A Simple Method for Porous Structure Characterization of Ultrafiltration Membranes from Permeability Data and Hydrodynamic Models: A Semi-Empirical Approach

Abstract

New approaches to the characterization of porous materials must satisfy principles of green analytical chemistry; in addition, they should be reproducible, versatile, and capable of providing relevant information for specific applications. Membrane characterization techniques often fail to meet some of these requirements. Specifically, hydrodynamic porous-based model methods (HPMMs) enable the simulation and evaluation of membrane properties, as well as the monitoring of changes in the response to controlled and uncontrolled modifications. Nevertheless, HPMMs are limited by the multifactorial relationships between their variables and by the generation of only single-value responses. Here, a semi-empirical approach to the characterization of membrane pore structure is proposed and evaluated using simple experimental measurements from pristine and modified membranes. The model enables the determination of the effective pore radius based on two size descriptors related to porosity and permeability, the construction of pore size distributions, and the estimation of structural parameters, such as the number of pores, pore size, and surface porosity. Furthermore, it allows for the simulation of Darcy-type flow behavior in both linear and nonlinear regimes. The model was evaluated on pristine and poly(vinyl alcohol)-modified poly(ethersulfone) ultrafiltration membranes (60–120 mmolL⁻¹) by diafiltration (100–400 kPa). Results demonstrate the usefulness of the model in characterizing membrane pore structure by using simple, fast, and non-destructive methods, thereby enabling advances in analytical diafiltration for membrane characterization.

1. Introduction

Ultrafiltration membranes (UFMs) represent a versatile technology with numerous applications. They are used in water treatment as pretreatment for other filtration operations (e.g., nanofiltration and reverse osmosis), for the removal and concentration of ions and other solutes in bioprocessing; the purification of drugs, polymers, and other substances; and operations in the food and beverage industry [1,2]. UFMs are manufactured from a variety of materials, including polysulfones, polyimines, cellulose acetate, and polyvinylidene fluoride, among others. Among commercial UFMs, poly(ether sulfone) membranes (PESms) are widely used due to their chemical resistance, mechanical stability, and ease of modification [3]. To design, optimize, and predict their performance over time, it is necessary to understand their porous structure, including their pore size distribution, effective pore size, surface porosity, pore number density, tortuosity, and active layer thickness [4,5,6]. These structural parameters govern critical functional properties, such as hydraulic permeability ($L_m$), solute rejection, fouling resistance, flux decline by effect of pressure ($P$), and mechanical durability; moreover, they provide valuable insights for the scaling of membrane processes to evaluate the impact of membrane modifications and for the development of new membranes [7,8,9,10,11].

The effective pore size, defined in terms of the effective mean pore diameter or effective mean pore radius ($r_p$) is the property most directly linked to the separation capacity of membranes. It is a key parameter for estimating which molecules or particles can be retained or pass through the membrane. In ultrafiltration (UF), membranes are typically characterized by the molecular exclusion cut-off, and in some cases, this parameter is related to the average pore radius, although their permeation properties are more accurately defined by pore size distribution [12]. In general terms, an excessively large pore size reduces solute rejection, while an excessively small pore size restricts flow and can increase fouling. Furthermore, pore size influences the critical blocking pressure and behavior under nonlinear regimes, for example, when cake compression or pore closure occurs at high pressures [13,14,15]. In the design of composite membranes and hierarchical structures, fine control of pore size at the active layer level is crucial to achieving high permeability and adequate rejection [16,17]. In turn, transport models require the assumption of a mean pore size to estimate aspects such as internal diffusion and convection rates in porous channels [18,19,20].

Surface porosity ($\epsilon$) is another structural property of UFMs and is defined as the fraction of porous area relative to the total membrane surface area.

$$\epsilon ={\displaystyle \frac{A_p^{net}}{A_m}}={\displaystyle \frac{V_p^{net}}{V_m}}$$

(1)

where $A_m$ is the membrane area, $A_p^{net}$ is the total surface area of the pores, $V_m$ is the membrane volume, and $V_p^{net}$ is the total volume of the pores. Assuming cylindrical pores,

$$\epsilon ={\displaystyle \frac{n_p\pi r_p^2}{A_m}}$$

(2)

where $n_p$ is the number of pores, $r_p$ is the pore radius, and $\pi =3.1416$. But also,

$$\epsilon =n_p{\left({\displaystyle \frac{r_p}{r_m}}\right)}^2$$

(3)

where $r_m$ is the radius of the membrane, which is assumed to have a disk shape [12]. Note that (3) can be used to correct the order of magnitude of the results, since they should be consistent with $\epsilon$ (scaling correction): $\epsilon ~1\times {10}^{-1}$, $r_m~1\times {10}^{-2}$, and $r_p~1\times {10}^{-9}$ [21]; therefore, it is expected that $n_p~1\times {10}^{-13}$.

Surface porosity defines the density of hydraulic channels available for the passage of fluids, commonly water and dissolved solutes; moreover, according to (1), it can be interpreted as the void space contained within the membrane. For equal pore size, greater surface porosity allows for greater hydraulic flow. Furthermore, when porosity is high, the internal volume available to distribute the flow increases, and the entry velocity in each individual pore is greater, thereby mitigating the polarization effects of fluid local concentration or particle cake formation [22,23]. The number of pores per unit area (${\delta }_p$) (or pore density) is a complementary measure used to compare membrane efficiency. Thus, two membranes can exhibit the same $\epsilon$ but different $n_p$ values, depending on whether the pores are larger but fewer or smaller but more numerous. The balance between ${\delta }_p$ and size is important to optimize flow within each channel and to minimize hydrodynamic losses induced by the local interconnection of adjacent pores. A greater number of well-distributed pores improves the utilization of the membrane’s effective area and reduces entry velocity in pores [18,24]. In addition, $\epsilon$ and ${\delta }_p$ influence the crossflow distribution within the support substrate in composite membranes and the uniformity of the local velocity profile, which is critical to avoiding dead zones (i.e., a membrane’s porous areas through which fluid does not pass; these areas act as non-permeable regions) or low-drag regions in porous-supported membranes [24,25].

The active layer thickness (${∆y}_m$) is the parameter influencing transport through the membrane, as it corresponds to the porous functional layer responsible for separation and flow, that is, the portion of the membrane that exerts actual hydraulic resistance. A lower thickness favors higher flow for the same pore size, reducing hydraulic resistance. However, too low a thickness value can compromise mechanical integrity or fouling resistance [1,2]. In addition, $∆y$ influences how local pressure and flux velocity are distributed within the pore; thus, in long pores, the effects of internal pressure drop (i.e., gradients within the pore channel) can be significant, whereas in shorter pores, the pressure drop can be considered uniform [26,27,28]. In asymmetric membranes, the transition interface between the support and active layers also contributes to the total thickness and can introduce hidden resistance, thereby limiting effective permeability and affecting ion rejection [29,30,31].

On the other hand, tortuosity ($\tau$) is a measure of how indirect, or nonlinear, the path of a water molecule passing through the porous medium is, compared with a straight path defined from one side of the membrane to the other. It is described as the ratio of the internal path, or effective pore path ($y_p$), to the membrane thickness.

$$\tau ={\displaystyle \frac{y_p}{{∆y}_m}}$$

(4)

High tortuosity implies that pores bend, bifurcate, or branch extensively, thereby increasing hydraulic resistance for the same pore size. Even with relatively large pores and good porosity, high tortuosity can significantly restrict flow [32]. Tortuosity also influences solute diffusion within the pore, concentration polarization and the resistance resulting from local diffusion. In membrane transport models, the inclusion of a tortuosity factor is essential to correcting effective diffusion within the pore relative to that in the free solution; moreover, in membranes with a hierarchical structure, the tortuosity of the supporting substrate can be as critical as that of the active layer [32,33,34].

Due to the micro- and nanoporous nature of membranes, characterized by relatively high spatial microheterogeneity, variable pore morphology, small thickness for proper functionality, and the changes they undergo during operation, the characterization of their structural parameters is inherently complex. For example, although techniques such as scanning electron microscopy (SEM), transmission electronic microscopy (TEM) or atomic force microscopy (AFM), and mercury porosimetry are widely used, they provide only information about the membrane in a non-operational state. In addition, although the geometric pore size can be determined by SEM or AFM, this value is unrealistic under operating conditions because permeation properties will be the result of a smaller pore size. Consequently, the information provided by SEM or TEM corresponds more closely to an average maximum pore size limit than to the effective mean pore radius [35,36,37,38].

Typical problems arising during analysis of membrane structural properties are related to their heterogeneity; for example, when point measurements are performed using SEM, TEM, or AFM, there is a risk that the analyzed area does not represent the region of interest under real operating conditions [37,38]. In addition, sample preparation steps such as cutting, drying, impregnation with contrast fluids, and the use of coating scan alter or collapse the original structure of the membrane. Many porous membranes undergo deformation under pressure-based measurement techniques (e.g., porosimetry and capillary pressure), which biases the results toward smaller apparent pore sizes. Some techniques, such as porosimetry, do not distinguish between pores that that contribute to actual flow and inaccessible internal cavities, dead zones, or non-functional regions [24]. When tracers or gases are used, diffusion or permeation may be affected by adsorption, internal diffusion retardation, or wall effects, which are directly related to the nature and chemical composition of membranes. In imaging-based techniques, many pores or their internal connections in the nanometer range are not adequately captured, leading to underestimated results [39,40]. Another aspect that often goes unnoticed is the difference between the standard conditions under which measurements should be performed and the conditions under which the membranes are used. This leads to a lack of information on the effect of in situ variables such as pH, temperature, humidity, ionic strength, and others [13,41]. This means that many results are nominal or referential and therefore do not provide a faithful description of the membranes under the conditions of interest. Different methods for determining various structural parameters of membranes are reported in

Table 1. Methods for determining membrane structural parameters [7,11,13,24,26,27,33,40].

Parameter	Method	Comments
Effective pore size	Mercury porosimetry	It can cover a wide range of pore sizes (from nm to μm). The membrane must be ruptured to perform the analysis; mercury, which is toxic, is used; the method does not distinguish between connected and closed pores; and high pressure may compress the material. It is not directly applicable in situ; therefore, it does not offer any information about the membrane in operation.
	Relative permeability method	This method is non-destructive and applicable to dry or wet membranes. It requires assumptions about pore geometry (cylindrical and straight), makes it difficult to separate parallel effects of pores of different sizes, is affected by gas/liquid adsorption effect, and in liquids, permeation can be in a nonlinear regime due to polarization or saturation.
	Electron microscopy (SEM/TEM) in combination with image analysis	This technique provides direct visual information and allows for the analysis of size distribution, morphology, and irregular shapes. In SEM, membrane collapse or structural change during pretreatment (coating with gold) may occur. Observation is limited to the surface (it does not provide information on pore depth), small pores may not be visible, and image segmentation is subjective. Moreover, it does not provide information on membrane behavior during operation.
	Atomic force microscopy (AFM) by tipping	The dimensions of open pores on the surface can be estimated. However, this technique presents difficulties like those of SEM analysis, and the scan is slow and covers a small area. It does not provide information about membrane behavior during operation.
Surface porosity and number of pores	Mass measurement of accessible pore volume	For a known liquid density, the surface pore volume is obtained. Liquid may not penetrate all pores (due to meniscus effects), and surface retention, weighing error, and inability to distinguish closed pores are possible.
	Microscopy combined with pore counting using image analysis	From SEM micrographs, visible pores in a known area can be counted, and using their diameters, the pore density (number/area) can be calculated. However, this approach is subject to sampling bias and failure to count small pores, does not account for small undetectable pores, neglects depth effects, and represents a 2D projection of a 3D structure. It does not provide information about membrane behavior during operation.
	Flow porosimetry (using gas/liquid permeation)	Assuming that each pore behaves as a capillary, permeation can be estimated from the number of effective pores contributing to the total flow. By knowing the flow rate, the mean pore diameter, and the tortuosity, the number of active pores can be inferred. However, this method requires a geometric model and assumptions of uniformity and can overestimate “ghost” pores that are not actually permeable.
Tortuosity	Diffusion of tracers in the pore	The effective diffusion of a solute through the pore is measured and compared with its diffusion in free solution. However, solute interaction with the pore wall, absorption, local diffusion in boundary layers, internal concentration gradients, and crossflow conditions may alter the diffusion of tracers.
	3D tomography images	This technique allows for the reconstruction of the 3D internal structure of membranes with adequate resolution, the plotting of simulated flow paths (Monte Carlo line method), and the calculation of the average true length relative to geometric length (thickness). However, it offers limited resolution for nanometer-sized pores and insufficient contrast between the polymer phase and void, and it requires long processing times. In addition, artifacts may arise during volume reconstruction and segmentation of 3D data.
	Empirical modeling using hydraulic permeability and porosity data	A porous medium-type relationship must be assumed (e.g., modified Kozeny–Carman equation). This approach uses simplified models that assume uniform, cylindrical pores without confluence, neglects edge effects, and is sensitive to errors in various parameters.
Active layer thickness	Cross-section microscopy (SEM)	The method requires preparing a cross-section of the membrane (e.g., by freeze-fracture and ultramicrotomy), after which the thickness of the active layer is measured from SEM micrographs. However, local damage or deformation may occur during sectioning, and membrane layers may be indistinguishable because of low contrast or inadequate sample alignment.
	Tracer penetration profiling (diffusion labeling)	This approach requires introducing a labeled solute (fluorescent or radiolabeled) on one side and monitoring its penetration depth over a controlled time frame. The maximum useful penetration depth is then taken as an estimate of the active layer thickness. However, inhomogeneous diffusion, drag effects, wall interactions, concentration gradients, and insufficient diffusion time may occur.
	Methods based on acoustic or ellipsometrical responses	For thin membranes, ellipsometry or other optical techniques can be used to measure thin-layer thickness. These methods require optically flat membranes, knowledge of the refractive index, sufficient contrast, and proper calibration, and they are not always suitable for thick porous membranes.

.

Note that membrane characterization using conventional methods such as mercury porosimetry, liquid–liquid displacement porosimetry, and SEM can deviate from several principles of green analytical chemistry. The main advantages of the proposed method are the use of water as the working fluid and the possibility of reusing the membrane (i.e., it is a non-sample-destructive method). This contrasts with techniques such as mercury porosimetry, which uses mercury, a well-known highly toxic element. Other techniques, such as liquid–liquid displacement porosimetry, use solvent mixtures like isobutanol, methanol, isopropanol, chloropentane, pentanol, and octanol to wet the surface, which can cause membrane swelling, distort the results, and even destroy the sample. These solvents have a higher relative toxicity than water, and their difficult recovery increases the environmental footprint of the analysis. In addition, techniques such as SEM have high energy consumption, specifically to maintain vacuum conditions. They also require the vaporization of metals for coating operations, which must be vaporized; these include Au, Pd, and Pt. In both porosimetry and microscopic-based techniques, samples cannot be reused, which increases waste generation. Furthermore, other aspects, such as the environmental costs of manufacturing and maintaining these techniques, should be highlighted.

On the other hand, equations describing permeability ($L_m$), $\epsilon$, and flow behavior, although well established, involve significant oversimplification, usually related to geometric assumptions [42,43,44]. Recently, new approaches based on more complex mathematical methods in conjunction with computational tools have been proposed, and although these are interesting, like high-resolution microscopic techniques such as SEM, TEM, and AFM, although they are useful for investigations, they are often impractical for industrial processes due to their high cost, which is frequently a limiting factor [45,46,47,48]. Therefore, semi-empirical approaches offer a lower-cost alternative that can minimize the effects of oversimplification while providing a more realistic description of the processes, since data are acquired under real operating conditions. Currently, many semi-empirical models have been developed; a recent review describes around 63 different models, developed for a wide range of specific conditions and phenomena [49]. Their complexity varies, and their usefulness depends on the researcher’s interest. They all start from the same basic set of laws or mathematical relationships, generally Darcy’s law, the Hagen–Poiseuille equation, and the inverse relationship between hydraulic permeability and resistance, among others. However, these models mainly focus on describing and predicting membrane flux, fouling, permeability, and resistance. They are, therefore, not usually applied to the structural characterization of membranes, but rather to the analysis of transport, retention, and fouling processes (mass transport coefficient, pore blockage, filter cake formation, flux prediction, and concentration polarization). The restriction on the use of these models for structural membrane characterization is due to the multifactorial relationships among membrane structural variables. In particular, the relationship among $L_m$, $\epsilon$, $r_p$, and $n_p$ is particularly complex due to their strong interdependence.

In this study, a semi-empirical approach to structural membrane characterization based on simple experimental measurements is proposed and evaluated using both pristine and modified membranes.

2. Details on the Modeling of Porous Membrane Structure

As a starting point, the concept of the Hagen–Poiseuille equivalent membrane (HPEM) is introduced. This membrane is flat, with evenly distributed cylindrical pores and a permeability equal to that of the real membrane. Under this premise, every membrane has an equivalent membrane that, although it may differ in shape, has structural characteristics that result in the same permeability. This concept also allows for the definition of a reference point from which comparisons can be made. Although this is not developed further in this document, a reference HPEM could, for example, be defined as one having $L_m=8\mu ∆y=1 \mathrm{s}{\mathrm{m}}^{-1}$ (or $\mathrm{L}{\mathrm{m}}^{-1}\mathrm{s}{\mathrm{P}\mathrm{a}}^{-1}$) at 25 °C.

According to the Young–Laplace equation, the pressure ($P$) necessary to force the passage of fluid through a membrane depends on $r_p$, the surface tension ($\gamma$), and the contact angle ($\theta$) at the liquid–solid interface [24]; for water, this is given by

$$P={\displaystyle \frac{2{\gamma }_w\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_w\right)}{r_p}}$$

(5)

where ${\gamma }_w$ and ${\theta }_w$ are the surface tension and contact angle of water, respectively. Furthermore, since membrane pore sizes are very small, $P$ must exceed a minimum pressure ($P_0$), which depends on the membrane permeability characteristics; here, $P_0$ is defined to be related to a dry membrane prior to hydration. By replacing $r_p$ from (5) into (2), it follows that

$$\epsilon =\left({\displaystyle \frac{\pi n_p}{A_m}}\right){\left({\displaystyle \frac{2{\gamma }_w\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_w\right)}{P_0}}\right)}^2$$

(6)

In addition, the Young–Laplace radius ($r_p^{YL}$) is defined to be the radius related to $L_m$ through $P_0$. Thus,

$$r_p^{YL}={\displaystyle \frac{2{\gamma }_w\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_w\right)}{P_0}}$$

(7)

This definition is useful because it simplifies the equations, as it is a recurring term in many of them; in addition, it is necessary later for model prediction and interpretation. Thus,

$$\epsilon =\left({\displaystyle \frac{\pi n_p}{A_m}}\right){\left(r_p^{YL}\right)}^2$$

(8)

On the other hand, for the HPEM, $L_m$ is given by

$$L_m={\displaystyle \frac{\epsilon r_p^2}{8\mu ∆y}}$$

(9)

By combining (7) and (9), we obtain

$$L_m={\displaystyle \frac{\epsilon }{8\mu ∆y}}{\left({\displaystyle \frac{2{\gamma }_w\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_w\right)}{P_0}}\right)}^2$$

(10)

$$L_m={\displaystyle \frac{\epsilon }{8\mu ∆y}}{\left(r_p^{YL}\right)}^2$$

(11)

where $\mu$ is the viscosity of the fluid. Thus, by experimental determination of $P_0$, $L_m$, and ${\theta }_w$, $\epsilon$ can be easily estimated.

According to Darcy’s law, which is commonly represented by a flux-versus-pressure plot $\left(P\right)$, or $J_v:\left(P\right)$ plot, the volumetric flux ($J_v$) is directly proportional to $P$ at low pressures, that is, $J_v=L_{m}P$ (linear regimen); however, as $P$ increases beyond the critical value of $P$ ($P_c$), linearity is lost (nonlinear regimen), and the flux decreases until it reaches a plateau. Therefore, from Darcy’s formulation, $L_m=\left(d{J}_v/dP\right)$.

Hence, if (10) is generalized for any value of $P$, it is concluded that

$$L_{m}P=J_v={\displaystyle \frac{\epsilon {\left(k_{\theta }^{YL}\right)}^2}{8\mu ∆y}}\left({\displaystyle \frac{1}{P}}\right)$$

(12)

Obviously, the equality does not hold for the same values of $P$, except at $P=1$, because the left side of (12) defines a linear flow behavior (linear regime of the $J_v:\left(P\right)$ plot) and therefore the values of $P\le P_c$, whereas the right side defines a nonlinear behavior (nonlinear regime of the $J_v:\left(P\right)$ plot), corresponding to $P\ge P_c$. However, Equation (12) is shown to explicitly illustrate the deductive reasoning underlying the previous conclusion. Here, $k_{\theta }^{YL}=2{\gamma }_w\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_w\right)$. By re-writing (12), we have

$$J_v=\left\{\begin{array}{cc}{L}_{m}P& if P_0\le P\le P_c\\ {\displaystyle \frac{\epsilon {\left(k_{\theta }^{YL}\right)}^2}{8\mu ∆y}}\left({\displaystyle \frac{1}{P}}\right)& if P\ge P_c\end{array}\right.$$

(13)

From (13), $L_m$ in each regime can be defined: $L_m=\left(d{J}_v/dP\right)$ (linear regime) and $L_m=\left(d{J}_v/d\left(1/P\right)\right)$ (nonlinear regime). By integrating the above, we have

$$J_v=\left\{\begin{array}{cc}{L}_{m}P+J_0& if P_0\le P\le P_c\\ L_m\ln \left({\displaystyle \frac{P}{P_c}}\right)+J_v^c& if P\ge P_c\end{array}\right.$$

(14)

The integration limits used are, for the linear regimen, $J_0\to J_v$ and $P_0\to P$, and for the nonlinear regimen, $J_v^c\to J_v$ and $P_c\to P$. Implicitly $∆y$ was assumed to be constant. The above is a valid approximation when water is used as the working fluid, so there is no concentration polarization or fouling. In addition, it is assumed that membrane compression is absent, which is valid if pressures above $P_c$ are kept relatively low. Thus, (14) is completely consistent with experimental data, and its importance is demonstrated in the Section 4.

On the other hand, $\epsilon$ can be calculated from (11), whereas two expressions are possible for $r_p$: one derived from (9) (Hagen–Poiseuille equation) and another defined by (7) (Young–Laplace equation). By introducing the Hagen–Poiseuille radius ($r_p^{HP}$), we obtain

$$r_p^{HP}={\left({\displaystyle \frac{8\mu ∆y{L}_m}{\epsilon }}\right)}^{1/2}$$

(15)

Also, two expressions are possible to obtain $n_p$. From (8),

$$n_p={\displaystyle \frac{\epsilon A_m}{\pi {\left(r_p^{YL}\right)}^2}}$$

(16)

and by combining (2) and (9),

$$n_p={\displaystyle \frac{8\mu ∆y{A_{m}L}_m}{\pi {\left(r_p^{YL}\right)}^2{\left(r_p^{HP}\right)}^2}}$$

(17)

From (17), it can be seen that $r_p$ is the geometric mean ($r_p^g$) calculated from $r_p^{LY}$ and $r_p^{HP}$. Thus,

$${\left(r_p^g\right)}^2=\sqrt{{\left(r_p^{YL}\right)}^2{\left(r_p^{HP}\right)}^2}$$

(18)

In other words, the conclusion that the geometric mean, rather than any other parameter, should be the descriptor of the mean value of effective pore radius arises directly from the underlying assumptions of the model, rather than from physical or statistical interpretations. By defining two radii instead of one and showing that they converge to the same expression, it is interesting to note that the use of the geometric mean appears as a natural descriptor in the model. Finally, it is concluded that

$$n_p={\displaystyle \frac{8\mu ∆y{A_{m}L}_m}{\pi {\left(r_p^g\right)}^4}}$$

(19)

However, the results indicate that a better description of $n_p$ is obtained using $r_p^{HP}$ derived from (17).

Note that it is implicitly assumed that the flow, although not strictly laminar, occurs in pores that are sufficiently small for inertial effects to be negligible. However, this assumption can be verified by determining the Reynolds number. In UF membranes, pore size is at the nanometer scale; therefore, even at high flow rates, the Reynolds number is less than 1 (in our case, Re ~ 2.3 × 10⁻⁷).

The approach proposed here is based on deriving the result through sequential approximations, which are based on scale criteria and congruency between equations. Further details are provided in the Section 3.

3. Materials and Methods

3.1. Reagents and Equipment

Ultrafiltration membranes of poly(ether sulfone) (PESms) were used for filtration experiments (Millipore, NH, USA; molecular weight cut-off of 10 kDa). These membranes are asymmetric membranes made from a thin layer of PES on a microporous layer of polypropylene. Deionized water (Milli-Q, Germany) was used in hydraulic permeability tests. Pristine membranes were used in all experiments. Poly(vinyl alcohol) (PVA; $30 \mathrm{k}\mathrm{D}\mathrm{a}$; Aldrich, MO, USA) was used as a surface modification agent. Permeability tests were performed using a dead-end UF system including one stirred-cell UF module (Amicon, MA, USA; maximum volume of 50–60 mL), one feed tank with the capacity of 1 L, one pressure source (N₂ was used), one pressure gauge, and one control valve for the flow of water or gas from the reservoir to the filtration cell (see Figure 1).

3.2. Hydraulic Permeability Tests and Membrane Surface Modification by Diafiltration

Hydraulic permeability tests were performed according to the following sequence: (i) The membrane was hydrated for 2–3 h before use. (ii) The membrane was installed in the filtration unit, and water was passed from the reservoir to the filtration cell for removing additives. (iii) Flow’s measurements started from the highest pressure and subsequently decreased to the next pressure (working pressures were 100, 200, 300, and 400 kPa, and the stirring rate was 250 rpm); for this purpose, UF was performed using continuous diafiltration by the washing method (i.e., water is fed from the reservoir to the filtration cell while maintaining a constant volume at all times) [50]. (iv) A mass of water was collected in previously weighed tubes, and the permeate time was measured with a stopwatch. (v) Filtration was stopped, the water inside the cell was removed, and 50 mL of a PVA solution were added (60.0 mmol/L with respect to repeating units; in addition, the polymer concentration was sufficiently low to ensure a diluted regime). (vi) Filtration was restarted in washing mode; thus, precipitation, filtration cake formation, and gelation of polymer were avoided. The purpose of this step was to slightly modify the membrane surface to alter its permeation properties by decreasing $r_p$ and/or $n_p$. (vii) After collecting ~50 mL of permeate, filtration was stopped, the polymer solution was removed from the cell, the system was rinsed with deionized water without disassembly, and steps (i)–(iv) were repeated for each pressure. (viii) The entire sequence from steps (i) to (vii) was repeated until three replicates were completed, and each replicate was performed using a pristine membrane. (ix) Finally, the entire sequence from steps (i) to (viii) was repeated using PVA solutions with concentrations of 80.0, 100.0, and 120.0 mmol/L at a pressure of 400 kPa. This pressure was selected because it lies within the working pressure range of UF equipment, which has a maximum operating pressure of 550 kPa. Furthermore, it is the maximum pressure used in the permeability tests. At higher pressures, deformation of the internal porous structure of the membrane and PVA fouling within the pores may occur. This is reflected in the reduction in surface porosity (see Section 4.2.3 and Section 4.3 and Supplementary Information in Table S1).

3.3. Data Processing

Some parameters were obtained from previous publications and used in modeling calculations as comparative information (see

Table 2. Parameters of PESms.

Parameter	Symbol	Unit	Value	Technique	Ref.
Effective area	$A_m$	${\mathrm{m}}^2$	0.0034	Direct	[45]
Effective pore radius	$r_p$	$\mathrm{n}\mathrm{m}$	$3.15\pm 0.25$	Solute retention test	[45]
Active layer thickness	$∆y$	$\mathrm{m}$	$1.075\times {10}^{-4}$	SEM	[45]
Number of pores	$n_p$	-	$7.55\times {10}^9$	SEM	[45]
Surface mean pore radius	$r_p$	$\mathsf{\mu }\mathrm{m}$	$140.0 \pm 0.31$	SEM	[45]
Contact angle (H2O/PESm)	${\theta }_w$	°	$75.6$	Sessile drop	[46]
Water viscosity (25 °C)	$\mu$	$\mathrm{P}\mathrm{a}\mathrm{s}$	0.891	Database	[51]
Water flux rate (25 °C)	$J_w$	${\mathrm{m}}^3{\mathrm{s}}^{-1}$	$9.578\times {10}^{-8}$	Calculated	-

). In addition, $J_v:\left(P\right)$ plots were obtained for each membrane before and after contact with PVA solutions. A linear correlation was used for pristine membranes, whereas a logarithm correlation was applied to the modified membranes, as described in (14). Thus, the membrane structure was established before modification.

Determination of Membrane Structural Characteristics

The calculation sequence for pristine membranes can be performed in two ways: one, here named “forward scaling”, is based on the use of $P_0$ to find $r_p^{YL}$ (i.e., $P_0\to r_p^{YL}$, followed by the re-scaling method applied to $\epsilon$, $n_p$, and $r_p^{HG}$), and another, here named “backward scaling”, is based on the use of $P_0$ to find $r_p^{YL}$, followed by the re-scaling of $r_p^{YL}$ and then the correction of $P_0$ ($P_0^{\mathrm{*}}$) (i.e., $P_0\to r_p^{YL}$, followed by the re-scaling method applied to $\epsilon$, $n_p$, and $r_p^{HG}$, and finally, $r_p^{YL}\to P_0$).

(A)

Forward-scaling method steps are described below:

• $L_m$ and $J_0$ are obtained from the linear correlation of the $J_v:\left(P\right)$ plot.
• $P_0$ is obtained from $P_0=\left|J_0\right|/L_m$.
• $r_p^{YL}$ is obtained from (7).
• The first approximation of $\epsilon$ is obtained with (10); however, it is necessary to introduce a scaling factor ($s_{\epsilon }$) due to (3). The new equation is given by

  $$\epsilon ={\displaystyle \frac{8v_w{s}_{\epsilon }{L}_m}{{\left(r_p^{YL}\right)}^2}}$$

  (20)

  where $v_w$ is the water flux rate at 25 °C and $s_{\epsilon }={10}^1 \mathrm{o}\mathrm{r} {10}^2$. $s_{\epsilon }$ is required to satisfy that $0<\epsilon <1$.
• The first approximation of $r_p^{HP}$($r_{p,0}^{HP}$) is obtained with (15).
• The first approximation of $n_p$ ($n_p^0$) is obtained with (16) or (17).
• According to (2) (scaling correction), a scaling factor ($s_n$) may be introduced. Thus, $s_n$ can be easily determined as

  $$s_n={1\times 10}^m$$

  (21)

  where $n_p=s_n{n}_p^0$ and $m=12-n$ or $m=13-n$, with $n$ being the exponent obtained for $n_p^0$ ($n=\pm 1$).
• $r_p^{HP}$ is re-calculated by modifying (15) to be

  $$r_p^{HP}={\displaystyle \frac{1}{{\left(r_p^{YL}\right)}^2}}\sqrt{{\displaystyle \frac{8v_w{A_{m}L}_m}{\pi s_n{n}_p^0}}}$$

  (22)
• $s_n$ values satisfying expected conditions are sought. In our case, for PESms with a cut-off of 10 kDa, $r_p$ cannot be lower than 1 nm, but also it should be lower than 8–10 nm. From the data analysis, it was concluded that this condition corresponds to a change in $n$ of $\pm 1$. Note that in item 7, a first value which can be an upper limit ($r_{p,u}^{HP}$) or lower limit ($r_{p,l}^{HP}$) for $r_p^{HP}$ was found.
• $r_p^{HP}$ and $r_p^{YL}$ must satisfy (18), being an implicit characteristic in (9). Thus, geometric $r_p$ or mean surface pore radius ($r_p$) is defined as the geometric mean according to (18).

(B)

Backward-scaling method steps are described below:

• $L_m$ and $J_0$ are obtained from the linear correlation of the $J_v:\left(P\right)$ plot.
• $P_0$ is obtained from $P_0=\left|J_0\right|/L_m$.
• $r_p^{YL}$ is obtained from (7) modified by $s_{YL}$, i.e., $r_p=s_{YL}{r}_p^{YL}$, with $s_{YL}=1\times {10}^{-3}$.
• $\epsilon$ is obtained with (20) but using $s_{\epsilon }=1\times {10}^{-5}$.
• The first approximation of $r_p^{HP}$($r_{p,0}^{HP}$) is obtained with (15).
• The first approximation of $n_p$ ($n_p^0$) is obtained with (16) or (17).
• According to (2) (scaling correction), in (21), $n_p=s_n{n}_p^0$, and $m=12-n$ or $m=13-n$, with $n$ being the exponent obtained for $n_p^0$ ($n=1$).
• $r_p^{HP}$ is re-calculated using (22).
• $s_n$ values satisfying expected conditions are sought. From the data analysis, it was concluded that this condition corresponds to a change in $n$ of $\pm 0.1$.
• According to this procedure, the values of $r_p^{HP}$, $n_p$, and $\epsilon$ are the same as those obtained by forward scaling. But, in this path, $r_p^{YL}=r_p^{HP}=r_p$; however, the relationship between $r_p$ and $P_0$ cannot be adequately predicted. Thus, while forward scaling absorbs this difference, causing $r_p^{YL}\ne r_p^{HP}$, by backward-scaling, this difference is transferred to $P_0$.

3.4. Impact of Experimental Error on Results

To evaluate the impact of experimental errors on the prediction of $P_0$, a dataset obtained from one of the linear models was used as a reference. The use of one linear model ensures that all errors present in the simulated data arise from the simulation itself and are not errors inherited or transmitted from experimental measurements; in addition, it allows the simulation to work in the same scale order as the empirical data. Next, a generating function of random error ($F_{ϵ}$) was used to introduce stochastic perturbations into the variables. Thus, the effect of random error on the results was analyzed by controlling a maximum and minimum threshold (${ϵ}_{max}$ and ${ϵ}_{min}$, respectively). Here, $f_{ϵ}$ is given by

$$F_{ϵ}\left({ϵ}_n,n\right)={\displaystyle \frac{{ϵ}_n}{\left|{ϵ}_n\right|}}\left(\left|{ϵ}_n\right|\left({ϵ}_{max}-{ϵ}_{min}\right)\right)$$

(23)

where ${ϵ}_n$ is the random error resulting from the difference of two consecutive pseudo-random events (${ϵ}_1$ and ${ϵ}_2$, respectively), i.e., ${ϵ}_n={ϵ}_2-{ϵ}_1$, with ${ϵ}_1$ and ${ϵ}_2$ being random numbers ranging between 0.000 and 1.000; therefore, ${ϵ}_n\in \left[-1, 1\right]$. In addition, ${ϵ}_n{\left|{ϵ}_n\right|}^{-1}$, $\left({ϵ}_{max}-{ϵ}_{min}\right)$, and $\left|{ϵ}_n\right|$ determine randomly the sign (i.e., direction), scale, and direction, respectively. Thus, $f_{ϵ}$ generates a random perturbation on a variable with a percentage relative absolute error ranging between ${ϵ}_{max}$ and ${ϵ}_{min}$. Finally, for one $x$ variable, random perturbation ($y$) is given by

$$y=x+\left({\displaystyle \frac{f_{ϵ}}{100}}\right)x$$

(24)

where $f_{ϵ}$ is the random error defined by $F_{ϵ}\left({ϵ}_n,{ϵ}_{max}\right)$.

For the simulation, random errors of the variables under analysis (i.e., $P$, $w$, and $t$) were simulated using (23) and (24), and subsequently, the new values were introduced into the linear model to obtain the magnitude of the error on $L_m$, $J_0$, and $P_0$. By applying (23) and (24), the Excel Microsoft function “= random ()” was used to produce the values of ${ϵ}_1$ and ${ϵ}_2$. The thresholds evaluated were (${ϵ}_{max}\left(\%\right)=1,2 \mathrm{a}\mathrm{n}\mathrm{d} 5$ with respect to $P$ (in kPa), $w$ (in kg), and $t$ (in $cs$) and ${ϵ}_{min}\left(\%\right)=0$ in all cases). Percentage relative absolute deviation (${ϵ}_{rad}$) and mean square deviation (${ϵ}_{msd}$) were used as error indicators; they were obtained by averaging ${ϵ}_{rae}$ and ${ϵ}_{msd}$ over 20 simulation cycles; in addition, the maximum possible error was set as the largest value observed within the observation window (${\overline{ϵ}}_{obs}$ and ${ϵ}_{obs}^{max}$, respectively).

3.5. Pore Distribution Modeling

To simplify the notation, $r_{p,l}^{HP}\to r_{p,1}$, $r_{p,u}^{HP}\to r_{p,2}$, and $\overline{r_p}=\left(r_{p,2}+r_{p,1}\right)/2$. Two distributions were evaluated. In the first case, the normal distribution ($\mathsf{\Lambda }$) is given by

$$\mathsf{\Lambda }={\displaystyle \frac{1}{{\left(2\pi \right)}^{0.5}{\sigma }_r}}exp\left[-{\displaystyle \frac{{(r_p-\overline{r_p})}^2}{2{{\sigma }_r}^2}}\right]$$

(25)

where ${\sigma }_r$ is the standard deviation. For the normal distribution, two approaches were performed: (i) considering only values of $r_p$ to describe the distribution and (ii) considering $r_p$ and $n_p$. In this case, the values of $n_p$ were used to define the weight ($w_n$); thus, $w_{n1}=n_{p,1}/\left(n_{p,1}+n_{p,2}\right)$, and $w_{n2}=n_{p,2}/\left(n_{p,1}+n_{p,2}\right)$.

3.6. Darcy’s Curve Modeling for Linear and Nonlinear Regimes

In (14), the behavior of $J_v$ was shown to be described by two regimes: linear and nonlinear regimes. Thus, the nonlinear flux ($J_v^{nl}$) is described as

$$J_v^{nl}={\varphi }_m\ln \left(P\right)+\left[J_v^c-{\varphi }_m\ln \left(P_c\right)\right]$$

(26)

$$J_v^{nl}={\varphi }_m\ln \left(P\right)+J_{v,obs}^c$$

(27)

with

$${\varphi }_m={\displaystyle \frac{\epsilon {\left(k_{\theta }^{YL}\right)}^2}{8\mu \mathsf{\Delta }y}}$$

(28)

$$J_{v,obs}^c=J_v^c-{\varphi }_m\ln \left(P_c\right)$$

(29)

where ${\varphi }_m$ is the slope and is understood to be a constant, while $J_{v,obs}^c$ is the observed critic volumetric flux obtained for $P_c$.

The procedure to obtain $P_0$, $P_0^{nl}$, and $P_c$ is described below:

• The $J_v:\left(P\right)$ plot is modeled using (14) and the cubic function given by

  $$J_v^{nl}=\sum _{n=0}^{n=3}{m}_n{P}^n$$

  (30)

  where $m_n$ denotes the coefficient with respect to the power $n$. Thus, from (14), we obtain $P_0^{nl}$ at $J_v^{nl}=0$, whereas from (30), we obtain $P_0$ at $J_v^{nl}=0$ using numeric methods.
• From the above analysis, two points in the linear regimen are found and used for linear modeling.
• The linear model is projected onto (14), since it is tangent to it. Thus, $P_c$ is obtained by replacing the respective value of $J_v^{nl}$(${=J}_{v,obs}^c$) in (14).

4. Results and Discussion

4.1. Analysis of Hydraulic Permeability Test of PESms

Examples of three hydraulic permeability tests on pristine membranes are shown in Figure 2. The models describing all results are summarized in

Table 3. Linear models and $P_0$ for all PESms included in this work.

ID	${\mathit{L}}_{\mathit{m}}\times {10}^{-7}\left(\mathbf{s}{\mathbf{m}}^{-1}\right)$	${\mathit{J}}_0\times {10}^{-3}\left(\mathbf{L}{\mathbf{m}}^{-2}{\mathbf{s}}^{-1}\right)$	${\mathit{r}}^2(\mathbf{n}.\mathbf{d}.)$	${\mathit{P}}_0\left(\mathbf{P}\mathbf{a}\right)$
PESm-1	2.71	2.43	0.9965	8964.5
PESm-2	2.59	3.13	0.9968	12,086.3
PESm-3	2.81	1.31	0.9962	4671.0
PESm-4	3.23	1.07	0.9972	3300.1
PESm-5	3.27	2.77	0.9974	8472.3
PESm-6	3.25	3.15	0.9974	9675.2
PESm-7	2.73	1.12	0.9495	4121.3
PESm-8	3.58	1.31	0.9943	3649.0
PESm-9	3.53	1.17	0.9992	3299.0
PESm-10	1.87	1.32	0.9906	7045.9
PESm-11	1.49	2.37	0.9944	15,958.3
PESm-12	1.41	2.12	0.9872	15,055.1
$\overline{x}$	2.71	1.94	0.9910	8024.8
$s$	0.75	0.81	0.0136	4511.8
$CV (\%)$	27.8	41.86	1.3700	56.22

ID (identification); $r^2$ (linear correlation coefficient); $\overline{x}$ (mean value); $s$ (standard deviation); $CV (\%)$ (coefficient of variation); n.d. (no dimension).

by using linear correlation parameters. In general, the behavior of the $J_v:\left(P\right)$ plot is as expected in all cases; that is, the membranes operate in a linear regime described by Darcy’s law. Results evidence that the selection of membranes with alike nature and the same molecular weight cut-off does not warrant the reproducibility of results in terms of $P_0$ because PESms are characterized by an inherent variability in terms of $P_0$, and consequently, for small samples, the impact on reproducibility can be significant.

Results of random error evaluation on quality in the determination of $L_m$ and $P_0$ are shown in

Table 4. Simulation results of random error in determining $L_m$.

Error	${\mathit{ϵ}}_{\mathit{m}\mathit{a}\mathit{x}}(\%)$	$\mathit{P}\left(\mathbf{k}\mathbf{P}\mathbf{a}\right)$		$\mathit{t}\left(\mathbf{s}\right)$		$\mathit{w}\left(\mathbf{k}\mathbf{g}\right)$
		${\overline{\mathit{ϵ}}}_{\mathit{o}\mathit{b}\mathit{s}}$	${\mathit{ϵ}}_{\mathit{o}\mathit{b}\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$	${\overline{\mathit{ϵ}}}_{\mathit{o}\mathit{b}\mathit{s}}$	${\mathit{ϵ}}_{\mathit{o}\mathit{b}\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$	${\overline{\mathit{ϵ}}}_{\mathit{o}\mathit{b}\mathit{s}}$	${\mathit{ϵ}}_{\mathit{o}\mathit{b}\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$
${ϵ}_{rad}(\%)$	5	4.8	8.3	0.07	0.2	0.35	0.85
	2	0.9	2.9	0.04	0.1	0.47	0.97
	1	0.4	1.8	0.02	0.04	0.34	0.81
${ϵ}_{msd} (\mathrm{n}.\mathrm{d}.)$	5	5.4	10.7	37.0	3.4	439.3	19.1
	2	2.8	3.7	57.9	0.9	306.1	20.1
	1	0.6	1.7	33.2	0.90	1917	24.3

n.d. (no dimension).

and

Table 5. Simulation results of random error in determining $P_0$.

Error	${\mathit{ϵ}}_{\mathit{m}\mathit{a}\mathit{x}}(\%)$	$\mathit{P}\left(\mathbf{k}\mathbf{P}\mathbf{a}\right)$		$\mathit{t}\left(\mathbf{s}\right)$		$\mathit{w}\left(\mathbf{k}\mathbf{g}\right)$
		${\overline{\mathit{ϵ}}}_{\mathit{o}\mathit{b}\mathit{s}}$	${\mathit{ϵ}}_{\mathit{o}\mathit{b}\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$	${\overline{\mathit{ϵ}}}_{\mathit{o}\mathit{b}\mathit{s}}$	${\mathit{ϵ}}_{\mathit{o}\mathit{b}\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$	${\overline{\mathit{ϵ}}}_{\mathit{o}\mathit{b}\mathit{s}}$	${\mathit{ϵ}}_{\mathit{o}\mathit{b}\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$
${ϵ}_{rad}(\%)$	5	48.0	93.1	2.1	5.3	20.7	53.5
	2	12.6	27.0	4.4	42.7	15.7	44.1
	1	7.8	18.2	0.45	0.9	11.6	37.2
${ϵ}_{msd} (\mathrm{n}.\mathrm{d}.)$	5	2.6	5.5	108.2	279.1	65.61	3013.2
	2	0.7	2.3	46.2	109.3	2424.8	2638
	1	0.15	1.2	1.53	59.45	1324	96.1

n.d. (no dimension).

. In general, $L_m$ was the most stable parameter in all cases, as for this parameter, the impact of random errors on the measurements of $P$, $t$, and $w$ is minimal within the maximum threshold evaluated. In contrast, the value of $P_0$ showed greater dependence on random errors that may arise during the hydraulic permeability tests (see Table 5). In addition, one example of perturbations on linear models using (23) and (24) are shown in Figure 3. According to the results, pressure is the variable of greatest concern, since the error in $P_0$ exceeded the defined threshold in some simulations. These results highlight the need for special care in pressure measurements, which in practice can be resolved by using manometers with the indicated accuracy. Furthermore, since the $L_m$ values are robust and the $P_0$ values can be obtained from carefully conducted experiments, the differences in the values shown in Table 3 cannot be entirely explained as results of internal variability within the experiments. Therefore, the variability in results is attributed to inherent variability of PESms, including variations in their porosity, pore shape and size, active layer thickness, and tortuosity.

4.2. Structural Characteristics of Pristine PESms

4.2.1. Results of Forward-Scaling Method

The values of $r_p^{YL}$, $r_p^{YL}$, $\epsilon$, $r_{p,0}^{HP}$, $n_p^0$, $n_p$, and $r_p^{HP}$ describing all sequence of steps for the characterization of PESm-1, PESm-2, and PESm-3 are shown in

Table 6. Results of porous structure characterization for three PESms by forward-scaling method.

Parameter			Membrane
			PESm-1	PESm-2	PESm-3
$\mathrm{Young}–\mathrm{Laplace} r_p$	$r_p^{YL}$	$\left(\mathsf{\mu }\mathrm{m}\right)$	3.99	2.96	7.66
Surface porosity	$\epsilon$	(n.d.)	13.0	22.7	36.0
$\mathrm{Scaling} \mathrm{factor} \mathrm{for} \epsilon$	$s_{\epsilon }$	(n.d.)	101	101	101
$\mathrm{Hagen}–\mathrm{Poiseuille} r_p$ (first approx.)	$r_{p,0}^{HP}$	$\left(\mathsf{\mu }\mathrm{m}\right)$	1.26	0.94	0.77
Number of pores (first approx.)	$n_p^0\times {10}^6$	(n.d.)	8.85	27.95	6.74
$\mathrm{Scaling} \mathrm{factor} \mathrm{for} n_p$ for lower limit	$s_n$	(n.d.)	107	106	107
$\mathrm{Lower} \mathrm{limit} \mathrm{of} n_p$	$n_{p,l}\times {10}^{13}$	(n.d.)	8.85	2.79	6.74
$\mathrm{Lower} \mathrm{limit} \mathrm{of} r_p^{HP}$	$r_{p,l}^{HP}$	$\left(\mathrm{n}\mathrm{m}\right)$	1.26	2.96	2.42
$\mathrm{Scaling} \mathrm{factor} \mathrm{for} n_p$ for upper limit	$s_n$	(n.d.)	106	105	106
$\mathrm{Upper} \mathrm{limit} \mathrm{of} n_p$	$n_{p,u}\times {10}^{12}$	(n.d.)	8.85	2.79	6.74
$\mathrm{Upper} \mathrm{limit} \mathrm{of} r_p^{HP}$	$r_{p,u}^{HP}$	$\left(\mathrm{n}\mathrm{m}\right)$	3.99	9.37	7.66
Width of pore size distribution	$∆r_p$	$\left(\mathrm{n}\mathrm{m}\right)$	2.73	6.41	5.24
$\mathrm{Mean} r_p^{HP}$	$\overline{r_p}$	$\left(\mathrm{n}\mathrm{m}\right)$	1.37 ± 1.93	3.21 ± 4.53	2.62 ± 3.71
Surface mean pore radius	$r_p$	$\left(\mathsf{\mu }\mathrm{m}\right)$	98.6	130.1	189.3
$\mathrm{Standard} \mathrm{deviation} \mathrm{of} r_p$	${\sigma }_r$	$\left(\mathrm{n}\mathrm{m}\right)$	±39.1	±51.5	±75.0

n.d. (no dimension).

. To simplify the analysis, results and discussion are shown for one membrane, are then compared with a small set of data, and are then generalized for all sets of PESms used.

For PESm-1, $r_p^{YL}=3.99 \mathsf{\mu }\mathrm{m}$, which is small compared with commercial microfiltration membranes (e.g., ~12.3 μm for polypropylene membranes) and large compared with UF membranes. In addition, for ${ϵ}_{rad}=2\%$, and considering ${ϵ}_{obs}^{max}$, $r_p^{YL}$ ranges between 2.07 and 4.57 μm. From (7) it can be seen that $r_p^{YL}$ is derived directly from the fluid properties and $J_v:\left(P\right)$ plot, suggesting that the membrane is highly hydrophilic, and it is possible that the apparent radius may result from lower resistance to the passing of water. Typically, PESms contain various additives to improve their surface hydrophilicity, with the addition of poly(vinyl pyrrolidone) between 2 and 10% being common [52]; consequently, a decrease in ${\theta }_w$ is related to an increase in $L_m$, and an HPEM with a larger pore radius is modeled. Therefore, $r_p^{YL}$ is an apparent radius directly related to $L_m$ rather than to the real size of the pores. According to this argument, permeability characteristics are transferred to a single value, which corresponds to maximum value of $r_p$; that is, an HPEM with a pore size described by $r_p^{YL}$ will have the experimentally measured $L_m$, from which $P_0$ is derived; however, there may be another HPEM with the same $L_m$ but with a smaller $r_p$, which is possible physically by increasing $n_p$. In this way, the analytical problem consists of reducing the pore size through progressive approximations until the minimum possible value is reached; this must be consistent with both equations and scale requirements.

An illustration of how the model obtains the results is shown in Figure 4. For PESm-1, $\epsilon$ = 13%; therefore, PESm-1 can be considered a relatively dense membrane. The value of $\epsilon$ obtained is reasonable in terms of its magnitude, so it is important to highlight that its value is obtained from fluid properties ($\mu$ and $\gamma$), membrane properties ($∆y$), the balance of adhesive and cohesive forces between water and the membrane material (${\theta }_w$), and the experimental information obtained from the $J_v:\left(P\right)$ plot ($L_m$ and $J_0$). Furthermore, the scale factor is derived directly from (2). The values of $r_p^{YL}$ suggest that it the order $L_m$: PESm-3 > PESm-1 > PESm-2 should be expected, which is consistent with the experimental values obtained. However, although the same order is concluded from $\epsilon$, the relationship between the values in both cases is not linear. That is, $L_m\ne k{r}_p^{YL}$ and $L_m\ne k\epsilon$, with $k$ = constant. This line of analysis is discussed in more detail later. On the other hand, $r_p^{HP}$ is related to $\epsilon$ in terms of total pore volume with respect to membrane volume.

By re-scaling, two values satisfy (20) at the same time and comply with the expected conditions defined by lower and upper limits ($r_{p,1}^{HP}$ and $r_{p,2}^{HP}$ such that $r_{p,1}^{HP}<r_{p,2}^{HP}$); consequently, the solution obtained for the problem of the determination of $r_p^{YL}$ is not a single point but a range, because of all values of $r_p^{HP}$ within $r_{p,1}^{HP}\le r_p^{HP}\le r_{p,2}^{HP}$ are satisfactory. Therefore, in terms of pore size distribution, it is suggested that a membrane’s heterogeneity descriptor is given by

$${\mathsf{\Delta }r}_p=r_{p,2}^{HP}-r_{p,1}^{HP}$$

(31)

where ${\mathsf{\Delta }r}_p$ is the width of the pore size distribution. Thus, from (31), it is concluded that the pore size distributions for PESm-1, PESm-2, and PESm-3 show great differences in their porosities (with a variation coefficient of 39.2%) (see Table 6).

In addition, it is seen that $r_p^{HP}$ is not a good descriptor of $L_m$, contrary to what is observed for $r_p^{YL}$; but also, $r_p^{HP}$ is a good descriptor of $\epsilon$, contrary to what is observed for $r_p^{YL}$. The mean surface pore radius for PESm-1, PESm-2, and PESm-3 is close to previously reported values [45] (see Table 2).

Two $r_p$ values to describe the membrane pore size might seem strange; however, it should be remembered that size and shape descriptors are generally concepts of equivalence, not absolute values, especially in systems characterized by a shape distribution [53,54]. Thus, $r_p^{YL}$ corresponds to the radius that a membrane must have to show a permeability equal to that experimentally measured for the real membrane; consequently, it is a parameter allowing one to describe the maximum surface porosity (here defined to be equal to ${\epsilon }_0$). On the other hand, $r_p^{HP}$ is the radius describing a membrane with the same permeability and porosity estimated for the real membrane but with a different pore distribution, so that the scaling conditions and defined pore size limits are satisfied. Consequently, this method requires prior information to properly calibrate the pore size thresholds.

4.2.2. Results of Backward-Scaling Method

This method is based on applying a scaling factor to $r_p^{YL}$, which in principle breaks the direct association of its value with the experimental data defined by Equation (7). The introduction of $s_{YL}$ triggers an increase in the value of $s_{\epsilon }$ to satisfy the scaling correction deduced from (2). However, regardless of these changes, the relationship between the variables remains the same and the results are the same as those obtained using the forward-scaling method, except for $r_p^{YL}$. This is explained because the values of $n_p$ and $r_p^{HP}$ are derived from $\epsilon$, which is re-scaled in both cases between 0 and 1 and remains numerically identical. Clearly, the scaling factors are the parameters that are changing. For PESm-1, PESm-2, and PESm-3, the respective calculation sequence is summarized in

Table 7. Results of porous structure characterization for three PESms by backward-scaling method.

Parameter			Membrane
			PESm-1	PESm-2	PESm-3
$\mathrm{Young}–\mathrm{Laplace} r_p$	$r_p^{YL}$	$\left(\mathrm{n}\mathrm{m}\right)$	3.99	2.96	7.66
$\mathrm{Scaling} \mathrm{factor} \mathrm{for} r_p^{YL}$	$s_{YL}$	(n.d.)	10−3	10−3	10−3
Surface porosity	$\epsilon$	(n.d.)	0.13	0.23	0.36
$\mathrm{Scaling} \mathrm{factor} \mathrm{for} \epsilon$	$s_{\epsilon }$	(n.d.)	10−5	10−5	10−4
$\mathrm{Hagen}–\mathrm{Poiseuille} r_p$ (first approx.)	$r_{p,0}^{HP}$	$\left(\mathsf{\mu }\mathrm{m}\right)$	1.26	0.94	0.77
Number of pores (first approx.)	$n_p^0\times {10}^{12}$	(n.d.)	8.85	27.95	6.74
$\mathrm{Scaling} \mathrm{factor} \mathrm{for} n_p$ for upper limit	$s_n$	(n.d.)	100	10−1	100
$\mathrm{Upper} \mathrm{limit} \mathrm{of} n_p$	$n_{p,u}\times {10}^{12}$	(n.d.)	8.85	3.07	6.74
$\mathrm{Upper} \mathrm{limit} \mathrm{of} r_p^{HP}$	$r_{p,u}^{HP}$	$\left(\mathrm{n}\mathrm{m}\right)$	3.99	9.37	7.66
$\mathrm{Lower} \mathrm{limit} \mathrm{of} n_p$	$n_{p,l}\times {10}^{12}$	(n.d.)	88.5	2.79	67.4
$\mathrm{Scaling} \mathrm{factor} \mathrm{for} n_p$ for lower limit	$s_n$	(n.d.)	101	100	101
$\mathrm{Lower} \mathrm{limit} \mathrm{of} r_p^{HP}$	$r_{p,l}^{HP}$	$\left(\mathrm{n}\mathrm{m}\right)$	1.26	2.96	2.42

n.d. (no dimension).

.

Using this method, $r_p^{YL}$ is equal to $r_{p,u}^{HP}$, and one characteristic is that this equality holds when $s_n={10}^0$. Thus, when $P_0$ is re-calculated for the full consistency of $r_p^{YL}$ using (7), it is necessary to introduce two intermediate concepts: apparent $P_0$, which is obtained from the $J_v:\left(P\right)$ plot, and effective $P_0$ ($P_0^{*}$). These concepts are necessary to avoid the overmagnification of $P_0$; therefore, $P_0^{*}$ is a scaling correction. Consequently,

$$P_0=P_0^{\mathrm{*}}{s}_{YL}$$

(32)

By comparison of the results in Table 6 and Table 7, it is concluded that two methods converge completely when $s_{YL}$ is introduced into the forward-scaling method. Consequently, the overall effect is that while the forward-scaling method first finds $r_{p,1}^{HP}$ and then $r_{p,2}^{HP}$, the backward-scaling method does the opposite, first estimating $r_{p,2}^{HP}$ and then $r_{p,1}^{HP}$.

4.2.3. Evaluation of Generalization of Modeling: Linear Regime

Model behavior was carried out for a set of 12 PES membranes (see

Table 8. Results of porous structure characterization for PESms (from PESm-1 to PESm-12).

Membrane	$\mathit{\epsilon }$	${\mathit{r}}_{\mathit{p}}^{\mathit{Y}\mathit{L}}$	${\mathit{n}}_{\mathit{p},1}$	${\mathit{r}}_{\mathit{p},1}^{\mathit{H}\mathit{P}}$	${\mathit{n}}_{\mathit{p},2}$	${\mathit{r}}_{\mathit{p},2}^{\mathit{H}\mathit{P}}$	$\overline{{\mathit{r}}_{\mathit{p}}}$	${\mathit{\sigma }}_{\mathit{r}}$
	$(\%)$	$\left(\mathsf{\mu }\mathbf{m}\right)$	$(\mathbf{n}.\mathbf{d}.)$	$\left(\mathbf{n}\mathbf{m}\right)$	$(\mathbf{n}.\mathbf{d}.)$	$\left(\mathbf{n}\mathbf{m}\right)$	$\left(\mathbf{n}\mathbf{m}\right)$	$\left(\mathbf{n}\mathbf{m}\right)$
PESm-1	13.0	3.99	88.5	1.26	8.8	3.99	2.63	1.93
PESm-2	22.7	2.96	27.9	2.96	2.8	9.37	6.16	4.53
PESm-3	36.6	7.66	6.7	2.42	67.4	7.66	5.04	3.71
PESm-4	21.0	10.85	1.9	3.43	19.3	10.85	7.14	5.24
PESm-5	14.0	4.23	8.5	4.23	85.0	13.36	8.79	6.46
PESm-6	18.2	3.70	14.4	3.70	1.4	11.70	7.70	5.66
PESm-7	27.7	8.69	3.9	2.75	39.8	8.69	5.72	4.20
PESm-8	28.5	9.81	3.2	3.10	32.1	9.81	6.46	4.74
PESm-9	23.0	10.85	2.1	3.43	21.1	10.85	7.14	5.25
PESm-10	55.5	5.08	23.3	1.61	23.3	5.08	3.34	2.46
PESm-11	22.6	2.24	48.7	2.24	48.7	7.09	4.67	3.43
PESm-12	19.1	2.38	36.5	2.38	36.5	7.52	4.95	3.64

Surface porosity ($\epsilon )$; number of pores ($n_{p,1}\times {10}^{11}$ and $n_{p,2}\times {10}^{10}$); Young–Laplace radius ($r_p^{YL}$); lower, upper, and mean Hagen–Poiseuille radii ($r_{p,1}^{HP}$, $r_{p,2}^{HP}$, and $\overline{r_p}$, respectively, where $r_{p,2}^{HP}=r_p^{YL}{s}_{YL}$); standard deviation of $\overline{r_p}$ (${\sigma }_r$); surface mean pore radius ($r_p$); n.d. (no dimension).

). In addition, the $J_v:\left(P\right)$ plots for PESm-1 to PESm-12 are shown in Figure 5. The linear modeling for each membrane is shown in

Table 9. Model parameters obtained from (14) and (30) for three PESms modified using 60 mmol/L PVA (MPESm-1, MPESm-2, and MPESm-3): logarithmic model (a) and power model (b).

(a) Logarithmic model: $J_v^{nl}=m_1\mathrm{l}\mathrm{n}\left(P\right)+m_0$
Parameter	unit	MPESm-1	MPESm-2	MPESm-3
$P(\times {10}^3)$	$\mathrm{P}\mathrm{a}$	100–400	100–400	100–400
$m_1(\times {10}^{-3})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	4.5619	4.2498	4.0243
$m_0(\times {10}^{-2})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	−4.7720	−4.4031	−4.1328
$r^2$	(n.d.)	0.9989	0.9981	0.9987
(b) Power model: $J_v^{nl}=m_3{P}^3+m_2{P}^2+m_{1}P+m_0$
Parameter	unit	MPESm-1	MPESm-2	MPESm-3
$m_3(\times {10}^{-19})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2{\mathrm{P}}^3$	2.0517	2.4305	2.1197
$m_2(\times {10}^{-13})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2{\mathrm{P}}^2$	−1.9439	−2.2051	−1.9615
$m_1(\times {10}^{-8})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2\mathrm{P}$	7.5424	7.9157	7.2396
$m_0(\times {10}^{-4})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	−9.8046	−1.0545	−4.8687
$r^2$	(n.d.)	1.0000	1.0000	1.0000

n.d. (no dimension).

. Values of $\epsilon$ ranged from 13.0 to 55.5%, reflecting a wide variability in porosity characteristics. Similar observations in terms of variability among membranes were observed for the other parameters.

Regarding modeling, it showed the same behavior in all samples, which allowed us to define all parameters, and for pore radius, in all cases, we observed two values. Therefore, by direct comparison, it is observed that $r_p^{YL}$ is equal to $r_{p,2}^{HP}$ when $s_{YL}$ is used (i.e., $r_{p,2}^{HP}=s_{YL}{r}_{p,2}^{YL}$).

Thus, from the backward-scaling method, it is concluded that membranes are defined by two pore sizes acting as upper and lower limits ($r_p^{YL}$ and $r_p^{HP}$, respectively). The above reflects several aspects of the model: (i) the differentiation of two pore radii is still necessary regardless of the scaling method used; (ii) the backward-scaling and forward-scaling methods are two different routes to achieve the same results; (iii) since $r_p^{HP}$ corresponds to the lower limit and $r_p^{YL}$ corresponds to the upper limit, the description of $r_p$ as geometric mean between both radii is still necessary, which is consistent with (18); (iv) for the application of this modeling approach, prior information is required to determine the thresholds that delimit the solution set (i.e., information that can be used as calibration is required).

From the ordered values of $r_p^{YL}$, a coherent relationship can be obtained from $\epsilon /n_p$ as a function of $r_p^{YL}$ (see Figure 6A). Thus, the corresponding modeling is given by

$${\displaystyle \frac{\epsilon }{n_p}}=\alpha {\left(r_p^{YL}\right)}^2$$

(33)

where $\alpha$ is a constant. This behavior is predicted by (3); consequently, from (33), it is concluded that ${\left(\epsilon /n_p\right)}^{1/2}$ in function of $r_p$ should be described by a linear model (see Figure 6B).

In addition, the combinations of other variables do not show any trends. The above suggests that $\alpha$ should be analyzed in more detail, and since ${\alpha }^{0.5}$ appears as a result of the overall behavior of all membranes, it is concluded that it must be associated with a common characteristic associated with all of them. By comparison with (3), it is concluded that ${\alpha }_1=\sqrt{10}/r_m$ and ${\alpha }_2=10/r_m$; but also, ${\alpha }_2/{\alpha }_1=\sqrt{10}$. The relationship with $\sqrt{10}$ is a consequence of the scale factor used. By defining an experimental $\epsilon$ to be ${\epsilon }_e$, it is seen that ${\epsilon }_e=\epsilon s_{\epsilon }$, since $s_{\epsilon }=10$, and by combining this with (3), we obtain ${\alpha }_1$. From these relationships the prediction of $r_m$ is completely satisfactory: the calculated $r_m$ from $A_m$ is 0.0328976 m, while the $r_m$ obtained from ${\alpha }_1$ and ${\alpha }_2$ is 0.0328979 m (the sequence of decimals was extended in order to show the difference in the calculation; thus, the error was close to 0.001%).

4.2.4. Simulation of Pore Size Distribution

From the results shown in Table 8, pore size distribution can be simulated in different ways: (i) by assuming a normal distribution, based on the central limit theorem because $n_p$ takes very large values (see examples for five PESms in Figure 7A) and (ii) by assuming a normal distribution but assigning a weight based on $n_{p,1}$ and $n_{p,2}$. In this case, the results show that small pores described by $r_{p,1}^{HP}$ are much higher in number compared with the large pores defined by $r_p^{YL}$; thus, it is seen that the former accounts for approximately 90% (see examples for five PESms in Figure 7B). A comparison of the effect of applying these weights is shown in Figure 7C. Clearly, distribution is shifted toward smaller pore size values.

4.3. Structural Characteristics of Modified PESms

Darcy’s Curve Modeling for Linear and Nonlinear Regimes

Darcy’s curve simulation was based on (14); thus, this modeling allows us to establish the linear regime parameters necessary for characterizing the porous structure using information from the nonlinear regime. Note that in these experiments, because of the modification of membranes with PVA, in all cases, a loss of linearity was observed.

The loss of linearity can be attributed to different factors or a combination thereof. The first factor is the accumulation of solutes on the membrane surface. Note that as the flow rate increases, solute accumulation does not necessarily increase linearly; consequently, the relationship given by (14) is no longer proportional. However, considering that the membrane was modified with PVA, the swelling of the polymer may also play a role in nonlinearity. This phenomenon is related to a second factor. The high hydrophilicity of PVA and its macromolecular nature promotes gel formation; therefore, permeability is no longer constant, since at higher pressures, the density of the gel formed will be greater (i.e., the fouling formed is not rigid). Another aspect that cannot be overlooked is the fact that the linearity predicted by Darcy’s law is fulfilled under laminar flow, which is not necessarily maintained with the increase in pressure and the deposition of PVA both inside and on the surface of the membrane.

Figure 8 shows different stages of the model exemplified for one membrane. Figure 8A shows experimental data (white circles) for MPESm-1 and how these data can be fitted using two different models: an experimental power model defined by (30) and a logarithmic model, derived from (12) and (13) and shown in (14). In all cases, $n=3$ in the power model; however, it might be expected that $n=3$ would change for membranes with a larger pore size (e.g., microfiltration membranes). Figure 8B shows how each model is associated with two parameters: the potential model, at $J_v=0$, allows us to establish $P_0$, while the logarithmic model, at $J_v=0$, allows us to establish $P_0^{nl}$. Figure 8C illustrates the position of $P_c$, which is an important parameter because it allows us to differentiate between the linear and nonlinear regimes; it was seen that $P_c>P_0^{nl}>P_0$. In addition, $P_c$ is implicitly contained in both models, closely related to the maximum and minimum fluxes observed in the linear and nonlinear models, respectively. Finally, Figure 8D shows the complete simulation of Darcy’s curve for that specific membrane (MPESm-1).

To illustrate the consistency of modeling against a wide set of PESms, $J_v:\left(P\right)$ plots were modeled for all modified membranes. Two examples are shown in Figure 9 and Figure 10. Note that these membranes were modified using different concentrations of PVA; therefore, different changes with respect to pristine membranes are expected. From the figures, it can be seen that the length of the linear segment ($\mathsf{\Gamma }$) varies between the different membranes; therefore, it is suggested that this length is a characteristic related to porosity features.

Furthermore, for the same membrane, different relationships can be established between the different parameters, for example,

$${\mathsf{\Gamma }}^2={\left({\displaystyle \frac{P_c-P_0}{∆y}}\right)}^2+{\left(J_{v,obs}^c\right)}^2$$

(34)

$$\tan \left(\theta \right)={\displaystyle \frac{J_{v,obs}^c}{P_c-P_0}}={\displaystyle \frac{J_{v,obs}^{nl}}{P_0^{nl}-P_0}}$$

(35)

where $\theta$ is the angle formed between the linear segment and the pressure axis, and $J_{v,obs}^{nl}$ is the flow on the linear segment corresponding to pressure $P_0^{nl}$. Note that in (34), $∆y$ is introduced to achieve correct dimensional congruency (the above implies that $\mathsf{\Gamma }$ is a type of flow, but also, from the procedure approach, the graph should be re-scaled by replacing $P$ with $P/∆y$.). Model parameters for all modified membranes are summarized in Table 9,

Table 10. Model parameters obtained from (14) and (30) for three PESms modified using 80 mmol/L PVA (MPESm-4, MPESm-5, and MPESm-6): Logarithmic model (a) and power model (b).

(a) Logarithmic model: $J_v^{nl}=m_1\mathrm{l}\mathrm{n}\left(P\right)+m_0$
Parameter	unit	MPESm-4	MPESm-5	MPESm-6
$P(\times {10}^3)$	$\mathrm{P}\mathrm{a}$	100–400	100–400	100–400
$m_1(\times {10}^{-3})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	5.9920	5.4462	5.1940
$m_0(\times {10}^{-2})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	−6.4632	−5.7950	−5.5084
$r^2$	(n.d.)	0.9981	0.9923	0.9995
(b) Power model: $J_v^{nl}=m_3{P}^3+m_2{P}^2+m_{1}P+m_0$
Parameter	unit	MPESm-4	MPESm-5	MPESm-6
$m_3(\times {10}^{-19})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2{\mathrm{P}}^3$	1.8459	3.9528	2.0589
$m_2(\times {10}^{-13})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2{\mathrm{P}}^2$	−2.1245	−3.7015	−2.1102
$m_1(\times {10}^{-8})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2\mathrm{P}$	9.4940	1.2733	8.6308
$m_0(\times {10}^{-4})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	−3.3023	−4.8747	−2.0570
$r^2$	(n.d.)	1.0000	1.0000	1.0000

n.d. (no dimension).

,

Table 11. Model parameters obtained from (14) and (30) for three PESm modified using 100 mmol/L PVA (MPESm-7, MPESm-8, and MPESm-9).

Logarithmic model: $J_v^{nl}=m_1\mathrm{l}\mathrm{n}\left(P\right)+m_0$
Parameter	unit	MPESm-7	MPESm-8	MPESm-9
$P(\times {10}^3)$	$\mathrm{P}\mathrm{a}$	100–400	100–400	100–400
$m_1(\times {10}^{-3})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	2.7484	2.5731	2.3600
$m_0(\times {10}^{-2})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	−2.8724	−2.6556	−2.3870
$r^2$	(n.d.)	0.9945	0.9938	0.9968
Power model: $J_v^{nl}=m_3{P}^3+m_2{P}^2+m_{1}P+m_0$
Parameter	unit	MPESm-7	MPESm-8	MPESm-9
$m_3(\times {10}^{-19})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2{\mathrm{P}}^3$	7.8321	1.2187	6.2268
$m_2(\times {10}^{-13})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2{\mathrm{P}}^2$	−9.7024	−1.2731	−7.7447
$m_1(\times {10}^{-8})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2\mathrm{P}$	4.4580	4.9867	3.6424
$m_0(\times {10}^{-4})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	−7.2466	−8.5269	3.2163
$r^2$	(n.d.)	1.0000	1.0000	1.0000

n.d. (no dimension).

and

Table 12. Model parameters obtained from (14) and (30) for three PESm modified using 120 mmol/L PVA (MPESm-7, MPESm-8, and MPESm-9): Logarithmic model (a) and power model (b).

(a) Logarithmic model: $J_v^{nl}=m_1\mathrm{l}\mathrm{n}\left(P\right)+m_0$
Parameter	unit	MPESm-10	MPESm-11	MPESm-12
$P(\times {10}^3)$	$\mathrm{P}\mathrm{a}$	100–400	100–400	100–400
$m_1(\times {10}^{-3})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	4.0264	4.3364	3.9303
$m_0(\times {10}^{-2})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	−4.2405	−4.6148	−4.1327
$r^2$	(n.d.)	0.9827	0.9838	0.9998
(b) Power model: $J_v^{nl}=m_3{P}^3+m_2{P}^2+m_{1}P+m_0$
Parameter	unit	MPESm-10	MPESm-11	MPESm-12
$m_3(\times {10}^{-19})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2{\mathrm{P}}^3$	3.0831	1.9611	0.7886
$m_2(\times {10}^{-13})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2{\mathrm{P}}^2$	−2.8313	−1.7534	−0.9775
$m_1(\times {10}^{-8})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2\mathrm{P}$	9.4953	6.6204	5.0443
$m_0(\times {10}^{-4})$	$\mathrm{L}/\mathrm{s}{\mathrm{m}}^2$	−3.2024	−1.1414	−2.0558
$r^2$	(n.d.)	1.0000	1.0000	1.0000

n.d. (no dimension).

, in addition, the values of $P_c$ and $J_{v,obs}^c$ are shown for all membranes in

Table 13. Parameters describing the linear regimen for modified PESms. The linear model is described by $J_v=L_{m}P+J_0$ with $P_0\le P\le P_c$.

	Membrane	${\mathit{L}}_{\mathit{m}}$	${\mathit{J}}_0$	${\mathit{P}}_0$	${\mathit{P}}_{\mathit{c}}$	${\mathit{J}}_{\mathit{v},\mathit{o}\mathit{b}\mathit{s}}^{\mathit{c}}$
${\mathit{C}}_{\mathit{P}\mathit{V}\mathit{A}}$		$\left(\mathbf{s}/\mathbf{m}\right)$	$\left(\mathbf{L}/\mathbf{s}{\mathbf{m}}^2\right)$	$\left(\mathbf{P}\mathbf{a}\right)$	$\left(\mathbf{P}\mathbf{a}\right)$	$\left(\mathbf{L}/\mathbf{s}{\mathbf{m}}^2\right)$
$\left(\mathbf{m}\mathbf{m}\mathbf{o}\mathbf{l}/\mathbf{L}\right)$		$\times {10}^{-8}$	$\times {10}^{-2}$	$\times {10}^3$	$\times {10}^3$	$\times {10}^{-3}$
60	MPESm-1	5.685	−7.652	13.459	80.0	3.78
	MPESm-2	6.004	−8.290	13.808	75.0	3.67
	MPESm-3	5.643	−3.866	6.8510	75.0	3.85
80	MPESm-4	7.176	−2.719	37.889	84.0	3.31
	MPESm-5	7.949	−2.755	34.662	70.0	2.81
	MPESm-6	9.404	−3.563	25.368	55.0	1.61
100	MPESm-7	3.660	−6.172	16.866	75.0	2.13
	MPESm-8	4.168	−7.462	17.904	62.0	1.84
	MPESm-9	3.414	−8.273	2.4234	75.0	2.48
120	MPESm-10	12.060	−4.561	37.814	43.0	0.63
	MPESm-11	4.725	−8.547	18.089	89.0	3.35
	MPESm-12	4.093	−1.681	4.1073	98.0	3.84

.

The parameters describing the linear regimen for modified PESms are shown in Table 13. The results show that the $L_m$ and $J_0$ values are one order of magnitude lower than those obtained for the pristine membranes in both cases; that is, $L_m$ decreases from 10⁻⁷ to 10⁻⁸, while for $J_0$, the change is from 10⁻³ to 10⁻⁴ (see Table 3 and Table 13). On the other hand, the $P_0$ values for the pristine membrane are in the range of ~3300 Pa to ~16,000 Pa, while for the modified membranes, $P_0$ is in the range of ~2000 to ~70,000 Pa. This suggests that although phenomena such as pore blockage may have occurred, since $P_0$ increased appreciably in some cases, another phenomenon directed the changes toward an increase in $L_m$. In this regard, the polymer’s hydrophilicity suggests that the membrane became sufficiently hydrophilic in some cases to decrease flow resistance, whereas in others, pore clogging increased resistance. Consequently, both processes can occur (blocking and hydrophilicity). Furthermore, no specific behavior was identified in relation to PVA concentrations.

With the backward-scaling method and $P_0$, the values of $r_p^{YL}$, $\epsilon$, and $n_p$ were calculated according to the procedure previously described in the methodology. However, since there was no predefined criterion for calibrating the threshold and membranes were modified by using PVA, the information used for pristine membranes is not necessarily adequate. Thus, in (21) and (22), $s_n$ was varied by changing $n$ from $-1$ to $3$. As a result of this procedure, a set of values were obtained for each membrane in function of $n$. Because $r_p^{YL}$ defines the maximum pore size related to $L_m$, a first criterion to establish the correct value is that $r_p^{YL}>r_p^{HP}$. On the other hand, since $r_p^{YL}$ satisfies (20), the $L_m$ calculated from (22) and using $r_p^{HP}$ must be consistent with the $L_m$ obtained from the experimental data. Thus, by comparing $L_m$ (Table 13) with the $L_m$ obtained from (202), it was possible to determine $r_p^{HP}$. In addition, from this procedure, it was observed that $L_m$ (in Table 13) < $L_m$ (from $r_p^{HP}$), which is interpreted as a partial contribution of $r_p^{HP}$ to the total value of $L_m$. Results are summarized in

Table 14. Porous structure characterization of MPESms.

PVA	MPESm	$\mathit{\epsilon }$	${\mathit{r}}_{\mathit{p}}^{\mathit{Y}\mathit{L}}$	${\mathit{n}}_{\mathit{p}}^{\mathit{Y}\mathit{L}}$	${\mathit{r}}_{\mathit{p}}^{\mathit{H}\mathit{P}}$	${\mathit{n}}_{\mathit{p}}^{\mathit{H}\mathit{P}}$	$\overline{{\mathit{r}}_{\mathit{p}}}$	$\pm {\mathit{\sigma }}_{\mathit{r}}$
				$\times {10}^{13}$		$\times {10}^{13}$
$\left(\mathbf{m}\mathbf{M}\right)$		$(\mathbf{\%})$	$\left(\mathbf{n}\mathbf{m}\right)$	$(\mathbf{a}.\mathbf{d}.)$	$\left(\mathbf{n}\mathbf{m}\right)$	$(\mathbf{a}.\mathbf{d}.)$	$\left(\mathbf{n}\mathbf{m}\right)$	$\left(\mathbf{n}\mathbf{m}\right)$
60	1	6.0	2.67	0.9	2.26	0.9	2.47	0.29
	2	7.0	2.61	1.1	2.15	1.1	2.38	0.32
	3	16.0	0.53	61.5	0.28	61.5	0.40	0.18
80	4	61.0	0.95	73.1	0.90	73.1	0.93	0.03
	5	57.0	1.04	56.8	0.34	56.8	0.69	0.49
	6	36.0	1.42	19.3	0.64	19.3	1.03	0.55
100	7	6.0	2.13	1.5	1.44	1.5	1.79	0.49
	8	8.0	2.01	2.1	1.28	2.1	1.64	0.52
	9	12.0	1.48	5.8	0.70	5.8	1.09	0.56
120	10	10.0	0.95	12.2	0.91	12.2	0.93	0.03
	11	9.0	1.99	2.5	1.25	2.5	1.62	0.52
	12	41.0	0.88	57.6	0.77	57.6	0.82	0.08

mM (mmol/L); n.d. (no dimension).

, and a comparison between pore size distribution functions of pristine and modified membranes is shown in Figure 11.

For PES membranes, $L_m$ depends on the pore size (described by $r_p$), $n_p$, and $\epsilon$, while for MPESms, in addition to the aforementioned parameters, fouling will also have an influence, which is a function of the PVA concentration during filtration and the relative size of PVA chains with respect to the pore size (the smaller the difference between $r_g$ and $r_p$ is, the greater the fouling expected inside the membrane pores is). Thus, for example, the increase in $L_m$ when the PVA solution concentration is increased from 60 to 80 mmol/L reflects a decrease in $r_p$ but also an increase in $\epsilon$ and $n_p$. However, it should be noted that the variability of the results is highly dependent on the accuracy of measurements during hydraulic permeability tests and the membrane’s inherent variability; therefore, a clearly defined behavior cannot be expected from the entire membrane assembly. One case that reflects the above is MPESm-10 (see Table 14), which has $L_m=12.060\times {10}^{-8}$ s/m, a relatively smaller $r_p$ (0.93 nm), and a high $n_p$ (i.e., high $\epsilon$) compared, for example, with MPESm-11. However, PESm-10 shows the highest $r_p^{YL}$, which is almost double the $r_p^{YL}$ obtained for PESm-11 (see Table 8).

Figure 11A shows that membranes can be distinguished from one another by the mean value, as well as by the intensity and width of the distribution. In contrast, Figure 11B compares two modified membranes with and without taking into consideration the weight (here, weight is given by the relative contribution fraction of the different $r_p$ to $L_m$). The results show that while the difference is not significant for MPESm-9, it is significant for MPESm-1. Furthermore, the rightward shift in the distribution suggests that $L_m$, in both cases, is controlled by a larger pore size fraction. Finally, Figure 11D,E illustrate the comparison of two membranes before and after modification. In both examples, the modification leads to a reduction in $r_p$, with variations different in each case.

5. Remarks and Conclusions

This document presents the development of a model based on the HPEM concept, which is derived from the Young–Laplace equation, Darcy’s law, and the Hagen–Poiseuille permeability equation. The initial strategy supporting the model is not new, and one of the main disadvantages usually mentioned in this type of model is the need to assume geometric aspects to simplify the pore shape and the obtaining of a single-value result, which is usually referred to as an effective value. However, it should be noted that the membrane modeling approach using permeability, porosity, and flow laws allows one to describe properties that cannot be obtained by other techniques, particularly those related to real operating conditions. Furthermore, compared with traditional techniques such as SEM and mercury porosimetry, this analytical approach avoids the use of solvents and reagents (a valuable aspect from a green analytical chemistry perspective), reduces equipment and personnel costs, and enables the study of a membrane without its destruction; but also, the behavior of membranes can be studied under real operation conditions, including hydration, ionic strength, and fluids of different nature characterized by a polarity, viscosity, or surface tension different from that of water. This is particularly valuable in studies about the changes that membrane undergo due to fouling or because of controlled modifications aimed at altering or improving their properties.

On the other hand, while the use of permeability, porosity, and flow laws for modeling porous structures may initially seem unoriginal, the approach developed here is. Firstly, the difficulties inherent in the $\epsilon r_p^2$ product (or $n_p{r}_p^4$, if preferred) result from a natural partitioning of the $r_p$ concept due to two well-established principles. Accordingly, the concepts of $r_p^{YL}$ (derived from the Young–Laplace capillarity equation) and $r_p^{HP}$ (derived from the HP permeability equation) are introduced, and considering the definition of $\epsilon$, the concept of $r_p^g$ is also introduced. Note that the geometric mean appears naturally from the recognition that two radii derived from different physical laws exert different effects on membrane properties. Thus, although these radii can be numerically equivalent, they do not necessarily have to be. In this way, our results show that when $r_p^{YL}$ is consistent with surface porosity, $r_p^{HP}$ is consistent with permeability, and vice versa. On the other hand, the introduction of two radii leads to a specific mathematical procedure that, considering only a single radius, is meaningless. The fact that two radii are defined in (17), instead of only one, allows for the separation of variables to obtain $r_p^{HP}$, while $r_p^{YL}$ is assumed as a known value. If this is not performed, the result is that the magnitude of the radius’s contribution corresponds to the fourth power, leading to drastic changes in the order of magnitude of the variables.

The proposed model is based on the premise that the void space within the membrane, which contributes to its permeability properties, can be molded into a shape without losing its spatial characteristics, since though these characteristics are distorted by the change in shape, the associated total space remains unchanged. This should be understood as follows: changes in the total pore volume can be distributed across a finite number of pores with a maximum size, in a manner consistent with the hydraulic permeability measurements. Thus, once this value is established, the analytical problem consists of finding $n_p$, $r_p$, and $\epsilon$, which are parameters that should be compatible with the membrane’s characteristics. To achieve this, prior information is needed to calibrate the thresholds within which the results will be measured, but also, scaling corrections are needed. This procedure is supported in (2) and information obtained from the literature. Although calibration using thresholds and scaling factors may appear to be a disadvantage, it is a common strategy that allows us to correlate the results with those obtained by other techniques.

One feature of the model is the introduction of scaling factors, which depend on the available information and the results obtained. To visualize the scope of the results appropriately, greater clarity on these scaling factors should be given. Thus, the model proposed here is based on the following premises: the values taken of the model parameters must (i) satisfy a set of interrelated equations, (ii) be consistent with other analytical techniques, and (iii) agree with experimental measurements of hydraulic permeability. Since the calculation equations are not new, the novelty lies in the calculation strategy employed, which, if preferred, can be considered a calculation algorithm based on solving a system of equations using iterative numerical methods. Consequently, the scale factors are themselves iterations of the calculation rather than physical parameters of the model (in other words, they are adjustment variables).

The relationship among the four main parameters ($\epsilon$, $n_p$, $r_p$, and $r_m$) is introduced in (3). In particular, whereas $r_m$ is known from direct measurements, $\epsilon$ is determined from its fractional definition, such that $0<\epsilon <1$. Consequently, if $\epsilon >1$ is obtained from physical equations, then an overestimation of $\epsilon$ is concluded, and the need of one adjustment parameter to achieve a congruent result is evident ($s_{\epsilon }$). Thus, by backward-scaling, the subsequent numerical problem is to find the value of $s_{\epsilon }$. Observing the results, $s_{\epsilon }$ is easily defined. The strategy described here assumes that overestimation does not occur in its numerical value but in the magnitude of its scale; consequently, with $s_{\epsilon }$, only the scale is modified. Essentially, this corresponds to the value by which $s_{\epsilon }$ must be multiplied for the result to be consistent with its fractional definition. However, by backward-scaling, $r_p^{YL}$ is calibrated, or re-scaled if preferred, using known information which can be obtained by any other technique (here, SEM from previous publications was used). Thus, the magnitude of $r_p^{YL}$ changes from micrometer to nanometer. Due to changes in $r_p^{YL}$, $s_{\epsilon }$ changes to 10⁻⁴ or 10⁻⁵ since its values must be consistent with (3) and its fractional definition. The above is explained by the interdependence of variables.

In addition, the model assumes cylindrical pores as a strategy for establishing equivalent porosity. That is, regardless of the pore shape, the membrane’s volumetric porosity remains unchanged. Note that strictly speaking, it is the surface porosity that is affected by the restriction of a cylindrical shape. However, the model does not address the membrane’s internal structure. We believe that tortuosity is a first approximation (with the simplest being the increase in the mean path length relative to the active layer thickness, assuming a cylindrical pore shape). However, the internal structure in terms of pore shape has not yet been incorporated into the model. Preliminary analyses allow us to conclude that the internal structure must be pre-defined in terms of shape categories (gaps formed by fibers, tubular bifurcations, etc.); thus, specific tortuosity models must be constructed for each category.

Finally, as shown in the results, values for $n_p$, $r_p$, and $\epsilon$, as well as $r_p^{YL}$ and $r_p^{HP}$, were consistently established; in addition, it is demonstrated that it is possible to construct simple distribution models that facilitate the easy description of changes occurring in the membrane. This work also developed a strategy for simulating Darcy’s curve beyond the linear regime. This allows for the definition of, among other things, critical values and characteristic shapes (from models and modeling parameters), which make it possible to apply the model even when changes experienced by the membrane cause flow behavior outside the linear range. Likewise, this procedure allowed us to establish the linear range, which is useful for redirecting experiments both at the operational level and in terms of the nature of the changes. In addition, in the field of simulation and computational analysis, the model is easily programmable and adaptable to scenario planning.

Regarding the results of the PESm and MPESm study, it can be concluded that these membranes exhibit significant variability in their porous structure. Therefore, the results obtained from the average values of different membranes, based on a small sample, can have such high variation that the effects of a modification process can be easily masked, or the overall behavior of the results becomes erratic and difficult to reproduce.