Relating uncharged solute retention of polyelectrolyte multilayer nanofiltration membranes to effective structural properties

description fitted to experimental data. Here, the studied PEM systems showed distinctive differences in both film thickness (PDADMAC/PSS: 44nm/BL, PAH/PAA: 689nm/BL) and mean pore size (PDADMAC/PSS: 0.44 nm, PAH/PAA: 0.27 nm). Within the range of layer numbers, the pore size of both PEMs in the layer dominated regime was independent of layer number. This indicates that there is an optimum layer number for PEM NF membranes regarding uncharged solute retention. Surprisingly, despite forming much thicker layers the PAH/PAA system closes off the support membrane pores at a higher bilayer number.


Introduction
Nanofiltration (NF) is a pressure-driven membrane process with its main application in the treatment of waste and drinking water [1][2][3], but also potential application in other fields such as food- [4,5], pharmaceutical- [6] and textile-industry [7].What makes NF highly competitive in these applications is its unique separation capability at relatively mild conditions and low energy consumption compared to conventional or alternative processes.
Separation in NF is mainly based on the size and charge of solutes [1,2].With nominal pore sizes in the range of 1 nm, NF membranes are capable to retain neutral solutes with a molecular weight in a range of 100-5000 Da.The rejection of charged solutes is governed by two additional mechanisms.The first, called Donnan Exclusion, is caused by functional groups at the membrane surface that become charged in an aqueous solution.The second, called Dielectric Exclusion, is caused by a difference in dielectric properties between the membrane and solvent phase.As a result of these mechanisms, NF membranes typically have high retention towards multivalent ions and * Corresponding author.E-mail address: r.g.h.lammertink@utwente.nl (R.G.H. Lammertink).low to moderate retention towards monovalent ions.These separation characteristics are desirable in water purification processes such as water softening [3,8] and removal of organic micropollutants [3,9].
There are, however, certain drawbacks of NF that hinder its widespread application.Amongst these are fouling, limited separation capability and short membrane lifetime [10].In addition to altering process conditions, membrane characteristics can be adjusted to tackle these challenges.This includes the development of membranes with high chemical and mechanical stability, resulting in a long membrane lifetime and resistance to chemical cleaning.A lot of research is also dedicated to enhancing the separation capability of NF membranes by fine-tuning pore size distribution and charge [11,12].
The market of NF membranes is dominated by Thin Film Composite (TFC) type membranes [13,14].The TFC design combines the need for membranes to withstand high pressures and allow high water permeation by dividing these functionalities into two layers: a thick layer (order of magnitude 100 μm) with large pores is coated with a thin layer (order of magnitude 100 nm) that has NF properties.https://doi.org/10.1016/j.memsci.2022 The discovery of TFC membranes is deservedly considered a milestone in membrane technology, as it improved membrane performance (originally of RO membranes) considerably [15].However, there is still room for improvement in the current commercial TFC NF membranes.For instance, does the material of the dense separation layer (typically polyamide) have a low resistance towards chlorine [16], which is present in standard chemicals used for membrane cleaning.Also, the separation performance of these membranes is limited by an upper bound for the trade-off between permeability and selectivity.To overcome this upper bound, multiple characteristics of the separation layer have to be optimized simultaneously [12].Therefore to make significant advances in NF membranes, new methods with fine control over membrane properties are required.One promising alternative to fabricate NF membranes is based on Polyelectrolyte Multilayers (PEMs) using the Layer-by-Layer (LbL) method, a concept already introduced almost 20 years ago [17,18].Here, maintaining the TFC design, a dense polymeric network with NF properties is formed on top of a support membrane by alternatingly coating polyanions and polycations.The main advantage of the LbL method is that the chemistry and structure of the polymeric network can be tuned in a fairly simple and straightforward way by a variety of parameters, such as type of polyelectrolytes, ionic strength and pH of coating solution [19,20].This way membranes with high chemical and physical stability, advanced separation characteristics and extremely thin separation layers can be fabricated [21].The high versatility of LbL further allows for functionalization of the membrane surface to reduce membrane fouling for example [22].The high potential of this novel method resulted in the recent commercialization of hollow fiber (HF) NF membranes based on PEMs.
Compared to established membrane fabrication methods, the LbL method is relatively young.As a result, although there has been a lot of progress in the field, the knowledge on fine-tuning membrane properties for specific applications is far from complete.For this method to reach its full potential, a more complete understanding of the influence of fabrication conditions on PEM structure as well as the influence of PEM structure on membrane performance is required.
One limitation in understanding the mentioned relations is the structural characterization of the PEM based selective layer.As characteristic dimensions are in the nanometer range, the availability of adequate analytical methods is limited.In addition, the significant influence of water on the PEM structure [23,24] requires measurements in a liquid environment.Therefore directly obtaining the internal structural properties of PEMs is challenging.Just to mention a few state-of-the-art characterization techniques that are used to determine properties of PEMs that are relevant for NF: streaming potential (surface charge) [25], neutron reflectometry (thickness and swelling) [26] and radiolabeling (net charge) [27].
An alternative characterization technique to determine properties relevant to membrane performance, that has been used for conventional polyamide-based NF membranes [28][29][30][31], is the indirect determination by relating the filtration performance to an effective structure via the use of transport models.These models can further be used to predict the performance of an NF process.
Clearly, to exploit the full potential of PEM based NF membranes a more elaborate understanding of the interplay between structure and transport processes is required.In this work, we utilize theoretical transport models that account for distribution in effective membrane pore size to describe polyethylene glycol (PEG) transport through two PEM based NF membranes (PDADMAC/PSS and PAH/PAA).The fitted model parameters and their relation to actual structural properties of the PEM are assessed by comparison to supplementary characterization methods as well as previous observations in the literature.This method allows us to gain further understanding of the relationship between structural properties to membrane properties of PEMs and is a step towards rational membrane design.

Membrane fabrication
Before dip coating, the HF UF support membranes (MWCO ∼10 kDa, Permeability ∼150 LMHbar, inner diameter 0.7 mm, negative charge) were immersed in 10% EtOH solution in H 2 O followed by rinsing in H 2 O, to remove added compounds for dry storage.The rinsed fibers were subsequently dip coated using the Layer-by-Layer (LbL) method to build polyelectrolyte multilayers on top of the support.In the first step of this method, the fibers are immersed in the polycation solution (0.1 gL −1 , 50 mM NaCl, pH 5.5) for 15 min.In the next step, which is repeated three times, the fibers are rinsed in 50 mM NaCl solution for 5 min, to wash off any residual polycation solution.The following steps include the immersion in a polyanion solution (0.1 gL −1 , 50 mM NaCl, pH 5.5) for 15 min, again, followed by three times rinsing in 50 mM NaCl solution for 5 min.This sequence completes one bilayer (BL).Here, two polyelectrolyte systems PDADMAC/PSS (up to 10 BL) and PAH/PAA (up to 14 BL) were studied.The pH value of the coating solutions was adjusted with 1 M NaOH/HCl solutions.After the LbL procedure, the membranes were immersed in a 15% glycerol in H 2 O solution for at least 4 h before drying.

Membrane performance evaluation
To evaluate the performance of the fabricated membranes, single fiber modules with an active fiber length of about L=18 cm (active membrane area of A  =3.95 cm 2 ) were prepared in quadruplicate.Using a crossflow filtration setup (see Supporting Information S1), previously described in [32], 16 membrane modules are tested simultaneously.The temperature of the feed solution is regulated at 20 ± 1 • C. The crossflow velocity within the fibers is set to 1 ± 0.05 ms −1 to limit the extent of concentration polarization.After rinsing the modules with H 2 O to remove any residual glycerol, the pure water permeability is measured at constant transmembrane pressures (TMP) of 2, 3 and 4 bar (estimated variation ±0.05 bar).The pure water permeability   (Lm −2 h −1 bar −1 ) can be calculated using the following equation: where   (g) is the permeate mass collected within the time  (h),   (998.21gL −1 ) is the density of water at 20 • C,   (m 2 ) is the active membrane area and TMP (bar) the applied transmembrane pressure.Using Darcy's law, the pure water permeability can be alternatively displayed as the hydraulic resistance  ℎ (m −1 ): where  (1.0016 ⋅10 −3 Pas) is the viscosity of water at 20 • C and   (now expressed in ms −1 Pa −1 ) the pure water permeability.The MgSO 4 retention (feed concentration of 5 mM) was measured at a constant TMP of 3 bar and 5 bar for the PDADMAC/PSS and PAH/PAA membranes, respectively.The PEG retention (1 gL −1 ) was measured at three different TMP values (PDADMAC/PSS: 1, 2, 3 bar; PAH/PAA: 2, 3.5, 5 bar) for each membrane.The retention  (-) is defined as: where   (molL −1 ) and   (molL −1 ) are the solute concentration in the permeate and feed, respectively.

Analytical methods
MgSO 4 concentrations were determined via conductivity measurements using a portable conductivity meter (Cond TM 3210, WTW TM GmbH).PEG concentrations were determined via Gel Permeation Chromatography (GPC).The GPC setup (Agilent 1200/1260 Infinity GPC/ SEC series) uses two columns (Suprema 8 × 300 mm, 1000 Å−30 Å, 10 μm, Polymer Standards Service GmbH) in series and a refractive index detector (Agilent 1100 HPLC G1362 A RID Detector).The flow rate of the eluent (50 mgL −1 NaN 3 in H 2 O) is 1 mLmin −1 .Information on the deconvolution method of the detector signal is given in the Supporting Information S2.

Salt retention
To transform the retention of MgSO 4 (a process parameter) into the salt permeability  (ms −1 ) (an intrinsic membrane parameter) the film model with a mass transfer correlation is applied, as described in [33]: Here   (ms −1 ) is the permeate flux and  * (ms −1 ) is the mass transfer coefficient corrected for suction, which is determined via [34,35]: (5) Here, ℎ (-) is the dimensionless Sherwood number,  (ms −1 ) is the mass transfer coefficient at no flux,  (m) is the HF inner diameter,   (0.75 ⋅10 −9 m 2 s −1 [36]) is the diffusion coefficient of MgSO 4 at 20 • C,  = ∕ (-) is the dimensionless Reynolds number,  = ∕(  ) (-) is the dimensionless Schmidt number,  (m) the HF length and  =   ∕ the ratio of flux   (ms −1 ) to mass transfer coefficient at no flux  (ms −1 ).The applied mass transfer correlation is valid for the laminar flow regime (Re<2300) and low recovery values.For high recovery values an additional correction factor should be applied [33].For very small Reynolds numbers (Re<0.16)Eq. ( 5) gets negative.Operation conditions of hollow fiber membranes, however, do not approach this limit.

PEG retention
To transform the retention of the PEG mix, which is again a process parameter, to an intrinsic membrane property, we apply a simplified version of the well known Donnan Steric Pore Model and Dielectric Exclusion (DSPM&DE) model [37,38].Here, we represent the membrane by a bundle of straight cylindrical pores.Although the actual membrane (or PEM) structure is likely far from this commonly applied idealized geometry [29,30,[37][38][39][40], this simplified theoretical description serves the purpose of this work well, which is to describe membrane transport, monitor selective layer build-up and qualitatively compare different PEM systems.As the PEG solutes are uncharged in solution, the exclusion mechanism of the DSPM&DE model reduces to steric exclusion, while the transport within the pores is described by advection and diffusion.To model the retention of PEG molecules based on steric exclusion, one needs to approximate their size in solution.
Here we choose to use the hydrodynamic radius  , (m) based on Stokes-Einstein equation [41]: with the Boltzmann constant   (1.381 ⋅10 −23 m 2 kgs −2 K −1 ) and the solution temperature  (293.15K).Here,  ,∞ (m 2 s −1 ) is the solute diffusion coefficient in infinite dilution which is approximated using a correlation [42] (corrected for temperature and viscosity): with   (Da) being the molecular weight of the PEG molecule.As we did not find a range of molecular weights of PEG for which this formula is valid, we compared it to literature data of diffusion coefficient measurements in the range of interest.A graph comparing the correlation to measurement values can be found in the Supporting Information S3.
For the relevant range of this study (MW ≤ 3000 Da) the correlation fits reasonably well.Also shown is the alternative correlation used to estimate the influence of error in PEG diffusion coefficient.

Single pore size
In the standard form, the DSPM&DE model assumes uniform pores with a singular effective pore radius, to describe the steric exclusion.Therefore, the membrane is represented by two parameters: the effective membrane thickness   (m) (which combines pore length, tortuosity and porosity) and the effective pore radius   (m).These can be determined by fitting the model to pure water permeability and uncharged solute retention.Details on the fitting procedure can be found in the Supporting Information S4.The system of equations describing the membrane transport after simplifications reduces to: (ms −1 ) is the permeate flux,  (1.0016 ⋅10 −3 Pas) is the viscosity of water at 20 • C,   (Pa) is the effective transmembrane pressure accounting for osmotic pressure (in the case of pure water measurements   =TMP),   (-) is the intrinsic membrane retention (in the ideal case of no concentration polarization),  (-) is the observed retention (as defined in Eq. ( 3)),    (-) is the Peclet number of transport through the membrane,  *  (ms −1 ) is the corrected mass transfer coefficient (determined as described in Eq. ( 5) and ( 6)),   (-) is the steric partitioning coefficient,   (-) is the ratio of solute Stokes radius to effective pore radius (  =  , /  ),  , (m 2 s −1 ) is the hindered diffusion within the nanopore,  , (-) the hindrance factor for convective transport,  ,∞ (m 2 s −1 ) the solute diffusion coefficient in infinite dilution, and  , (-) the hindrance factor for diffusive transport.

Pore size distribution
To extend the model from a single pore radius to a pore size distribution, we introduce a discrete log-normal distribution function to describe the relative abundance of a range of pore sizes, in a similar way as it was presented by [28].The continuous log-normal distribution is describe by: where   (-) is the relative amount by number of pores,   (m) the pore size,   (m) the mean pore size and  (m) is the standard deviation in pore size distribution.To allow for numerical solution, this equation has to be discretized by integrating over a range of pore size: Here  , (-) is the discretized distribution of relative number and  (0.01 nm) is the discretization step size (determined from a mesh refinement study shown in the Supporting Information S5).In addition, the pore size distribution function is truncated, which has been shown to increase fit quality [28], and reduces computation time.For the truncation, as well as for fitting, the distribution is transformed to a distribution based on relative flux   , (-): The truncation is made at a value of 5% of the maximum value of   , .After that, the distribution is normalized to assure that the sum over   , equals 1.To calculate the retention, we calculate the retention and flux for each pore size and calculate the overall retention  (-) via: (22) with  , (ms −1 ) the permeate flux through a discrete pore radius and   (-) the retention by a discrete pore radius.For each discrete pore radius, the single pore size model can be applied, where the effective membrane thickness has to be updated according to: , (m 3 ) is the effective membrane thickness updated for the pore size distribution and   (ms −1 Pa −1 ) the pure water permeability.The permeate flux through a discrete pore radius  , is calculated via: (24) The parameter  , accounts for fractions of the different pore sizes to the overall flux and cannot be directly compared to measured thickness values.To estimate the real thickness of the membrane and also compare it to single pore size results, the effective thickness  , for the pore size distribution was again transformed using the pore radius with maximum contribution to the relative flux:

Imperfections
To account for the presence of imperfections causing leakage of feed solution, e.g.module related leakage or pinholes, we assume an additional non-selective flux in parallel to the PEM: Here, the permeate flux   (ms −1 ) is composed of the permeate flux through the active separation layer  , (ms −1 ) and the nonselective flux through imperfections  , (ms −1 ).The relative amount of non-selective flux  (%) of the complete permeate flux is introduced: The observed retention of the membrane module can then simply be calculated via: Here,  , (molm −3 ) is the permeate concentration of the selective layer and   (molm −3 ) is the feed concentration for the non-selective flux.
The concentration   at the membrane surface is derived from the film model (Eqs.( 5), ( 6) and ( 29)) assuming only selective transport.

Results and discussion
In the first part of this section (4.1 and 4.2) we present experimental results of the conducted filtration measurements and discuss layer number dependent membrane properties.In the second part of this section (4.3-4.6)we focus on the different theoretical membrane transport models considered in this work and analyze the implied intrinsic PEM properties.In the third part of this section (4.7) we summarize and discuss the structural properties of the two PE systems considering experimental observations of this study as well as previous observations in the literature.

Selective layer build-up
To monitor the growth of the PEM layer on top of the UF support, fibers were taken out during the LbL coating procedure for each BL until the final layer was reached.As it has been shown by de Grooth et al. [43], there are typically two regimes observed when coating porous UF support membranes with PEM layers to reach a membrane with NF properties (illustrated in Fig. 1).In the ''pore dominated regime'', the initial layers do not fully cover the pores of the support membrane, yet.After fully closing the pores of the support, the PEM layer dominates the transport through the membrane (''layer dominated regime'').It is essential to reach the ''layer dominated regime'' when fabricating PEM based NF membranes.In this regime, the observed filtration performance can directly be related to the intrinsic properties of the PEM.For fabrication purposes (also on a commercial scale) assuring full closure of support pores also increases the reproducibility of membrane properties.Last but not least, the selectivity increases significantly once the pores of the support are fully closed, which is required for NF applications.
One membrane characteristic that indicates pore closing is the pure water permeability.Fig. 2 displays the pure water permeability of the membrane as a function of the BL number for both PEM systems.Also shown is the hydraulic resistance, which is inversely proportional to the pure water permeability (Eq.( 2)).For both systems, the pure water permeability decreases with an increasing number of BLs.Moreover, a decrease in the slope of permeability with BL number is observed and the permeability seems to slowly go towards a plateau value.However, one can observe that at high BL numbers the resistance of the PEM systems still increases with layer number.This we attribute to the ''layer dominated regime'' in which, when considering linear multilayer growth [27,44,45], the layer thickness should increase linearly with layer number and therefore so does the hydraulic resistance.The initial steep decrease in permeability is partially caused by the high sensitivity of permeability to effective membrane thickness (A∝  −1  ).In the case of PDADMAC/PSS, the increase in hydraulic resistance with BL number decreases after 4 BLs.This indicates the transition between two different layer growth regimes of the PEM NF membrane, where the increase of hydraulic resistance is higher during pore closing and lower but linear for the layer dominated regime.For PAH/PAA, the increase of hydraulic resistance with BL number does not reduce at a distinct BL number in the observed range, so no clear transition can be determined.It even increases slightly after 9 BLs, which will be discussed later.One other option to monitor pore closing that should be mentioned here, which was also shown in [43], is to use the influence of the ending layer (''odd-even effect'') on the swelling behavior of the PEM.This is caused by the great influence that the ending layer of a PEM can have on the overall multilayer properties [24,46].The difference in swelling of positive and negative ending PEMs has exactly opposite effects on pure water permeability in the ''pore dominated'' and ''layer dominated'' regimes.Since the transition between the two regimes is not the main focus of this work, but rather the properties of the PEM layer itself, we chose to only consider even BL numbers (negative ending layer).Membranes with a negative surface charge are typically favored in water treatment, due to their advantageous fouling properties [47][48][49][50][51].
When comparing the two PEM systems, two important differences stand out.First, PAH/PAA based membranes have a significantly higher hydraulic resistance than PDADMAC/PSS based membranes (one order of magnitude).Second, the change in slope of hydraulic resistance over BL number, which we attribute to the closing of the support membrane pores, only appears for the PDADMAC/PSS system at around 4-6 BLs.
One additional characterization method often used to quantify membrane performance is the MgSO 4 retention, which is displayed in Fig. 3 as a function of the PEM BL number.For both PEM systems an increase in MgSO 4 retention within the first few BLs is observed, which stagnates rather abruptly after a certain BL number.The initial sharp increase in MgSO 4 retention, especially for PDADMAC/PSS, is related to the pore closing of the support membrane, which by itself has a very low MgSO 4 retention.To translate the observed retention to an intrinsic membrane property, that in addition accounts for water permeability, we fitted the MgSO 4 retention to a simple solution-diffusion transport model, while accounting for the effect of concentration polarization.The effective selectivity of water over salt is hereby represented by the ratio of the pure water permeability A over the salt permeability B. We hypothesize, that A/B behaves differently within the two regimes of coating porous support membranes.In the ''pore dominated'' regime, the reduction in salt permeability B due to surface exclusion and hindered transport within the membrane leads to an increase in selectivity A/B.Although, pure water permeability A experiences a similar decrease in this regime (see Fig. 2), the decrease in salt permeability B is dominating.In the ''layer dominated'' regime, there is a constant increase in layer thickness with BL number.If one assumes thickness independent intrinsic properties of the PEM film according to the solution-diffusion model, the selectivity A/B should stay constant as both A and B are proportional to the inverse of layer thickness [52].As shown in Fig. 3, the A/B value as a function of the BL number behaves as predicted for both PEM systems: in the initial layers there is an increase of A/B and after a certain BL number the A/B value stagnates within the range of the 95% confidence intervals.
Here, the point of full pore closure is determined at 6 BL and 9 BL for the PDADMAC/PSS and PAH/PAA systems, respectively.Ion transport through PEM NF membranes is governed by both electrostatic and steric interactions.Thus, ion retention measurements alone do not suffice to unambiguously identify the extent of both effects (here it is reasonable to describe solute transport by using a permeability B).

PEG retention
To isolate the size based hindrance and exclusion mechanisms of the membrane, which can further be related to the structural properties  of the PEM, we measured the filtration performance for PEG solutions with a range of molecular weights at three different TMPs.For reasons of clarity, in the following, we only show a selected set of results that suffices to demonstrate the qualitatively observed behavior of neutral solute retention.An overview of the observed retention for the full set of measured PEG can be found in the Supporting Information S6.
The dependency of PEG retention on BL number is illustrated in Fig. 4 by means of the PEG 200 Da retention.For a fixed BL number, it is evident that the observed retention increases with permeate flux.This dependency of retention on permeate flux is typical for dense membranes (RO and NF) with significant solute transport via diffusion [33].The PEG retention as a function of BL number also indicates the two regimes of PEM membrane fabrication.In the ''pore dominated'' regime, both flux and uncharged solute retention change strongly with BL number.Due to an increase in hydraulic resistance, as already seen in Fig. 2, the permeate flux decreases.At the same time, the closing of the support membrane pores leads to increased steric exclusion of solutes and by that higher observed retention.When reaching the ''layer dominated'' regime the retention of PEG stagnates and the flux slightly decreases with BL number.From observed PEG retention measurements, however, it is difficult to pinpoint the transition to the ''layer dominated'' regime.
To compare the two PEM systems directly and independent of support membrane properties, we examine the PEG retention in the ''layer dominated'' regime, here displayed in Fig. 5.The significant difference in hydraulic resistance leads to lower permeate fluxes for the PAH/PAA compared to the PDADMAC/PSS system (one order of magnitude).At the same time, it can be seen that the retention of PEG ranging from EG to PEG 200 Da is much higher for the PAH/PAA system.These observations suggest a denser structure, here related to mesh/pore size, of PAH/PAA compared to PDADMAC/PSS PEMs coated under these conditions.Both systems reach close to full retention of PEG starting from a molecular weight of 400 Da.This indicates a MWCO below 400 Da for both systems (with MWCO PAH/PAA < MWCO PDADMAC/PSS ).Lastly, it can also be noticed that the retention of PEGs with molecular weight higher than 600 Da do not reach full retention for the PAH/PAA system, which could be falsely interpreted as imperfections of the PEM system.As will be shown later, the amount of imperfections (in the ''layer dominated'' regime) for both PEM systems is very similar and does likely result from leakage related to the membrane module preparation.The effect of imperfections in the PAH/PAA modules is however amplified due to the much lower permeability.
The experimentally observed retention of PEG, similar to the MgSO 4 retention, is a filtration process parameter.This means, that there are several factors that in addition to the intrinsic membrane properties influence the measured retention.Among these, as already seen, are permeate flux and solute size.One way to allow direct comparison would be to keep all process parameters identical for both PEM systems.Alternatively, one can reduce the retention observed at multiple fluxes and solute sizes to an intrinsic membrane property by means of a transport model.There are several potential benefits in the introduction of an intrinsic film property: the model can be applied to predict the retention of uncharged molecules with similar solute-membrane interactions as PEG within the range of process conditions considered (decreasing accuracy for extrapolation), the intrinsic property simplifies direct comparison of PEMs (or other separation films) and the intrinsic property can be directly translated to structural properties of the PEM being studied.Therefore, we fitted the observed retention of PEGs simultaneously at all permeate fluxes and solute sizes for each BL and PEM system using a pore model.The results using different intrinsic membrane properties will be shown in the following sections.

Effective pore radius
The first, most basic, theoretical representation of the membrane structure assumes parallel cylindrical pores with identical sizes (length and radius).This results in a set of two fitting parameters: effective pore radius and effective membrane thickness.Here, the term ''effective'' is not only used to emphasize the theoretical nature of these quantities (due to the breakdown of continuum theory below 1 nm [53], these quantities at most indicate the order of magnitude and allow qualitative comparison), but also since the effective membrane thickness is a combination of pore length, membrane porosity and membrane tortuosity.
Fitting the model to the PEG retention measurements and pure water permeability results in an effective pore radius and thickness for each membrane, which is displayed in Fig. 6.To check for the potential influence of PEG on the membrane properties we compare the permeability in both measurements.We find no significant variation in permeability for 10 BLs of PDADMAC/PSS (pure water permeability of 8.3 ± 0.2 LMHbar −1 , permeability during PEG filtration corrected for osmotic pressure during PEG filtration 8.4 ± 0.1 LMHbar −1 ) and for 14 BLs of PAH/PAA (pure water permeability of 0.57 ± 0.03 LMHbar −1 , permeability during PEG filtration corrected for osmotic pressure during PEG filtration 0.58 ± 0.05 LMHbar −1 ).The dashed lines in the figure highlight the ''layer dominated'' regime as determined in the previous sections.From the effective pore radius, it can be seen, that in the initial BL the closing of the pores leads to a sharp decrease in pore radius (here the pore radius of the support membrane with a MWCO of 10 kDa is estimated to be around 2-3 nm).Once, the BL number reaches the point of transition to the ''layer dominated'' regime the effective pore radius becomes constant.The effective membrane thickness does follow a similar trend in the ''pore dominated'' regime.The reduction in effective thickness during pore closing correlates well to the PEM layer becoming the active separation layer (which is thinner than the active separation layer of the support membrane).Once the ''layer dominated'' regime is reached, there is a steady increase in effective membrane thickness for both PEM systems.An increase in effective thickness with BL number is expected from a continuous increase in hydraulic resistance, also observed in other studies [21,25,43,54].One thing to point out, is the higher uncertainty for fitting parameters in the pore dominated regime.We attribute this to both the higher absolute uncertainty at higher permeate fluxes and the high sensitivity of membrane properties to small variations in PEM properties in the region of pore closing.Comparing the two intrinsic membrane properties of the two PEM systems in the ''layer dominated'' regime, two differences that have been indicated in the experimental results become very apparent.First, the effective pore radius, and with that, the mesh size of the PE network of the PAH/PAA system (0.31 nm) is smaller than the one of the PDADMAC/PSS system (0.5 nm).Secondly, the effective membrane thickness, which is proportional to the actual film thickness and the inverse of porosity, is higher and has a steeper increase for the PAH/PAA system.This indicates that the PAH/PAA network has a higher incremental thickness per BL.In order to improve the prediction capability of the transport model and at the same time the (effective) physical representation of the PEM layer, the model is extended to account for distribution in pore radius.

Pore size distribution
For commercial TFC NF membranes often a pore size distribution (following a log-normal distribution) is applied to model the transport of uncharged molecules [28][29][30][31].These types of distributions have also been determined experimentally for the surface pores using atomic force microscopy [28,30,55,56].It is reasonable to assume that the porous network of a PEM can be described by a similar distribution in mesh size.It is known that the internal structure does not consist of fully separated alternating layers, but rather strongly interpenetrating and less ordered layers [23,26,[57][58][59][60][61].Extrinsic charge compensation by ions interrupts the dense network of ionic crosslinks likely causing a larger mesh size with a more heterogeneous distribution [23,43,59].The ionic network is by itself also more dynamic than covalently crosslinked networks and thereby allows for (small) structural changes over time.Using NMR cryoporometry, Chávez et al. [62] observed a distribution in pore size within a PEM layer.Therefore, we extend the theoretical pore model to account for a log-normal distribution in pore radius.
The resulting best fits of the pore size distribution for a selected set of 6 different BL numbers for both PEM systems are shown in   [43,54].Two Polyelectrolyte Multilayer systems: PDADMAC/PSS (left) and PAH/PAA (right).Error bars display the 95% confidence intervals.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Fig. 7.In both systems, similar behavior in the variation of pore size distribution during layer growth can be observed.At low BL numbers (''pore dominated'' regime), the membrane is represented by a quite broad distribution in pore size.This can be explained by the only partially closed pores of the support UF membrane with an estimated pore radius in the range of 2-3 nm.With increasing BL numbers, the pore size distribution narrows and shifts to a lower mean pore radius.Both PEM systems progress towards a constant mean pore radius of 0.44 nm and 0.27 nm (corresponding to the highest relative flux at pore radii of 0.49 nm and 0.29 nm) for the PDADMAC/PSS and PAH/PAA system, respectively.As one would expect for narrow distributions such as these, the obtained mean pore sizes in the ''layer dominated'' regime correspond well with the ones obtained from fitting an average pore radius.The decrease in the mean pore size of the PEM systems is accompanied by a decrease in the spread of pore size distribution.
Regarding the transition to the ''layer dominated'' regime, again the theoretical model description matches very well with previous observations, here indicating pore closure at 6 BL and 9-10 BL for the PDADMAC/PSS and PAH/PAA system, respectively.
In addition to the pore size distribution, an effective thickness was fitted to the PEG retention and pure water permeability measurements.Here, the effective thickness is compared to thicknesses obtained in reflectometry measurements performed on silicon wafers [43,54] (see Fig. 8).After reaching the layer dominated regime a monotonous increase in effective layer thickness can be observed, which again fits nicely with the idea that a continuous thin film of PEM is built on top of the support membrane and is dominating transport.When comparing the effective thickness to the thickness measured via reflectometry a few things have to be considered.
Firstly, during the fabrication of PEMs typically two growth regimes can be observed [27,44,45].In the initial layers at low total thickness, the incremental thickness per layer increases with film thickness.In this regime, the amount of available ionic sites of the previously incorporated polyelectrolyte and with that essentially the buffer capacity of the layer increases with layer thickness, resulting in a seemingly exponential layer growth (thermodynamic limit).After a certain PEM thickness, which is dependent on the PE system and coating conditions, the less mobile polyelectrolyte does not fully compensate for the ionic sites inside the multilayer in one coating step, resulting in a more linear increase in layer thickness with layer number (kinetic limit).This can also be seen in the reflectometry measurements of the two PEM systems.Here, again we assume that the layer growth in the ''layer dominated'' regime of the membranes is linear.Therefore the average incremental thickness of the last 4 BLs from reflectometry measurements is used as a reference.
Secondly, the measured film property in reflectometry concerns the adsorbed polyelectrolyte mass per area.This, however, does not account for the water incorporated in the multilayer and the resulting change in layer thickness due to swelling.To transfer the measured mass density to a thickness, we assume a polyelectrolyte density of 1100 kgm −3 for both PEM systems [63].In addition, to account for the thickness variation caused by swelling we assume a swelling degree of 35% for both systems [64][65][66][67].Lastly, the effective membrane thickness fitted in the model already accounts for membrane porosity (< 1) and tortuosity (> 1), meaning that the actual layer thickness is lower than this value.Therefore, one expects the thickness obtained from reflectometry to be lower than the effective thickness obtained from the model fit.
When comparing the estimated thickness values from both methods, the thickness increment per BL obtained from reflectometry is about one order of magnitude lower for both PEM systems.In qualitative comparison, the thickness obtained for the different systems matches nicely in both characterization methods shown here.The PAH/PAA system forms thicker layers than the PDADMAC/PSS system (also clearly visible in the FESEM images shown in the Supporting Information S7).

Pore size distribution and imperfections
Another extension to the theoretical model concerns non-selective permeate flux (e.g. through imperfections in the PEM layer or module related leakage).In addition to improving the quality of the fit to the PEG retention data, one can obtain information on the pore closing behavior of the two PEM systems (where an open support pore with maximum retention of 6% towards PEG 3000 can be considered as non-selective) as well as the susceptibility of the PEM systems to imperfections.Lastly, considering the relative flux through imperfections one can estimate their influence on the observed retention.
Both the absolute amount of imperfections and the relative impact on the observed retention are displayed in Fig. 9 by means of the permeability through imperfections (to account for applied TMP) and the relative proportion of non-selective flux, respectively.The permeability of imperfections in the layer dominated regime is constantly low and very similar for both PEM systems (around 0.03 LMHbar).Thus, it is very likely the residual non-selective flux is caused by module related leakage and not by imperfections of the PEM system.From the amount of imperfections the transition from the ''pore dominated'' regime to the ''layer dominated'' regime can easily be spotted and fits very well with the previously determined 6 BL for PDADMAC/PSS and 9-10 BL for PAH/PAA.
To evaluate the impact of the non-selective flux on the observed retention one can simply look at the relative amount compared to the overall permeate flux.Due to the relatively high permeability of the PDADMAC/PSS system, the relative amount of non-selective flux is negligible and therefore not expected to affect the observed retention.The permeability of the PAH/PAA system, however, is one order of magnitude lower resulting in a more significant relative amount of non-selective flux (>3%).From the relative non-selective flux one can roughly estimate the maximally obtainable retention of the membrane module for a solute that is fully retained by the PEM layer using Eq. ( 28) (Ret < 97%).This outcome also explains why the PAH/PAA system, in contrast to the PDADMAC/PSS system and despite the smaller effective pore radius, does not reach full retention even for the 3000 Da PEG (see Fig. 5).
Lastly, one can consider the effect of constant intrinsic PEM properties with an increase of BL number in the ''layer dominated'' regime.With constant module related non-selective flux and continuously increasing hydraulic resistance, the relative proportion of non-selective flux will monotonously increase with increasing BL number.This trend is already indicated for the PAH/PAA system since the relative amount is already significant.As a result, the observed retention would continue to decrease with the BL number.The fitted pore size distributions are similar to the ones obtained without considering imperfections and are provided in the Supporting Information S8.Again, the mean pore radius in the ''layer dominated'' regime is 0.44 nm and 0.27 nm for the PDADMAC/PSS and the PAH/PAA system, respectively.However, there is a decrease in the spread of the pore size distribution  for both systems from 98 pm to 96 pm and from 45 pm to 41 pm for PDADMAC/PSS and PAH/PAA, respectively.

Quality of fits
To quantify the benefit of using a pore size distribution to describe the PEM structure and predict PEG retention, we compare the quality of fits for the different theoretical model approaches by means of Mean Squared Error (MSE) (see Fig. 10).Graphs of fits using different model approaches for selected layer numbers can be found in the Supporting Information S9.
Here, again, it is very useful to distinguish the different regions of membrane structure for both PEM systems.Both PEM systems behave qualitatively very similar in the ''pore dominated'' regime.The introduction of a pore size distribution (which is quite broad in this regime) reduces the MSE by one order of magnitude.We attribute this to the influence of pore size distribution of the support membrane, which still influences the transport through the selective layer considerably.
The transition to the ''layer dominated'' regime is also visible in the quality of fit by a reduction in MSE by one order of magnitude.This shows the difficulty to describe membrane transport in the ''pore dominated'' regime as a transition region between the bare support membrane and the PEM based thin film composite membrane.This transition region is characterized by a very heterogeneous membrane structure that is difficult to control and describe.
In the ''layer dominated'' regime both systems show a distinct difference in the quality of fit.The use of a log-normal pore size distribution while accounting for non-selective permeate flux reduces the MSE by one order of magnitude.However, a closer independent look at both structural effects reveals the dominating factors for both systems.
For the PDADMAC/PSS system, the pore size distribution is relatively broad and the relative impact of non-selective flux is almost negligible.Hence, accounting for imperfections only marginally reduces the MSE while accounting for the spread in pore size significantly reduces the MSE to (almost) the same value when accounting for both effects at the same time.
For the PAH/PAA system, the pore size distribution is relatively narrow and the relative impact of non-selective flux is considerable.Due to the narrow distribution in pore size, the fit quality does not improve as much compared to PDADMAC/PSS when adding a pore size distribution.The high relative impact of imperfections, however, results in a significant reduction of MSE when accounting for imperfections.When accounting for both effects at the same time, the MSE is again reduced by one order of magnitude compared to the single pore size model.A table with optimization parameters for the two selected systems, 10 BLs PDADMAC/PSS and 14 BLs PAH/PAA, for all models is shown in the Supporting Information S10.
It should be mentioned that the chosen correlation to define the size of PEG molecules depending on their molecular weight influences the effective structural representation of the PEMs to some extent.More details can be found in the Supporting Information S11.

Implications for the Polyelectrolyte Multilayer structure
When combining the results of the characterization techniques (both experimental and theoretical) applied in this study, we can deduce and compare the structural properties of the two PEM systems.
By fitting theoretical transport models with different effective structural PEM properties to PEG retention measurements, we can qualitatively compare the mesh size of the ionic crosslinking network of the PEM.Here, since we are interested in the intrinsic properties of the PEMs, it is important to consider membrane performance in the ''layer dominated'' regime.The higher retention of uncharged molecules, as well as, the higher hydraulic resistance of PAH/PAA over PDADMAC/PSS has also been observed in previous studies [32,68].PDADMAC/PSS and PAH/PAA have a mean pore radius of 0.44 nm and 0.27 nm, respectively.Here, the order of magnitude matches well to the range of average pore size estimated based on uncharged solute transport in previous studies [39,69].The difference between the two systems is in line with a previous study by Krasemann et al. [70], where it was shown that the charge density of the PEM system affects the mesh size of its ionic crosslinking network.The charge densities (defined for the repeating unit of the polyelectrolyte as the number of ion pairs/number of carbon atoms) of PDADMAC/PSS and PAH/PAA are 0.0625 and 0.167, respectively.Thus, the mesh size and with that the effective pore radius of PAH/PAA is expected to be smaller.
As the theoretical depiction of the PEM film (even when accounting for distribution in pore size) is still a simplification and the order of magnitude in pore size reaches the limit of continuum theory, one might wonder about the reliability and meaning of obtained effective pore radii.Therefore, we compared the obtained results on effective pore size distribution to results obtained by Chávez and Schönhoff [62], where an alternative approach was applied to determine pore size distributions in PEMs.Here, nuclear magnetic resonance cryoporometry was applied on PAH/PSS based PEMs coated on top of silica particles.Mean pore radii in the range of 0.6-0.75nm were measured.Even though a different PEM system was studied the order of magnitude in pore size is comparable to the ones determined in this study.Therefore, we are convinced that indirectly determining pore radii via uncharged solute retention to the least gives a good estimate of the order of magnitude of PEM mesh size.
Another very relevant property regarding mesh size of PEMs is its dependency on BL number, as this will have crucial consequences on membrane design.In the study of Chávez and Schönhoff [62] the spread in pore size distribution did not vary with BL number, although the mean pore radius of the distribution slightly increased with BL number.This was attributed to the fact that, frequently, outer layers are less densely packed in PEMs.In our study, the effective pore size distribution in the ''layer dominated'' regime did not vary with BL number (within the range of uncertainties).These observations suggest there is an optimum BL number in membrane fabrication for uncharged solute retention, as after reaching the layer dominated regime additional BLs only increase the hydraulic resistance while not changing (or even increasing) the effective pore size.
In addition to pore radius, the fit of effective membrane thickness can give relevant information on the PEM systems.There are inherent limitations to directly comparing the effective thickness to the actual PEM thickness.One reason for this is that the effective thickness also includes tortuosity and porosity of the membrane.To unambiguously determine these three quantities, additional information is required.
Despite these inherent limitations, we have shown by the use of supplementary techniques (such as Reflectometry or FESEM) that the effective membrane thickness gives a very good qualitative indication of the PEM thickness.
When comparing the effective thickness increment per BL number in the ''layer dominated'' regime, a striking difference between the two PEM systems was observed.Here, PAH/PAA (689 nm/BL) has a more than ten times higher value in effective thickness than PDADMAC/PSS (44 nm/BL).Although one can certainly expect differences in tortuosity and porosity between those systems, this clearly indicates that PAH/PAA forms thicker layers under the selected coating conditions.This qualitative difference between the two PEM systems has been confirmed by additional measurement techniques.
The significantly different behavior in layer formation between PAH/PAA and PDADMAC/PSS under the chosen coating conditions is caused by the susceptibility of PE towards protonation.Both PAH (pKa 8-9 in solution [71]) and PAA (pKa 6.5 in solution [71]) are weak PEs, meaning their degree of charge is a function of pH.Both PDADMAC and PSS are strong PEs, meaning their degree of charge does not depend on pH in the used range.It has been shown that changes in the degree of charge in PEs can majorly influence the thickness of PEMs formed by those [71][72][73][74][75][76].The group of Rubner has intensively studied the influence of pH (both during and after fabrication) on PEMs made of PAH/PAA [57,71,73,77].It has been shown that the film thickness of PAH/PAA multilayers is very susceptible to solution pH during layer fabrication, where slight changes in pH can lead to a significant increase in layer thickness.In a pH range of 4.5 to 6, both layer thickness and surface roughness were substantially increased [73].This was attributed to conformational changes of the PE chains to a more coiled one.This explains the very high film thicknesses observed in the present study, as the PEMs were coated at pH 5.5.
In contrast, the PEM build of PDADMAC and PSS (both PEs fully charged) forms significantly thinner layers.At the applied salt concentration (50 mM NaCl) the PEs are expected to adsorb in a flat configuration, resulting in compact layers [78].
Considering the just discussed intrinsic properties of both PEM systems, the observed closing behavior of the support pores seems rather contradictory.Even though PAH/PAA clearly forms much thicker PEM films, which in the declared ''layer dominated'' regime has a smaller effective pore size, the transition to the ''layer dominated'' regime is observed at higher BL numbers (9-10 BLs) than for PDADMAC/PSS (6 BLs).
Here, we want to propose a slightly adapted mechanism for closing the pores of the support membrane and forming a dense transport dominating separation layer.The hypothesis is, that these two processes do not necessarily happen simultaneously, but can be consecutive.
Although the PAH/PAA system fills the surface pores of the support membrane more quickly and forms thicker layers (explaining the much higher hydraulic resistance), the membrane properties, and with that, the intrinsic properties of the PEM, are still BL number dependent.Only after 9 BLs do the intrinsic properties of the PEM become independent of BL number.For the PDADMAC/PSS system, this clearly happens at a much lower layer thickness.Therefore, this transition does not depend on PEM thickness alone.One potential explanation for the observed behavior could be an increasing density of the PAH/PAA layer with an increasing BL number (often the outermost layers of a PEM are less densely linked [23]).An indication of a distinct transition to a more dense selective layer is the increase in hydraulic resistance increment as a function of BL number after 9 BLs.An alternative explanation could be that PAH/PAA multilayers coated in this pH regime are more prone to imperfections, which in addition have a higher impact on membrane performance.The relatively high amount of imperfections could be caused by a high surface roughness [73] and with that heterogeneity of the film as well as the relatively high rigidity of the PEM (with at the same time high mobilities of PEs inside the PEM [79]).To clearly identify the cause for the delayed transition to the ''layer dominated'' regime further studies are required.

Conclusion and outlook
In this work, we demonstrate the use of theoretical transport descriptions in combination with filtration measurements of uncharged molecules to be a simple but insightful technique to characterize PEM based NF membranes.The technique allows to properly predict membrane performance and provides indications on the intrinsic structure of PEMs.Thus, it can be used to systematically study the influence of coating conditions on PEM properties.
The two PEM systems studied in this work, PDADMAC/PSS and PAH/PAA, showed very distinctive differences in structural properties.Both systems were represented by a relatively narrow distribution in pore radius in the order of 0.3-0.5 nm.However, the spread in pore radius represented by a log-normal distribution of PAH/PAA (0.04 nm) is smaller than of PDADMAC/PSS (0.1 nm).Accordingly, PAH/PAA clearly formed denser (0.27 nm) and thicker (689 nm/BL) films in comparison to PDADMAC/PSS (0.44 nm, 44 nm/BL).These differences match very well with the previous studies on these PEM systems by other groups as well as with our FESEM measurements.
Another useful feature of this technique is, that it reveals the transition of a support dominated membrane (UF) to a PEM layer dominated membrane (NF) with an increasing number of coating layers.Here, unexpectedly the PDADMAC/PSS system transitions at a smaller number of bilayers.This could indicate a fundamentally different mechanism of separation layer formation for both PEM systems.However, more studies are required to identify the exact mechanisms.
It was also shown, that (within the range of BL numbers studied) the intrinsic network structure of the PEMs is independent of the BL number.This has important implications for the fabrication of NF membranes.With an increasing layer thickness at a constant effective pore size, the energy efficiency of the membrane decreases.Thus, we propose there is an optimum separation layer thickness for the retention of uncharged molecules.
Finally, the observations of this study again underline the potential of PEMs as selective layers for NF membranes.Compared to the dominantly applied polyamide-based thin film composite membranes, similar ranges in pore size can be obtained with superior control of film thickness and structure.Especially promising is the use of asymmetric PEM films for NF applications, to avoid the considerable penalty in hydraulic resistance that is inherent for dense PEM systems during the closure of the underlying support pores to form a selective layer.

Declaration of competing interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Dr. J de Grooth currently holds a part-time position at NX Filtration B.V., a membrane manufacturer.The other authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 2 .Fig. 3 .
Fig. 2. Pure water permeability [LMH/bar] (left -axis, blue) and hydraulic resistance [1/m] (right -axis, red) as a function of bilayer number.Two Polyelectrolyte Multilayer systems: PDADMAC/PSS (left) and PAH/PAA (right).Mind the scale for the hydraulic resistance.Error bars display the 95% confidence intervals.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6 .
Fig. 6.Fitted effective pore radius [nm] (left -axis, blue) and effective membrane thickness [μm] (right -axis, red) as a function of bilayer number [-].Two Polyelectrolyte Multilayer systems: PDADMAC/PSS (left) and PAH/PAA (right).Dashed line highlights ''layer dominated'' regime.Error bars display the 95% confidence intervals.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7 .
Fig. 7. Log-normal pore size distribution fit.Relative flux [-] as a function of pore radius [nm].Two Polyelectrolyte Multilayer systems: PDADMAC/PSS (left) and PAH/PAA (right).Discretization size of 0.01 nm.Discrete pore size distributions fitted using a spline function to guide the eye.

Fig. 8 .
Fig. 8. Log-normal pore size distribution fit.Effective thickness [μm] (left -axis, blue) and thickness measured with Reflectometry [nm] (right -axis, red) as a function of bilayer number [-].Reflectometry data from [43,54].Two Polyelectrolyte Multilayer systems: PDADMAC/PSS (left) and PAH/PAA (right).Error bars display the 95% confidence intervals.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9 .
Fig. 9. Log-normal pore size distribution + imperfections fit.Absolute non-selective flux through imperfections by means of permeability [LMH/bar] (left) and relative non-selective permeate flux [%] (right) as a function of bilayer number [-].Two Polyelectrolyte Multilayer systems: PDADMAC/PSS (red) and PAH/PAA (blue).Error bars display the 95% confidence intervals.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)