Temperature effects on sieving characteristics of thin-film composite nanofiltration membranes: pore size distributions and transport parameters

Journal of Membrane Science 223 (2003) 69–87

Temperature effects on sieving characteristics of thin-film composite nanofiltration membranes: pore size distributions and transport parameters Ramesh R. Sharma a , Rachana Agrawal a , Shankararaman Chellam a,b,∗ a

Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77204-4003, USA b Department of Chemical Engineering, University of Houston, Houston, TX 77204-4004, USA Received 13 February 2003; received in revised form 13 June 2003; accepted 17 June 2003

Abstract Crossflow filtration experiments were performed to measure transport of water and hydrophilic neutral organic solutes spanning a range of molecular sizes across two commercial thin-film composite nanofiltration (NF) membranes in the temperature range 5–41 ◦ C. Non-viscous contributions to activation energies of pure water permeation across these polymeric membranes were calculated to be 3.9 and 6.4 kJ mol−1 . Analysis of solute rejection using a phenomenological model of membrane transport revealed that sizes of pores that contributed to rejection followed a lognormal distribution at any given temperature. Additionally, increasing temperature increased mean pore radii and the molecular weight cutoff suggesting changes in the structure and morphology of the polymer matrix comprising the membrane barrier layer. Consistent with the free volume theory of activated gas transport, activation energies of neutral solute permeability in aqueous systems also increased with Stokes radius and molecular weight indicating their hindered diffusion in membrane pores. All activation energies for pore diffusion calculated in this study were greater than just the viscous contribution to bulk diffusion demonstrating hindered transport across the nanofiltration membranes. Finally, similar to gas transport across zeolites and rubbers, the activation energy and the Arrhenius pre-exponential factor for hindered diffusion coefficients increased with solute size and were highly correlated with each other. © 2003 Elsevier B.V. All rights reserved. Keywords: Thin-film composite membranes; Nanofiltration; Transport phenomena; Activation energy; Hindered transport

1. Introduction Polymeric thin-film composite nanofiltration (NF) membranes are being increasingly implemented during municipal and industrial water purification for removing hardness, synthetic organic compounds, natural organic matter and disinfection by-product ∗ Corresponding author. Tel.: +1-713-743-4265; fax: +1-713-743-4260. E-mail address: [email protected] (S. Chellam).

precursors, mono- and multivalent ions, etc. [1–3]. Partially due to their recent invention and the growing interest in their implementation, characterizing the separation capabilities of NF membranes is an active research area [4–10]. Consistent with the definition of NF membranes as those that remove solutes smaller than approximately 2 nm [11], their pore diameters have been reported to be in the range ∼0.8–2.4 nm [4,5,9,12,13]. Because NF membrane pores are only a few times the size of water and dissolved solute molecules, transport across

0376-7388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0376-7388(03)00310-7

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these membrane pores can be expected to be substantially hindered and subsequently an activated process. The activation energies for pure water permeation (including viscous contribution) across reverse osmosis (RO) membranes and inorganic NF membranes have been reported to be in the range 18–30 kJ mol−1 [14–18]. On the other hand, only viscosity correction is necessary to correlate temperature effects on pure water permeability of micro- and ultrafiltration (UF) membranes [16]. To date, activation energies for water transport across thin-film composite NF membranes have not yet been published, even though they can be expected to be lower than that of RO membranes but greater than just viscosity correction. Pore size distributions of UF [19–21] and NF membranes [4,8] have been determined using a variety of techniques. One important shortcoming of microscopy and gas adsorption/desorption is that they measure pores that contribute to permeation (active pores) as well as those that dead-end within the thickness of the thin barrier layer (passive pores). Further, separating the support layers in asymmetric UF and thin-film composite NF membranes prior to N2 adsorption/desorption is difficult [22]. Additionally, atomic force microscopy has been reported to overestimate the pore radius compared to calculations based on polyethylene glycol retention by a factor of 2–9 [23]. Further, activation energies for membrane transport in aqueous systems can be obtained only from water and solute transport measurements. For these reasons, we have used molecular transport measurements to obtain information corresponding only to those pores that contribute to membrane selectivity. Even though activation energies of salt transport across RO membranes have been reported in the range 20–30 kJ mol−1 [15,18], corresponding values for organic compounds have not been measured presumably due to their very high rejections. Charge interactions strongly influence electrolyte rejection from aqueous streams complicating interpretation of their activated transport and calculations of pore size distributions [5,9,24]. To overcome this difficulty, we employ several neutral organic solutes of varying molecular weight and Stokes radius to evaluate sieving characteristics of polymeric NF membranes. Unlike average pore sizes of inorganic NF membranes that have been reported to be constant in the temperature range 20–60 ◦ C [17], sizes of pores in

the active layer of polymeric NF membranes can be expected to depend on operating temperature. Paradoxically, even though temperature should be an important parameter in determining selectivity, its effects on the separation characteristics of thin-film composite NF membranes have not yet been reported. The primary objective of this research was to investigate changes in water permeability and sieving capabilities of polymeric NF membranes with temperature. A phenomenological model of membrane transport was employed to analyze pure water permeability and rejection of several neutral organic solutes in the temperature range 5–41 ◦ C. Changes in pore size distributions with temperature and the apparent activation energy associated with hindered transport of water and neutral solutes of different molecular dimensions across two commercially available polymeric thin-film composite NF membranes are reported. 2. Theoretical work 2.1. Membrane transport model Pure water and neutral solute transport across NF membranes were analyzed using a phenomenological model [25,26]: Jv = Lp PTM
 dc Js = −Px + (1 − σ)Jv c dx

(1) (2)

where Lp is the pure water permeability, σ the reflection coefficient, P the solute permeability, PTM the transmembrane pressure, c solute concentration, and x is the coordinate perpendicular to membrane surface. Osmotic pressure contribution by the neutral organic solutes employed (see Section 3) has been neglected because it was less than 2.5 kPa (<2% of the lowest pressure applied). According to Eq. (2), σ and P are the measures of solute transport due to convection and diffusion, respectively. By integrating Eq. (2) across the membrane thickness x, the intrinsic rejection (R) has been expressed in terms of permeate and membrane surface solute concentrations (Cp and Cm ) as [25]: R=1−

Cp σ(1 − F) = Cm 1 − σF

(3)

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where F = exp (−((1 − σ)/P)Jv ). The concentration at the membrane surface can be calculated using film theory in terms of the mass transfer coefficient, k, and the bulk concentration Cb [27,28]:
 Cm − Cp Jv = exp (4) Cb − C p k 2.2. Parameter estimation The intrinsic membrane transport parameters σ and P, and the mass transfer coefficient k were obtained using non-linear regression by combining Eqs. (3) and (4) and using the experimental measurements of solute permeate concentrations and volumetric water flux in a range of pressures for a constant feed flow rate and feed concentration. The sum of squares of the residuals (S) for all experimental data points was minimized using the Solver tool in Microsoft Excel with 5% tolerance and 10−6 precision to determine σ, P, and k:

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2.3. Pore size distribution calculations Phenomenological coefficients obtained from solute rejection measurements were linked to membrane morphological parameters using models of hindered transport [9,19,21,30]. Three hydrodynamic approaches namely, steric hindrance pore model [27], model of Bungay and Brenner [31], and the slit pore model [32] were used to obtain the pore radius and the ratio of porosity to membrane thickness (Ak /x) by non-linear least squares regression [30]: σ = 1 − φKic P = D∞ φKid

(7) Ak Ak = De x x

(8)

where n is the number of observations, Rexpt the experimentally measured rejection, and Rtheory is the theoretical rejection calculated by combining Eqs. (3) and (4). The confidence intervals for σ and P were determined by propagating errors produced during parameter estimation. The joint (1 − α) (α = 0.05) likelihood region for σ and P corresponds to the contour with level [29]:   ˆ 1 + pF(p, n − p, α) , SF = S(Θ) (6) n−p

where φ is the equilibrium partitioning coefficient given as (1 − λ)2 for steric hindrance pore model and Bungay and Brenner’s model, and (1 − λ) for slit pore model (λ is the ratio of the solute radius to pore radius), Kic and Kid are the hindrance factors for the convective and diffusive transport, D∞ and De are the bulk and hindered diffusion coefficients, respectively. Pore size (distributions) of NF and UF membranes at room temperature have been reported earlier using this approach [12,33] even though these models are not strictly applicable to highly constricted pores that are tortuous and varying in cross-sectional area. As suggested previously [33], the number-based lognormal pore size distribution was obtained after incorporating it into the corresponding hindered transport model and then fitting it to experimental measurements of the reflection coefficient: ∞ φKic no exp[−(ln(r/Rmp )/ln Sg )2 ]r 4 dr σ = 1 − 0 ∞ 2 4 0 no exp[−(ln(r/Rmp )/ln Sg ) ]r dr (9)

where α is the significance level, SF denotes the value of the sum of squares contour defining the (1 − α) ˆ denotes the optimal parameter estimate, region, Θ p is the number of parameters, F(p, n − p, α) the cumulative Fisher F distribution corresponding to significance level α with p and n − p degrees of freedom for the numerator and denominator, respectively. Typically, a high initial guess compared to the optimal value for the parameter converged to the upper confidence limit when the other parameters were held at their optimal values.

where Rmp is the most probable or modal value of the pore radius r, and no the number of pores at the maximum in the distribution function and Sg is the geometric standard deviation for the lognormal distribution. Note that the steric hindrance pore model [27], model of Bungay and Brenner [31], and the slit pore model [32] only differ in the analytical expressions for Kic in Eq. (9). Reflection coefficients of various solutes at different temperatures were also used to deduce the geometric mean radius (¯r) and standard deviation (Sp ) of the pores assuming that

S=

n 

(Rexpt − Rtheory )2

(5)

i=1

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solute molecules permeate completely through every pore that is larger than its diameter [12]:   r∗ 1 1 (ln(r) − ln(¯r ))2 ∗ dr σ(r ) = exp − √ 2Sp2 0 Sp 2π r (10) The upper limit of the integration in Eq. (10) is r∗ , the solute Stokes radius, so as to include only pores that reject solutes completely based on size exclusion. For convenience, Eq. (10) is referred to as the “lognormal model” in this manuscript even though hindered transport models depicted in Eq. (9) also assume a lognormal distribution of pore sizes. The mean and variance of the different mathematical representations of the lognormal distribution given in Eqs. (9) and (10) were related using relationships available in the literature [34]. 2.4. Temperature effects on water and solute permeability As in previous studies of temperature effects on reverse osmosis and inorganic membrane transport parameters [10,15,17,35], the water and solute permeability were modeled using Arrhenius equation:
 Ei Ni = Ai exp − (11) RT where Ei is the apparent activation energy for transport of species i through the membrane, and Ai is the pre-exponential factor. Ni represents either the permeability or hindered diffusivity of solute i, or viscosity-corrected water permeability (Lp µ). The best-fit values and standard deviations of the apparent activation energy as well as the pre-exponential factor were estimated using linear regression.

3. Experimental work

recycled to the 20 l feed tank to keep the feed concentration constant and to limit the necessary feed water volume. For these studies, a stainless steel pressurized cell (model SEPA-CF, Osmonics, Minnetonka, MN) using a 19 cm × 14 cm flat membrane sheet (effective filtration area 155 cm2 ) was employed. Feed and permeate spacers were employed to enhance back-diffusion and to reduce concentration polarization. The use of a positive displacement gear pump (model 74011-11, Cole-Parmer, Chicago, IL) for feed water minimized pressure fluctuations. Using inert materials such as Teflon or stainless steel for all wetted components including tubing, connectors, gears, and the membrane cell reduced adsorptive or reactive losses in the apparatus. A flow meter (model P-32046-16, Cole-Parmer, Chicago, IL) was used to monitor the retentate flow rate. Additionally, both permeate and retentate flow rates were manually measured using a stopwatch and a weighing balance. For pure water permeability measurements, the permeate water was continuously collected on a weighing balance (Ohaus Navigator N1H110, Fisher Scientific, Houston, TX), which had a full scale of 8100 g and a least count of 0.5 g. The in-built RS232 port in the balance was directly connected to the serial port of a personal computer to obtain a digital signal corresponding to the weight. Online data acquisition using LabVIEW (Version 5.1, National Instruments, Austin, TX) software was used to continuously monitor the pressure and temperature. A 0.5–5.5 V pressure transducer (PX303-200G5V, Omega Engineering Co., Stamford, CT) was employed to continuously monitor the pressure of the feed water, which was excited using 24 V DC from an external power supply. The transducer was calibrated using a high accuracy (±1%) glycerin filled pressure gauge (U-68022-05, Cole Parmer, Vernon Hills, Illinois). This pressure gauge was also used to manually record pressure readings at the retentate side. The temperature of the feed water was recorded using a 30.48 × 10−2 m (12 in.), rugged probe (TJ120 CPSS 116G, Omega Engineering Co., Stamford, CT).

3.1. Bench scale experimental apparatus 3.2. Membranes and solutes employed Crossflow filtration experiments were conducted in a temperature controlled room (10-8-R-NR, Tafco, Pittsburgh, PA) maintained to ±0.5 ◦ C of the desired temperature. Similar to some previous investigations [13,14], both retentate and permeate streams were

Two commercially available polyamide thin-film composite membranes designated as “DL” (Osmonics, Minnetonka, MN) and “TFCS” (Koch Fluid Systems, San Diego, CA) by the respective manufacturers were

R.R. Sharma et al. / Journal of Membrane Science 223 (2003) 69–87

employed. Following manufacturer recommendations, membrane sheets were kept in the dark at room temperature as received from them. All membrane coupons employed in this study were selected from the same lot and the manufacturers confirmed that their membranes could be operated in the range 5–41 ◦ C without irreversible damage to their performance. Experiments were performed at 5, 15, 23, 35, and 41 ◦ C. Highly water-soluble solutes viz. methanol, ethanol, ethylene glycol, t-butyl alcohol, dextrose, and sucrose (ACS grade, EM Science, Gibbstown, NJ), and xylose, glycerol, raffinose 5-hydrate, and ␣-cyclodextrin (ACS grade, Sigma Aldrich Company, St. Louis, MO) spanning a wide range of molecular sizes were used to comprehensively characterize membranes. These hydrophilic neutral solutes have been previously used to characterize NF membranes because their specific chemical interactions with membranes are expected to be small [8,9,17]. Also, employing these monomers of different molecular weight and Stokes radius reduces complications associated with conformational effects on rejection of long chain polymers such as polyethylene glycols during crossflow filtration [30,36]. Neutral organic solute concentrations were analyzed using a total organic carbon (TOC) analyzer (TOC5050A, Shimadzu Instrument Co., Columbia, MD). All results reported herein represent an average of four or five injections with less than 2% coefficient of variation. Samples collected in glass vials with Teflon lined caps (03-340-121, Fisher Scientific, Houston, TX) were stored at 4 ◦ C and brought to room temperature just prior to analysis, and measured within 24 h to minimize possible volatilization or adsorptive losses. 3.3. Crossflow filtration procedure Fresh membrane coupons were first soaked in ultrapure water that was replenished 3–4 times over a 24 h period. This coupon was then placed in the membrane holder and ultrapure water was passed through it for 24 h to remove biocides and other chemicals used for membrane preservation. Following membrane setting, the transmembrane pressure was changed in random order and flux was monitored to obtain the pure water permeability at the start of each experimental run. Each experimental run comprised of several measure-

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ments of a particular solute rejection at a variety of fluxes but at a constant temperature. Pure water transport was measured again after the conclusion of each run to ensure that membrane permeability remained constant. In all cases, 20 l feed water was prepared containing a single solute at a target concentration of 20 mg l−1 as TOC. All experiments with neutral organic solutes were conducted at a feed water recovery <1% to reduce changes in bulk concentration. The crossflow velocity was maintained constant at 9.6 and 19.2 cm s−1 for DL and TFCS membrane, respectively. The permeate concentrations of all solutes were measured as a function of volumetric flux by varying pressure (100–760 kPa) in random order to reduce systematic errors in experimental measurements. The water flux and permeate solute concentrations were measured continuously upon changing the operating pressure. In all cases, steady state (defined in this study as <2% change in the permeate water quality collected over a span of 1 h and <2% change in allowable mass balance closure error) was reached in 5–10 h. Once permeate flux and solute rejection reached steady state, five separate 10 ml samples of permeate water were collected. Five to seven different pressures (permeate fluxes) were used for each solute–membrane–temperature combination, which was completed over ∼40 h. Thus, each phenomenological coefficient corresponds to 25–35 measurements of solute rejection allowing increased precision of parameter estimates. To accurately monitor rejection, a minimum of six measurements of feed water concentrations of each solute was made over the ∼40 h duration of experimentation at each temperature. During this time, feed water concentrations of ethylene glycol, dextrose, sucrose, xylose, glycerol, raffinose 5-hydrate, and ␣-cyclodextrin remained essentially constant (<2% variation). Similar results were obtained for methanol, ethanol, and t-butyl alcohol at 5 and 15 ◦ C whereas a 2–5% loss was measured over ∼40 h at 23 and 35 ◦ C. To compensate for this, rejections were calculated using paired (matched) samples of feed and permeate water collected at the same time. However, because >10% loss of these compounds at 41 ◦ C was measured presumably due to volatilization, only experimental data between 5 and 35 ◦ C were included for methanol, ethanol and t-butyl alcohol.

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4. Results and discussions 4.1. Experimental reproducibility and model parameter uncertainty During long-term operation with a membrane coupon using various solutes at different temperatures, pressures, and crossflow velocities, the membrane characteristics could possibly change over time either due to the adsorption, fouling, or other surface damage caused by trace contaminants. Changes in water permeability or neutral solute retention by a particular NF membrane coupon with time (for one set of operating conditions) indicates that its sieving characteristics (thickness, surface porosity, pore size distribution, etc.) have changed during the course of the experiments necessitating replacement of the coupon. To examine possible changes in membrane performance with operating time, water permeability tests were conducted before and after experimentation with each solute and temperature. Non-viscous contributions to water transport were probed by analyzing pure water permeability data after multiplying it by the corresponding solution viscosity (Lp µ). Fig. 1 depicts viscosity-corrected water permeability data for DL and TFCS membranes at variety of temperatures obtained from duplicate tests conducted prior to and at the conclusion of each solute rejection experiment (the error bars denote the 95% confidence intervals). As seen, there were no statistically discernable changes in the viscosity-corrected water permeability (Lp µ) at 95% confidence level over the entire course of experimentation for all solutes and at all temperatures. Thus, water permeability of these membrane coupons did not change following filtration using various solutes and fouling was negligible under experimental conditions employed once steady state was reached. Selected experiments were also repeated with different solutes at various temperatures and pressures to quantify experimental variability and possible degradation of the coupon. During these quality control studies, rejection of t-butyl alcohol, glycerol, dextrose, and NaCl was measured periodically at different temperatures to examine possible changes in membrane selectivity over the complete duration of experimentation (Fig. 2). These solutes were chosen because their rejection was highly sensitive to the pressures in the

Fig. 1. Reproducibility of viscosity-corrected pure water permeability over the duration of experiments. The error bars denote 95% confidence intervals.

entire range employed herein. Thus, small changes in membrane selectivity could be better discerned using these solutes thereby increasing the stringency of our quality assurance protocols. Fig. 2a and d depicts rejections for these duplicate experiments as filled and empty symbols, which can be observed to follow each other very closely. Additionally, the 95% confidence intervals of the reflection coefficient and permeability calculated from each of these experiments for all solutes investigated

R.R. Sharma et al. / Journal of Membrane Science 223 (2003) 69–87 Fig. 2. Tests of experimental reproducibility for both membranes using different solutes and temperatures over the entire duration of this study. The empty and filled symbols in solute rejection graphs (a and d) denote results of duplicate experiments. The error bars in parameter estimates (b, c, e, and f) denote 95% confidence intervals.

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overlapped for both membranes (Fig. 2b, c, e, and f). Hence, there were no statistical differences in pure water permeability and solute rejection by both membranes allowing a quantitative comparison of results obtained over the entire course of experimentation. 4.2. Pure water permeability In all cases, permeate flux increased linearly with pressure indicating that membrane compaction effects were negligible in the pressure range employed. The

viscosity-corrected water permeability (Lp µ) of the TFCS membrane increased by 20% (from 10.3×10−15 to 12.4 × 10−15 m) and that of the DL membrane increased by 32% (from 7.8 × 10−15 to 10.3 × 10−15 m) when the temperature changed from 5 to 41 ◦ C suggesting activated permeation of pure water. Fig. 3 is an Arrhenius plot of Lp µ normalized to 23 ◦ C in the temperature range 5–41 ◦ C. A negative slope, statistically different from zero, was observed for both membranes demonstrating that viscous transport alone does not completely explain water

Fig. 3. Arrhenius plots of viscosity-corrected water permeability across two commercially available thin-film composite NF membranes. The solid line depicts the best-fit curve whereas the dashed lines denote the 95% confidence bands. The optimal estimates and 95% confidence intervals of the non-viscous contributions to activation energy are also given.

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permeability through these NF membranes. The observed optimal value and 95% confidence interval of the non-viscous activation energy for pure water transport across the DL membrane was 6.4 ± 0.9 kJ mol−1 as compared to 3.9 ± 0.5 kJ mol−1 of the TFCS membrane. The higher activation energy for water permeation across the DL membrane compared to the TFCS membrane is consistent with its lower permeability. Non-viscous contributions to activation energies of water permeation across the NF membranes employed in this study are comparable to those reported for aromatic thin-film composite polyamide RO membranes [14,15], but lower than those for asymmetric RO membranes [18]. Based on these data, the temperature correction factors (normalized to 25 ◦ C) valid in the temperature range 5 ≤ T ≤ 41 ◦ C are: TFCS membrane :
 Jv,T 1 1 = exp 2617 − Jv,25 ◦ C 298 273 + T DL membrane : 
 Jv,T 1 1 = exp 2918 − Jv,25 ◦ C 298 273 + T where Jv,T is the volume flux of water at any temperature T (in ◦ C) and Jv,25 ◦ C is the corresponding volume flux at 25 ◦ C. Membrane manufacturers employ similar expressions for temperature correction factors of RO membranes where the temperature should be expressed in ◦ C [16,37]. The statistically significant differences in activation energies for water permeability of the two NF membranes studied demonstrate the need to develop separate expressions for individual membranes to accurately normalize water productivity observed over a range of temperatures. 4.3. Solute reflection coefficient and permeability Permeate volume flux and solute rejection measurements were made using neutral solutes of molecular weights in the range 46–594 g mol−1 for DL membrane and 32–973 g mol−1 for TFCS membrane. As reported previously [4,5,9,13,36,38,39], neutral solute rejection increased with flux. Therefore, rejections were plotted as functions of inverse permeate flux so that consistent with Eq. (3), the y-axis intercept corresponds to the reflection coefficient (Jv → ∞ ⇒

Fig. 4. Estimation of phenomenological transport coefficients at 23 ◦ C using neutral solutes of different molecular dimensions for the two thin-film composite polymeric nanofiltration membranes employed.

R → σ). Fig. 4 depicts this behavior at 23 ◦ C for both DL and TFCS membranes. Similar results were obtained at 5, 15, 35, and 41 ◦ C (data not shown). Mass transfer coefficients obtained by fitting Eqs. (3) and (4) to these data were very high (∼104 ␮m s−1 ) indicating that concentration polarization was negligible under the experimental conditions investigated and that the observed rejection can be treated as the intrinsic rejection. Rejections of the

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highest molecular weight solutes employed in this study were >90% and independent of permeate flux at all temperatures (raffinose and sucrose for the DL membrane and sucrose, raffinose, and ␣-cyclodextrin for the TFCS membrane). These results can be interpreted from Eq. (3), where as P → 0, F → 0 regardless of Jv , and therefore, R → σ. On the other hand, experimental rejections of the lowest molecular weight solutes investigated (ethanol for the DL membrane, and methanol for the TFCS membrane) were independent of permeate flux and statistically indistinguishable from zero (with 95% confidence) at 41, 35, 23 and 15 ◦ C. This behavior can also be interpreted from Eq. (3) where σ → 0 ⇒ R → 0. Mathematically, any non-zero value of permeability will still result in zero observed rejection when σ → 0. Note that statistically significant rejections of ethanol by the DL membrane and methanol by the TFCS membrane were obtained at 5 ◦ C resulting in non zero estimates of phenomenological coefficients. 4.4. Pore size distributions obtained from hydrodynamic considerations Even though phenomenological coefficients quantitatively captured variations in uncharged solute rejection by NF membranes with temperature and molecular size, they do not incorporate any mechanistic interpretation of changes in membrane morphological and structural parameters. Pore size distributions were also derived to better interpret changes in membrane selectivity towards neutral solutes with temperature and Stokes radius. Pore size distributions obtained by fitting Eqs. (9) and (10) to experimental reflection coefficients of neutral solutes having different molecular dimensions at 23 ◦ C are shown in Fig. 5. Consistent with the definition of NF membranes, mean pore radii obtained using hindered transport models [27,31–33] and the lognormal distribution [12] of the DL and TFCS membranes were in the range 0.26–0.73 nm at 23 ◦ C. These results also agree with average pore radii reported for commercial thin-film composite NF membranes [9,12,13] using molecular transport measurements at room temperature. For both membranes, excellent fits of the lognormal model (Eq. (10)) to experimental reflection coefficients were obtained (0.88 < R2 < 0.99) at

Fig. 5. Comparison of pore size distributions for two NF membranes obtained by fitting various models to experimental reflection coefficients at 23 ◦ C.

all temperatures. Additionally, the lognormal model resulted in a lower sum of squared errors and higher correlation coefficients compared to hindered transport models (Eq. (9)) indicating that the sieving characteristics of NF membranes were better described assuming a lognormal distribution of pore sizes. Recently, it has been suggested that a lognormal distribution of pore sizes is not theoretically necessary [40]. However, other density functions with non-negative domains for the pore size distribution such as the Weibull, gamma, and Rayleigh distributions revealed worse fits (higher sum of squared errors and lower correlation coefficients) to experimental reflection coefficients compared to the lognormal distribution.

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Thus, in agreement with previous studies on microand ultrafiltration membranes [33,41–43], and NF membranes [12], the lognormal distribution may be most suitable for modeling pore size distributions of thin-film composite NF membranes. Further, the mean pore radius calculated using the lognormal model for the DL and TFCS membranes was 0.30 and 0.26 nm, respectively at 23 ◦ C. Models based on steric hindrance in cylindrical pores, slit pore, and Bungay and Brenner’s calculations revealed mean pore radii of 0.52, 0.51, and 0.73 nm and 0.42, 0.40, and 0.57 nm, for the DL and TFCS membranes, respectively at 23 ◦ C. Thus, a purely sieving model based on a lognormal distribution of pore sizes (Eq. (10)) resulted in a lower value of the mean pore radius compared to hindered transport models (Eq. (9)) for both membranes at all temperatures. Similar results have been reported for another commercial NF membrane at room temperature [12] even though no explanation was provided for this observation. According to the lognormal model (Eq. (10)), rejection occurs only in pores smaller than the solute. Neglecting hindered convection and diffusion in pores larger than the solute, which should also contribute to sieving, skews the mean pore size to smaller values compared to those obtained using hydrodynamic models. Excellent fits of the simple lognormal model to experimental reflection coefficients allows its use to quantitatively compare changes in pore size distribution of polymeric NF membranes with changes in temperature. However, mechanistic interpretations of hindered transport incorporated in the hydrodynamic models [27,31–33] are not possible using the purely sieving lognormal model. 4.5. Effects of temperature on pore size distributions and molecular weight cut-off Reflection coefficients along with best-fits of the lognormal pore size distribution model (Eq. (10)) for DL and TFCS membranes obtained at 5, 15, 23, 35 and 41 ◦ C are compared in Fig. 6. The reflection coefficients of solutes having an intermediate molecular size (ethanol, dextrose, xylose, t-butyl alcohol, glycerol, and ethylene glycol) increased with decreasing temperature. In contrast, reflection coefficients of extreme molecular size solutes investigated (methanol, sucrose, raffinose, and ␣-cyclodextrin) were not sub-

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Fig. 6. Lognormal fits to experimental reflection coefficients with solute Stokes radius in the temperature range 5–41 ◦ C. Insets depict the change in mean pore radius and molecular weight cut-off (MWCO) with increase in temperature for both polymeric NF membranes investigated.

stantially influenced by temperature. The average pore size of the DL membrane increased by 21% whereas that of the TFCS membrane increased only by 12% when the temperature increased from 5 to 41 ◦ C (see insets in Fig. 6). Additionally, average pore radius of the DL membrane increased at a constant rate with temperature whereas only a 4% increase was observed between 5 and 35 ◦ C for the TFCS membrane followed by an 8% increase when the temperature increased from 35 to 41 ◦ C. This might be caused by changes in the structure of the polymer constituting the active layer of the TFCS membrane near 40 ◦ C, even though

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the manufacturer recommended its use at these temperatures. Traditionally, sieving characteristics of UF (and to some extent NF) membranes have been specified using the molecular weight cut-off (MWCO) [1,20,21], which corresponds to the lowest molecular weight of a solute that is rejected to 90% by the membrane under consideration [11,20,21]. The simplistic expectation is that solutes with molecular weight greater than the molecular weight cut-off will be rejected to greater than 90% and those with lower molecular weight will be rejected to a lesser extent under the same operating conditions. Fig. 6 can also be interpreted as a special case of the MWCO curve at infinitely high flux (R → σ as Jv → ∞), if the abscissa in Fig. 6 is changed from Stokes radius to molecular weight. Under these conditions, as depicted in the insets of Fig. 6, the MWCO of the DL membrane increased as 192, 226, 268, 277, and 377 g mol−1 when temperature increased as 5, 15, 23, 35, and 41 ◦ C, respectively. Similarly the MWCO of the TFCS membrane increased as 170, 175, 180, 196 and 246 g mol−1 when temperature increased as 5, 15, 23, 35, and 41 ◦ C, respectively. These MWCO values are in the manufacturer-specified ranges for NF membranes [1,2]. The large increase in MWCO for both membranes (∼100% for DL and 45% for TFCS) compared to only a 20 and 12% increase in their mean pore size when temperature was increased from 5 to 41 ◦ C indicates that temperature influenced MWCO more than the average pore size. 4.6. Dependence of permeability on solute and temperature

Fig. 7. Arrhenius plots of permeability for solutes of varying molecular dimensions for two commercially available polymeric NF membranes. The optimal estimates of the activation energies along with the standard deviations are also given.

The temperature dependency of solute permeabilities across the DL and TFCS membranes was quantified using the Arrhenius relationship (Fig. 7). The rejection of raffinose, sucrose, and ␣-cyclodextrin by the membranes evaluated was very high and invariant with flux (P → 0). Further, ethanol rejection by the DL membrane and methanol rejection by the TFCS membrane were very low and invariant with flux (σ → 0 and P indeterminable). Therefore, Arrhenius parameters for these solute–membrane combinations could not be calculated. In all other cases, statistically significant values of the activation energy were obtained at the 95% confidence level suggesting solute

permeation across these NF membranes is an activated process. In the absence of chemical interactions between the solute and water with the membrane matrix, only geometric (steric) factors contribute to selectivity. Activation energy of free diffusion is controlled by the temperature dependence of inverse viscosity, that can be calculated to be 17.9 kJ mol−1 in the range 5–41 ◦ C [18]. Therefore, apparent activation energy values higher than 17.9 kJ mol−1 indicate hindered diffusion. Chemical effects including van der Waals interactions, adsorption, and hydrogen bonding, may result

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in large discrepancies between the apparent activation energies of diffusion within membrane pores and the corresponding solute permeability across the membrane. The negative enthalpy associated with water and solute adsorption on membrane pores effectively reduces the activation energy of permeability [18]. Therefore, activation energies of solute permeation smaller than 17.9 kJ mol−1 does not necessitate free diffusion within the pores. Positive values of the activation energies of permeation for all solutes depicted in Fig. 7 suggest diffusion-dominated transport. Fig. 8 depicts a general trend of increasing activation energy with the ratio of solute size to the pore size corresponding to 99th percentile reflection coefficient at 23 ◦ C, (λ0.99 )23 ◦ C , as well as solute Stokes radius for both membranes. Normalizing the solute size by an appropriate membrane pore diameter allows the simultaneous comparison of activation energies from both membranes investigated. Similar results were also obtained for other reflection coefficient

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percentiles (data not shown). The activation energy of permeation for both membranes increased 10-fold from ∼5 to ∼55 kJ mol−1 when Stokes radius approximately doubled from 0.20 nm (ethanol) to 0.37 nm (dextrose). The free volume theory has been used to interpret activation energy for gas transport across nanoporous membranes and zeolites, wherein energy is required for a molecule to break the attractive forces with an adjacent molecule in order to make a jump into the next cavity [44]. Analogously, the activation energy associated with aqueous phase separations by NF membranes can also be expected to increase with the molecular size because higher energies would be necessary to affect a jump (i.e. diffuse) and the best-fit sigmoidal curves in Fig. 8 only empirically describe this increase. Thus, our experimental observation of increase in activation energy of permeation with the solute radius is consistent with the notion of hindered diffusion of the solute in the pores of the thin barrier layer of the thin-film composite membranes. A

Fig. 8. Increasing activation energy of solute permeation with 99th percentile of the ratio of molecular size to membrane pore radius (λ0.99 )23 ◦ C and Stokes radius (inset). The empty symbols correspond to TFCS membrane and filled symbols are data obtained for the DL membrane. The error bars represent the standard deviation.

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similar observation of increasing activation energy with solute size has been reported for transport across inorganic NF membranes [17] even though detailed mechanistic interpretations were not provided. 4.7. Dependence of effective diffusivity on solute and temperature To further investigate hindered transport of solutes across NF membranes hindrance factors for diffusion (φKid ) were calculated using hydrodynamic models for idealized geometries wherein the effective diffusion coefficient (De ) is a unique function of the ratio of solute radius to pore radius [30]. Fig. 9 is an Arrhenius plot of the hindered diffusivity for various solutes, calculated using Bungay and Brenner’s model

[31] for both NF membranes employed in this study. The straight-lines denote fits obtained using the activation energy of diffusion and pre-exponent factor as adjustable parameters. Best-fit values of the activation energy of diffusion along with the standard deviation are also shown in Fig. 9. Activation energies for pore diffusion of all solutes are higher than the bulk diffusion activation energy (17.9 kJ mol−1 ) for both membranes, demonstrating that solute transport inside pores (0 < λ < 1) is hindered. Similar results were also obtained with the steric hindrance pore model [27] and the slit pore model [32] (data not shown). The energy barrier for pore diffusion of weakly interacting neutral solutes such as those employed in this work depends largely on the hydrodynamic drag. Diffusing molecules can more easily overcome this

Fig. 9. Arrhenius plots of solute hindered diffusion coefficients in pores of two thin-film composite nanofiltration membranes. The best-fit values of the activation energies and the corresponding standard deviations in kJ mol−1 are also given.

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Fig. 10. Increase of hindered diffusion activation energy (a) and pre-exponential factor (b) in Arrhenius expression with solute Stokes radius for both polymeric thin-film composite membranes. The straight-line relationship between ln(pre-exponential factor) and activation energy for both membranes investigated is depicted in (c). The error bars denote one standard deviation.

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energy barrier at elevated temperatures because they possess higher kinetic energies [45]. As depicted in Fig. 10a, both membranes exhibited an increase in the pore diffusion activation energy with solute Stokes radius, which is also consistent with the results depicted in Fig. 8. These results are also similar to gas transport across zeolites and nanoporous inorganic membranes, where the activation energy of diffusion increases with the kinematic diameter of the diffusant [46,47]. Further, the activation energy for gas transport can be theoretically expressed in terms of interand intra-molecular interactions using kinetic theory [48]. However, quantitatively extending the kinetic molecular theory to aqueous systems is difficult [44], complicating the molecular interpretation of experimental data. Nevertheless, our results follow the same trends as observed in previous studies of the transport of gas molecules of different sizes across nanoporous zeolites and dense inorganic membranes. The pre-exponential factors in the Arrhenius equation for hindered diffusion of various solutes across the TFCS and DL membranes also increased with the size of the diffusing molecule (Fig. 10b). As seen in Fig. 10a and b, the activation energy and the natural log of pre-exponential factor followed similar trends (same functional form) with solute Stokes radius for both membranes investigated. An excellent straight-line relationship (correlation coefficient > 0.99) was determined between natural log of preexponential factor and activation energy using both membranes (Fig. 10c). Thus, the pre-exponential factor is correlated to the activation energy in aqueous membrane separations suggesting they may not be independent of each other. Even though a quantitative mechanistic interpretation of this behavior in aqueous systems cannot be made, a similar relationship has been reported for gas transport across various polymers [48], wherein the pre-exponential factor was related to both the entropy of activation and jump length. It is important to note that representing the hindered diffusion coefficient in the Arrhenius form gives the pre-exponential factor and activation energy only as empirical fitting parameters. Mechanistically, the pre-exponential factor has been reported to be a function of the solute velocity in the membrane, diffusional length, and geometric factors associated with the pores [46]. A closed form expression for the pre-exponential factor has also been derived for water

transport across a hydrophobic polymer [49], which is not representative of our experiments. However, because the relationship between the activation energy and the pre-exponential factor cannot be derived a priori, we have depicted both of them in Fig. 10a and b, respectively. Interestingly, data from both membranes investigated fit the same line suggesting that each solute molecule possesses similar jump length and frequency in both the membranes. Further as observed in Fig. 10a and b, the increase in activation energy and pre-exponential factor with solute size for both membranes was not identical presumably due to differences in their pore size and its distribution, degree of packing and stiffness of the polymer chains, and on the cohesive energy of the polymer constituting the active layer of these thinfilm composite membranes. Additionally, the degree of crystallanity, crosslinking, and additives such as fillers and plasticizers may also contribute to these differences [48].

5. Conclusions Statistically significant activation energies of water permeation demonstrate the need to incorporate temperature correction factors for normalizing water productivity data obtained using thin-film composite NF membranes over a range of feed water temperatures. Further, unlike the procedure commonly employed during municipal water purification (Jv,T /Jv,25 ◦ C = 1.03T −25 , T in ◦ C) [16,50], temperature correction expressions specific to individual thin-film composite membranes should be employed because non-viscous contributions to activation energies of water transport across the two membranes investigated varied substantially (∼65%). Lognormal pore size distributions obtained for these thin-film composite nanofilters validate the widespread use of this density function in modeling membrane sieving. Additionally, an increase in average pore size and a corresponding increase in MWCO with increasing temperature indicated thermal expansion of the polymer constituting the active layer of these thin-film composite membranes. It should be emphasized that all activation energies reported herein are apparent values because the enthalpy of water and solute sorption on the membranes

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were not measured. Additionally, in contrast to apparent activation energies reported for polymeric RO and inorganic NF membranes, values reported in this manuscript for polymeric NF membranes also include contributions from increasing average pore size with temperature. Environmental contaminants such as natural organic matter can be expected to associate more strongly with polymeric membranes [51] compared to the hydrophilic sugars and alcohols employed in this work. Thus, determining activation energy of natural organic matter transport across polymeric NF membranes such as those commonly employed in water purification should incorporate explicit measurements of the enthalpy of its sorption in addition to its permeability.

Acknowledgements David Paulson and Peter Eriksson of Osmonics Inc. and Randolph Truby and Tom Stocker of Koch Membrane Systems Inc. generously donated membrane samples. We also appreciate the insightful and constructive comments by two anonymous reviewers of an earlier version of the manuscript. This research has been funded by grants from the National Science Foundation CAREER program (BES-0134301) and the State of Texas as part of the program of the Texas Hazardous Waste Research Center (082UHH2816). The contents do not necessarily reflect the views and policies of the sponsors nor does the mention of trade names or commercial products constitute endorsement or recommendation for use.

Nomenclature Ai Ak Cb Cm Cp D∞

pre-exponential factor membrane porosity bulk phase solute concentration (ML−3 ) membrane phase solute concentration (ML−3 ) permeate solute concentration (ML−3 ) free solute diffusion coefficient (L2 T−1 )

De Ed Ei Ep Ew

Js Jv Jv,T k Kic Kid Lp n no Ni

p P PTM r r¯ r∗ R Rexpt Rmp Rtheory S SF Sg Sp T x

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effective/hindered solute diffusion coefficient in pore (L2 T−1 ) activation energy of diffusion (ML2 T−2 mol−1 ) activation energy of species i (ML2 T−2 mol−1 ) activation energy of permeation (ML2 T−2 mol−1 ) Non-viscous contribution to activation energy of water transport (ML2 T−2 mol−1 ) solute flux (ML−2 T−1 ) volumetric pure water flux (LT−1 ) volumetric pure water flux at temperature T in ◦ C (LT−1 ) mass transfer coefficient (LT−1 ) convective hindrance factor diffusive hindrance factor pure water permeability (M−1 L2 T) total number of observations in Eq. (6) number of pores at the maximum of distribution function permeability (LT−1 ) or diffusivity (L2 T−1 ) of solute i, or viscosity-corrected pure water permeability (L) number of parameters in Eq. (6) solute permeability (LT−1 ) transmembrane pressure (ML−1 T−2 ) pore radius (L) geometric mean pore radius (L) upper limit of solute radius in Eq. (10) (L) solute rejection experimentally measured rejection pore radius at the maximum in the distribution function (L) theoretically calculated rejection by combining Eqs. (3) and (4) sum of squares of the residuals value of the sum of squares contour defining the (1 − α) likelihood region geometric standard deviation (L) standard deviation of pore radius (L) temperature (K or ◦ C) membrane thickness (L)

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Greek letters α significance level λ ratio of solute radius to pore radius (λ0.99 )23 ◦ C ratio of solute radius to pore radius corresponding to 99th percentile reflection coefficient at 23 ◦ C ˆ Θ optimal parameter estimate µ solvent viscosity (ML−1 T−1 ) σ reflection coefficient (asymptotic rejection) φ solute equilibrium partitioning coefficient

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