Three independent ways to obtain information on pore size distributions of nanofiltration membranes

Available online at www.sciencedirect.com

Journal of Membrane Science 309 (2008) 17–27

Three independent ways to obtain information on pore size distributions of nanofiltration membranes J.A. Otero a , O. Mazarrasa a , J. Villasante a , V. Silva b , P. Pr´adanos b , J.I. Calvo b , A. Hern´andez b,∗ a

Grupo de Ingenier´ıa de Procesos de Filtraci´on con Membranas (IPFM), Depto. de Ingenier´ıa Qu´ımica y Qu´ımica Inorg´anica, E.T.S. de Ingenieros Industriales y Telecomunicaci´on, Universidad de Cantabria, 39005 Santander, Spain b Grupo de Superficies y Materiales Porosos (SMAP), Dpto. de F´ısica Aplicada, Facultad de Ciencias, Universidad de Valladolid, Valladolid 47071,Spain Received 20 July 2007; received in revised form 26 September 2007; accepted 30 September 2007 Available online 6 October 2007

Abstract Three independent methods are used to get the pore size distribution of nanofiltration membranes. Two membranes from PCI – AFC-40 and AFC-80 – have been studied. The surface pore size distribution has been studied by AFM images of the membrane surface. A steric pore flow model with friction has been used for different neutral solutes to obtain the effective pore size seen by several organic uncharged solutes. This model is improved by taking into account the modifications of viscosity for confined geometries in narrow pores and by using the Hagen–Poiseuille model to isolate the pore radius as the only parameter of the model. The retentive fractions of pores for the solutes used, which sizes are over the effective pore size seen by them, have been considered to obtain the pore size distribution. Finally a liquid–liquid displacement technique has been used to directly get the open pore size distribution. A fairly good agreement has been obtained by all the three methods used. © 2007 Elsevier B.V. All rights reserved. Keywords: Nanofiltration; Pore size; AFM; Solute retention; Liquid–liquid displacement porosimetry

1. Introduction Transport through nanofiltration membranes merge size and electrical effects, as total charge density, dielectric exclusion, etc., with solution diffusion mechanisms. Actually it joins factors typically relevant in reverse osmosis with those usually controlling ultrafiltration. The pore size of nanofiltration membranes is typically near and frequently below 1 nm in diameter. They have fixed charges developed by dissociation of appropriate groups present in the membrane materials. Due to these charges and sizes, a nanofiltration membrane retains multivalent complex ions and transmits relatively well small uncharged solutes and low charged ions. The low energy consumption and the high fluxes attained by the process, makes nanofiltration very useful in fractionation and selective removal of solutes from complex process streams [1].

∗

Corresponding author. Tel.: +34 983 423134; fax: +34 983 423136. E-mail address: [email protected] (A. Hern´andez).

0376-7388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2007.09.065

Transport of solutes takes place by convection due to the applied pressure difference and by diffusion due to the concentration gradient that appears across the membrane. A sieving-friction mechanism explains the retention of uncharged solutes [2,3]. In order to take into account both the steric and electrical interactions on charged solutes, a space charge model is commonly used. The so-called Donnan steric pore model (DSPM) proposed by Bowen et al. [4], in 1996 has resulted particularly useful. This model is based on the extended Nernst–Planck equation, but includes steric or sieving effects along the Donnan equilibrium to give the equilibrium partition of ions between the solutions in and outside the membrane. More recent developments also include a description of dielectric exclusion whose effects are very relevant for such narrow pores. Non-uniformity of membranes can also be taken into account by introducing information on actual pore size distribution, leading to slight modifications on retention if such distributions are wide enough [5]. Attending to the extremely relevant influence of pore size and pore size distribution for nanofiltration membranes, it seems

18

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27

quite interesting an adequate elucidation of such porometric properties [6]. Here the structural characteristics of two PCI nanofiltration membranes (AFC-40 and AFC-80) will be studied. These membranes have been selected as far as they are quite similar giving different salt retentions. A = 1.0, E = 1.0,

as kc = A + Bλ + Cλ2 + Dλ3

(3)

kd = E + Fλ + Gλ2 + Hλ3 The values of the constants are:

B = 0.054, C = −0.988 and D = 0.441 F = −2.30, G = 1.154 and H = 0.224 or

A = −6.830, E = −0.105, Pore size distributions by AFM of AFC-80 have been obtained by Bowen and Doneva [7], and ourselves [8,9]. Organic solutes have been used to get average pore sizes from fittings of the model for the nanofiltration transport of non-charged solutes through AFC-80 [8,9], and AFC-40 [9,10]. Such pore size distribution will be studied here in detail by atomic force microscopy (AFM). In fact AFM gives us the pore size distribution appearing on the membrane surface. The actual effective pore sizes acting in nanofiltration will be studied and the corresponding pore size distributions elucidated by fitting the experimental results on retention versus flux for aqueous solutions of uncharged solutes. This will be done by accurately using the non-charged nanofiltration model with the help of the retention porosimetric method usually applied for ultrafiltration membranes. Finally liquid–liquid displacement porosimetry will be used for the first time to get pore size distributions for nanofiltration membranes. It is worth noting that these two last techniques refer to the actual flow passing through the pores rather than to the pores apparently present on the membrane surfaces. Nevertheless, results obtained from these three independent methods will be compared.

2.1. Nanofiltration transport 2.1.1. Pore hindrance Attending solely to size effects, including friction for example, transport is hindered in such a way that, the convective and diffusive hindrance factors (kc and kd ) can be defined as u , uw

kd =

Dp D∞

(1)

These hindrance factors are: the solute speed in the pore, u, divided by the average solvent speed, uw , and the solute diffusion coefficient in the pore, Dp , divided by the bulk diffusivity, D∞ , respectively. These hydrodynamic drag coefficients can be correlated with the solute to pore radii fraction [11] λ=

r rp

C = −12.518

F = 0.318,

G = −0.213

D=0

and and

H =0

(2)

0 < λ ≤ 0.8

(4)

0.8 < λ ≤ 1

(5)



according to a detailed study of Deen [12], and Bowen et al. [13,14]. These Eqs. (3–5) are valid for cylindrical pores with nontotally developed velocity profiles. When the pores are relatively narrow and long, these profiles should in fact be totally developed and kc should be multiplied by (2 − Φ) with [13] Φ=

cp (in) cm (in) = = (1 − λ)2 cm cp

(6)

where m refers to the pore entrance (x = 0) and p to the pore end (x = δ) and “in” refers to the interior of the membrane. This parameter corresponds to the solubility equilibrium at the interfaces. Of course, concentrations for not relatively diluted solutions should be substituted by activities (cγ) with activity coefficients appropriately evaluated. 2.1.2. Transport of uncharged solutes When uncharged solutes are considered, the so-called steric pore flow models can be used [12,14–16]. In this case, only diffusive and convective flows contribute to the solute flux J = −Dp

dc dc Jw Jw = −kd D∞ + kc c + kc c dx Ak dx Ak

dc JV ∼ + kc c = −kd D∞ dx Ak

2. Theory

kc =

B = 19.348,



(7)

where Jw is the solvent flux (m/s) and JV the solution flux (m/s), both referred to the unit of membrane area, while the molar flux of the i-th species, J (mol/m2 s), is evaluated for an square meter of pore section. Ak is the membrane porosity as the percentage of open area per unit membrane surface. Given that in terms of cp (the concentration of the permeate) the solute flux is: J = cp (JV /Ak ), Eq. (7) can be written as   dc kc 1 1 = c − cp J V (8) dx Ak D∞ k d kd This equation can be integrated from the concentration at the pore entrances, cm , to the permeate concentration (at the pore ending), cp , and used to evaluate the actual retention coefficient defined as R = 1 − (cp /cm ) to give R=1−

kc φ 1 − exp(−Pe)(1 − kc φ)

(9)

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27

With the Peclet number given by     δ kc JV δ kc Jw  Pe = kd D ∞ A k kd D ∞ A k

(10)

According with the Hagen–Poiseuille model, for pure water     rp2 δ 1 = (11) Ak Lpw 8η with Lpw the water permeability. Thus Eq. (10) can be written as   kc rp2 1 (12) JV Pe  D∞ Lpw kd 8η On the other hand, inside a pore, as those typically appearing in nanofiltration membranes, which are similar in size to the water molecules adsorbed on the pore walls (dw = 0.28 nm), the viscosity could go from the bulk value in the center of the pore to a value 10 times bigger at the pore walls. Thus a profile like  2   η dw dw −9 = 1 + 18 (13) η0 rp rp can be assumed [17], being η0 the bulk viscosity. This profile is shown in Fig. 1. Of course this dependency of η with rp should be taken into account in Eqs. (9) and (12), in such a way that Eqs. (9) and (12) give R = R(JV ; r, rp , D∞ , Lpw , η)

(14)

In this way a knowledge of R as a function of JV for known r, D∞ , Lpw and η allows to obtain rp . 2.1.3. Concentration polarisation Of course the retention coefficient to fit to Eqs. (9) and (12) is the so-called true retention coefficient as far as the directly measurable (observable) retention coefficient is R0 = 1 −

cp cf

(15)

19

with cf the concentration in the feed that differs from the concentration directly in contact with the membrane in the retentate side, cm , due to concentration polarisation. By taking into account the film layer model for concentration polarisation cm = cp + [cf − cp ]eJv /Km

(16)

where the mass transfer coefficient, Km , can be calculated according to the Chilton–Colburn or Dittus–Boelter correlation [18], for turbulent regimes (in our conditions we have Re = 31200 ± 100) [19] Sh = 0.023(Re)0.8 (Sc)1/3

(17)

with Re =

vρdh , η

Sc =

η , ρD∞

Sh =

Km dh D∞

(18)

being v the speed through the channel whose hydraulic diameter is dh (12.7 mm in our case). The fluid is assumed to have a density, ρ and a viscosity, η. The diffusion coefficient in the channel has been taken as D∞ and density and viscosity as those corresponding to pure water. In this way the true retention coefficient can be evaluated. 2.2. Retention pore size distribution In order to evaluate the pore size distributions of a partially retaining membrane, it could be assumed that the retention is due to a pure sieving mechanism. As a consequence, for each solute there is a fraction of totally retaining pores while the rest of them allow a free pass of the molecules [20–22]. Then we can write the mass balance for each solute as JV cp = JV,t cm

(19)

where JV is the total volumetric flux and JV,t is the volumetric flux transmitted through the non-rejecting (transmitting) fraction of pores. On the other hand the ratio of the transmitted volumetric flux and pure water flux, Jw,t , passing through the transmitting pores is JV,t η(cm ) = Jw,t η(0)

(20)

η(cm ) and η(0) are the solution and solvent viscosities. But for low cm this ratio can be approximated by 1 in such a way that Eq. (19) can be rewritten as [23], Jw,t = Jv (1 − R),

Jw,r = Jv R

(21)

Therefore Jw,t and Jw,r (pure water flux passing through the retentive pores, Jw = Jw,t + Jw,r ) can be evaluated once JV for each R is known. Then, by using again the mass balance Jv cp = Js ,

Fig. 1. Dependence of the water viscosity as a function of the pore radii as a consequence of the confinement of water according to Eq. (13).

Jv − Js = Jw

(22)

where cp has to be expressed as a volume fraction, JS the solute flow per unit of membrane area (J = JS /Ak ) and the total pure water flow, Jw , can be obtained by this equation. Thus Jw,t /Jw versus the solute molecular weight or size of the solute used gives the accumulated fraction of flux passing through the pores

20

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27

of a size bigger than that of the solute. Thus if many solutes with different sizes are used the cumulative pore size distribution should be obtained. To reproduce well the experimental data, a logical curve [21,22], with horizontal asymptotes at Jw,t /Jw = 1 and 0, seems appropriate Jw,t 1 = Jw 1 + (rp /B)C

(23)

permeability of the membrane (just by dividing flow by pressure), are obtained. Thus a pore size distribution of permeability contributions can be evaluated. Assuming cylindrical pores, the Hagen–Poiseuille equation can be used to correlate the volume flow, JV , and the number of pores per surface unit, N, having a given pore radius, r. For each pressure step, pi , the corresponding volume flow measured is correlated with the number of pores thus opened by [26]: i 4 p Nk πrpk i

where B and C are constants to be evaluated by fitting Eq. (23) to experimental results. In this way d(Jw,t /Jw )/drp can be obtained.

JVi =

2.3. Liquid–liquid displacement porosimetry

where η is the dynamic viscosity of the displacing fluid, Nk is the number of pores in the class kth per unit membrane surface and δ is the pore length. This could be used to estimate size distributions in number of pores. This procedure should need an adequate consideration of viscosity according to Eq. (13). It should be remembered that Hagen–Poiseuille’s law in that previous simple form only holds for the convective flow through cylindrically shaped straight pores. Thus, the so-obtained pore size distributions should be more model-dependent.

The liquid–liquid porosimetry (LLDP) is a method that can be used to provide information on the pore size distribution of membranes with small pores. The procedure is based on the same principles of the air–liquid displacement or extended bubble point technique, both methods using the correlation between the applied pressure and the pore radius open to flux as given by the Washburn equation [24]: p =

2γ cos θ rp

(24)

being γ the surface tension and θ the contact angle between the permeating interface and the pore material. The high potentiality of this technique in order to evaluate the active pores in the nanometer and subnanometer range makes it a very promising technique to study the pore size distribution of some nanofiltration membranes. Some advantages of LLDP are the following [25]: • it tests the membrane in the wet state, so can give information very close to the normal operating conditions of the membrane, • it also evaluates only the open pores, not any closed or bottle ended ones, • it does not use too high pressures, thus avoiding mechanical over-stress of the membrane during the test that could result in membrane damage or structure collapse; and • finally, it operates quickly which makes analysis simple and easily manageable. A pair of immiscible liquids with low interfacial tension are used which means that pore sizes can be measured at relatively low pressures. The procedure consists in filling the membrane with one liquid, the wetting liquid, and then displacing it with the other one. By monitoring the pressure and the flow through the membrane, the corresponding pore radius opened at a given applied pressure can be calculated using the Cantor equation, provided that contact angle between the liquid–liquid interface and the membrane material can be assumed to be zero 2γ p = (25) rp By increasing the applied trans-membrane pressure stepwise, corresponding pore radius and flow values, represented as the

k=1

8ηδ

(26)

3. Experimental 3.1. Membranes and chemicals Two nanofiltration membranes made out of aromatic polyamide have been used. They are AFC80PCI and AFC40PCI (Paterson Candy International-Ltd., UK), made by the thin-film composite (TFC) method on a porous polysulfone substrate. According to the manufacturers, the recommended working temperature is below 60 ◦ C, the maximum applied pressure is 6.0 MPa and pH must be in the 2–11 range. Several non-charged solutes have been used, namely: ethanol, cyclohexanone, galactose, maltose, lactose, rafinose and ␣cyclodextrine (pro an´alisis, E. Merck, Darmstadt, Germany); aqueous were at 500 ppm were used. All solutions prepared from demineralized, by ion-exchange, and reverse osmosis treated water. In the LLDP experiments methanol and isobutanol (2 methyl1-propanol) from Riedel-de-Ha¨en (Seelze, Germany) with 99.5 and 99% of purity, respectively, were used as received without further purification. 3.2. Atomic force microscopy AFM was performed by using a MultiMode Scanning Probe Microscope Nanoscope IIIA from Digital Instruments provided with an AS-10 E Scanner. For the AFM exploration the samples were in ambient air and the tapping mode (intermittent contact) was used with an EBD (electron beam deposited) sharpened tip made by Nanotools® with length of 1000 nm, a point angle less than 10◦ and an end curvature radius below 5 nm, according to the manufacturer specifications. This tip oscillated at a natural frequency of 300 kHz. The AFM images were processed by using a fast Fourier transform (FFT) filtering procedure [27].

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27

Image analysis was carried out by means of Jandel® ScanPro sofware (version 3.00.0030), in order to study the pore size distribution. Each photograph was digitized with a resolution of 1024 × 768 pixels, assigning to each one a grey level ranging from 0 (black) to 255 (white). Then, a clear-field equalization was applied to each image field to eliminate parasite changes in grey levels due to uneven illumination. This procedure has been explained in detail elsewhere [28]. Once the bending effects were eliminated, the image grey spectrum was spanned to get the maximum contrast and definition. Then the images were redefined according to an assigned grey threshold level under which every pixel was assigned to 1 and the rest to 0. The resulting binary picture was improved by scraping isolated pixels, in such a way that all the remaining 1’s in the matrix were assumed to belong to a pore. Finally the pore borders were smoothed in order to reduce the influence of the finite size of pixels and low definition. Of course a correct selection of threshold grey level is fundamental for correctly and accurately identifying pores. Customarily the grey spectrum is analysed and the threshold centred in the peak to peak valley of the almost bimodal distributions obtained. Unfortunately sometimes the spectra are so flat that this technique needs to be complemented to make a correct threshold selection [29]. Inspection by eye facilitates the process of selection of several reasonable threshold candidates, in within the bottom of the histogram valley, whose outcomes are averaged. When the membrane roughness is relatively low, the pore entrances are of course easily detected. 3.3. Filtration experiments Experiments have been performed in a pilot plant, designed by the IPFM group in the Universidad de Cantabria, described elsewhere [8]. This plant is provided with a tubular module manufactured by PCI provided with 18 tubes (0.0125 m in diameter with a length of 1.2 m and 0.864 m2 of membrane area) connected in series. Total recirculation was used with both permeate and concentrated returning to the feed tank to stationary conditions. Pressures up to 5 MPa were used with a feed cross-flow around

21

1000 L/h at a temperature of 25 ◦ C. Feed and permeated concentrations were measured by the total organic carbon (TOC) technique. 3.4. LLDP experiments The porosimeter used in this work consists in an automated device constructed in the SMAP laboratory in the University of Valladolid and described elsewhere [25]. Experimentally, the method usually involves measurements carried out at controlled flow; each test is driven by increasing stepwise the flow through the membrane supplied by a very stable and precise syringe pump (ISCO 500D), and waiting at each step until the pressure of the system reaches its equilibrium value, measured by a pressure transducer. AFC tubular membranes were cut to be used in the measurement cell, then several pieces of each membrane were cut to the appropriate cell radius and immersed into the LLDP wetting phase for half an hour under vacuum to assure complete membrane wetting. According to the expected pore size (in the nanofiltration range) the most adequate porosimetric mixture has been considered to be a (25/7/15, v/v) water–methanol–isobutanol mixture. The mixture was prepared by pouring appropriate amounts of water, methanol and isobutanol into a separator funnel and shaking it vigorously. The mixture was then allowed to stand overnight. Then the separated alcohol-rich phase was drained off and used as the wetting liquid and the aqueous-rich phase was used as the displacing liquid. All experiments were performed at 15 ◦ C to avoid the high volatility of the alcoholic compounds. It is difficult to know the actual interfacial tension between the alcohol and water-rich phases. In order to bypass these difficulties the pore sizes of several ultrafiltration membranes were measured with a binary mixture of isobutanol and water which has a known, interfacial tension [25]. Then these pore size distributions are compared with those obtained with the ternary mixture. In this way an equivalent interfacial tension for the ternary mixture of 0.44 mN/m has been obtained. This is the value used here to evaluate the pore size distributions of our membranes according to Eq. (24).

Fig. 2. Topographic AFM pictures of the membranes studied after FFT filtering.

22

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27

Fig. 3. Pore size distribution as obtained from AFM pictures for both AFC-40 and AFC-80. Data corresponding to AFC-80 were presented by us elsewhere [8].

4. Results and discussion 4.1. AFM pore size distributions Two examples of AFM pictures after FFT filtering are shown in Fig. 2a (AFC-40) and 2b (AFC-80). In our case, both the membranes have low roughness. The experimental average roughness is Rq = 0.08 ± 0.06 nm for AFC-40 and Rq = 0.16 ± 0.09 nm for AFC-80, both measured from 10H10 nm scanned areas. Images have been obtained with several scan areas. As a matter of fact, roughness depends on the scan area as far as it has a fractal behaviour. In spite of that, pore sizes do not depend significantly on the scan size unless you use too big or small areas. In order to get statistically significant pore size distributions 11 pictures resulting from 10 nm × 10 nm scanning areas of different portions of the active layer of each studied membrane have been required. The pore radii distribution obtained for the active layer of the membranes are shown in Fig. 3. Note that both the distributions are quite similarly broad. The pore size distribution for the AFC80 membrane is much broader and with a longer tail for the large pores than that corresponding to the AFC-40 membrane. Pore size distributions obtained by AFM for the AFC-80 membrane obtained by Bowen and Doneva [7], are less detailed but substantially quite alike to those shown here.

Fig. 4. Observed and true retention for maltose though the AFC-40 membrane. The pure water flux is presented vs. the applied pressure in the insert.

Two examples of the observed and true retention versus the volume flow are shown in Figs. 4 and 5, for maltose through the AFC-40 and AFC-80 membranes, respectively. The corresponding pure water flux versus applied pressure is included as an insert. Note that the differences between observed and true retention coefficients are quite significant for high enough fluxes when concentration polarization is higher. In Figs. 6 and 7 the pore sizes, as obtained by using the nanofiltration model for each uncharged solute, are compared with the Stokes’ radii of the solutes used. Note that in average there is a correlation close to rp = r + αdw

(27)

This means that a molecule of a given solute cannot approach the walls of a pore closer than a certain interaction length. In our case, this distance of nearest approximation for the molecule of solute and the pore walls resulted to be 0.14 nm thus leading to α = 1/2. This length could be interpreted as referring to the hydration layer of the pore walls that adds to that of the molecules of the solute already taken into account in the Stokes’ radius.

4.2. Retention of neutral solutes The diffusivity taken for all the neutral solutes used are shown in Table 1. The theoretical values obtained from different correlations are shown in this table along with the experimental diffusivities found in literature. The molecular radii are the Stokes’ ones [45], as obtained from the average experimental diffusivity when available and otherwise from the average theoretical diffusivity by using the Stokes–Einstein correlation. Viscosity, in all cases has been taken as the pure water one and molar volumes are evaluated by using the additive volumes method of Le Bas [46].

Fig. 5. Observed and true retention for maltose though the AFC-80 membrane. The pure water flux is presented vs. the applied pressure in the insert.

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27

23

Table 1 Data for the non-ionic solutes used DT (×10−9 m2 /s)

Mw (g/mol)

DE (×10−9 m2 /s) Average

1.382

1.23

0.20

98.14

0.899 (a) 0.933 (b) 0.964 (c) 0.899 (d)

0.911

–

–

0.23

180.16

0.794 (a) 0.733 (b) 0.796 (c) 0.742 (d)

0.756

0.690 (j) 0.713 (i)

0.701

0.35

342.30

0.534 (a) 0.528 (b) 0.540 (c) 0.504 (d)

0.527

0.47

342.30

0.534 (a) 0.528 (b) 0.540 (c) 0.504 (d)

0.494

0.50

Rafinose

504.44

0.441 (a) 0.418 (b) 0.524 (c) 0.396 (d)

0.417

0.420 (m) 0.415 (n) 0.435 (k) 0.423 (h)

0.423

0.58

␣Cyclodextrine

972.86

0.346

0.326 (o) 0.312 (p)

0.319

0.77

Cyclohexanone

Galactose

Maltose

Lactose

46.07

Average 1.24 (f) 1.28 (g) 1.14 (h) 1.25 (i)

Ethanol

1.383 (a) 1.402 (b) 1.460 (c) 1.361 (d)

r (nm)

0.346 (e)

0.522

0.522

0.569 (i) 0.520 (k) 0.492 (h) 0.500 (l) 0.490 (l) 0.492 (h)

The theoretical diffusivities have been evaluated according to different correlations: (a) Scheibel [30], (b) Hayduk and Laudie [31], (c) Wilke and Chang [32], (d) Wilke and Chang 1974 [31] and (e) Wilke and Chang 1982 [33]. The corresponding averages have been evaluated from (a), (b) and (d) references as far as (c) gives values outlying the others. The experimental diffusivities have been obtained from different sources: (f) Johnson and Babb [34], (g) Loncin and Merson [35], (h) Coulson and Richardson [36], (i) Mayerhoff et al. [37], (j) Perry and Chilton [38], (k) Monteiro and Penhoat [39], (l) Dembczynski and Jankowki [40], (m) Brandrup and Immergut [41], (n) Feber [42], (o) Danielsen et al. [43] and (p) Naidoo et al. [44]. The Stokes’ radii have been evaluated from the average experimental diffusivity according to the Stokes’ equation. For cyclohexanone, the Stokes’ radius has been evaluated from the average theoretical diffusivity.

From the retention data, using Eqs. (20) and (21), we can obtain Jw,t /Jw as a function of rp corresponding to all the solutes used. These cumulative distributions of flux passing by the different pore sizes are shown for both the membranes in Fig. 8.

This cumulative distribution can be fitted to Eq. (22) and thus gives the differential distributions shown in Fig. 9. In this figure, the corresponding peak values of the AFM pore size distributions for both the membranes are also shown. It is clear that

Fig. 6. Effective pore radii obtained from the steric-friction model vs. the Stokes’ radii for the uncharged solutes used for the AFC-40 membrane.

Fig. 7. Effective pore radii obtained from the steric-friction model vs. the Stokes’ radii for the uncharged solutes used for the AFC-80 membrane.

24

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27

Fig. 8. Cumulative pore size distribution as obtained from retention in nanofiltration of uncharged solutes and both the membranes studied.

a very good accordance is shown. Note also that here, like in the AFM results, the AFC-80 membrane presents a wider and longer large pores tail than the AFC-40 membrane. The pore radii for the AFC-40 membrane was investigated by using polyethyleneglycols by Szymczyk et al. [10], leading to an effective pore radius of 0.54 nm. We used elsewhere [8], cyclohexanone for the AFC-80 membrane, leading to a radius of 0.58 nm. In both cases without taking into account the corrections of viscosity for narrow pores. When these results are compared with those shown here it seems clear that the corrections for viscosity extend the accessible pore range to lower radii. 4.3. LLDP pore size distributions The corresponding differential permeability pore size distributions as obtained by using Eq. (24) for both the studied membranes are shown in Fig. 10. This technique also reveals that AFC-80 has a wider and more asymmetric pore size distribution than AFC-40. Nevertheless, for AFC-40 the accordance with the pore size distributions obtained by both AFM and retention experiments is not as good as for AFC-80.

Fig. 9. Pore size distribution as obtained from retention in nanofiltration of uncharged solutes for both the solutes. The peak values for the AFM pore size distributions are also shown for comparison.

Fig. 10. Pore size distribution as obtained from liquid–liquid displacement porosimetry. The peak values for the AFM pore size distributions are also shown for comparison. A shadowed area is shown covering the portion of the distributions that should be uncertain attending to the size of isobutanol molecule.

The technique assumes that isobutanol can enter into the pores as far as it is present in both the wetting and pushing phases. Actually the Stokes’ radius of isobutanol can be evaluated from its diffusivity D∞ = (0.72–0.78) × 10−9 m2 /s [47], by the Stokes–Einstein relation to give a Stokes’ radius r = 0.31–0.34 nm. This corresponds to an average pore radius, according to Eq. (27) of 0.47 nm. The corresponding area of the pore size distributions obtained by LLDP that should be un-accessible by isobutanol is shaded in Fig. 10. 4.4. Comparison Both the membranes result to have very similar pore size distributions obtained by any of the three techniques presented here. Nevertheless their salt retentions are actually very different as shown for NaCl in Fig. 11 with, as a way of example, very similar galactose retentions. There, it is clear that for equal fluxes the NaCl retention is higher for AFC-80 than for AFC-40. This behaviour for salt retention agrees with data presented in the literature [48], and with the manufacturer specifications. This

Fig. 11. Example of the NaCl true retention for both the membranes studied along with the corresponding very similar retention for galactose.

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27

different retention of charged species should be attributed mainly to the different charge densities of both the membranes. As far as the AFC40 membrane should be the less charged, as corresponds to its lower salt retention, it should be more hydrophobic. On the other hand the uncharged solutes used in the retention experiments have different dipolar moments (2.8 debye for cyclohexanone; 1.7 debye for ethanol [49], and very nearly zero for the rest of the solutes used). These differences in hydrophilicity of the membrane material and the solute molecules seem to have no relevant consequences on the pore size distribution as obtained from retention experiments. On the other hand it is seen that the pore size distribution obtained from liquid–liquid displacement techniques for AFC80 membrane agrees better than for AFC-40 to those obtained from the other methods. The greater asymmetry of the AFC80 distribution could be a relevant factor. Moreover, due to its hydrophobicity, the AFC-40 membrane could possibly adsorb more isobutanol at the pore walls and many small pores could not be accessible to the water-rich pushing fluid giving pore size distributions slightly displaced to big pores. 5. Conclusions Three independent ways have been tested to get information on the pore size distributions of nanofiltration membranes. The accordance between them is rather fair. Nevertheless, it is important to point out that the pore size distributions obtained from retention experiments and liquid–liquid displacement porosimetry refer to effective, active or open, pores; while atomic force microscopy gives information on the pore openings present at the membrane surface. It is crucial to note that an adequate consideration of the actual viscosity of water when confined in narrow pores is fundamental to get pore size distributions by using the nanofiltration transport model for uncharged solutes. It is also very helpful to eliminate from the model the δ/Ak parameter by correlating it with rp though the pure water permeability as far as it makes the fitting procedure much more robust. In any case it seems clear that the pore size seen by each solute is easily correlated with the corresponding Stokes’ radius pore size by the addition of an interaction length. Concerning the results of liquid–liquid displacement porosimetry it is worth noting that it is the first time that nanofiltration membranes have been analysed by this technique. This is a promising method to be extended to the nanofiltration range, even presenting some difficulties. Specifically the wetting-displacing system liquids could be optimized attending to the maximum size of the species penetrating the membrane pores. In spite of the clear limitations of isobutanol, results seem to be better than expected.

and VA112/A06), ITACYL (BU-03-C3-2) and SODERCANGobierno de Cantabria (Spain) Plan de Gobernanza Tecnol´ogico del I Plan Regional de I + D + i 2006-2010 de Cantabria (PGT(32/2006)) for financing this work.

Nomenclature Ak B c cf cm cp C dh dw Dp D∞ J JS Jv Jv,t

Jw Jw,r Jw,t kc kd Km Lpw Nk p Pe r rp R Ro Rq Re Sc Sh v x

membrane porosity parameter in Eq. (23) (m) concentration (mol/m) feed concentration (mol/m) membrane concentration in contact with the highpressure interface (mol/m) permeate concentration (mol/m) parameter in Eq. (23) hydraulic diameter of the channel (m) size of the water molecule (m) diffusion coefficient of the solute in the pore (m2 /s) bulk diffusivity (m2 /s) molar flux (mol m−2 s−1 ) solute flux (based on membrane area) (m/s) solution volume flux (based on membrane area) (m/s) solution volume flux transmitted through nonrejecting fraction of pores (based on membrane area) (m/s) water flux (based on membrane area) (m/s) water flux transmitted through rejecting fraction of pores (based on membrane area) (m/s) water flux transmitted through non-rejecting fraction of pores (based on membrane area) (m/s) hindrance factor for convection hindrance factor for diffusion mass transfer coefficient (m/s) water permeability (m s−1 Pa−1 ) number of pores in the class kth per unit membrane surface (m−1 ) applied pressure (Pa) Peclet number Stokes’ radius of the solute (m) pore radius (m) true retention observed retention average roughness (m) Reynolds number Schmidt number Sherwood number fluid velocity in the channel (m/s) transversal distance in the membrane (m)

Acknowledgements Authors would like to thank Ministerio de Educaci´on y Ciencia-Plan Nacional de I + D + i (Spain) (projects, CTQ2006-13012/PPQ; PPQ2001-0774; MAT2005-04976 and PPQ2006-01685), Junta de Castilla y Le´on (VA116/A06

25

Greek symbols α parameter in Eq. (27) γ surface tension (N/m) δ membrane thickness (m)

26

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27

η η0 Θ λ ρ φ

viscosity in the pore (Pa s) bulk viscosity (Pa s) contact angle pore radii fraction density (kg/m) steric partitioning term in Eq. (6)

References [1] W.R. Bowen, A.W. Mohammad, Diafiltration by nanofiltration: prediction and optimization, AIChE J. 44 (1998) 1799. [2] B. Van der Bruggen, C. Vandecasteele, Removal of pollutants from surface water and groundwater by nanofiltration: overview of possible applications in the drinking water industry, Environ. Pollut. 122 (3) (2003) 435. [3] B. Van der Bruggen, C. Vandecasteele, Modelling of the retention of uncharged molecules with nanofiltration, Water Res. 36 (5) (2002) 1360. [4] W.R. Bowen, H. Mukhtar, Characterisation and prediction of separation performance of nanofiltration membranes, J. Membr. Sci. 112 (1996) 263. [5] C. Labbez, P. Fievet, F. Thomas, A. Szymczyk, A. Vidonne, A. Foissy, J. Pagetti, Evaluation of the “DSPM model” in a titania membrane: measurement of charged an uncharged solute retention, electrokinetic charge, pore size, and water permeability, J. Colloid Interface Sci. 262 (2003) 200. [6] W.R. Bowen, J.S. Welfoot, Modelling of membrane nanofiltration—pore size distribution effects, Chem. Eng. Sci. 57 (8) (2002) 1393. [7] W.R. Bowen, T.A. Doneva, Atomic force microscopy studies of nanofiltration membranes: surface morphology, pore size distribution and adhesion, Desalination 129 (2000) 163. [8] J.A. Otero, G. Lena, J.M. Colina, P. Pr´adanos, F. Tejerina, A. Hern´andez, Characterisation of nanofiltration membranes. Structural analysis by the DSP model and microscopical techniques, J. Membr. Sci. 279 (2006) 410. [9] J.A. Otero, R. Guti´errez, I. Arn´aez, P. Pr´adanos, L. Palacio, A. Hern´andez, Structural and functional study of two nanofiltration membranes, Desalination 200 (2006) 354. [10] A. Szymczyk, N. Fatin-Rouge, P. Fievet, C. Ramseyer, A. Vidone, Identification of dielectric effects in nanofiltration of metallic membranes, J Membr. Sci. 287 (2007) 102. [11] S. Bandini, D. Vezzani, Nanofiltration modeling: the role of dielectric exclusion in membrane characterization, Chem. Eng. Sci. 58 (2003) 3303. [12] W.M. Deen, Hindered transport of large molecules in liquid-filled pores, AIChE J. 33 (1987) 1409. [13] W.R. Bowen, A.O. Sharif, Transport through microfiltration membranes: particle hydrodynamics and flux reductions, J. Colloid Interface Sci. 168 (1994) 414. [14] W.R. Bowen, A.V. Mohammad, N. Hilal, Characterisation of nanofiltration membranes for predictive purposes. Use of salts, uncharged solutes and atomic force microscopy, J. Membr. Sci. 126 (1997) 91. [15] S. Nakao, S. Kimura, Analysis of solutes rejection in ultrafiltration, J. Chem. Eng. Jpn. 14 (1981) 32. [16] M.N. de Pinho, V. Semi˜ao, V. Geraldes, Integrated modelling of transport processes in fluid/nanofiltration membrane systems, J. Membr. Sci. 206 (2002) 189. [17] W.R. Bowen, J.S. Welfoot, Modelling the performance of membrane nanofiltration—critical assessment and model development, Chem. Eng. Sci. 57 (2002) 1121. [18] V. Gekas, B. Halstr¨om, Mass transfer in the concentration polarization layer under turbulent cross flow. 1. Critical literature review and adaptation of existing Sherwood correlations to membrane operations, J. Membr. Sci. 30 (1987) 153. [19] P. Pr´adanos, J.I. Arribas, A. Hern´andez, Hydraulic permeability, mass transfer, and retention of PEGs in cross-flow ultrafiltration through a symmetric microporous membrane, Sep. Sci. Technol. 27 (1992) 2121. [20] M.S. Le, J.A. Howell, Alternative model for ultrafiltration, Chem. Eng. Res. Des. 62 (1984) 373.

[21] N.A. Ochoa, P. Pr´adanos, L. Palacio, C. Pagliero, J. Marchese, A. Hern´andez, Pore size distribution based on AFM imaging and retention of multidisperse polymer solution. Characterisation of polyethersulfone UF membranes with dopes containing different PVP, J. Membr. Sci. 187 (2001) 227. [22] J. Marchese, M. Ponce, N.A. Ochoa, P. Pr´adanos, L. Palacio, A. Hern´andez, Fouling behaviour of polyethersulfone UF membranes made with different PVP, J. Membr. Sci. 211 (2003) 1. [23] R. Nobrega, H. de Balmann, P. Aimar, V. Sanchez, Transfer of dextran through ultrafiltration membranes: a study of rejection data analyzed by gel permeation chromatography, J. Membr. Sci. 45 (1989) 17. [24] J.I. Calvo, A. Hern´andez, P. Pr´adanos, L. Mart´ınez, W.R. Bowen, Pore size distributions in microporous membranes. II. Bulk characterization of track-etched filters by air porometry and mercury porosimetry, J. Colloid Interface Sci. 176 (1995) 467. [25] J.I. Calvo, A. Bottino, G. Capannelli, A. Hern´andez, Comparison of liquid–liquid displacement porosimetry and scanning electron microscopy image analysis to characterise ultrafiltration track-etched membranes, J. Membr. Sci. 239 (2004) 189–197. [26] A. Hern´andez, J.I. Calvo, P. Pr´adanos, F. Tejerina, Pore size distributions in microporous membranes. A critical analysis of the bubble point extended method, J. Membr. Sci. 112 (1996) 1. [27] M. Vilaseca, E. Mateo, L. Palacio, P. Pr´adanos, A. Hern´andez, A. Paniagua, J. Coronas, J. Santamar´ıa, AFM characterization of the growth of MFI-type zeolite films on alumina substrates, Micropor. Mesopor. Mater. 71 (1–3) (2004) 33. [28] A. Hern´andez, J.I. Calvo, P. Pr´adanos, L. Palacio, M.L. Rodr´ıguez, J.A. de Saja, Surface structure of microporous membranes by computerized SEM image analysis applied to anopore filters, J. Membr. Sci. 137 (1997) 89. [29] L. Zeman, L. Denault, Characterization of microfiltration membranes by image analysis of electron micrographs. Part I. Method development, J. Membr. Sci. 71 (1992) 221. [30] E.G. Scheibel, Liquid diffusivities, Ind. Eng. Chem. 46 (1954) 2007. [31] W. Hayduk, H. Laudie, Prediction of diffusion-coefficients for nonelectrolytes in dilute aqueous-solutions, AIChE J. 20 (3) (1974) 611. [32] C.R. Wilke, P.C. Chang, Correlation of diffusion coefficients in dilute solutions, AIChE J. 1 (1955) 264. [33] Ch.J. Geankoplis, Procesos de Transporte y Operaciones Unitarias, Continental, S.A. de C.V, M´exico DC, 1982. [34] P.A. Johnson, A.L. Babb, Liquid diffusion of non-electrolytes, Chem. Rev. 56 (1956) 387. [35] M. Loncin, R.L. Merson, Foof Engineering—Principles and Selected Applications, Acadenic Press, New York, 1979. [36] J.M. Coulson, J.F. Richardson, Chemical Engineering. 1. Fluid Flow Heat Transfer and Mass Transfer, 6th ed., Butterworth–Heinemann, Oxford, 2000. [37] J. Meyerhoff, G. John, K.H. Bellgardt, K. Schugerl, Characterization and modeling of coimmobilized aerobic/anaerobic mixed cultures, Chem. Eng. Sci. 52 (1997) 2313. [38] R.H. Perry, C.H. Chilton, Chemical Engineers Handbook, 5th ed., McGraw-Hill, New York, 1973. [39] C. Monteiro, C. Penhoat, Translational diffusion of dilute aqueous solutions of sugars as probed by NMR and hydrodynamic theory, J. Phys. Chem. A 105 (2001) 9827. [40] R. Dembczynski, T. Jankowski, Characterisation of small molecules diffusion in hydrogel-membrane liquid-core capsules, Biochem. Eng. J. 6 (2000) 41. [41] J. Brandrup, E.H. Immergut (Eds.), Polymer Handbook, 3rd ed., John Wiley & Sons, Inc., New York, 1989. [42] J.J. Faber, Filtration and diffusion across the immature placenta of the anaesthetized rat embryo, Placenta 20 (1999) 331. [43] J. Danielsson, J. Jarvet, P. Damberg, A. Gr¨aslund, Two-site binding of ␤cyclodextrin to the Alzheimer a␤ (1–40) peptide measured with combined PFG-NMR diffusion and induced chemical shifts, Biochemistry 43 (2004) 6261. [44] J.K. Naidoo, J.Y-J. Chen, J.L.M. Jansson, G. Widmalm, A. Maliniak, Molecular properties related to the anomalous solubility of ␤-cyclodextrin, J. Phys. Chem. B. 108 (2004) 4236.

J.A. Otero et al. / Journal of Membrane Science 309 (2008) 17–27 [45] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, New York, 1992. [46] G. Le Bas, The Molecular Volume of Liquid-Chemical Compounds, Longmans, New York, 1915. [47] M. Nilsson, A.M. Gil, I. Delgadillo, G. Morris, Improving pulse sequences for 3D Dosy:Cosy-Idosy, Chem. Commun. (2005) 1737.

27

[48] E.M. van Voorthuizen, A. Zwijnenburg, M. Wessling, Nutrient removal by NF and RO membranes in decentralized sanitation systems, Water Res. 39 (2005) 3657. [49] D.R. Lide (Ed.), Handbook of Chemistry and Physics, 75th ed., CRC Press, Boca Raton, 1994.