore walls is covered by the deposited solute. Solute permeability

Fig. 9. The permeability (a) and selectivity (b) versus time and the selectivitypermeability relationship for the glass fiber membrane at various solution pH (c). The arrow denotes the evolution of the selectivity-permeability tradeoff during microfiltration and the lines represent the connection of experiment data.

decreases until the monolayer is complete, at which point the rejection coefficient is equal to that for solute transport through a pore with radius (rp  2rs). It should be noted that the adsorption/deposition rate under monolayer restriction is initially higher in the pore entrance region as compared to the rest of the pore because the solute concentration at the pore entrance is higher. However, as the filtration proceeds, the probability of solute capture in the pore entrance region becomes lower than that that deeper inside the pore (monolayer exclusion). This results in a leveling of the profile of deposited solute along the length of the pore, leading to a virtually uniform monolayer deposition as the process approaches steady-state conditions (Fig. 1(i)).

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The rough estimates made with Poiseuille equation and analytical formulas for actual sieving coefficient Sa for a hard sphere in a cylindrical pore [8] are as follows: 2

Sa = ( 1 − λ ) (1 + 2λ − λ2)(1.0 + 0.054λ − 0.988λ2 + 0.441λ 3)

(1)

for 0 < λ < 0.8,

(1 − λ ) 2 (1 + 2λ − λ2) ⎣⎡ 1.0 + 0.117 (1 − λ ) ⎤⎦ 2 ⎣⎡ 1.0 − 1.217 (1 − λ ) ⎤⎦

Sa =

(2)

for λ > 0.8, and

ψ = (βSa exp [ − Nλ ])−1,

(3)

here, the concentration polarization effects can be described using a stagnant film model [16,37], which relates the observed (S0 ¼Cpermeate/Cfeed) and actual sieving coefficients (Sa ¼Cpermeate/Cwall),

Sa

S0 =

(1 − Sa ) exp

β=

( )+S k J

1 S0 = Sa ( 1−Sa ) exp

a

(4)

( )+S k J

a

k=

χ

A=

Re 0.567

Sc 0.33D∞

rP

,

(6)

(7)

ν , D∞

(8)

The BSA diffusion coefficient D∞ evaluated from the dextran molecular weight using the correlation presented by Granath and Kvist:

log D∞ = − 4.1154 − 0.47752 log MW

(9)

In Eqns. 1 to 3, λ = rsolute/rpore, z = 0 , β was the concentration polarization factor, β = Cwall/Cfeed , and its value cannot be higher than the actual sieving coefficient Sa = Cpermeate/Cwall , where the permeate value is based only upon the size exclusion effect (before solute adsorption evolves down the pore) [8,38]. u is the stirring speed, ν is the kinematic viscosity, J is permeate flux. The parameter χ ¼0.27. The exponential term in Eq. (3), exp [ − Nλ ] accounts for the decline in the concentration of solute inside the pore due to their adsorption on the pore wall. In this study, the exp [ − Nλ ] was calculated by binomial fitting (based on Sa as the abscissa and exp [ − Nλ ] as the ordinate, the fitting methods see Fig. S1 of supporting information). The time difference can be calculated from the equation [16]:

Δt =

πlp rp02 1 c0 G 0 ε0 JNλ B ( σ ) Sa ( σ )

where, B (σ ) =

⎛ Φ (hm ) − Φ (h2 ) ⎞ kThm /rs γ exp ⎜ − ⎟ ⎠ ⎝ 6πμrs (1 + hm /rs) 2πkT kT

1−

σ ε0

=

(11)

where, a is particle radius, m; hm is separation distance between the surface of collector and the point of maximum (energy barrier), m; T is temperature, K; γ = −

urp Re = , ν Sc =

equal to 3.98 nm in all model calculations, which corresponds to the highest experimental value of the selectivity, 35, obtained in the filtration experiment conducted at pH ¼ 4.78. In the equation for ψ the product (β exp [ − Nλ ]) is assumed to be close to 1, which is close to the monolayer adsorption condition developed in a previous study in view of high values of Sa (high concentration of solute in the pore), high axial velocity, and relatively slow decrease in the membrane permeability with time [8]. As such, a small portion of BSA passing into the pore is adsorbed, forming a uniform monolayer over the pore length, as described mathematically in a recent study [8]. In this case, the selectivity first increases slowly due to the reduction of the wall surface area available for solute deposition, but then rapidly increases due to the reduction in the pore diameter at the pore entrance and the corresponding increase in steric rejection. The initial period of pore constriction is manifested by the most dramatic drop in the permeability. The adsorbed layer formed on the pore wall grows as the filtration proceeds until the final flow diameter in the pore becomes equal to the solute diameter (rejection becomes equal to 100%). Mathematically, this process is related to the value of exp [ − Nλ ], which is a function of the adsorption coefficient and particle size. The behavior of exp [ − Nλ ] can be explained as follow. The adsorption coefficient (A) in our case can be written as [18]:

(5)

where

393

d2Φ dh2 h m

; Φ (hm ) is interaction potential

energy at the point of maximum (energy barrier), J; Φ (h2 )is interaction potential energy, J. The adsorption coefficient (A) is a function of the particle size and zeta potential. In addition, both particle size and zeta potential are

(

influenced by solution pH. For BSA filtration, exp −

Φ (hm ) −Φ (h2 ) kT

) in

Eq. (11) has the maximum value at the pH of 4.78 and the minimum value at the pH of 10.0, which follows an order of pH¼4.784pH¼3.784pH¼5.784pH¼10. On the contrary, the particle size follows an order of pH¼4.78opH¼3.78opH¼ 5.78opH¼ 10. Therefore, the adsorption coefficient (A) should be followed an order of pH¼4.784pH¼3.784pH¼5.784pH¼10. Because Nλ is inversely proportional to the pore size and proportional to the adsorption coefficient; therefore, the value of exp [ − Nλ ] follows an order of pH¼ 4.78opH¼ 3.78opH¼5.78opH¼10 and the changes of the calculated permeability follows an order of pH¼4.784pH¼3.784pH¼5.784pH¼10. 4.1.2. Multilayer adsorption inside membrane pores For the PVDF membrane used in the study, the pore size is approximately five times greater than BSA. Therefore, BSA molecule can enter the pores and then pass into the permeate or approach the pore wall and deposit on the pore wall, possibly in multiple layers. The adsorption of solute on membrane pore wall

(10) rp

rp0

, lp is the membrane thickness (pore

length), m, c0 is the volume fraction of suspended solutes in the feed, G0 is the initial volumetric permeate flow rate through one cylindrical pore, m3 s  1 and ε0 is the initial membrane porosity. The selectivity ( ψ ) and time ( Δt ) was calculated using the flow diameters at the points where the permeability curves (Fig. 7a). The calculations performed by the model simulation method described above yield selectivity results that deviate less than 0.8% from the experimental values (Fig. 7(b)). Herein, the BSA size is taken to

Table 2. Experiment condition and model simulation results. Experiment conditions and calculation results pH C/C0a α

3.78 0.952( 7 0.004) 0.018( 7 0.0016)

4.78 0.917( 7 0.003) 0.032( 7 0.0013)

5.78 0.954( 70.004) 0.017( 70.0014)

10.0 0.962( 70.004) 0.014( 70.0015)

a The C/C0 values were based on the intersection point of two tangent lines for the second inflection point of breakthrough curve (see Fig. S1 of supporting information).

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leads to pore constriction and changes to the electrokinetic properties of the membrane. The similar (as above) numerical analysis is used herein to simulate the experimental data and the selectivity results are shown in Fig. 8b (solid lines). In the equation for ψ the product (β exp [ − Nλ ]) is assumed to be close to 0.5, which is very close to the condition for multilayer adsorption [16]. In this case, only a very small portion of the solutes passing through the pore is adsorbed, uniformly forming a multilayer of adsorbed solutes. The second layer starts to deposit at the pore inlet before the remote clean surface of the pore wall is covered by solutes, which was shown mathematically in previous studies [16,17]. Similar to the PVC membrane, the evolution of the selectivity depends upon the interplays among three simultaneous processes: (1) the increase in steric/electrostatic rejection due to the reduced pore mouth flow diameter caused by particle deposition on the pore wall; (2) the reduction in the permeation flux and, hence, solute mass flux due to the narrowed pore flow diameter; and (3) the reduction in intrapore area available for solute deposition (mostly determined by electrostatic phenomena). The adsorption of solute on the pore wall increases with increasing Nλ , which is a function of the filter coefficient λ 0 and accounts for the ability of the pore wall (pore structure and property, particle size etc.), or the surface of the layer of deposited particles, to capture suspended solute [15]. The value of exp [ − Nλ ] follows an order of pH¼4.78opH¼3.78opH¼5.78o pH¼10 through the analysis of the filtration by the PVC membrane. Therefore, the value of Nλ should follow an order of pH¼4.784pH¼3.784pH¼5.784pH¼10. The adsorption of solute on pore wall has three consequences. First, it reduces the current pore diameter, causing a decrease in membrane permeability and, hence, in the mass flux of solute entering the pore. Second, it reduces the area available for solute adsorption/deposition, which decreases the ability of the membrane pores to capture solute. Third, the reduction in the pore diameter causes an increase in the purely steric rejection of the solute by the membrane. The interplay between the above three processes determines the behavior of the membrane permeability and selectivity. The ability of solute in adsorbing onto the pore wall depends upon the filter coefficient λ 0 . The more solutes adsorbing onto the pore wall (or on the adsorption layer), the greater pore constriction effect occurs. Consequently, membrane selectivity increases and permeability decreases. In addition, the adsorption of solute onto the pore wall is proportional to Nλ . Therefore, the selectivity and permeability should follow an order of pH ¼4.78 4pH ¼ 3.78 4pH ¼5.78 4pH ¼10 and pH ¼4.78 o pH ¼3.78 opH ¼ 5.78o pH ¼10, respectively. The resulting variation in membrane permeability was experimentally observed in the microfiltration of BSA by the PVDF membrane (Fig. 8a). 4.1.3. Dynamic sorption to the membrane (depth filtration) The deep bed filtration approaches were used to calculation the attachment efficiency (α) because the structure of the glass fiber membrane does not really have a network of 0.5 mm pores and more like a packed column. The detailed were as follow [39]:

α=−

2 dc C ln 3 L (1 − ε) η C0

(12)

where C/C0 is the fractional breakthrough, or the ratio of effluent BSA concentration (C) divided by the influent concentration (C0), L the thickness of the filter bed, ε is the porosity, α the attachment efficiency, η the collector efficiency, and dc the collector diameter. The collector efficiency is calculated as follows [40] and the calculated results are shown in Table 2. (The parameter used in calculation was listed in Table S2 of supporting information)

⎛ 4 ⎞1/8 η = As ⎜ NA ⎟ NR15/8 + 0.00338As NG6/5 NR−2/5 + 4As1/3 P e−2/3 ⎝3 ⎠

(13)

NA, NR, NG and Pe, are dimensionless quantities that describe BSAfilter media collisions due to London-van der Waals interactions, interception, sedimentation and diffusion, respectively.

NA =

Ha 3πμd2p vF

(14)

NR =

dP dc

(15)

NG =

Pe =

(

)

g ρp −ρ d2p 18μvF

(16)

3πμvF dc dp kBT

(17)

where dp and ρp are the equivalent diameter and density of the BSA, vF the filtration rate, μ and ρ the kinematic viscosity and fluid density, T the temperature, Ha the Hamaker constant, g the gravitational constant, and kB the Boltzmann constant. As accounts for the influence of neighboring collectors on flow and is calculated as follow:

As =

(

)

2 1−γ 5

2−3γ + 3γ 5 − 2γ 6

(18)

1/3

where γ = ( 1 − ε) . When membrane pore size (Fig. 5c) is considerably larger than the hydrodynamic size of BSA molecules even after BSA adsorption (Fig. 3), the electrostatic repulsion at pore entrances is negligible and BSA can penetrate into the membrane and be adsorbed around the spherical surface of fibers, thus decreasing the slit gap (much smaller than 0.5 mm) between them and hence increasing the hydraulic resistance. The adsorption front moves to the bottom of the filter, at the same time, BSA adsorption only negligibly reduces membrane pore dimension. As a result, the loss of membrane permeability is caused by increased hydraulic resistance as water molecules flow over adsorbed BSA molecules. Therefore, the more BSA adsorbed, the greater would hydraulic resistance increases and permeability drops. As the filtration continues, the adsorption capacity of the membrane for BSA will be gradually saturated, and consequently, the membrane loses its selectivity to BSA. Because the selectivity directly results from adsorption, solution chemistry may affect membrane selectivity by altering the physico-chemical interactions between BSA and the porous membrane that are relevant to BSA adsorption. As seen in Table 2, the attachment efficiency (α) follows an order of pH ¼4.78 4 pH ¼3.78 4pH ¼5.78 4pH ¼ 10. Therefore, the selectivity and permeability should follow an order of pH ¼4.78 4 pH ¼3.78 4pH ¼5.78 4pH ¼ 10 and 4.78 opH ¼ 3.78 opH ¼ 5.78o pH ¼10, respectively. 4.2. Evolution of the selective-permeability relationship for different membranes 4.2.1. The PVC membrane In present study, the PVC membrane possesses narrow pore size distribution (Fig. 5(a)) that is comparable to the diameter of BSA molecule (Fig. 3). As predicted by the filtration model presented above, pore constriction induced by monolayer BSA adsorption and physical sieving at pore entrances play an important

J. Ma et al. / Journal of Membrane Science 514 (2016) 385–397

role in BSA filtration. During the filtration experiment, BSA entered the membrane pores and then passed into the permeate or adsorbed onto the pore wall. The adsorption of BSA on membrane pore wall led to pore constriction until the monolayer was completely formed, and at this time, the pore size was smaller than BSA and the physical sieving became an important mechanism for BSA rejection. Therefore, the initial selectivity was closed to unity for the PVC membrane due to the sieving effect occurred after a certain time of filtration (pore constriction), which explains that the initial selectivity was independent of solution pH and remained relatively stable during the beginning stage of filtration (Fig. 7). Due to the different in electrostatic interactions based on the measured values of zeta potentials as a function of pH by using the adsorption coefficient expression, the selectivity for PVC membrane follows an order of pH ¼4.78 4pH ¼3.78 4 pH ¼5.78 4pH ¼ 10. Concurrently, the continuous decrease in permeability is attributable to the adsorption of a BSA layer on the pore surface of the membrane, and at this time, the size exclusion (or cake/gel filtration) should be the dominant mechanism in the ultrafiltration process. At the first stage, the adsorption of BSA on membrane pore wall led to pore constriction and the permeability decreased rapidly. At the second stage, the monolayer was completely formed. At this case, the physical sieving became an important mechanism for BSA rejection and the permeability decreased slightly at long time. The numerical estimates of permeability follows an order of pH ¼ 4.78 opH ¼3.78 opH ¼ 5.78 opH ¼10. Consequently, the highest selectivity and lowest permeability occurred at a pH of 4.78 and the lowest selectivity and highest permeability appeared at a pH of 10.0 (Fig. 7). These results are consistent with the finding obtained in previous studies. Fane et al. [41,42] reported that the permeability of the fouling layer was lower at the pHIEP, as the proteins have small sizes and thus formed a densely packed layer. Therefore, at pH of 4.78, the permeability was the lowest. At different pH, the size of BSA was different and followed an order of pH ¼10.0 4pH ¼ 5.78 4pH ¼3.78 4 pH ¼4.78 (Fig. 3). Smaller protein size resulted in more densely packed layer, and thus lower permeability. With increasing pH, the BSA molecules unfold and form different isomerization [43,44]. The unfolding of BSA molecules can make the deposited layer on the membrane surface more porous [45], and the porous structure was beneficial to water permeation. 4.2.2. The PVDF membrane During the filtration by the PVDF membrane, BSA molecules passed into membrane pores and adsorbed onto the pore walls, uniformly forming a multilayer of adsorbed solutes during the beginning stage of the filtration (Fig. 1(ii)). At the same time, the diameter of the pore entrance was reduced and the steric/electrostatic rejection increased. Therefore, the initial selectivity followed a decreasing order of pH ¼ 4.784 pH ¼3.78 4pH ¼ 5.78 4pH ¼10.0, which was consistent with increasing of Nλ . In addition, Nλ is a function of the filter coefficient ( λ 0 ) and accounts for the ability of the pore wall (pore structure and property, particle size etc.), or the surface of the layer of deposited particles, to capture solute. In this study, the charge of membrane was the key factor to influence the λ 0 , which further influenced the Nλ . Concurrently, the Nλ was consistent with decreasing of negative charges on BSA (Fig. 3) and the PVDF membrane (Fig. 6b). Hence, the initial selectivity was consistent with decreasing of negative charges on BSA (Fig. 3) and the PVDF membrane (Fig. 6(b)). As the filtration proceeded, BSA adsorption further led to pore constriction until multilayers of BSA were completely formed and the filtration evolved into the second stage. In the second stage, the BSA was removed by size exclusion and then cake/gel filtration. Therefore, the selectivity of the PVDF membrane increased while the permeability decreased (Fig. 8).

395

Because of greater negative surface charges and stronger intermolecular repulsion in the first stage and size exclusion in the second stage, noticeable increase in selectivity took place at higher permeability with increasing pH except for pH ¼ 3.78 (Fig. 8). The highest selectivity and lowest permeability occurred at the pH of 4.78. The reason is that the BSA at pH of 4.78 (the IEP of BSA) possessed no net charge and the weak intermolecular interactions favored multilayer adsorption of BSA molecules [46]. In this case, the size exclusion (or the sieving action) became more obvious during the second stage of filtration. 4.2.3. The glass fiber membrane The pore size of the GF membrane (0.5 mm) was far greater than the hydrodynamic diameter of BSA molecules (Figs. 3 and 5c). But, the structure of this filter does not really have a network of 0.5 mm pores due to the slit gap between adjacent fibers was much smaller than 0.5 mm (Fig. 4b). The filtration was dominated by the adsorption of BSA onto the filter, which was also controlled by the zeta potentials of BSA and the membrane [23]. In this study, all chemical factors influencing BSA attachment were incorporated into the attachment efficiency (α) and the membrane depended solely upon adsorption to remove BSA (Fig. 1(iii)). During the beginning stage of the filtration, the solutes were adsorbed onto the glass fibers, thus decreasing the slit gaps among them. As the adsorption front gradually moved towards the bottom of the filter, the hydraulic resistance to membrane filtration increased and the permeability decreased. In this case, membrane selectivity showed a rapid drop as soon as the front came close to filter bottom. Due to the difference in adsorption coefficient at various pH, the selectivity and permeability followed an order of pH ¼ 4.78 4pH ¼ 3.78 4pH ¼5.78 4pH ¼10 and 4.78 opH ¼ 3.78 opH ¼5.78 o pH ¼10, respectively. The filtration entered the second stage as the adsorption capacity of the GF membrane approached depletion. At this time, the selectivity approached unity and the final membrane permeability was determined by the amount of BSA adsorbed onto the membrane. Consequently, the lowest permeability occurred at the pH of 4.78 where the highest attachment efficiency existed. 4.3. Implications to membrane water treatment For the purpose of water and wastewater treatment, an ideal membrane would have both high selectivity for contaminants and high permeability to water (providing the potential for very high filtration rates). Therefore, there is a practical need to overcome the trade-off between membrane selectivity and permeability for full-scale porous membrane filtration systems. Based upon the results of this study, two approaches may be undertaken to achieve this goal. One approach is the selection/preparation of membrane with appropriate pore structure. Massive efforts have been made in the development of novel membranes with high permeability and selectivity; these include reducing the thickness of the selective layer, increasing the porosity of the membrane, and enhancing the affinity of the solute to the pore wall via surface modification [47]. Furthermore, Kanani et al. [13] studied the effect of pore geometry on permeability–selectivity relationship and found that membranes with slit-shaped pores have higher performance. Moreover, the fibrous GF membrane studied here had a potential advantage over other porous membranes due to its extremely high initial permeability (Table 1) and decent selectivity within the range of its adsorption capacities (Fig. 9c). However, practical value of fibrous membranes will rely upon significantly increase in the adsorption capacities of the membranes, to a level that allows extended time of filtration to be feasible. The other approach to balance the permeability-selectivity tradeoff is to carefully control feed water condition for membranes. For

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the PVDF membrane used in this study, BSA filtration at a pH of 10.0 resulted in relatively highly selectivity and permeability (Fig. 8c), which provided a simple means to enhance membrane performance for BSA filtration. Considering the complex nature of realistic water or wastewater, future studies are warranted to determine the suitable chemical conditions for the filtration of realistic wastewater effluents.

5. Conclusion Protein filtration has important implications in water and wastewater treatment. In this study, membrane filtration experiments were carried out with a PVDF, MF membrane; a PVC, UF membrane; and a glass fiber, MF membrane by using BSA as a surrogate for proteinaceous contaminants in wastewater. Different selectivity-permeability relationships were identified with these commercially available membranes that have been or possess potentials to be employed in wastewater treatment. Combination of experimental and model simulation results reveals that BSA filtration by the UF membrane was caused by pore constriction due to solute adsorption on pore walls, size exclusion and cake/gel filtration, while that by the glass fiber membrane was primarily protein adsorption. The PVDF membranes showed a shift of protein rejection by adsorption to that by pore constriction at the early stage which was followed by size exclusion and then cake/gel filtration then after. Consequently, pH strongly affected the evolution of membrane permeability by changing the zeta potential of solutes and the studied membranes. Also, the selectivity was less significantly affected by pH for the polymeric membranes than the glass fiber membrane. Moreover, the change in solute size produced by the change in pH did not noticeably affect the selectivity and permeability of the polymeric membranes but exerted some effects on those of the glass fiber membrane. These findings suggest that temporal evolution of the selectivity-permeability relationship may vary for differently structured membranes during water and wastewater treatment. Consequently, there are two possible approaches to tackle the selectivity and permeability tradeoff for membranes used in fullscale application: (1) optimization of membrane pore structure, by reducing the thickness of the selective layer, increasing the porosity of the membrane, and/or enhancing the affinity of the solute to the pore wall via surface modification; (2) control of feed water condition (such as pH), for instance, by enhancing repulsive solute-membrane interactions for membranes whose performance are affected by pore constriction.

Acknowledgements Mr. Chenhao Gong is acknowledged for performing the zeta potential measurement for BSA. The author would also like to express sincere gratitude to the two anonymous reviewers who carefully reviewed our manuscript and provided insightful and constructive comments on the work. This work was financially supported by the National Natural Science Foundation of China (Grant No. 21447004) and the Special Fund for State Joint Key Laboratory of Environmental Simulation and Pollution Control of China (Grant No. 2070403GK).

Appendix A. Supplementary material Supplementary data associated with this article can be found in

the online version at http://dx.doi.org/10.1016/j.memsci.2016.05.022.

Nomenclature A c0 C C0 Cf, Cfeed,

adsorption coefficient volume fraction of suspended solutes in the feed effluent concentration (g L  1) influent concentration (g L  1) solute concentration in the bulk feed solution (kg m  3) Cp, Cpermeate solute concentration in the permeate (kg m  3) Cw, Cwall solute concentration at the upstream surface of the membrane (kg m  3) solute diffusion coefficient in the free solution D∞ outside the pore (m2 s  1) dp equivalent diameter of solute G0 initial volumetric permeate flow rate through one cylindrical pore (m3 s  1) g the gravitational constant (m s  2) Ha Hamaker constant (J) hm is separation distance between the surface of collector and the point of maximum (energy barrier) J permeate flux (m s) kB Boltzmann constant (J K  1) lp membrane thickness (pore length) (m) Lp membrane permeability (m s  1 Pa  1) Nλ = λ 0 lp dimensionless factor accounting for the efficiency of solute capture by pore walls transmembrane pressure (Pa) ΔP rp0, rpore,z ¼ 0 initial pore radius (m) rp pore radius (m) rs,rsolute solute radius (m) Sa actual solute sieving coefficient (at the upstream surface of the membrane) S0 ratio of solute permeate concentration to feed concentration time (s) Δt T temperature (K) u stirring speed Greek letters

α η

dimensionless attachment efficiency dimensionless the collector efficiency β = Cw/Cf concentration polarization factor initial membrane porosity ε0 ε membrane porosity λ = rs/rp ratio of solute radius to pore radius initial filter coefficient (m  1) λ0 filter coefficient (m  1) λf fluid viscosity (Pa s) μ ν kinematic viscosity vF filtration rate ρp density of solution (g cm  3) ρ fluid density (g cm  3) σ specific deposit (volume fraction occupied by deposited particles) interaction potential energy at the point of maxΦ (h m ) imum (energy barrier) (J) interaction potential energy (J) Φ (h2 ) solute partition coefficient φ selectivity ψ

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