3.1. The Cation Rejection and Permeability Behavior of the Membrane
Rejection of total salts, shown as average E
C rejection values (
Figure 3), in the dependence of nanofiltration pores fineness, i.e., MWCO differentness, shows mutual electrolyte retention levels of the three investigated membrane configurations.
Figure 3 shows changes of average investigated cation rejection at different MWCO conditions.
Correlation of total ions rejection and membrane effective pore radii (
Figure 4) showed that a very small range in membrane MWCO values of 50 Da only, with regard to membrane types used, derived permeates of versatile chemical composition.
In order to define mutual coherence of causal physical units, i.e., membrane pressure and surface area, as well as feed and permeate flow rates, with consequent chemical composition and molar concentration of permeate ionic solutes, the membrane ion rejection affinity dimension -MIRA in bar/% was established as shown in Equation (15):
MIRA is normalized TMP with regard to permeate solute moiety and represents the pressure value necessary for a solute rejection change of 1%. Bringing MIRA data and permeate flux utilization contributes to the estimation of the nature of the processes at the membrane surface and in the membrane pores. As the MIRA is lower, the affinity toward to cation removal is higher, from which it follows that more energy (pressure) is needed to permeate the ion.
The plot of MIRA vs. flux efficiency results in a straight line and can be presented as follows in Equation (16):
The plots of MIRA vs. FE for all investigated cations are presented in
Figure 5. Slopes (k), intercepts (n), and linear regression measure of strength of the association (r
2) of all obtained plots, as well as average relative permeabilities, are presented in
Table 5.
Extremely low TMP per rejection percent was found for all investigated kosmotropes and depends mostly on the membrane effective pore size, cation charge densities, and hydrated radii.
Flux and TMP increase had an influence on MIRA enlargement for all investigated cations, but in a different order with regard to membrane effective pore radii. Cations with higher charge density (Mg
2+(aq) and Fe
2+(aq)) were preferentially rejected in the order of membrane r
E as: 0.34 nm < 0.36 nm < 0.33 nm (
Figure 5b,d). The MIRA increase for cations with lower CD–Ca
2+(aq) and Mn
2+(aq) (
Figure 5a,c) was observed in the r
E sequence as follows: 0.34 nm < 0.33 nm < 0.36 nm.
Membrane rejection affinity to monovalent sodium was high for r
E of 0.33 nm and 0.34 nm and to ammonium ion in 0.33 nm experiments and depended primarily on charge density. Pore effective radii of 0.36 nm for Na
+(aq), and 0.34 nm for NH
4+(aq) showed a dominant influence of charge density on MIRA. The extremely high MIRA values for sodium ion at membrane MWCO of 250 Da (
Figure 5e) and for ammonium ion at membrane MWCO of 217 Da (
Figure 5f) indicated a distinct permeation process of these cations. Na
+(aq), and NH
4+(aq) ions manifested typical chaotrope behavior through a very low charge density and small hydrated radius. FE increase contributed to monovalent ion MIRA values declination for specific membrane MWCO that was opposite to the divalent cations MIRA trend. This phenomenon probably argues that monovalent ions steric hindrance appears as a consequence of the similarity of hydrated radii to effective membrane pore radii.
Linear regression measure of strength of the association of all obtained plots was above 0.9 value, which indicates a strong MIRA vs. FE linear relationship. Strong linear relationship was evident due to the way that experimental points were set up. Namely, FE values are influenced by the J
p values and MIRA values are proportional to the TMP. Adjustment of concentrate and recirculate flow rates at each experimental point was conducted which contributed to the different feed water pressure thus affecting TMP. Feed water and permeate fluxes were also influenced by the changes at every experimental point. Obtained data in this paper (
Figure 5 and
Table 5) comply with linear correlation of pressure vs. flux parameters in other experiments using nanofiltration membranes [
51,
52].
Obtained k and n values depend primarily on membrane system MWCO, cation hydration potential, charge density, and hydration radius. This was indicated by the standard deviation (SD) values which suggested the dispersion of a k (SDk) and n (SDn) set of values (
Table 6).
Slope k (
Table 5), defined as ion rejection coefficient IRC, represents the rejection capability of a membrane and indicates the prevailed nanofiltration partitioning mechanism near the membrane surface. Positive values of the IRC indicate satisfactory rejection efficiency of the membrane process, and its negative values show very low rejection affinity and high permeability of the membranes for the individual solutes. Negatively signed IRC attributed to negative membrane rejection behavior for some cations (
Figure 5e,f).
Within the 217 Da series, low divalent ions SDk indicated high rejection of Ca2+(aq), Mg2+(aq), Fe2+(aq), and Mn2+(aq). IRC value for Na+(aq) had a positive sign, but was lower than IRC for divalent ions, showing significantly lower rejection. NH4+(aq) ions dominantly permeated through the membrane which was demonstrated by a negative IRC value.
The 200 Da MWCO experimental series provided positive IRC values for all investigated ions with very low SDk and high rejection of divalent ions. Na+(aq) and NH4+(aq) ions with higher SDk were majority rejected.
Results obtained in 250 Da series indicated high SDk values for both monovalent and divalent cations. Positive values of IRC for divalent cations indicated dominant rejection of those ions for the difference of monovalent cations with negative signed IRC that were highly permeated.
Therefore, the obtained results consequently mean that IRC and its SDk values indicate the nature of the ion separation mechanism at the membrane surface. As SDk values are lower, the predominant separation process is a size-based exclusion at the pore opening, i.e., repelling of ions is valuably proportional to membrane system effective pore radii, i.e., MWCO.
Intercept n value (
Table 5) represents a solute permeability indicator (SPI) that shows an ion transport mechanism through the membrane. Negatively signed SPI is associated with less permeable cations and positively signed SPI describes cations of high permeability within defined MWCO membrane configuration. In addition, it is discernible that lower SDn values indicate weaker cation permeation. Obtained results provided the conclusion that the SPI represents a measure of the relative permeability of a particular membrane to a particular solute. The smallest permeation was registered at the membrane system MWCO of 200 Da. The most probable mechanism of the smaller part of the divalent solvents moving through the narrow capillaries of the membrane was a combination of diffusion and electrostatic convection, while the majority of monovalent cations permeated across the membrane pores by diffusion, convection, and electromigration, including significant steric hindrance. The steric hindrance effect was considered only for the sodium and ammonium as ions with the largest Stokes radii (
Table 4) [
53] and related to their diffusivity in groundwater solution. The dependences of S
F and S
D vs. SPI are presented in
Figure 6.
Enormous enhancement of the steric hindrance factor values for sodium and ammonium ions indicated their high permeation. The SPI values were proportional to S
F (r
2 = 0.9567; r
2 = 0.9812; r
2 = 0.7473) and S
D values (r
2 = 0.9752; r
2 = 0.9925; r
2 = 0.7943) expressed through a strong linear regression, respectively, for NF3-90, NF3-70 and NF90-70-90 series (refer to
Supplementary Data). These high correlations provided good proportion of SPI to q for the series NF3-90, NF3-70, and NF90-70-90 as follows r
2 = 0.8589; r
2 = 0.9273; r
2 = 0.6142, respectively. Therefore, the lowest q, and consequently lowest S
D and S
F for sodium and ammonium ions, indicate the minimal rejection of these cations. The highest regression coefficients obtained for 250 Da MWCO filtration indicated dominant cation permeation for this pore fineness. High coherence between SPI and steric hindrance factors can predict intensity of different cation permeations and different membrane effective radii. The SPI values enable calculation of the most probable average membrane pores effective radius in a very simple way at the base of the hydraulic and concentration experimental parameters cognition. The dominant size exclusion process is in the case of NF3-90 series where the membrane pore effective radius of 0.33 nm was smaller than hydrated radii of all investigated cations.
It was discovered that the ammonium ion was most permeable probably due to temporary rearrangement of the water molecules in the hydration shells near the membrane surface. This steric transformation is a consequence of a small, hydrated radius and affinity of narrow hydrophilic membrane pores to permeate ammonium ion. Similar to ammonium ions, sodium-hydrated ions expressed excellent permeability, electrostatically promoted by negatively charged membrane surface.
3.2. The Membrane Separation Process Thermodynamics
Setting out from the proposals of nonequilibrium thermodynamics which were originally defined by Onsager [
54], the authors of this paper assumed steady state during the cations partitioning at the membrane surface. A state of equilibrium of solute ion molar concentrations between the feed and the permeate stream can occur. Constant TMP, J
f, J
p, C
f, and temperature during the 30-min lasting time at every investigated point ensured constant permeate solute molar concentration, neglecting concentration polarization. This assumption is very plausible when the experimental conditions are perceived as aged isolated systems with sufficient duration to secure thermodynamic equilibrium [
54]. Sodium and ammonium ions favorably permeated (
Table 5) because of smaller hydrated radii and smaller hydration energies than investigated divalent cations (
Figure 2). This assumption has recently been investigated by Kolev and Freger. Kolev and Freger [
55] showed where dynamic simulation of molecular ion uptake by membranes indicated highly localized ions at charged sites and absence of their free movement in the membrane phase. Particularly, divalent ions make a very strong binding to membrane fixed charges, and their uptake, with regard to binding to fixed charges, was extremely low leading to possible saturation [
56].
High hydration energies and larger hydrated radii of calcium, magnesium, manganese, and iron ions contributed to their repel at the membrane surface, as also found by Richards et al. [
45]. These monovalent and divalent cations’ different behaviors can also be explained through the adsorption equilibrium process onto a porous membrane charged interface. It was found previously that the adsorption is an ascendant mechanism for separation of cations from the water solution [
57]. Energy transferred during phase transformations can be explained by Equation (10). Apparent separation equilibrium constant for all experimental points was higher than one, thus indicating a spontaneous separation process. Effects of the membrane ion rejection affinity to changes of the Gibbs free energy at the membrane active layer are presented in
Figure 7.
The average values of the Gibbs free energy changes are found to be proportional to the cations charge density values, in the following order: NH4+ < Na+ < Ca2+ < Mn2+ < Mg2+ < Fe2+. Linear correlation of the plots of CD vs. average Gibbs free energy changes showed that regression coefficients r2 were 0.7453, 0.8929, and 0.9089 for NF3-90, NF90-70-90, and NF3-70 series, respectively. These highly correlated dependencies indicated that influence of the cations charge density to ΔG0 amounts increased with declination of the membrane effective pore radii. The average Gibbs free energy changes were lower than the HFEs for all investigated cations for all applied membrane configurations. With higher ΔG0, cation rejection was more exothermic and valuably enlarged. Low ΔG0 values near the endothermic zone indicated expressed ion permeation. Besides, when MIRA is lowest, the spontaneity and exothermicity of the cation rejection process is higher. In addition, high permeation of sodium and ammonium ions was indicated by increased MIRA values at very low ΔG0 numbers near zero.