membranes Article

Permeability-Selectivity Analysis of Microfiltration and Ultrafiltration Membranes: Effect of Pore Size and Shape Distribution and Membrane Stretching Muhammad Usama Siddiqui, Abul Fazal Muhammad Arif * and Salem Bashmal Mechanical Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia; [email protected] (M.U.S.); [email protected] (S.B.) * Correspondence: [email protected]; Tel.: +966-138-602-579 Academic Editor: Enrico Drioli Received: 23 May 2016; Accepted: 22 July 2016; Published: 6 August 2016

Abstract: We present a modeling approach to determine the permeability-selectivity tradeoff for microfiltration and ultrafiltration membranes with a distribution of pore sizes and pore shapes. Using the formulated permeability-selectivity model, the effect of pore aspect ratio and pore size distribution on the permeability-selectivity tradeoff of the membrane is analyzed. A finite element model is developed to study the effect of membrane stretching on the distribution of pore sizes and shapes in the stretched membrane. The effect of membrane stretching on the permeability-selectivity tradeoff of membranes is also analyzed. The results show that increasing pore aspect ratio improves membrane performance while increasing the width of pore size distribution deteriorates the performance. It was also found that the effect of membrane stretching on the permeability-selectivity tradeoff is greatly affected by the uniformity of pore distribution in the membrane. Stretching showed a positive shift in the permeability-selectivity tradeoff curve of membranes with well-dispersed pores while in the case of pore clustering, a negative shift in the permeability-selectivity tradeoff curve was observed. Keywords: permeability-selectivity; membrane stretching

1. Introduction Microfiltration and ultrafiltration are widely used techniques in applications ranging from wastewater treatment to biomedical applications and the food industry. In water desalination, for example, the use of ultrafiltration membranes instead of conventional pre-treatment improves the Reverse Osmosis (RO) feed quality and provide stable permeability. Microfiltration and ultrafiltration membranes are generally made using organic polymers, such as polytetrafluoroethylene (PTFE), polyethylene terephthalate (PET), polyvinylidene fluoride (PVDF), polypropylene (PP), polyethylene (PE), polysulfone (PS), and polyether sulfone (PES) and are prepared by techniques, such as track-etching, stretching, and phase inversion. The microstructures of phase-inversion and track-etched membranes are shown in Figure 1.

Membranes 2016, 6, 40; doi:10.3390/membranes6030040

www.mdpi.com/journal/membranes

Membranes 2016, 6, 40 Membranes 2016, 6, 40

2 of 14 2 of 14

Figure 1. Microstructure Microstructure of an (a) phase-inversion PES membrane and (b) track-etched membrane [1].

Microfiltration and and ultrafiltration ultrafiltration are are pressure pressure driven driven processes processes that that work work by Microfiltration by removing removing particles particles larger than the pore size of the membrane through a sieving mechanism. The quality of separation is larger than the pore size of the membrane through a sieving mechanism. The quality of separation is normally expressed by the rejection factor or the separation factor of the membrane for a given solute. normally expressed by the rejection factor or the separation factor of the membrane for a given solute. The rejection rejectionfactor factorisisdefined definedasas1 1´−cperm cperm/c /cfeed feed while separation factor factorisisdefined definedasascfeed cfeed/c/cperm perm (where The while the the separation (where ccperm = permeate concentration and c feed = feed concentration). Ideally, a membrane should combine perm = permeate concentration and cfeed = feed concentration). Ideally, a membrane should combine high permeability and high high rejection rate. Practically, this is is not not the the case high permeability and rejection rate. Practically, this case as as permeability permeability and and rejection rejection rate cannot both be increased at the same time [2]. rate cannot both be increased at the same time [2]. In order order to to understand understand and and predict predict the the performance performance of of microfiltration microfiltration and and ultrafiltration ultrafiltration In membranes, several made to to formulate formulate models models for for the the membrane membrane performance performance membranes, several efforts efforts have have been been made under the influence of fouling [3–6], pore size distribution [2,7], and pore shape [8,9]. Variation in under the influence of fouling [3–6], pore size distribution [2,7], and pore shape [8,9]. Variation in pore pore sizes has been found to affect membrane performance significantly. Effect of pore size sizes has been found to affect membrane performance significantly. Effect of pore size distribution in distribution membranes in track-etched membranes on the permeability-selectivity characteristics of track-etched on the permeability-selectivity characteristics of ultrafiltration membranes ultrafiltration membranes was studied by Mehta and Zydney [2] for circular pores and by Kanani et was studied by Mehta and Zydney [2] for circular pores and by Kanani et al. [9] for slot-shaped al. [9] for slot-shaped pores. Kanani et al. also found that using slot-shaped pores (very high aspect pores. Kanani et al. also found that using slot-shaped pores (very high aspect ratio) resulted in higher ratio) resulted in higher permeate compared circularratios. pores Increased for same rejection ratios. permeate flux compared to circularflux pores for sameto rejection aspect ratio hasIncreased also been aspect ratio has also been linked to reduced fouling rates in membranes [10]. linked to reduced fouling rates in membranes [10]. To take take advantage advantage of of improved improved membrane membrane performance performance when when pore pore aspect aspect ratio ratio is is high, high, studies studies To have also been conducted to see the effect of artificially increasing pore aspect ratio by uniaxial have also been conducted to see the effect of artificially increasing pore aspect ratio by uniaxial stretching ofofthethe membranes [11–13]. Morehouse et al. and [11,12] and carried out stretching membranes [11–13]. Morehouse et al. [11,12] Worrel [13]Worrel carried [13] out experimental experimental numerical studies to study the effect of uniaxial on the pore geometry and numericaland studies to study the effect of uniaxial stretching on thestretching pore geometry and performance and performance of PET track-etched membranes and phase-inversion PES and PVDF membranes, of PET track-etched membranes and phase-inversion PES and PVDF membranes, respectively. respectively. Bothan found that an the pore aspect ratio in resulted in improved permeate Both found that increase in increase the poreinaspect ratio resulted improved permeate flux, butflux, no but no definite trend in the rejection rate of the solutes was observed as the uniaxial strain applied to definite trend in the rejection rate of the solutes was observed as the uniaxial strain applied to the the membrane was increased. membrane was increased. Although models membrane performance account pore size distribution distribution have have Although models of of membrane performance that that take take into into account pore size been published for various pore geometries, they are not applicable to real membranes. As is evident been published for various pore geometries, they are not applicable to real membranes. As is evident from Figure Figure1,1,conventional conventional membrane preparation methods doresult not result in well-defined pore from membrane preparation methods do not in well-defined pore shapes. shapes. For example, the track-etched membrane Figure 1b, not hassize a pore size distribution For example, the track-etched membrane in Figurein1b, not only hasonly a pore distribution but also but also a pore aspect ratio distribution. Similarly, the process of membrane stretching will also a pore aspect ratio distribution. Similarly, the process of membrane stretching will also introduce introduce an aspect ratio distribution in the membrane. Currently, no model is available that an aspect ratio distribution in the membrane. Currently, no model is available that can predict can the predict the permeability-selectivity characteristics of microfiltration and ultrafiltration membranes permeability-selectivity characteristics of microfiltration and ultrafiltration membranes taking into taking into theaspore as well as pore ratio distributions. account theaccount pore size, wellsize, as pore aspect ratio aspect distributions. In the thecurrent currentwork, work, three major are carried out. aFirst, model study permeabilityIn three major taskstasks are carried out. First, modela to studyto permeability-selectivity selectivity trade-off of membranes with pore size and aspect ratio distributions is formulated is trade-off of membranes with pore size and aspect ratio distributions is formulated and is used toand study used to study the effect of aspect ratio and size distribution on the permeability-selectivity trade-off the effect of aspect ratio and size distribution on the permeability-selectivity trade-off of the membrane. of the membrane. Second, a finite element model for porous membranes is developed and is used to study the effect of membrane stretching on pore size and aspect ratio distributions and on the

Membranes 2016, 6, 40

3 of 14

Second, a finite element model for porous membranes is developed and is used to study the effect of membrane stretching on pore size and aspect ratio distributions and on the permeability-selectivity trade-off. Finally, the effect of porosity dispersion uniformity on the performance of stretched membrane is studied. 2. Permeability-Selectivity Analysis A methodology to carry out permeability-selectivity analysis of ultrafiltration and microfiltration membranes has been presented by Mehta and Zydney [2] for membranes with circular pores of variable sizes and later extended by Kanani et al. [9] for slot pores. The following development for membranes with circular pores was done by Mehta and Zydney [2] to determine the permeability of the solvent through a membrane with circular pores and is presented here for the sake of completeness with some additional intermediate steps. The methodology is then extended to include the effect of pore shape distribution. 2.1. Circular Pores with Size Distribution Under the assumption of a convection dominated process, the velocity of all impurities in the feed is the same as the solvent and the separation coefficient of the membrane for the large solute to be filtered is defined as: S α “ small (1) Slarge where Slarge is the selectivity of the larger solute that needs to be separated and Ssmall is the selectivity of the small solutes that pass through the membrane without any hindrance. The selectivity of the larger solute that needs to be filtered can be determined using the expression developed by Zeman and Wales [14]: ´ ¯ ´ ¯ Sa prq “ p1 ´ λq2 2 ´ p1 ´ λq2 exp ´0.7146λ2

(2)

where λ “ rs {r, r, and rs are the pore and solute radii. The permeability of the solvent through the membrane can be determined by assuming that the pores are perfect cylinders. Under this condition, the Hagen–Poiseuille equation is valid. The volumetric flow rate of the solvent through the membrane is: Q“

∆P Np πr4 8µδm

(3)

where Q is the flow rate (m3 /s), ∆P is the pressure drop (Pa) across the membrane, µ is the viscosity (Pa¨s) of the solvent, δm is the membrane thickness (m), and Np is the number of pores in the membrane. The volumetric flux through the membrane becomes: Jv “ Q{Amem

(4)

where Amem is the membrane area defined by: Amem “

Np πr2 ϕ

(5)

and ϕ is the porosity fraction in the membrane. Using Equations (3)–(5), the permeability of the solvent through the membrane can be defined using: Jv ϕr2 Lp “ “ (6) ∆P 8µδm

Membranes 2016, 6, 40

4 of 14

Equations (2) and (6) can be used to generate the permeability-selectivity curves for membranes. The effect of variation in the pore radius can also be included in the equations for permeability and selectivity. This is done by averaging the permeability and selectivity over the entire range of pore radii. Mochizuki and Zydney [7] carried out a theoretical analysis to determine the average permeability and selectivity of membranes. The average permeability and selectivity are given by Equations (7) and (8), respectively: r8 ϕ 0 n prq r4 dr r Lp “ (7) 8µδm 08 n prq r2 dr r8 Sa prq n prq r4 dr Sa “ 0 r 8 (8) 4 0 n prq r dr where n(r) is the probability density function of pore radii. 2.2. Elliptical Pores with Size and Aspect Ratio Distribution Using a similar approach to Mehta and Zydney [2], the permeability of a solvent through a membrane with elliptical pores was derived. The solution of the Hagen–Poiseuille equation for elliptical cross-section leads to the following equation for solvent flow rate: Q“

Np ∆P πa3 b3 4µδm a2 ` b2

(9)

where a and b are the major and minor axes half lengths of the elliptical cross section of the pore. The permeability of the solvent through the membrane is, therefore: Lp “

ϕ a2 b2 4µδm a2 ` b2

(10)

To take into account the variation in the pore sizes in the membrane, the area weighted average of Equation (10) is taken. The average permeability through the membrane is given by Equation (11): r8r8 a3 b3 ϕ 0 0 n paq n pbq a2 `b2 dadb r r Lp “ 4µδm 08 08 n paq n pbq ab dadb

(11)

where n(a) and n(b) are the probability density functions of the major and minor axes’ half lengths of the pores. To determine the average selectivity of the membrane, it is noted that the selectivity will depend on the minor axis length of the pore cross section. Setting λ “ rs {b and assuming Equation (2) is still valid for determining selectivity, the average selectivity of membranes with elliptical pores of variable sizes can be determined using: r8 Sa pbq n pbq b2 db Sa “ 0 r 8 (12) 2 0 n pbq b db 3. Effect of Pore Geometry on Membrane Performance In order to study the effect of pore aspect ratio on the permeability-selectivity performance of microfiltration and ultrafiltration membranes, a study was conducted in which the aspect ratio of the pores was changed while keeping the pore cross-section area constant. The parameters used in the model were solute radius ra = 3.65 nm for bovine serum albumin protein, membrane porosity fraction ϕ = 0.3, membrane thickness δ = 0.3 µm and solvent viscosity µ = 0.001 Pa¨s. The log-normal

Membranes 2016, 6, 40

5 of 14

pores was changed while keeping the pore cross-section area constant. The parameters used in the Membranes 2016, 6, 40 5 of 14 model were solute radius ra = 3.65 nm for bovine serum albumin protein, membrane porosity fraction ϕ = 0.3, membrane thickness δ = 0.3 μm and solvent viscosity μ = 0.001 Pa∙s. The log-normal probability distribution distribution function, function, given given by by Equation Equation (13) (13) was was used used to to describe describe the the pore size variation probability pore size variation within the the membrane membrane [7]. [7]. within , $ ˆ 1 2 2˙2 ”   2 ı 1{2 ’   2 / ’ ln x x 1   x   1 ` pσ{xq     / / ’   ln px{xq 1 2 . & ¯ı ” ´ n 2 ´ 1 { 2   n 0   0 2 n “x  ? 1 pσ{xq exp ´   x   exp ” ı  n pxq lnln1 ` 2 ’ x x2π2 22ln ln 11`  x  2 / ’  / / ’   pσ{xq  %







(13) (13)



where σσ isisthe thestandard standarddeviation deviationofofx, x, is mean the mean x,nand n0 ismaximum the maximum possible where x isx the valuevalue of x, of and possible value 0 is the x normalized value n(x).two The two parameters that control the distribution arex mean and the normalized of n(x).ofThe parameters that control the distribution are mean and the standard deviationdeviation σ{x.  x. standard Two conducted; first, to see of average aspect ratio on ratio permeability-selectivity Two studies studieswere were conducted; first, to the seeeffect the effect of average aspect on permeabilitytrade-off for different normalized standard deviations of pore sizes a and b, and second, to see the selectivity trade-off for different normalized standard deviations of pore sizes a and b, and second, to effect of normalized standard deviations of pore sizes a and b on the performance for different aspect see the effect of normalized standard deviations of pore sizes a and b on the performance for different ratios. The results are shown in Figures 2 and 3. Each point in in thethe figures was generated by by selecting an aspect ratios. The results are shown in Figure 2,3. Each point figures was generated selecting average pore radius r, calculating the average major and minor axes half lengths under the condition an average pore radius r, calculating the average major and minor axes half lengths under the that the pore the same for aspect and calculating permeability and selectivity condition thatarea the remains pore area remains theall same forratios all aspect ratios and calculating permeability and using Equations (11) and (12), respectively. selectivity using Equations (11) and (12), respectively.

(a)  aa / aavg avg   bb / bavg  0.1

(b)  a / aavg   b / bavg  0.2

(c)  aa / aavg avg   bb / bavg avg  0.3

(d)  aa / aavg   bb / bavg  0.5 avg avg

Figure 2. 2. Effect Effect of of pore pore aspect aspect ratio ratio on on permeability-selectivity permeability-selectivity trade-off. Figure trade-off.

Membranes 2016, 6, 40 Membranes 2016, 6, 40

6 of 14 6 of 14

(a) Aspect Ratio  1

(b) Aspect Ratio  2

(c) Aspect Ratio  3

(d) Aspect Ratio  5

Effect of pore size distribution on permeability-selectivity trade-off. Figure 3. Effect

As ratio improves thethe performance of As can can be be seen seen from fromFigure Figure2,2,increasing increasingthe theaverage averageaspect aspect ratio improves performance microfiltration andand ultrafiltration membranes for all distributions. In other words, for thefor same of microfiltration ultrafiltration membranes forsize all size distributions. In other words, the separation factor, a higher permeate flux can be obtained if the pore aspect ratio is increased. This same separation factor, a higher permeate flux can be obtained if the pore aspect ratio is increased. resultresult agrees with the findings of Worrel [13] and et al. [11,12] who found This agrees withexperimental the experimental findings of Worrel [13]Morehouse and Morehouse et al. [11,12] who the same effect of increasing aspect ratio of permeate flux. This strengthens confidence in found the same effect of increasing aspect ratio of permeate flux. This strengthens confidence in the the theoretical development carried carried out out in in the the current current work. work. theoretical development The effectofofpore poresize size distribution membrane performance is shown in Figure The The effect distribution onon thethe membrane performance is shown in Figure 3. The3. figure figure shows the detrimental effect of non-uniform pore size on the permeability-selectivity trade-off shows the detrimental effect of non-uniform pore size on the permeability-selectivity trade-off curve. curve. larger standard deviation poreresults sizes results in a separation lower separation factor for all HavingHaving a largera standard deviation of poreofsizes in a lower factor for all aspect aspect ratios. Intoorder to achieve theseparation same separation factor, pore sizetoneeds to be reduced, which ratios. In order achieve the same factor, the porethe size needs be reduced, which reduces reduces the permeability. the permeability. 4. Membrane MembraneStretching Stretching Previous studies show that membrane stretching can have a positive effect on membrane thethe other hand, the the effect of stretching on the factor factor could not be permeability [11–13]. [11–13].On On other hand, effect of stretching onseparation the separation could properly identified as experiments showed that that there were cases in inwhich not be properly identified as experiments showed there were cases whichthe theseparation separation factor increased, remained unchanged, or even reduced reduced [13]. [13]. In the current work, a finite element model of a microfiltration membrane was developed to analyze the effect of membrane stretching on the size and shape distributions of pores in the stretched membrane. Using Using the the results results of of the the finite element model, model, the effect of membrane stretching on the permeability-selectivity trade-off curve was also analyzed. This section presents the development of

Membranes 2016, 6, 40

7 of 14

permeability-selectivity trade-off curve was also analyzed. This section presents the development of the finite element model and the resulting pore sizes due to membrane stretching. The effect of membrane stretching on its performance is analyzed in the next section. 4.1. Finite Element Model for Membrane Stretching The membrane stretching problem consists of a finite element model of the representative volume element (RVE) of the porous membrane. The constitutive behavior of the membrane material is defined explicitly as viscoelastic-rate-independent plastic. The geometry of a 30% porous membrane with an average pore size of 0.1 µm was generated using an in-house code and is shown in Figure 4 along with Membranes 6, 40 mesh. The microscale model is defined by Equations (14)–(17). 7 of 14 the finite 2016, element

the finite element model and the resulting pore due of ´ ∇ ¨ p1 ` ∇sizes uq S “ FV to membrane stretching. The effect(14) membrane stretching on its performance is analyzed in the next section. S “ C : εel ` Sq (15) 4.1. Finite Element Model for Membrane Stretching εel “ ε ´ ε pl (16) The membrane stretching problem consists of a finite element model of the representative ı 1” ε “membrane. p∇uqT `The ∇uconstitutive ` p∇uqT ∇ubehavior of the membrane material (17) volume element (RVE) of the porous 2 is defined explicitly as viscoelastic-rate-independent plastic. The geometry of a 30% porous where u is the displacement field, S is the second Piola-Kirchhoff stress, Fv is the body load, C is the membrane with an average pore size of 0.1 μm was generated using an in-house code and is shown elasticity tensor, Sq is the relaxation stress due to viscoelasticity, and ε, εel and ε pl are the total, elastic, in Figure 4 along with the finite element mesh. The microscale model is defined by Equations (14)– and plastic strain tensors. (17).

Figure Membrane geometry Figure 4. 4. Membrane geometry with with finite finite element element mesh. mesh.

For viscoelasticity, the bulk modulus was assumed to be constant while the shear modulus was  1   u S The FV model consists of several spring-damper (14) assumed to be defined by the generalized  Maxwell model. branches in parallel each of which is defined by a shear modulus and a relaxation time. Considering m S  C : el  Sq using Equations (18) and (19): (15) parallel branches, the viscoelasticity model can be described



Sq “



ÿ   `   ˘ el pl ´ ε 2G vm eel,dev vm

(16) (18)

m

1 T T   τ ε.u  `εu“ ε u  u    vm vm vm el,dev 2

(17) (19)

where Gvm is the shear modulus inisbranch m, τvm is the relaxation time branch εvmC is where u is the displacement field, S the second Piola-Kirchhoff stress, Fv isofthe body m, load, is the the viscoplastic strain of branch m, and εel,dev is the deviatoric part of the elastic strain tensor. elasticity tensor, Sq is the relaxation stress due to viscoelasticity, and , el and  pl are the total,

elastic, and plastic strain tensors. For viscoelasticity, the bulk modulus was assumed to be constant while the shear modulus was assumed to be defined by the generalized Maxwell model. The model consists of several springdamper branches in parallel each of which is defined by a shear modulus and a relaxation time. Considering m parallel branches, the viscoelasticity model can be described using Equations (18) and (19):

Membranes 2016, 6, 40

8 of 14

The rate-independent plasticity of the membrane material was modeled using the bilinear isotropic hardening model. The model is described by Equations (20)–(23): .

.

ε p “ ε p,e f f Membranes 2016, 6, 40

BF BS

F “ σmises ´ σys

(20) 8 of 14

(21)

The rate-independent plasticity of the membrane ET,iso material was modeled using the bilinear ε p,e f f (20)–(23): (22) σ “ σ ` ys y0 isotropic hardening model. The model is described byET,iso Equations 1´ E ` ˘ . F ε. ε p,e f f ě 0, F σ,σys ď0, (23) p,e f f F “ 0 (20) p p ,eff

S

where ε p is the plastic strain tensor, ε p,e f f is the von Mises effective plastic strain, σmises is the von Mises stress, σy0 and σy are the initial and current and F mises yield  ys stress, and E and ET are the elastic(21) tangential moduli. ET ,iso of a larger membrane, periodic boundary Since the finite element model represents a part  ys   y 0   conditions are applied to the model using constraint equations. (22) ET ,iso p ,effFor a two-dimensional model, shown 1 in Figure 5, the constraint equations are given as Equations (24)–(28). Uniaxial stretching is applied to E the membrane by controlling the displacement of reference node 1.  p ,eff  0, F  ,   0,  p ,eff F  0 (23) Ñ Ñ ysÑ u 2 ´ u 1 ´ u re f1 “ 0 (24)



where



 p is the plastic strain tensor, Ñp ,eff Ñ is theÑvon Mises effective plastic strain, mises is the

u ´ u 1 ´ u re f2 “ 0 (25) von Mises stress, σy0 and σy are the initial4and current yield stress, and E and ET are the elastic and Ñ Ñ Ñ Ñ tangential moduli. (26) u 3 ´ u 1 ´ u re f1 ´ u re f2 “ 0 Since the finite element model represents a part of a larger membrane, periodic boundary Ñ Ñ Ñ u right,no ´ u le f t,noequations. (27) conditions are applied to the model using constraint edges edges ´ u reFor f 1 “a0two-dimensional model, shown in Figure 5, the constraint equations are given (24)–(28). Uniaxial stretching is applied Ñ Ñ as Equations Ñ u top,no edges ´ u bottom,no edges ´ u re f2 “ 0 (28) to the membrane by controlling the displacement of reference node 1.

Figure 5. Applying Applying periodic periodic boundary boundary conditions conditions to a microscale RVE. Figure

In the current work, the membrane material to be PET. The properties used in the u2  u1 was urefassumed 0 (24) 1 current work were determined by Hanks et al. [15] who experimentally determined the stress-strain response and the viscoelastic material properties u4  u1ofdense uref2 PET. 0 The properties used in the finite element (25) model are listed in Table 1.

u3  u1  uref1  uref2  0

(26)

uright ,noedges  uleft ,noedges  uref1  0

(27)

utop,noedges  ubottom,noedges  uref2  0

(28)

In the current work, the membrane material was assumed to be PET. The properties used in the current work were determined by Hanks et al. [15] who experimentally determined the stress-strain response and the viscoelastic material properties of dense PET. The properties used in the finite element model are listed in Table 1.

Membranes 2016, 6, 40

9 of 14

Membranes 2016, 6, 40

9 of 14

Table 1. Material properties of dense PET used in finite element model.

Table 1. Material properties of dense PET used in finite element model. Property Value Property Value Bulk modulus, K 0.162 GPa Bulk modulus, K 0.162 GPa Initial Shear modulus, Ginit 0.075 GPa Initial Shear modulus, Ginit 0.075 GPa Tangent modulus, ET 5.95 MPa Tangent modulus, ET 5.95 MPa Viscoelasticity [15] [15] Viscoelasticity Branch Shear Modulus Ratio, Gvm /G Relaxation TimeTime (s) (s) Branch Shear Modulus Ratio, Gvminit /Ginit Relaxation −5 11 0.0402 0.0402 10´5 10 22 0.0468 0.0468 10´4 10−4 33 0.0572 0.0572 10´3 10−3 0.1805 44 0.1805 10´2 10−2 0.0487 10´1 10−1 55 0.0487 0.0988 100 100 66 0.0988 7 0.0205 101 101 7 0.0205 8 0.1394 102 102 8 0.1394 0.0000 103 103 99 0.0000 10 0.1283 104 10 0.1283 104 11 0.0470 105 11 0.0470 105 12 0.1005 106 12 0.1005 106 ˝ C by applying the total strain over a period of The stretching stretching process process was wascarried carriedout outatat160 160°C The by applying the total strain over a period of 5 5 min. This was followed a holding time of min 10 min to allow stress relaxation followed by cooling min. This was followed byby a holding time of 10 to allow stress relaxation followed by cooling the the membrane to room temperature over a period five minutes.Finally, Finally,the theapplied appliedstrain strain load load was was membrane to room temperature over a period of of five minutes. membrane. The The membrane membrane temperature temperature and and applied applied released to remove any elastic strains within the membrane. strain load are shown in Figure 6.

Figure Figure 6. 6. Applied Applied temperature temperature and and strain strain load. load.

4.2. Finite Element Modeling Results 4.2. Finite Element Modeling Results For the membrane geometry presented in Figure 4, the cases of 15%, 30%, 40%, and 50% uniaxial For the membrane geometry presented in Figure 4, the cases of 15%, 30%, 40%, and 50% uniaxial stretch were solved and analyzed. A summary of pore sizes after membrane stretching is presented stretch were solved and analyzed. A summary of pore sizes after membrane stretching is presented in in Table 2, while Figure 7 shows the distribution of major and minor axes sizes. The deformed Table 2, while Figure 7 shows the distribution of major and minor axes sizes. The deformed geometries geometries for two of the solved cases (15% and 30% stretch) are shown in Figure 8. for two of the solved cases (15% and 30% stretch) are shown in Figure 8.

Membranes 2016, 6, 40 40 Membranes 2016, 6, 40

10 of 14 10 of 14

Table Table 2. 2. Size Size distribution distribution parameters parameters for for stretched stretched membranes. membranes. Table 2. Size distribution parameters for stretched membranes. Major Axis Size Major Axis Size

Avg. Major Axis Avg. Major Axis

Stretch Stretch Stretch

Unstretched Unstretched Unstretched 15% 15% 15% 30% 30% 30% 40% 40% 40% 50% 50% 50%

Size, aavg Axis (μm) Avg. Major Size, aavg (μm) Size, aavg (µm) 0.097 0.097 0.097 0.129 0.129 0.129 0.162 0.162 0.162 0.180 0.180 0.180 0.199 0.199 0.199

Avg. Minor Axis Avg. Minor Axis

Size, bavg Axis (μm) Avg. Minor Size, bavg (μm) Size, bavg (µm)

(a) (a)

0.097 0.097 0.097 0.095 0.095 0.095 0.087 0.087 0.087 0.084 0.084 0.084 0.079 0.079 0.079

Major Axis Size Distribution, Distribution, Majorσ/a Axis avgSize σ/aavgσ/aavg Distribution, 0.060 0.060 0.060 0.142 0.142 0.142 0.191 0.191 0.191 0.194 0.194 0.194 0.201 0.201 0.201

Major Axis Size Distribution, Distribution, Major σ/bAxis avg Size σ/bavg σ/bavg Distribution, 0.060 0.060 0.060 0.094 0.094 0.094 0.156 0.156 0.156 0.196 0.196 0.196 0.256 0.256 0.256

Average Pore Average Pore Aspect Ratio Average Pore Aspect Ratio Aspect Ratio

1 1 1.3741 1.374 1.374 1.866 1.866 1.866 2.256 2.256 2.256 2.732 2.732 2.732

(b) (b)

Figure Figure 7. 7. (a) (b) minor minor axes axes size size distribution distribution for for stretched stretched membranes. membranes. (a) Major Major and and (b)

(a) (a) Stretch=15% Stretch=15%

(b) (b) Stretch=30% Stretch=30%

Figure 8. geometries of stretched membranes (b) 30% stretch. Figure 8. Deformed Deformed membranes for for (a) (a) 15% 15% stretch stretch and and (b) (b) 30% 30% stretch. stretch. Figure 8. Deformed geometries geometries of of stretched stretched membranes for (a) 15% stretch and

As As expected, expected, increasing increasing the the uniaxial uniaxial strain strain applied applied to to the the membrane membrane increases increases the the pore pore aspect aspect AsThe expected, increasing the uniaxial strain applied to the membrane increases the stretch. pore aspect ratios. average pore aspect ratio increases from 1.374 for 15% stretch to 2.732 for 50% ratios. The average pore aspect ratio increases from 1.374 for 15% stretch to 2.732 for 50% stretch. The The ratios. Theprocess average pore aspect ratio increasesoffrom 1.374 for 15% stretch to 2.732 strains for 50%result stretch. stretching also affects the distribution pore sizes as increasing stretching stretching process also affects the distribution of pore sizes as increasing stretching strains result in in The stretching process also affects the distribution of pore sizes as increasing stretching strains result in larger larger standard standard deviations deviations in in pore pore sizes. sizes. The The maximum maximum normalized normalized standard standard deviation deviation was was observed observed larger standard deviations in pore sizes. The maximum normalized standard deviationAs was observed for for the the 50% 50% stretch stretch case case with with values values of of 0.201 0.201 and and 0.256 0.256 for for major major and and minor minor axes axes sizes. sizes. As was was shown shown for the 50% stretch case with values of 0.201 and 0.256 for major and minor axes sizes. As was in Figure 2,3, increasing the aspect ratio improves the membrane performance while a in Figure 2,3, increasing the aspect ratio improves the membrane performance while a wider wider shown in Figures 2 and 3, increasing the aspect ratio improves the membrane performance while a distribution distribution of of pore pore sizes sizes results results in in aa degradation degradation of of membrane membrane performance. performance. Since Since membrane membrane wider distributioninofan pore sizes results in a degradation of membrane performance. Since membrane stretching stretching results results in an increase increase in in pore pore aspect aspect ratios, ratios, as as well well as as the the width width of of the the distribution distribution of of sizes, sizes, stretching will results in an increase in pore aspect ratios, as well as the width of the distribution of sizes, not necessarily result in an improved permeability-selectivity trade-off. stretching will not necessarily result in an improved permeability-selectivity trade-off. stretching will not necessarily result in an improved permeability-selectivity trade-off. 5. 5. Effect Effect of of Stretching Stretching on on Membrane Membrane Performance Performance Using Using the the finite finite element element model model results results of of Section Section 44 and and the the permeability permeability and and selectivity selectivity models models formulated in Section 2, the effect of membrane stretching on its permeability-selectivity formulated in Section 2, the effect of membrane stretching on its permeability-selectivity trade-off trade-off

Membranes 2016, 6, 40

11 of 14

5. Effect of Stretching on Membrane Performance Using the6, finite element model results of Section 4 and the permeability and selectivity models Membranes 2016, 40 11 of 14 formulated in Section 2, the effect of membrane stretching on its permeability-selectivity trade-off was was analyzed. As was discussed in Section 4.2 and shown in Figure 7, there competing effects analyzed. As was discussed in Section 4.2 and shown in Figure 7, there areare twotwo competing effects of of stretching that modify membrane performance. First, stretching causes pore aspect ratios stretching that modify thethe membrane performance. First, stretching causes thethe pore aspect ratios to to increase, which improves performance. Second, the of width size distribution is also increase, which improves performance. Second, the width pore of sizepore distribution is also increased, increased, which degradesperformance. membrane Their performance. combined effect on membrane which degrades membrane combined Their effect on membrane performance for the performance for considered the stretching cases considered in Section 4.29.isEach shown in Figure Eachispoint in the stretching cases in Section 4.2 is shown in Figure point in the 9. figure generated figure is generated taking an initial radiusthe r and applying theand average major minor to axes by taking an initialby pore radius r andpore applying average major minor axesand stretches it. stretches to it. The normalized standard deviation assumed of to the be independent of the average The normalized standard deviation was assumed to bewas independent average pore size. The figure pore size. that30% the stretches cases of 15% 30% stretches a positive effect of the shows thatThe the figure cases ofshows 15% and showand a positive effect ofshow the permeability-selectivity permeability-selectivity trade-off curve. The performance starts to magnitudes. degrade at higher trade-off curve. The performance starts to degrade at higher stretch This isstretch easily magnitudes. easily observable at high selectivity values. The degradation of performance can observable atThis highisselectivity values. The degradation of performance can be associated with the large be associated withdistribution the large width ofispore size distribution which is numerically represented as the width of pore size which numerically represented as the normalized standard deviation normalized in Table 2. standard deviation in Table 2.

Figure trade-off. Figure 9. 9. Effect Effect of of membrane membrane stretching stretching on on permeability-selectivity permeability-selectivity trade-off.

Effect of Porosity Dispersion Quality on Membrane Performance Effect of Porosity Dispersion Quality on Membrane Performance The results in Figure 9 show that at higher stretch levels, the effect of size distribution dominates The results in Figure 9 show that at higher stretch levels, the effect of size distribution dominates the effect of pore aspect ratio. This results in a worse performance than the case of a lower stretch. In the effect of pore aspect ratio. This results in a worse performance than the case of a lower stretch. this section, a study is presented to examine the hypothesis that the pore size distribution after In this section, a study is presented to examine the hypothesis that the pore size distribution after stretching is related to the uniformity of pore dispersion. stretching is related to the uniformity of pore dispersion. To start, four porous membrane RVEs were generated with different pore dispersion To start, four porous membrane RVEs were generated with different pore dispersion uniformities. uniformities. The dispersion quality for each microstructure was quantified by first calculating the The dispersion quality for each microstructure was quantified by first calculating the nearest-neighbor nearest-neighbor distances d (calculated from center to center) for all pores and representing them as distances d (calculated from center to center) for all pores and representing them as the normalized the normalized average nearest-neighbor distance davg/ravg and its standard deviation σ/ravg. A higher average nearest-neighbor distance davg /ravg and its standard deviation σ/ravg . A higher average average nearest-neighbor distance and a lower standard deviation represent better dispersion. The nearest-neighbor distance and a lower standard deviation represent better dispersion. The four four membrane RVEs, along with the dispersion quality parameters, are shown in Figure 10, in which membrane RVEs, along with the dispersion quality parameters, are shown in Figure 10, in which case case (b) is the one studied in previous sections. The dispersion quality is increasing from case (a) to (b) is the one studied in previous sections. The dispersion quality is increasing from case (a) to (d). (d). The distributions of normalized nearest neighbor distances in the four RVEs are shown in Figure The distributions of normalized nearest neighbor distances in the four RVEs are shown in Figure 11. 11.

Membranes 2016, 6, 40 Membranes 2016, 6, 40 Membranes 2016, 6, 40

12 of 14 12 of 14 12 of 14 (a) d avg ravg  2.381 (a) d avg ravg  2.381  ravg  0.186  ravg  0.186

(c) d avg ravg  2.495 (c) d avg ravg  2.495  ravg  0.098  ravg  0.098

(b) d avg ravg  2.398 (b) d avg ravg  2.398  ravg  0.154  ravg  0.154

(d) d avg ravg  2.807 (d) d avg ravg  2.807  ravg  0.084  ravg  0.084

Figure 10. Porous membrane RVEs with controlled porosity dispersion. Figure with controlled controlledporosity porositydispersion. dispersion. Figure10. 10.Porous Porous membrane membrane RVEs RVEs with (a) d avg ravg  2.381 (a) d avg ravg  2.381  ravg  0.186  ravg  0.186

(c) d avg ravg  2.495 (c) d avg ravg  2.495  ravg  0.098  ravg  0.098

(b) d avg ravg  2.398 (b) d avg ravg  2.398  ravg  0.154  ravg  0.154

(d) d avg ravg  2.807 (d) d avg ravg  2.807  ravg  0.084  ravg  0.084

Figure 11. Normalized nearest neighbor distances in membrane RVEs with controlled porosity Figure 11. Normalized nearest neighbor distances in membrane RVEs with controlled porosity Figure 11. Normalized nearest neighbor distances in membrane RVEs with controlled porosity dispersion. dispersion. dispersion.

Membranes 2016, 6, 40 Membranes 2016, 6, 40

13 of 14 13 of 14

Using Using the the four four RVEs, the cases for 30% and 50% stretches were solved using finite element analysis and the the permeability-selectivity permeability-selectivitytrade-off trade-offcurves curveswere weregenerated generated which shown Figure which areare shown in in Figure 12. 12. figure the significant effect that dispersion porosity dispersion on the The The figure showsshows the significant effect that porosity quality canquality have oncan the have performance performance of stretchedAs membrane. Asquality dispersion quality isthe improved, theofselectivity stretched of stretched membrane. dispersion is improved, selectivity stretched of membranes membranes the same permeability. improves forimproves the samefor permeability.

(a) Stretch =30%

(b) Stretch =50%

Figure Figure 12. 12. Effect Effect of of porosity porosity dispersion dispersion on on permeability-selectivity permeability-selectivity trade-off trade-off for for (a) (a) 30% 30% stretch stretch and and (b) 50% stretch. (b) 50% stretch.

The results presented in this section are significant as they can help explain why previous The results presented in this section are significant as they can help explain why previous experimental studies with stretched membranes showed that the separation factor increased, stayed experimental studies with stretched membranes showed that the separation factor increased, stayed the same, or even reduced for stretched membranes [13]. the same, or even reduced for stretched membranes [13]. 6. 6. Conclusions Conclusions Permeability-selectivity Permeability-selectivity tradeoff tradeoff analysis analysis provides provides aa simple simple tool tool for for the the analysis analysis and and comparison comparison of the performance of microfiltration and ultrafiltration membranes. In the current work, of the performance of microfiltration and ultrafiltration membranes. In the current work, aa model model to to carry out permeability-selectivity analysis is formulated that takes into consideration the distribution carry out permeability-selectivity analysis is formulated that takes into consideration the distribution of of pore pore sizes sizes and and aspect aspect ratios. ratios. Using Using the the formulated formulated model, model, the the effect effect of of pore pore aspect aspect ratio ratio and and size size distribution on membrane performance was studied. It was found that increasing the pore aspect distribution on membrane performance was studied. It was found that increasing the pore aspect ratio ratio improves membrane performance while increasing width thedistribution pore size distribution improves membrane performance while increasing the widththe of the poreofsize deteriorates deteriorates the performance. the performance. The effectofofuniaxial uniaxial stretching membrane performance wasstudied also studied a finite The effect stretching on on membrane performance was also using ausing finite element element model of a porous membrane in conjunction with the permeability-selectivity model. the model of a porous membrane in conjunction with the permeability-selectivity model. For the For porous porous membrane modeled, improvement was observed in 15% 30% uniaxial stretching cases. membrane modeled, improvement was observed in 15% and 30%and uniaxial stretching cases. Further Further stretching deteriorated the membrane performance. The key factor in the deterioration of stretching deteriorated the membrane performance. The key factor in the deterioration of performance performance was found to be the width of the pore size distribution which became larger with stretch. was found to be the width of the pore size distribution which became larger with stretch. Porosity Porosity dispersion was key role in distribution pore size distribution of membranes. stretched membranes. dispersion was found to found play a to keyplay roleain pore size of stretched Using the Using the finite element model, it was determined that membranes with well-dispersed pores finite element model, it was determined that membranes with well-dispersed pores had less had size less size distribution around the average value. This minimized the negative effect of pore size distribution around the average value. This minimized the negative effect of pore size distribution on distribution on membrane As a result, membranes with well-dispersed had membrane performance. Asperformance. a result, membranes with well-dispersed porosity had betterporosity performance better performance improvement after stretching. improvement after stretching. Acknowledgments: The authors would like to acknowledge the support of King Fahd University of Petroleum Acknowledgments: The authors would like to acknowledge the support of King Fahd University of Petroleum and Minerals through DSR project # FT131003. and Minerals through DSR project # FT131003. Author Author Contributions: Contributions: Abul Abul Fazal Fazal Muhammad Muhammad Arif Arif and and Salem Salem Bashmal Bashmal conceived conceived and and defined defined the the problem; problem; Abul Fazal Muhammad Muhammad Arif Arif and Muhammad Muhammad Usama Usama Siddiqui Siddiqui developed developed the the methodology methodology and and case case studies; studies; Muhammad Usama Siddiqui implemented the model and run test cases; All authors analyzed the results;

Membranes 2016, 6, 40

14 of 14

Muhammad Usama Siddiqui implemented the model and run test cases; All authors analyzed the results; Muhammad Usama Siddiqui wrote the paper; Abul Fazal Muhammad Arif and Salem Bashmal reviewed and contributed in discussions and conclusions. Conflicts of Interest: The authors declare no conflict of interest.

References 1.

2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14.

15.

Etching the Tracks in a Track-Etched Membrane Filter/Sterlitech Blog. Available online: http://www. sterlitech.com/blog/2013/07/29/etching-the-tracks-in-a-polycarbonate-track-etched-membrane-filter/ (accessed on 27 April 2016). Mehta, A.; Zydney, A.L. Permeability and selectivity analysis for ultrafiltration membranes. J. Memb. Sci. 2005, 249, 245–249. Polyakov, Y.S.; Zydney, A.L. Ultrafiltration membrane performance: Effects of pore blockage/constriction. J. Memb. Sci. 2013, 434, 106–120. [CrossRef] Ho, C.; Zydney, A. A Combined pore blockage and cake filtration model for protein fouling during microfiltration. J. Colloid Interface Sci. 2000, 232, 389–399. [CrossRef] [PubMed] Bolton, G.; LaCasse, D.; Kuriyel, R. Combined models of membrane fouling: Development and application to microfiltration and ultrafiltration of biological fluids. J. Memb. Sci. 2006, 277, 75–84. [CrossRef] Duclos-Orsello, C.; Li, W.; Ho, C.C. A three mechanism model to describe fouling of microfiltration membranes. J. Memb. Sci. 2006, 280, 856–866. [CrossRef] Mochizuki, S.; Zydney, A.L. Theoretical analysis of pore size distribution effects on membrane transport. J. Memb. Sci. 1993, 82, 211. [CrossRef] Hanks, P.L.; Forschner, C.A.; Lloyd, D.R. Sieve mechanism estimations for microfiltration membranes with elliptical pores. J. Memb. Sci. 2008, 322, 91–97. [CrossRef] Kanani, D.M.; Fissell, W.H.; Roy, S.; Dubnisheva, A.; Fleischman, A.; Zydney, A.L. Permeability-selectivity analysis for ultrafiltration: Effect of pore geometry. J. Memb. Sci. 2010, 349, 405–410. [CrossRef] [PubMed] Knoell, T.; Safarik, J.; Cormack, T.; Riley, R.; Lin, S.W.; Ridgway, H. Biofouling potentials of microporous polysulfone membranes containing a sulfonated polyether-ethersulfone/polyethersulfone block copolymer: Correlation of membrane surface properties with bacterial attachment. J. Memb. Sci. 1999, 157, 117–138. [CrossRef] Morehouse, J.A. The Effect of Uni-Axial Stretching on Microporous Phase Separation Membrane; University of Texas at Austin: Austin, TX, USA, 2006. Morehouse, J.A.; Worrel, L.S.; Taylor, D.L.; Lloyd, D.R.; Freeman, B.D.; Lawler, D.F. The effect of uni-axial orientation on macroporous membrane structure. J. Porous Mater. 2006, 13, 61–72. [CrossRef] Worrel, L.S. Modification of Track-Etched Membrane Structure and Performance via Uniaxial Stretching; The University of Texas at Austin: Austin, TX, USA, 2005. Zeman, L.; Wales, M. Polymer solute rejection by ultrafiltration membranes. In Synthetic Membranes: Volume II; Turbak, A.F., Ed.; ACS Symposium Series; American Chemical Society: Washington, DC, USA, 2009; Volume 154, pp. 411–434. Hanks, P.L.; Kaczorowski, K.J.; Becker, E.B.; Lloyd, D.R. Modeling of uni-axial stretching of track-etch membranes. J. Memb. Sci. 2007, 305, 196–202. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).