- Email: [email protected]
Relating the pore size distribution of ultrafiltration membranes to dextran rejection
Journal of Membrane Science 340 (2009) 1–8
Contents lists available at ScienceDirect
Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Relating the pore size distribution of ultrafiltration membranes to dextran rejection S. Ranil Wickramasinghe a,∗ , Shane E. Bower a , Zhen Chen a , Abhik Mukherjee a , Scott M. Husson b a b
Department of Chemical and Biological Engineering, Colorado State University, Fort Collins, CO 80523-1370, USA Department of Chemical and Biomolecular Engineering, Clemson University, Clemson, SC 29634-0909, USA
a r t i c l e
i n f o
Article history: Received 1 January 2009 Received in revised form 25 April 2009 Accepted 29 April 2009 Available online 12 May 2009 Keywords: Dextran Log-normal distribution Molecular weight cutoff Rejection Ultrafiltration
a b s t r a c t The rejection of various molecular weight dextrans for two commercially available regenerated cellulose and one polyethersulfone membrane has been investigated. Experimentally determined rejection curves were compared to calculated rejection curves based on field-emission scanning electron microscopy imaging assuming that the membrane pore size distribution may be described by the log-normal distribution function. Results show relatively good agreement between calculated and experimental curves; however, the calculated curves generally predict a broader rejection curve. At lower rejection values, the actual rejection is less than calculated. Results also indicate that field-emission scanning electron microscopy, a routinely used imaging technique, may be used to estimate the rejection curve for a membrane, thus enabling more informed selection of an appropriate membrane for a given separation. The results of this work are relevant to ultrafiltration, which is used in the biotechnology industry for protein concentration and buffer exchange. The success of a given UF operation depends on efficient passage or rejection of the desired solute species in solution. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Today, ultrafiltration is used in the biotechnology industry for protein concentration and buffer exchange (diafiltration) [1]. In addition, numerous studies have shown that fractionation of proteins with molecular weight differences of less than an order of magnitude is feasible by combining electrostatic interactions with narrow pore size distribution membranes [2]. For these applications, the choice of an appropriate ultrafiltration membrane requires knowledge of the membrane rejection behavior for the protein(s) of interest. Commercially available ultrafiltration membranes display a pore size distribution [3] and generally contain tortuous interconnected pores [4]. Consequently, ultrafiltration membranes do not display a sharp molecular weight cutoff. In addition, electrostatic interactions between the solvent, solute and membrane affect the observed rejection of a given solute. The situation is further complicated by the fact that the observed membrane rejection depends on more than just the membrane properties. Mochizuki and Zydney [5] point out that solute properties such as shape, degree of branching and deformability affect transport through the membrane pores.
∗ Corresponding author. E-mail address: [email protected] (S.R. Wickramasinghe). 0376-7388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2009.04.056
Concentration polarization and gel layer formation also affect the degree of solute rejection. Consequently, membrane manufacturers rate a given ultrafiltration membrane by its nominal molecular weight cutoff (NMWCO), which generally refers to the smallest molecular weight species for which the membrane displays more than 90% rejection. Rejection is defined as one minus the quotient of solute concentration in the permeate divided by solute concentration in the retentate. Since the NMWCO depends on the solute species and operating conditions, it only applies to the test conditions as specified by the manufacturer. Rejection curves for membranes are determined using model solute species. Dextrans are used frequently [6–8]. They are available commercially at a number of different molecular weights. Though dextran deformability is important in in vivo studies aimed at determining glomerular permselectivity, under the experimental conditions used to determine rejection coefficients for UF membranes dextrans may be modeled as rigid spheres. Because dextrans are neutral polymers of d-glucopyranose, electrostatic interactions among solute, solvent, and membrane are avoided. In addition, dextrans have been shown not to adsorb onto cellulose-based membranes and to have limited adsorption onto polyethersulfone (PES) membranes [8]. Thus, dextran is a good model solute because its effects on rejection are minor. Dextran solutions are used in this work. Prediction of protein rejection for a given membrane is essential when designing an ultrafiltration step. Dextran rejection curves,
2
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
which represent rejection of model solutes under idealized conditions, provide a method for comparing the rejection behavior of different membranes. Here, we have conducted dead-end (normal flow) filtration experiments under conditions that minimize concentration polarization and gel layer formation (effects of operating conditions on membrane rejection) in order to obtain rejection curves for two commercially available regenerated cellulose (RC) membranes and one PES membrane. Numerous methods have been developed to estimate membrane pore size distributions. Nakao [3] and Zhao et al. [9] provide summaries of some of the more frequently used techniques. Importantly, Leypoldt [10] has studied the mathematical problem that results when attempting to obtain the membrane pore size distribution from the rejection curve. The problem is ill posed and extremely difficult to solve. In this work, field-emission scanning electron microscopy (FESEM) was used to image the three commercially available membranes. As described by Nakao [3], microscopic observation and appropriate image processing may be used to give direct visual information on membrane pore size and pore size distribution. Derjani-Bayeh and Rogers [11] indicate that the model chosen to describe the membrane pore size distribution will affect the calculated sieving coefficients. Here, the log-normal distribution was fit to the experimentally determined pore size distribution, as numerous previous investigators have suggested that this distribution often accurately describes the observed pore size distribution [12–17]. The flux through the membrane and the observed rejection both depend on the membrane pore size distribution [3]. Therefore, in order to predict the rejection curve based on the experimentally determined pore size distribution, a relationship is needed among flux, rejection and the membrane pore size distribution. Three different approaches have been used: irreversible thermodynamics derived by Kedem and Katchalsky [18] and Spiegler and Kedem [19]; Stefan–Maxwell multicomponent diffusion equations introduced to membrane transport by Peppas and Meadows [20] and Robertson and Zydney [21]; and the hydrodynamic model [22]. The hydrodynamic model is used in this work. The next section summarizes the theory behind this model. The aim of this work was to compare predicted rejection curves based on FESEM analysis of three commercially available membranes to the experimentally determined dextran rejection curves. While numerous investigators have studied and modeled rejection by ultrafiltration membranes, few investigators have directly compared predicted rejection curves based on microscopic analysis to experimentally determined dextran rejection curves. FESEM is used routinely to image membranes. Prediction of the rejection curve for a membrane based on FESEM images could greatly aid in the selection of a suitable membrane for a given separation. 2. Theory Here, we consider the transport of a rigid solute through cylindrical pores where the pore size distribution is described by the log-normal distribution. We further assume that the diameter of a given pore is constant through the membrane cross-section. The pores are assumed to be non-tortuous, and pore interconnectivity is ignored. Mochizuki and Zydney [5] and Deen [22] provide a summary of the relevant theoretical analysis. Briefly, the dextran flux through the membrane pores, N, is given by dCs N = Kc vCs − Kd D dz
(1)
where Cs is the radially averaged solute concentration in the pores, v is the radially averaged solution velocity in the pores (solvent flux), D is the diffusivity of a given molecular weight dextran molecule, Kc and Kd represent the hindrance factors for convective and diffusive
transport and z is the distance through the pore. Deen [22] provides a number of analytical expressions for the hindrance factors Kc and Kd . We assume that long-range interactions are negligible (dextran molecules are neutral and show minimal adsorbance onto the surface of membranes). Further, we make use of the ‘centerline approximation’, i.e., we only consider the axisymmetric case where the dextran molecules lie on the centerline of the cylindrical pores. Deen has shown that the centerline approximation results in reasonably accurate values for Kc and Kd . We use the expressions derived by Bungay and Brenner [23], Eqs. (2) and (3) below, as they apply to within 10% for all ratios of dextran to membrane pore radius. Kd =
6 Kt
(2)
Kc =
(2 − )Ks 2Kt
(3)
In Eqs. (2) and (3), is the solute partition coefficient between the pore and bulk solutions. When only steric interactions between the solute and pore wall are considered (as is the case here), = (1 − 2 )
(4)
where is the ratio of the solute to pore radius. The hydrodynamic radius values, rp in Å, of the various dextran molecules used in the rejection experiments were estimated from [24,25], rp = 0.488 MW0.437
(5)
where MW is the molecular weight of the dextran. Eq. (5) was determined from experimental quasielastic light scattering data. The Stokes–Einstein equation was used to relate the experimentally determined diffusion coefficient to the dextran molecular radius.
9 √ −5/2 n 1+ an (1 − ) Kt = 2 2(1 − ) 4
2
n=1
9 √ −5/2 n Ks = 2 2(1 − ) 1+ bn (1 − ) 4 2
n=1
+
4
an+3 n
(6)
n=0
+
4
bn+3 n
(7)
n=0
Table 1 gives the coefficients for an and bn . The Peclet number, Pe, is the ratio of rate of convection to diffusion. The membrane Peclet number is defined as in ref. [5] Pe =
K vı c m Kd
(8)
D
where ım is the thickness of the active or skin layer for ultrafiltration membranes. In this work, an applied pressure drop across the membrane leads to convective flow through the membrane. Consequently, the Peclet number is large (Pe 1), and we can ignore the diffusive contribution to solute flux through the membrane in Eq. (1). The rejection coefficient for the solute under these conditions is given by refs. [5,22]. R = 1 − Kc
(9)
Table 1 Coefficients for computing the hydrodynamic functions Ks and Kt in Eqs. (6) and (7) as given by Deen [22]. n
an
bn
1 2 3 4 5 6 7
−73/60 77,293/50,400 −22.5083 −5.6117 −0.3363 −1.216 1.647
7/60 −2,227/50,400 4.0180 −3.9788 −1.9215 4.392 5.006
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
3
Fig. 1. Experimental set up for water flux measurement.
The radially averaged fluid velocity (permeate flux), v, through the membrane pores (in Eq. (1)) is given by the sum of the flow through each pore of radius ri using the Hagen–Poiseuille equation (for laminar flow), P
v=
n(ri )ri2
i
(10)
8L
Table 2 Dextran standards and concentrations used in the feed solution. Dextran standard
MW (kDa)
Feed concentration (g L−1 )
Manufacturer Pharmacosmos (Holbaek, Denmark) Serva (Frankfurt, Germany) GE – Healthcare Biosciences (Piscataway, NJ) GE – Healthcare Biosciences GE – Healthcare Biosciences GE – Healthcare Biosciences GE – Healthcare Biosciences
T1
1
0.74
T4 T10
4 10
1.22 0.54
40 70 500 2000
0.74 0.34 0.27 3.65
where P is the constant applied pressure drop across the membrane, n(ri ) is the number of pores of radius ri , is the fluid viscosity and L the thickness of the membrane (length of the pore). Here, we assume the membrane pore size distribution is given by the log-normal distribution [17],
T40 T70 T500 T2000
n0 ∗ 2 −1/2 exp n(r) = √ [ln(1 + ( r ∗ ) )] r 2
Water fluxes were determined as a function of pressure in the range 69–345 kPa by measuring the mass of permeate collected for a specified time. Fig. 1 is a schematic diagram of the experimental set up.
⎛ 2 ⎞
∗ 2 −1/2 r ⎜ ln r ∗ 1 + r ∗ ⎟ ⎜ ⎟ 2 × ⎜− ⎟ ∗ ⎝ ⎠ 2 ln 1 + ∗
(11)
3.2. Dextran rejection
r
where n0 is the total number of pores of all radii. In this study, the continuous log-normal distribution was used to approximate the discrete pore size distribution determined by FESEM imaging. Consequently, the continuous variable r was used to approximate the discrete variable ri . The mean and standard deviation values from the log-normal distribution are given by r* and *. Eq. (11) is fitted to the experimentally derived pore size distribution. The total number of pores present, n0 , is determined from Eq. (10) where v is set equal to the measured average solvent flux. The rejection curve is then determined from Eq. (9). 3. Experimental Regenerated cellulose (RC) (Millipore PLCMK, NMWCO of 300 kDa and Millipore PLCXK, NMWCO 1000 kDa) and polyethersulfone (PES) (Millipore PBHK, NMWCO 100 kDa) ultrafiltration membranes were provided by Millipore (Bedford, MA). Membranes were prewetted with ethanol for 15 min and subsequently rinsed with Nanopure water for 15 min prior to testing.
The feed solution consisting of various molecular weight dextrans was prepared according to the concentrations given in Table 2. In a recent study, Zydney and Xenopoulos [26] have shown that using these concentrations yields a uniform concentration of dextrans over a wide range of MWs. The dextran standards were dissolved in 0.05 mol/L reagent grade KH2 PO4 (Sigma–Aldrich, St. Louis, MO). The pH of the buffer was adjusted to 7.0 ± 0.1 by adding HPLC grade NaOH (Sigma–Aldrich). The pH was measured using an Orion Model 420A pH meter (Beverly, MA). The same buffer was used as the mobile phase for the gel permeation chromatography (GPC) analysis described below. The membrane was equilibrated for 1 h under permeate recycle using a Masterflex peristaltic pump (Cat. No. 7520-35, Cole-Parmer, Vernon Hills, IL) at a flow rate of 0.1 mL/min (constant flux filtration), as shown in Fig. 2. These conditions have been shown
3.1. Water flux All experiments were conducted at room temperature. Water flux was measured using a Millipore model 8050 stirred ultrafiltration cell having a diameter of 44.5 mm and a maximum feed volume of 50 mL. Each membrane sample was loaded into the stirred cell, which was then filled with 40 mL of Nanopure water. The sample was stirred at 650 rpm using a VWR (West Chester, PA) model 371 hot plate stirrer. Nitrogen was used to pressurize the feed.
Fig. 2. Experimental set up for dextran rejection measurement.
4
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
to minimize concentration polarization and gel layer formation (effects of operating conditions on membrane rejection) [26]. All experiments were conducted in triplicate and average values are given. After equilibration, multiple permeate and feed samples were collected for GPC analysis. Feed and permeate samples (1 mL) were analyzed using an Agilent Technologies (Palo Alto, CA) Model 1050 HPLC system using an Agilent Technologies 1047A refractive index detector. The gel permeation column used was a Showa Denko (Tokyo, Japan) SB806M HQ. The main column was preceded by a Showa Denko SB-G guard column. Column temperature was maintained at 25 ± 1 ◦ C. The HPLC was calibrated using solutions of the narrow molecular weight dextran standards listed in Table 2. Dextran rejection curves were obtained from the GPC data using methods described by previous investigators [6,26]. Briefly, the variation of retention time with dextran molecular weight was determined from the chromatograms for individual, narrow molecular weight dextran standards. Though dextran rejection studies were conducted using a mixed feed (see Table 2 for concentrations) the retention times obtained for the individual dextran fractions corresponded exactly to those obtained in the mixed feed, justifying the use of a mixed feed. Rejection curves were determined as a function of molecular weight by fitting the chromatograms for the individual, narrow molecular weight dextran fractions to the curves for the feed and permeate samples and utilizing calibration data relating peak areas to the concentrations of the dextran fractions in the sample. 3.3. FESEM imaging In order to prevent collapse of pores, critical point drying was conducted using supercritical CO2 . The method involved soaking the samples sequentially in ethanol/water solutions containing 25%, 50%, and 75% ethanol (v/v). Finally the samples were soaked in absolute ethanol. Next, the samples were placed inside a high pressure stainless steel container. The container was flushed with supercritical CO2 at 37 ◦ C and 8500 kPa (85 bar), 3–5 times in order to replace all the ethanol in the membrane pores. The membrane samples were then attached with copper tape on top of aluminum stubs and sputtered with a 5 nm-thick gold layer using a Hummer VII sputtering system (Anatech Ltd., Alexandria, VA). A JEOL JSM-6500F field-emission scanning electron microscope (Peabody, MA) at an accelerating voltage of 5 kV was used to image the membrane surface. The thickness of the active layer was determined by taking additional cross-sectional micrographs. Critical point drying was carried out in the manner described above. The dried membrane samples were frozen in liquid nitrogen and cross-sections were obtained by slicing the samples with a sharp blade. The samples were attached to the sides of the sample holder with copper tape and sputtered with a 5 nm-thick gold layer prior to FESEM analysis.
Fig. 3. Variation of water flux with applied pressure for the three membranes tested.
containing the equivalent diameters of all the pores in the images. The discrete pore distribution function was then determined. A log-normal distribution was fit to the data. The calculated rejection curve was determined using the method described in Section 2. 4. Results and discussion Fig. 3 gives the variation of water flux as a function of applied pressure for the three membranes. The flux increases approximately linearly with applied pressure. Eq. (10) indicates that the liquid velocity and, therefore, permeate flux should increases linearly with applied pressure. Slight deviations from linearity at higher feed pressures could be due to compaction of the membrane. The water flux at a given pressure increases with increasing NMWCO of the membrane. In fact the Hagen–Poiseuille equation for laminar flow through a cylindrical tube predicts that the permeate flux increases with pore radius to the second power. The situation is more complicated for membranes as the membranes contain more than one pore with a distribution of radii. Further, as seen in the FESEM images (Figs. 4–6) and pore size distribution derived from these images (Figs. 7–9), the shape of the pore size distribution for the three membranes is not the same. In particular, as indicated by Table 3, the standard deviations of the distributions are not the same. In addition, the porosities of the membranes and the thickness of the active layer also vary. Consequently, while the permeate flux at the same applied pressure drop increases with increasing NMWCO, the increase is not predicted by the Hagen–Poiseuille equation for laminar flow. In general, the increase in permeate flux
3.4. Determination of pore size distribution The discrete pore size distribution of each membrane was determined by analyzing 10 FESEM images for each membrane. We ensured that the pore size distribution for any given membrane was in good agreement with the average results for the 10 membranes, thus increasing our confidence that the images we used are an accurate representation of the surface pore size distribution of the membrane we tested. Image analysis functions in MATLAB were used to define the pore circumference in the image. For each apparent through pore in the image, we calculated the equivalent diameter of a circle with an equal cross sectional area. A histogram with a 2 nm bin size was constructed using a dataset
Fig. 4. FESEM image of RC 300 kDa membrane. Magnification is 50,000×. The solid bar represents 200 nm.
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
5
Table 3 Membrane properties obtained using FESEM imaging, along with mean radius and standard deviation for the log-normal distribution function. Membrane
RC 300 kDa RC 1000 kDa PES 100 kDa
Thickness of active layer (ım , m)
6 14 85
Discrete distribution
Log-normal distribution
Number of pores corresponding to most probably size
Most probable pore size (nm)
* (nm)
r* (nm)
2231 773 2120
3 4 3
1.99 2.2 1.4
5 3.6 3.6
Fig. 5. FESEM image of RC 1000 kDa membrane. Magnification is 50,000×. The solid bar represents 200 nm.
with increasing NMWCO, even for the same base membrane, will not be predicted by the Hagen–Poiseuille equation, as changing the NMWCO often leads to changes in the shape of the pore size distribution and the membrane porosity. Figs. 4–6 give FESEM images of the active surface of all three membranes at a magnification of 50,000×. Figs. 7–9 give the membrane pore size distributions obtained from the FESEM images. Table 3 gives values for the modal pore size and the total number of pores that were measured for each membrane, as well as the thickness of the active layer, ım , as measured by FESEM imaging. Interestingly, the mean pore radius is similar for all three membranes. While the mean pore radius is nearly the same for all the
Fig. 6. FESEM image of PES 100 kDa membrane. Magnification is 50,000×. The solid bar represents 200 nm.
Fig. 7. Frequency distribution of pore radii in nm for the RC 300 kDa membrane.
membranes, the standard deviation is different. A log-normal distribution was fit to each of the frequency distributions given in Figs. 7–9. Table 3 gives the standard deviation, *, and the mean radius, r*, for the log-normal distribution functions that were used. Figs. 10–12 compare the experimentally determined and calculated rejection curves. In plotting the modeled rejection curves, we have calculated rejection values for the seven individual, narrow molecular weight dextran standards tested (see Table 2). Though we used a feed consisting of a mixture of different dextran standards (see Table 2), we find that the retention times for the individual, narrow molecular weight dextran standards correspond exactly to those in the mixed feed. We link the values of the rejection coefficients obtained for the seven different dextrans by straight line segments, given as a dashed line in Figs. 10–12. While it is possible
Fig. 8. Frequency distribution of pore radii in nm for the RC 1000 kDa membrane.
6
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
Fig. 9. Frequency distribution of pore radii in nm for the PES 100 kDa membrane.
Fig. 10. Comparison of experimental and calculated rejection coefficients for RC 300 kDa membrane plotted as (A) rejection coefficient versus dextran molecule weight (kDa) and (B) rejection coefficient versus Stokes radius (nm). The smooth curve gives the experimentally determined rejection curve while the dashed curve is the calculated rejection curve.
to calculate rejection coefficients for any molecular weight dextran, our aim is to compare experimental and predicted rejection curves. Hence we only calculate rejection coefficients for the actual dextran species tested experimentally. Figs. 10–12 show that the agreement between the experimental and calculated rejection curves is much poorer at lower values of the rejection coefficient. In plotting the modeled rejection curves (given by dashed lines), we assume the membrane pore size distribution may be estimated by the filtration surface pore size distribution. A log-normal distribution is fitted to the experimentally determined pore size distribution. However, the membranes tested here contain a few very large pores. When fitting a log-normal distribution, we ensure the modal value of the distribution is close to the measured modal pore size. Next, the standard deviation of the distribution is determined by minimizing the difference between the measured number of pores of a given size and the value predicted by the log-normal distribution. This procedure is repeated until the difference between the modal values of the actual and log-normal distribution and the difference between the measured number of pores of a given size and the number predicted by the log-normal distribution are both minimized. While this procedure accurately captures the shape of the measured pore size distribution, it underpredicts the number of larger pores present. Membrane pore tortuosity tends to enhance rejection of larger molecular weight dextrans but has less impact on smaller molecules. It is likely that lower molecular weight dextrans will pass through these pores leading to the lower than predicted rejection.
Fig. 11. Comparison of experimental and calculated rejection coefficients for RC 1000 kDa membrane plotted as (A) rejection coefficient versus dextran molecule weight (kDa) and (B) rejection coefficient versus Stokes radius (nm). The smooth curve gives the experimentally determined rejection curve while the dashed curve is the calculated rejection curve.
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
Fig. 12. Comparison of experimental and calculated rejection coefficients for PES 100 kDa membrane plotted as (A) rejection coefficient versus dextran molecule weight (kDa) and (B) rejection coefficient versus Stokes radius (nm). The smooth curve gives the experimentally determined rejection curve while the dashed curve is the calculated rejection curve.
One of the earliest attempts to predict the variation of solute rejection coefficient with solute molecular weight during ultrafiltration was by Ferry [27]. The focus of this study was to show that uniform pore size membranes were capable of partial rejection of spherical, monodisperse solutes if the solute and pore radii were similar. Geometric arguments were used to define the probability that a solute will either pass through a pore or be rejected depending on the relative location of the solute and pore centerlines. The predicted sieving (one minus rejection) curves were compared to experimental data for ultrafiltration of horse serum albumin [28], hemocyanin [29] and foot and mouth disease virus [30]. In a more recent study, Kassotis et al. [31] investigated the rejection of dextran molecules by polyacrylonitrile membranes. They included the effects on dextran rejection due to viscous drag on the molecules as they pass through the membrane pores as suggested by Zeman and Wales [7]. Mochizuki and Zydney [5] have conducted a detailed investigation of the effects of operating conditions and membrane pore size and pore size distribution on rejection of dextran molecules. A hydrodynamic model was used to predict sieving coefficients. As is the case in this study, hindrance factors derived by Bungay and Brenner were used for spherical dextran molecules passing through cylindrical pores. Mochizuki and Zydney define an effective solute to pore size ratio. The solute radius was calculated from the projected solute radius using the method described by Giddings et al. [32]. Unlike this study, where
7
we fit a log-normal pore size distribution to the experimentally determined pore size distribution, the pore radius was determined from hydraulic permeability data. In general, good agreement was obtained between the predicted and experimentally determined sieving coefficients. The aim of this study was to compare experimentally determined rejection coefficients to predicted rejection coefficients under conditions where concentration polarization and other effects due to operating conditions are minimized. For higher values of the rejection coefficient (i.e., higher molecular weight dextrans), we obtain good agreement between the predicted and experimentally determined rejection curves. From a practical perspective when designing an ultrafiltration process for protein concentration or buffer exchange, rejection values of at least 95% are required. In this range of rejection values, we obtain good agreement between predicted and experimentally determined rejection coefficients. Consequently, using the surface pore size distribution of a membrane may allow a priori estimation of the solute rejection and, hence, identification of an appropriate ultrafiltration membrane for a given application. It is important to remember that more than one pore size distribution may be used to predict a given rejection curve. We have chosen to fit the experimentally determined pore size distribution to a log-normal distribution as previous investigators have shown that this distribution has broad applicability. However, other distributions may give a better fit for specific cases. Further, we have attempted to describe the pore size distribution of asymmetric membranes using the surface pore size distribution as measured using FESEM images. These approximations will lead to differences between the measured and fitted pore size distributions. Nevertheless, our results indicate that fitting a log-normal distribution to the measured pore size distribution may be used to estimate the expected dextran rejection curve. Figs. 4–6 illustrate that distinguishing between a real through pore and a surface marking is often difficult. In this work, based on the ‘darkness’ of the marking on the FESEM image, we determined by eye whether to include a given pore in our analysis. If FESEM imaging is to be used on a routine basis to predict membrane rejection curves, an automated method based on a grey scale will be needed. In this work we studied two RC and one PES membranes. As indicated earlier, even for a given base membrane, e.g. RC membranes, as the modal pore size is increased, the membrane asymmetry and the shape of the pore size distribution often change due to changes in the manufacturing conditions. Though trends exist within each family of membranes, it is often difficult to predict the rejection curve of a given membrane based on the rejection curve for the same base membrane with a different NMWCO. Our results for RC and PES membranes indicate that the method we have developed for estimating the rejection curve for a membrane may be applied to different base membranes; though, using the rejection behavior of a given base membrane to predict the behavior for a different base membrane is not straightforward. While industrial scale ultrafiltration processes are run in tangential flow mode, we have investigated dextran rejection using normal flow filtration tests. Zydney and Xenopoulos [26] recently compared dextran rejection result for normal and tangential flow modes. They state that the results obtained for normal flow filtration are more reproducible and robust, as it is much easier to control the effects of operating conditions such as concentration polarization and gel layer formation. Thus, normal flow filtration is more appropriate when testing product quality or rapidly screening different membranes for a given protein purification.
8
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
5. Conclusions
References
Rejection curves for various molecular weight dextrans have been determined for three commercially available ultrafiltration membranes. Using a hydrodynamic model, rejection curves were predicted based on the surface pore size distribution of the membranes as determined by FESEM images. Good agreement between the predicted and experimentally determined rejection curves was obtained for higher values of the rejection coefficient. As it is these higher values of the rejection coefficient that are of most practical interest for designing a UF operation, predicting the rejection behavior of a membrane based on the membrane pore size distribution as determined by FESEM analysis may provide a method to screen membranes.
[1] R. van Reis, A.L. Zydney, Review bioprocess technology, J. Membr. Sci. 297 (2007) 16–50. [2] R. van Reis, A.L. Zydney, Protein ultrafiltration, in: M.C. Flickinger, S.W. Drew (Eds.), Encyclopedia of Biprocess Technology: Fermentation, Biocatalysis, and Bioseparations, John Wiley & Sons, Inc., New York, 1999. [3] S. Nakao, Review determination of pore size and pore size distribution 3. Filtration membranes, J. Membr. Sci. 96 (1994) 131–165. [4] S.S. Kulharni, E.W. Funk, N.L. Li, Membranes, in: W.S.W. Ho, K.K. Sirkar (Eds.), Membrane Handbook, van Nostrand Reinhold, New York, 1992. [5] S. Mochizuki, A.L. Zydney, Dextran transport through asymmetric ultrafiltration membranes: comparison with hydrodynamic models, J. Membr. Sci. 68 (1992) 21–41. [6] G. Tkacik, S. Michaels, A rejection profile test for ultrafiltration membranes and devices, Bio/technology 9 (1991) 941–946. [7] L. Zeman, M. Wales, Steric rejection of polymeric solutes by membranes with uniform pore size distributions, Sep. Sci. Technol. 16 (1981) 275–290. [8] H. Susanto, S. Franzka, M. Ulbricht, Dextran fouling of polyethersulfone ultrafiltration membranes-causes, extent and consequences, J. Membr. Sci. 296 (2007) 147–155. [9] C. Zhao, X. Zhou, Y. Yue, Determination of pore size and pore size distribution on the surface of hollow fiber filtration membranes: a review of methods, Desalination 129 (2000) 107–123. [10] J.K. Leypoldt, Determining pore size distributions of ultrafiltration membranes by solute sieving-mathematical limitations, J. Membr. Sci. 31 (1987) 289–305. [11] S. Derjani-Bayeh, V.G.J. Rodgers, Sieving variations due to the choice in pore size distribution model, J. Membr. Sci. 209 (2002) 1–17. [12] A.R. Cooper, D.S. van Derveer, Characterization of ultrafiltration membranes by polymer transport measurements, Sep. Sci. Technol. 14 (1979) 551–556. [13] K.J. Kim, A.G. Fane, C.D.J. Fell, T. Suzuki, M.R. Dickson, Quantitative microscopic study of surface characteristics of ultrafiltration membranes, J. Membr. Sci. 54 (1990) 89–102. [14] P. Aimar, M. Meireles, V. Sanchez, A contribution to the translation of retention curves into pore size distributions for sieving membranes, J. Membr. Sci. 54 (1990) 321–338. [15] M. Meireles, P. Aimar, V. Sanchez, Effects of protein fouling on the apparent pore size distribution of sieving membranes, J. Membr. Sci. 56 (1991) 13–28. [16] S. Mochizuki, A.L. Zydney, Theoretical analysis of pore size distribution effects on membrane transport, J. Membr. Sci. 82 (1993) 211–227. [17] A.L. Zydney, P. Aimar, M. Meireles, J.M. Pimbley, G. Belfort, Short communication: use of the log-normal probability density function to analyze membrane pore size distributions: functional forms and discrepancies, J. Membr. Sci. 91 (1994) 293–298. [18] O. Kedem, A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to nonelectrolytes, Biochim. Biophys. Acta 27 (1958) 229–246. [19] L.K. Spiegler, O. Kedem, Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient membranes, Desalination 1 (1966) 311–326. [20] N.A. Peppas, D.L. Meadows, Macromolecular structure and solute diffusion in membranes: an overview of recent theories, J. Membr. Sci. 16 (1983) 361–377. [21] B.C. Robertson, A.L. Zydney, A Stefan–Maxwell analysis of protein transport in porous membranes, Sep. Sci. Technol. 23 (1988) 1799–1811. [22] W.M. Deen, Hindered transport of large molecules in liquid-filled pores, AIChE J. 33 (1987) 1409–1425. [23] P.M. Bungay, H. Brenner, The motion of closely fitting sphere in a fluid-filled tube, Int. J. Multiphase Flow 1 (1973) 25–56. [24] D. Venturoli, B. Rippe, Ficoll and dextran vs globular proteins as probes for testing glomerular permselectivity: effects of molecular size, shape, charge, and deformability, Am. J. Physiol. Renal. Physiol. 288 (2004) F605–F613. [25] J.D. Oliver III, S. Anderson, J.L. Troy, B.M. Brenner, W.M. Deen, Determination of glomerular size-selectivity in the normal rat with Ficoll, J Am. Soc. Nephrol. 3 (1992) 214–228. [26] A.L. Zydney, A. Xenopoulos, Improving dextran tests for ultrafiltration membranes: effects of device format, J. Membr. Sci. 291 (2007) 180–190. [27] J.D. Ferry, Statistical evaluation of sieve constants in ultrafiltration, J. Gen. Physiol. 20 (1936) 95–104. [28] W.J. Elford, J.D. Ferry, The ultrafiltration of proteins through graded collodion membranes. I. The serum-proteins, Biochem. J. 28 (Part 1) (1934) 650–662. [29] W.J. Elford, J.D. Ferry, The ultrafiltration of proteins through graded collodion membranes. II. Haemocyanin (Helix), edestin, and egg albumin, Biochem. J. 30 (Part 1) (1936) 84–91. [30] I.A. Galloway, W.J. Elford, Filtration of the virus of foot-and-mouth disease through a new series of graded collodion membranes, Br. J. Exp. Pathol. 12 (1931) 407–425. [31] J. Kassotis, J. Schmidt, L.T. Hodgins, H.P. Gregor, Modeling of the pore size distribution of ultrafiltration membranes, J. Membr. Sci. 22 (1985) 61–76. [32] J.C. Giddings, E. Kucera, C.P. Russell, M.N. Myers, Statistical theory for equilibrium distribution of rigid molecules in inert porous networks. Exclusion chromatography, J. Phys. Chem. 72 (1968) 4397–4408.
Acknowledgements Funding for this work was provided by the NSF Industry/University Cooperative Research Center for Membrane Applied Science and Technology at the University of Colorado. The authors are grateful to Drs. Anthony Allegrezza, Neil Soice and Gabriel Tkacik for their valuable comments and insights.
Nomenclature an bn Cs D Kc Kd Kt Ks L n0 n(ri ) n(r) N Pe P r r* r
i
rp R
v z
coefficient in Eqs. (6) and (7) (dimensionless) coefficient in Eqs. (6) and (7) (dimensionless) radially averaged solute concentration in the pores (mol L−3 ) diffusivity of dextran molecule (L2 t−1 ) convective hindrance factor in Eq. (1) (dimensionless) diffusive hindrance factor in Eq. (1) (dimensionless) hydrodynamic function (see Eq. (2)) (dimensionless) hydrodynamic function (see Eq. (3)) (dimensionless) membrane thickness (L) total number of pores of all radii (dimensionless) number of pores of radius ri (dimensionless) number of pores of radius r given by the log-normal distribution function (dimensionless) solute flux (mol L−2 t−1 ) Peclet number (dimensionless) pressure drop (M L−1 t−2 ) pore radius (continuous function) (L) mean pore radius (L) pore radius (discrete function) (L) hydrodynamic radius of dextran molecule (L in Å) rejection coefficient (dimensionless) radially averaged solution velocity in the pores (L t−1 ) distance through the pore (L)
Greek letters ım thickness of the active or skin layer (L) ratio of the solute to pore radius (dimensionless) viscosity (M L−1 t−1 ) * standard deviation (dimensionless) solute partition coefficient between the pore and bulk solutions (dimensionless)
Contents lists available at ScienceDirect
Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Relating the pore size distribution of ultrafiltration membranes to dextran rejection S. Ranil Wickramasinghe a,∗ , Shane E. Bower a , Zhen Chen a , Abhik Mukherjee a , Scott M. Husson b a b
Department of Chemical and Biological Engineering, Colorado State University, Fort Collins, CO 80523-1370, USA Department of Chemical and Biomolecular Engineering, Clemson University, Clemson, SC 29634-0909, USA
a r t i c l e
i n f o
Article history: Received 1 January 2009 Received in revised form 25 April 2009 Accepted 29 April 2009 Available online 12 May 2009 Keywords: Dextran Log-normal distribution Molecular weight cutoff Rejection Ultrafiltration
a b s t r a c t The rejection of various molecular weight dextrans for two commercially available regenerated cellulose and one polyethersulfone membrane has been investigated. Experimentally determined rejection curves were compared to calculated rejection curves based on field-emission scanning electron microscopy imaging assuming that the membrane pore size distribution may be described by the log-normal distribution function. Results show relatively good agreement between calculated and experimental curves; however, the calculated curves generally predict a broader rejection curve. At lower rejection values, the actual rejection is less than calculated. Results also indicate that field-emission scanning electron microscopy, a routinely used imaging technique, may be used to estimate the rejection curve for a membrane, thus enabling more informed selection of an appropriate membrane for a given separation. The results of this work are relevant to ultrafiltration, which is used in the biotechnology industry for protein concentration and buffer exchange. The success of a given UF operation depends on efficient passage or rejection of the desired solute species in solution. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Today, ultrafiltration is used in the biotechnology industry for protein concentration and buffer exchange (diafiltration) [1]. In addition, numerous studies have shown that fractionation of proteins with molecular weight differences of less than an order of magnitude is feasible by combining electrostatic interactions with narrow pore size distribution membranes [2]. For these applications, the choice of an appropriate ultrafiltration membrane requires knowledge of the membrane rejection behavior for the protein(s) of interest. Commercially available ultrafiltration membranes display a pore size distribution [3] and generally contain tortuous interconnected pores [4]. Consequently, ultrafiltration membranes do not display a sharp molecular weight cutoff. In addition, electrostatic interactions between the solvent, solute and membrane affect the observed rejection of a given solute. The situation is further complicated by the fact that the observed membrane rejection depends on more than just the membrane properties. Mochizuki and Zydney [5] point out that solute properties such as shape, degree of branching and deformability affect transport through the membrane pores.
∗ Corresponding author. E-mail address: [email protected] (S.R. Wickramasinghe). 0376-7388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2009.04.056
Concentration polarization and gel layer formation also affect the degree of solute rejection. Consequently, membrane manufacturers rate a given ultrafiltration membrane by its nominal molecular weight cutoff (NMWCO), which generally refers to the smallest molecular weight species for which the membrane displays more than 90% rejection. Rejection is defined as one minus the quotient of solute concentration in the permeate divided by solute concentration in the retentate. Since the NMWCO depends on the solute species and operating conditions, it only applies to the test conditions as specified by the manufacturer. Rejection curves for membranes are determined using model solute species. Dextrans are used frequently [6–8]. They are available commercially at a number of different molecular weights. Though dextran deformability is important in in vivo studies aimed at determining glomerular permselectivity, under the experimental conditions used to determine rejection coefficients for UF membranes dextrans may be modeled as rigid spheres. Because dextrans are neutral polymers of d-glucopyranose, electrostatic interactions among solute, solvent, and membrane are avoided. In addition, dextrans have been shown not to adsorb onto cellulose-based membranes and to have limited adsorption onto polyethersulfone (PES) membranes [8]. Thus, dextran is a good model solute because its effects on rejection are minor. Dextran solutions are used in this work. Prediction of protein rejection for a given membrane is essential when designing an ultrafiltration step. Dextran rejection curves,
2
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
which represent rejection of model solutes under idealized conditions, provide a method for comparing the rejection behavior of different membranes. Here, we have conducted dead-end (normal flow) filtration experiments under conditions that minimize concentration polarization and gel layer formation (effects of operating conditions on membrane rejection) in order to obtain rejection curves for two commercially available regenerated cellulose (RC) membranes and one PES membrane. Numerous methods have been developed to estimate membrane pore size distributions. Nakao [3] and Zhao et al. [9] provide summaries of some of the more frequently used techniques. Importantly, Leypoldt [10] has studied the mathematical problem that results when attempting to obtain the membrane pore size distribution from the rejection curve. The problem is ill posed and extremely difficult to solve. In this work, field-emission scanning electron microscopy (FESEM) was used to image the three commercially available membranes. As described by Nakao [3], microscopic observation and appropriate image processing may be used to give direct visual information on membrane pore size and pore size distribution. Derjani-Bayeh and Rogers [11] indicate that the model chosen to describe the membrane pore size distribution will affect the calculated sieving coefficients. Here, the log-normal distribution was fit to the experimentally determined pore size distribution, as numerous previous investigators have suggested that this distribution often accurately describes the observed pore size distribution [12–17]. The flux through the membrane and the observed rejection both depend on the membrane pore size distribution [3]. Therefore, in order to predict the rejection curve based on the experimentally determined pore size distribution, a relationship is needed among flux, rejection and the membrane pore size distribution. Three different approaches have been used: irreversible thermodynamics derived by Kedem and Katchalsky [18] and Spiegler and Kedem [19]; Stefan–Maxwell multicomponent diffusion equations introduced to membrane transport by Peppas and Meadows [20] and Robertson and Zydney [21]; and the hydrodynamic model [22]. The hydrodynamic model is used in this work. The next section summarizes the theory behind this model. The aim of this work was to compare predicted rejection curves based on FESEM analysis of three commercially available membranes to the experimentally determined dextran rejection curves. While numerous investigators have studied and modeled rejection by ultrafiltration membranes, few investigators have directly compared predicted rejection curves based on microscopic analysis to experimentally determined dextran rejection curves. FESEM is used routinely to image membranes. Prediction of the rejection curve for a membrane based on FESEM images could greatly aid in the selection of a suitable membrane for a given separation. 2. Theory Here, we consider the transport of a rigid solute through cylindrical pores where the pore size distribution is described by the log-normal distribution. We further assume that the diameter of a given pore is constant through the membrane cross-section. The pores are assumed to be non-tortuous, and pore interconnectivity is ignored. Mochizuki and Zydney [5] and Deen [22] provide a summary of the relevant theoretical analysis. Briefly, the dextran flux through the membrane pores, N, is given by dCs N = Kc vCs − Kd D dz
(1)
where Cs is the radially averaged solute concentration in the pores, v is the radially averaged solution velocity in the pores (solvent flux), D is the diffusivity of a given molecular weight dextran molecule, Kc and Kd represent the hindrance factors for convective and diffusive
transport and z is the distance through the pore. Deen [22] provides a number of analytical expressions for the hindrance factors Kc and Kd . We assume that long-range interactions are negligible (dextran molecules are neutral and show minimal adsorbance onto the surface of membranes). Further, we make use of the ‘centerline approximation’, i.e., we only consider the axisymmetric case where the dextran molecules lie on the centerline of the cylindrical pores. Deen has shown that the centerline approximation results in reasonably accurate values for Kc and Kd . We use the expressions derived by Bungay and Brenner [23], Eqs. (2) and (3) below, as they apply to within 10% for all ratios of dextran to membrane pore radius. Kd =
6 Kt
(2)
Kc =
(2 − )Ks 2Kt
(3)
In Eqs. (2) and (3), is the solute partition coefficient between the pore and bulk solutions. When only steric interactions between the solute and pore wall are considered (as is the case here), = (1 − 2 )
(4)
where is the ratio of the solute to pore radius. The hydrodynamic radius values, rp in Å, of the various dextran molecules used in the rejection experiments were estimated from [24,25], rp = 0.488 MW0.437
(5)
where MW is the molecular weight of the dextran. Eq. (5) was determined from experimental quasielastic light scattering data. The Stokes–Einstein equation was used to relate the experimentally determined diffusion coefficient to the dextran molecular radius.
9 √ −5/2 n 1+ an (1 − ) Kt = 2 2(1 − ) 4
2
n=1
9 √ −5/2 n Ks = 2 2(1 − ) 1+ bn (1 − ) 4 2
n=1
+
4
an+3 n
(6)
n=0
+
4
bn+3 n
(7)
n=0
Table 1 gives the coefficients for an and bn . The Peclet number, Pe, is the ratio of rate of convection to diffusion. The membrane Peclet number is defined as in ref. [5] Pe =
K vı c m Kd
(8)
D
where ım is the thickness of the active or skin layer for ultrafiltration membranes. In this work, an applied pressure drop across the membrane leads to convective flow through the membrane. Consequently, the Peclet number is large (Pe 1), and we can ignore the diffusive contribution to solute flux through the membrane in Eq. (1). The rejection coefficient for the solute under these conditions is given by refs. [5,22]. R = 1 − Kc
(9)
Table 1 Coefficients for computing the hydrodynamic functions Ks and Kt in Eqs. (6) and (7) as given by Deen [22]. n
an
bn
1 2 3 4 5 6 7
−73/60 77,293/50,400 −22.5083 −5.6117 −0.3363 −1.216 1.647
7/60 −2,227/50,400 4.0180 −3.9788 −1.9215 4.392 5.006
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
3
Fig. 1. Experimental set up for water flux measurement.
The radially averaged fluid velocity (permeate flux), v, through the membrane pores (in Eq. (1)) is given by the sum of the flow through each pore of radius ri using the Hagen–Poiseuille equation (for laminar flow), P
v=
n(ri )ri2
i
(10)
8L
Table 2 Dextran standards and concentrations used in the feed solution. Dextran standard
MW (kDa)
Feed concentration (g L−1 )
Manufacturer Pharmacosmos (Holbaek, Denmark) Serva (Frankfurt, Germany) GE – Healthcare Biosciences (Piscataway, NJ) GE – Healthcare Biosciences GE – Healthcare Biosciences GE – Healthcare Biosciences GE – Healthcare Biosciences
T1
1
0.74
T4 T10
4 10
1.22 0.54
40 70 500 2000
0.74 0.34 0.27 3.65
where P is the constant applied pressure drop across the membrane, n(ri ) is the number of pores of radius ri , is the fluid viscosity and L the thickness of the membrane (length of the pore). Here, we assume the membrane pore size distribution is given by the log-normal distribution [17],
T40 T70 T500 T2000
n0 ∗ 2 −1/2 exp n(r) = √ [ln(1 + ( r ∗ ) )] r 2
Water fluxes were determined as a function of pressure in the range 69–345 kPa by measuring the mass of permeate collected for a specified time. Fig. 1 is a schematic diagram of the experimental set up.
⎛ 2 ⎞
∗ 2 −1/2 r ⎜ ln r ∗ 1 + r ∗ ⎟ ⎜ ⎟ 2 × ⎜− ⎟ ∗ ⎝ ⎠ 2 ln 1 + ∗
(11)
3.2. Dextran rejection
r
where n0 is the total number of pores of all radii. In this study, the continuous log-normal distribution was used to approximate the discrete pore size distribution determined by FESEM imaging. Consequently, the continuous variable r was used to approximate the discrete variable ri . The mean and standard deviation values from the log-normal distribution are given by r* and *. Eq. (11) is fitted to the experimentally derived pore size distribution. The total number of pores present, n0 , is determined from Eq. (10) where v is set equal to the measured average solvent flux. The rejection curve is then determined from Eq. (9). 3. Experimental Regenerated cellulose (RC) (Millipore PLCMK, NMWCO of 300 kDa and Millipore PLCXK, NMWCO 1000 kDa) and polyethersulfone (PES) (Millipore PBHK, NMWCO 100 kDa) ultrafiltration membranes were provided by Millipore (Bedford, MA). Membranes were prewetted with ethanol for 15 min and subsequently rinsed with Nanopure water for 15 min prior to testing.
The feed solution consisting of various molecular weight dextrans was prepared according to the concentrations given in Table 2. In a recent study, Zydney and Xenopoulos [26] have shown that using these concentrations yields a uniform concentration of dextrans over a wide range of MWs. The dextran standards were dissolved in 0.05 mol/L reagent grade KH2 PO4 (Sigma–Aldrich, St. Louis, MO). The pH of the buffer was adjusted to 7.0 ± 0.1 by adding HPLC grade NaOH (Sigma–Aldrich). The pH was measured using an Orion Model 420A pH meter (Beverly, MA). The same buffer was used as the mobile phase for the gel permeation chromatography (GPC) analysis described below. The membrane was equilibrated for 1 h under permeate recycle using a Masterflex peristaltic pump (Cat. No. 7520-35, Cole-Parmer, Vernon Hills, IL) at a flow rate of 0.1 mL/min (constant flux filtration), as shown in Fig. 2. These conditions have been shown
3.1. Water flux All experiments were conducted at room temperature. Water flux was measured using a Millipore model 8050 stirred ultrafiltration cell having a diameter of 44.5 mm and a maximum feed volume of 50 mL. Each membrane sample was loaded into the stirred cell, which was then filled with 40 mL of Nanopure water. The sample was stirred at 650 rpm using a VWR (West Chester, PA) model 371 hot plate stirrer. Nitrogen was used to pressurize the feed.
Fig. 2. Experimental set up for dextran rejection measurement.
4
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
to minimize concentration polarization and gel layer formation (effects of operating conditions on membrane rejection) [26]. All experiments were conducted in triplicate and average values are given. After equilibration, multiple permeate and feed samples were collected for GPC analysis. Feed and permeate samples (1 mL) were analyzed using an Agilent Technologies (Palo Alto, CA) Model 1050 HPLC system using an Agilent Technologies 1047A refractive index detector. The gel permeation column used was a Showa Denko (Tokyo, Japan) SB806M HQ. The main column was preceded by a Showa Denko SB-G guard column. Column temperature was maintained at 25 ± 1 ◦ C. The HPLC was calibrated using solutions of the narrow molecular weight dextran standards listed in Table 2. Dextran rejection curves were obtained from the GPC data using methods described by previous investigators [6,26]. Briefly, the variation of retention time with dextran molecular weight was determined from the chromatograms for individual, narrow molecular weight dextran standards. Though dextran rejection studies were conducted using a mixed feed (see Table 2 for concentrations) the retention times obtained for the individual dextran fractions corresponded exactly to those obtained in the mixed feed, justifying the use of a mixed feed. Rejection curves were determined as a function of molecular weight by fitting the chromatograms for the individual, narrow molecular weight dextran fractions to the curves for the feed and permeate samples and utilizing calibration data relating peak areas to the concentrations of the dextran fractions in the sample. 3.3. FESEM imaging In order to prevent collapse of pores, critical point drying was conducted using supercritical CO2 . The method involved soaking the samples sequentially in ethanol/water solutions containing 25%, 50%, and 75% ethanol (v/v). Finally the samples were soaked in absolute ethanol. Next, the samples were placed inside a high pressure stainless steel container. The container was flushed with supercritical CO2 at 37 ◦ C and 8500 kPa (85 bar), 3–5 times in order to replace all the ethanol in the membrane pores. The membrane samples were then attached with copper tape on top of aluminum stubs and sputtered with a 5 nm-thick gold layer using a Hummer VII sputtering system (Anatech Ltd., Alexandria, VA). A JEOL JSM-6500F field-emission scanning electron microscope (Peabody, MA) at an accelerating voltage of 5 kV was used to image the membrane surface. The thickness of the active layer was determined by taking additional cross-sectional micrographs. Critical point drying was carried out in the manner described above. The dried membrane samples were frozen in liquid nitrogen and cross-sections were obtained by slicing the samples with a sharp blade. The samples were attached to the sides of the sample holder with copper tape and sputtered with a 5 nm-thick gold layer prior to FESEM analysis.
Fig. 3. Variation of water flux with applied pressure for the three membranes tested.
containing the equivalent diameters of all the pores in the images. The discrete pore distribution function was then determined. A log-normal distribution was fit to the data. The calculated rejection curve was determined using the method described in Section 2. 4. Results and discussion Fig. 3 gives the variation of water flux as a function of applied pressure for the three membranes. The flux increases approximately linearly with applied pressure. Eq. (10) indicates that the liquid velocity and, therefore, permeate flux should increases linearly with applied pressure. Slight deviations from linearity at higher feed pressures could be due to compaction of the membrane. The water flux at a given pressure increases with increasing NMWCO of the membrane. In fact the Hagen–Poiseuille equation for laminar flow through a cylindrical tube predicts that the permeate flux increases with pore radius to the second power. The situation is more complicated for membranes as the membranes contain more than one pore with a distribution of radii. Further, as seen in the FESEM images (Figs. 4–6) and pore size distribution derived from these images (Figs. 7–9), the shape of the pore size distribution for the three membranes is not the same. In particular, as indicated by Table 3, the standard deviations of the distributions are not the same. In addition, the porosities of the membranes and the thickness of the active layer also vary. Consequently, while the permeate flux at the same applied pressure drop increases with increasing NMWCO, the increase is not predicted by the Hagen–Poiseuille equation for laminar flow. In general, the increase in permeate flux
3.4. Determination of pore size distribution The discrete pore size distribution of each membrane was determined by analyzing 10 FESEM images for each membrane. We ensured that the pore size distribution for any given membrane was in good agreement with the average results for the 10 membranes, thus increasing our confidence that the images we used are an accurate representation of the surface pore size distribution of the membrane we tested. Image analysis functions in MATLAB were used to define the pore circumference in the image. For each apparent through pore in the image, we calculated the equivalent diameter of a circle with an equal cross sectional area. A histogram with a 2 nm bin size was constructed using a dataset
Fig. 4. FESEM image of RC 300 kDa membrane. Magnification is 50,000×. The solid bar represents 200 nm.
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
5
Table 3 Membrane properties obtained using FESEM imaging, along with mean radius and standard deviation for the log-normal distribution function. Membrane
RC 300 kDa RC 1000 kDa PES 100 kDa
Thickness of active layer (ım , m)
6 14 85
Discrete distribution
Log-normal distribution
Number of pores corresponding to most probably size
Most probable pore size (nm)
* (nm)
r* (nm)
2231 773 2120
3 4 3
1.99 2.2 1.4
5 3.6 3.6
Fig. 5. FESEM image of RC 1000 kDa membrane. Magnification is 50,000×. The solid bar represents 200 nm.
with increasing NMWCO, even for the same base membrane, will not be predicted by the Hagen–Poiseuille equation, as changing the NMWCO often leads to changes in the shape of the pore size distribution and the membrane porosity. Figs. 4–6 give FESEM images of the active surface of all three membranes at a magnification of 50,000×. Figs. 7–9 give the membrane pore size distributions obtained from the FESEM images. Table 3 gives values for the modal pore size and the total number of pores that were measured for each membrane, as well as the thickness of the active layer, ım , as measured by FESEM imaging. Interestingly, the mean pore radius is similar for all three membranes. While the mean pore radius is nearly the same for all the
Fig. 6. FESEM image of PES 100 kDa membrane. Magnification is 50,000×. The solid bar represents 200 nm.
Fig. 7. Frequency distribution of pore radii in nm for the RC 300 kDa membrane.
membranes, the standard deviation is different. A log-normal distribution was fit to each of the frequency distributions given in Figs. 7–9. Table 3 gives the standard deviation, *, and the mean radius, r*, for the log-normal distribution functions that were used. Figs. 10–12 compare the experimentally determined and calculated rejection curves. In plotting the modeled rejection curves, we have calculated rejection values for the seven individual, narrow molecular weight dextran standards tested (see Table 2). Though we used a feed consisting of a mixture of different dextran standards (see Table 2), we find that the retention times for the individual, narrow molecular weight dextran standards correspond exactly to those in the mixed feed. We link the values of the rejection coefficients obtained for the seven different dextrans by straight line segments, given as a dashed line in Figs. 10–12. While it is possible
Fig. 8. Frequency distribution of pore radii in nm for the RC 1000 kDa membrane.
6
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
Fig. 9. Frequency distribution of pore radii in nm for the PES 100 kDa membrane.
Fig. 10. Comparison of experimental and calculated rejection coefficients for RC 300 kDa membrane plotted as (A) rejection coefficient versus dextran molecule weight (kDa) and (B) rejection coefficient versus Stokes radius (nm). The smooth curve gives the experimentally determined rejection curve while the dashed curve is the calculated rejection curve.
to calculate rejection coefficients for any molecular weight dextran, our aim is to compare experimental and predicted rejection curves. Hence we only calculate rejection coefficients for the actual dextran species tested experimentally. Figs. 10–12 show that the agreement between the experimental and calculated rejection curves is much poorer at lower values of the rejection coefficient. In plotting the modeled rejection curves (given by dashed lines), we assume the membrane pore size distribution may be estimated by the filtration surface pore size distribution. A log-normal distribution is fitted to the experimentally determined pore size distribution. However, the membranes tested here contain a few very large pores. When fitting a log-normal distribution, we ensure the modal value of the distribution is close to the measured modal pore size. Next, the standard deviation of the distribution is determined by minimizing the difference between the measured number of pores of a given size and the value predicted by the log-normal distribution. This procedure is repeated until the difference between the modal values of the actual and log-normal distribution and the difference between the measured number of pores of a given size and the number predicted by the log-normal distribution are both minimized. While this procedure accurately captures the shape of the measured pore size distribution, it underpredicts the number of larger pores present. Membrane pore tortuosity tends to enhance rejection of larger molecular weight dextrans but has less impact on smaller molecules. It is likely that lower molecular weight dextrans will pass through these pores leading to the lower than predicted rejection.
Fig. 11. Comparison of experimental and calculated rejection coefficients for RC 1000 kDa membrane plotted as (A) rejection coefficient versus dextran molecule weight (kDa) and (B) rejection coefficient versus Stokes radius (nm). The smooth curve gives the experimentally determined rejection curve while the dashed curve is the calculated rejection curve.
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
Fig. 12. Comparison of experimental and calculated rejection coefficients for PES 100 kDa membrane plotted as (A) rejection coefficient versus dextran molecule weight (kDa) and (B) rejection coefficient versus Stokes radius (nm). The smooth curve gives the experimentally determined rejection curve while the dashed curve is the calculated rejection curve.
One of the earliest attempts to predict the variation of solute rejection coefficient with solute molecular weight during ultrafiltration was by Ferry [27]. The focus of this study was to show that uniform pore size membranes were capable of partial rejection of spherical, monodisperse solutes if the solute and pore radii were similar. Geometric arguments were used to define the probability that a solute will either pass through a pore or be rejected depending on the relative location of the solute and pore centerlines. The predicted sieving (one minus rejection) curves were compared to experimental data for ultrafiltration of horse serum albumin [28], hemocyanin [29] and foot and mouth disease virus [30]. In a more recent study, Kassotis et al. [31] investigated the rejection of dextran molecules by polyacrylonitrile membranes. They included the effects on dextran rejection due to viscous drag on the molecules as they pass through the membrane pores as suggested by Zeman and Wales [7]. Mochizuki and Zydney [5] have conducted a detailed investigation of the effects of operating conditions and membrane pore size and pore size distribution on rejection of dextran molecules. A hydrodynamic model was used to predict sieving coefficients. As is the case in this study, hindrance factors derived by Bungay and Brenner were used for spherical dextran molecules passing through cylindrical pores. Mochizuki and Zydney define an effective solute to pore size ratio. The solute radius was calculated from the projected solute radius using the method described by Giddings et al. [32]. Unlike this study, where
7
we fit a log-normal pore size distribution to the experimentally determined pore size distribution, the pore radius was determined from hydraulic permeability data. In general, good agreement was obtained between the predicted and experimentally determined sieving coefficients. The aim of this study was to compare experimentally determined rejection coefficients to predicted rejection coefficients under conditions where concentration polarization and other effects due to operating conditions are minimized. For higher values of the rejection coefficient (i.e., higher molecular weight dextrans), we obtain good agreement between the predicted and experimentally determined rejection curves. From a practical perspective when designing an ultrafiltration process for protein concentration or buffer exchange, rejection values of at least 95% are required. In this range of rejection values, we obtain good agreement between predicted and experimentally determined rejection coefficients. Consequently, using the surface pore size distribution of a membrane may allow a priori estimation of the solute rejection and, hence, identification of an appropriate ultrafiltration membrane for a given application. It is important to remember that more than one pore size distribution may be used to predict a given rejection curve. We have chosen to fit the experimentally determined pore size distribution to a log-normal distribution as previous investigators have shown that this distribution has broad applicability. However, other distributions may give a better fit for specific cases. Further, we have attempted to describe the pore size distribution of asymmetric membranes using the surface pore size distribution as measured using FESEM images. These approximations will lead to differences between the measured and fitted pore size distributions. Nevertheless, our results indicate that fitting a log-normal distribution to the measured pore size distribution may be used to estimate the expected dextran rejection curve. Figs. 4–6 illustrate that distinguishing between a real through pore and a surface marking is often difficult. In this work, based on the ‘darkness’ of the marking on the FESEM image, we determined by eye whether to include a given pore in our analysis. If FESEM imaging is to be used on a routine basis to predict membrane rejection curves, an automated method based on a grey scale will be needed. In this work we studied two RC and one PES membranes. As indicated earlier, even for a given base membrane, e.g. RC membranes, as the modal pore size is increased, the membrane asymmetry and the shape of the pore size distribution often change due to changes in the manufacturing conditions. Though trends exist within each family of membranes, it is often difficult to predict the rejection curve of a given membrane based on the rejection curve for the same base membrane with a different NMWCO. Our results for RC and PES membranes indicate that the method we have developed for estimating the rejection curve for a membrane may be applied to different base membranes; though, using the rejection behavior of a given base membrane to predict the behavior for a different base membrane is not straightforward. While industrial scale ultrafiltration processes are run in tangential flow mode, we have investigated dextran rejection using normal flow filtration tests. Zydney and Xenopoulos [26] recently compared dextran rejection result for normal and tangential flow modes. They state that the results obtained for normal flow filtration are more reproducible and robust, as it is much easier to control the effects of operating conditions such as concentration polarization and gel layer formation. Thus, normal flow filtration is more appropriate when testing product quality or rapidly screening different membranes for a given protein purification.
8
S.R. Wickramasinghe et al. / Journal of Membrane Science 340 (2009) 1–8
5. Conclusions
References
Rejection curves for various molecular weight dextrans have been determined for three commercially available ultrafiltration membranes. Using a hydrodynamic model, rejection curves were predicted based on the surface pore size distribution of the membranes as determined by FESEM images. Good agreement between the predicted and experimentally determined rejection curves was obtained for higher values of the rejection coefficient. As it is these higher values of the rejection coefficient that are of most practical interest for designing a UF operation, predicting the rejection behavior of a membrane based on the membrane pore size distribution as determined by FESEM analysis may provide a method to screen membranes.
[1] R. van Reis, A.L. Zydney, Review bioprocess technology, J. Membr. Sci. 297 (2007) 16–50. [2] R. van Reis, A.L. Zydney, Protein ultrafiltration, in: M.C. Flickinger, S.W. Drew (Eds.), Encyclopedia of Biprocess Technology: Fermentation, Biocatalysis, and Bioseparations, John Wiley & Sons, Inc., New York, 1999. [3] S. Nakao, Review determination of pore size and pore size distribution 3. Filtration membranes, J. Membr. Sci. 96 (1994) 131–165. [4] S.S. Kulharni, E.W. Funk, N.L. Li, Membranes, in: W.S.W. Ho, K.K. Sirkar (Eds.), Membrane Handbook, van Nostrand Reinhold, New York, 1992. [5] S. Mochizuki, A.L. Zydney, Dextran transport through asymmetric ultrafiltration membranes: comparison with hydrodynamic models, J. Membr. Sci. 68 (1992) 21–41. [6] G. Tkacik, S. Michaels, A rejection profile test for ultrafiltration membranes and devices, Bio/technology 9 (1991) 941–946. [7] L. Zeman, M. Wales, Steric rejection of polymeric solutes by membranes with uniform pore size distributions, Sep. Sci. Technol. 16 (1981) 275–290. [8] H. Susanto, S. Franzka, M. Ulbricht, Dextran fouling of polyethersulfone ultrafiltration membranes-causes, extent and consequences, J. Membr. Sci. 296 (2007) 147–155. [9] C. Zhao, X. Zhou, Y. Yue, Determination of pore size and pore size distribution on the surface of hollow fiber filtration membranes: a review of methods, Desalination 129 (2000) 107–123. [10] J.K. Leypoldt, Determining pore size distributions of ultrafiltration membranes by solute sieving-mathematical limitations, J. Membr. Sci. 31 (1987) 289–305. [11] S. Derjani-Bayeh, V.G.J. Rodgers, Sieving variations due to the choice in pore size distribution model, J. Membr. Sci. 209 (2002) 1–17. [12] A.R. Cooper, D.S. van Derveer, Characterization of ultrafiltration membranes by polymer transport measurements, Sep. Sci. Technol. 14 (1979) 551–556. [13] K.J. Kim, A.G. Fane, C.D.J. Fell, T. Suzuki, M.R. Dickson, Quantitative microscopic study of surface characteristics of ultrafiltration membranes, J. Membr. Sci. 54 (1990) 89–102. [14] P. Aimar, M. Meireles, V. Sanchez, A contribution to the translation of retention curves into pore size distributions for sieving membranes, J. Membr. Sci. 54 (1990) 321–338. [15] M. Meireles, P. Aimar, V. Sanchez, Effects of protein fouling on the apparent pore size distribution of sieving membranes, J. Membr. Sci. 56 (1991) 13–28. [16] S. Mochizuki, A.L. Zydney, Theoretical analysis of pore size distribution effects on membrane transport, J. Membr. Sci. 82 (1993) 211–227. [17] A.L. Zydney, P. Aimar, M. Meireles, J.M. Pimbley, G. Belfort, Short communication: use of the log-normal probability density function to analyze membrane pore size distributions: functional forms and discrepancies, J. Membr. Sci. 91 (1994) 293–298. [18] O. Kedem, A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to nonelectrolytes, Biochim. Biophys. Acta 27 (1958) 229–246. [19] L.K. Spiegler, O. Kedem, Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient membranes, Desalination 1 (1966) 311–326. [20] N.A. Peppas, D.L. Meadows, Macromolecular structure and solute diffusion in membranes: an overview of recent theories, J. Membr. Sci. 16 (1983) 361–377. [21] B.C. Robertson, A.L. Zydney, A Stefan–Maxwell analysis of protein transport in porous membranes, Sep. Sci. Technol. 23 (1988) 1799–1811. [22] W.M. Deen, Hindered transport of large molecules in liquid-filled pores, AIChE J. 33 (1987) 1409–1425. [23] P.M. Bungay, H. Brenner, The motion of closely fitting sphere in a fluid-filled tube, Int. J. Multiphase Flow 1 (1973) 25–56. [24] D. Venturoli, B. Rippe, Ficoll and dextran vs globular proteins as probes for testing glomerular permselectivity: effects of molecular size, shape, charge, and deformability, Am. J. Physiol. Renal. Physiol. 288 (2004) F605–F613. [25] J.D. Oliver III, S. Anderson, J.L. Troy, B.M. Brenner, W.M. Deen, Determination of glomerular size-selectivity in the normal rat with Ficoll, J Am. Soc. Nephrol. 3 (1992) 214–228. [26] A.L. Zydney, A. Xenopoulos, Improving dextran tests for ultrafiltration membranes: effects of device format, J. Membr. Sci. 291 (2007) 180–190. [27] J.D. Ferry, Statistical evaluation of sieve constants in ultrafiltration, J. Gen. Physiol. 20 (1936) 95–104. [28] W.J. Elford, J.D. Ferry, The ultrafiltration of proteins through graded collodion membranes. I. The serum-proteins, Biochem. J. 28 (Part 1) (1934) 650–662. [29] W.J. Elford, J.D. Ferry, The ultrafiltration of proteins through graded collodion membranes. II. Haemocyanin (Helix), edestin, and egg albumin, Biochem. J. 30 (Part 1) (1936) 84–91. [30] I.A. Galloway, W.J. Elford, Filtration of the virus of foot-and-mouth disease through a new series of graded collodion membranes, Br. J. Exp. Pathol. 12 (1931) 407–425. [31] J. Kassotis, J. Schmidt, L.T. Hodgins, H.P. Gregor, Modeling of the pore size distribution of ultrafiltration membranes, J. Membr. Sci. 22 (1985) 61–76. [32] J.C. Giddings, E. Kucera, C.P. Russell, M.N. Myers, Statistical theory for equilibrium distribution of rigid molecules in inert porous networks. Exclusion chromatography, J. Phys. Chem. 72 (1968) 4397–4408.
Acknowledgements Funding for this work was provided by the NSF Industry/University Cooperative Research Center for Membrane Applied Science and Technology at the University of Colorado. The authors are grateful to Drs. Anthony Allegrezza, Neil Soice and Gabriel Tkacik for their valuable comments and insights.
Nomenclature an bn Cs D Kc Kd Kt Ks L n0 n(ri ) n(r) N Pe P r r* r
i
rp R
v z
coefficient in Eqs. (6) and (7) (dimensionless) coefficient in Eqs. (6) and (7) (dimensionless) radially averaged solute concentration in the pores (mol L−3 ) diffusivity of dextran molecule (L2 t−1 ) convective hindrance factor in Eq. (1) (dimensionless) diffusive hindrance factor in Eq. (1) (dimensionless) hydrodynamic function (see Eq. (2)) (dimensionless) hydrodynamic function (see Eq. (3)) (dimensionless) membrane thickness (L) total number of pores of all radii (dimensionless) number of pores of radius ri (dimensionless) number of pores of radius r given by the log-normal distribution function (dimensionless) solute flux (mol L−2 t−1 ) Peclet number (dimensionless) pressure drop (M L−1 t−2 ) pore radius (continuous function) (L) mean pore radius (L) pore radius (discrete function) (L) hydrodynamic radius of dextran molecule (L in Å) rejection coefficient (dimensionless) radially averaged solution velocity in the pores (L t−1 ) distance through the pore (L)
Greek letters ım thickness of the active or skin layer (L) ratio of the solute to pore radius (dimensionless) viscosity (M L−1 t−1 ) * standard deviation (dimensionless) solute partition coefficient between the pore and bulk solutions (dimensionless)