3.2. Estimated Values of Total Volume Membrane Charge Density
The total volume charge densities of the ceramic TiO
2 membrane as a function of pH for all the experimentally investigated solutions of asymmetric salts (namely: Cu(NO
3)
2, Pb(NO
3)
2, Fe(NO
3)
3, Zn(NO
3)
2), and the influence of pH on the retention of heavy metals, are presented in
Figure 4,
Figure 5,
Figure 6 and
Figure 7. For all ions, the trends of the retention curves were the same as the charge density curves in terms of their qualitative manner. All
R = f(pH) and
Xd = f(pH) curves possess the S-shape, with the inflexion in the range of pH between 4.9 and 6.0. In the case of asymmetric salt, Labbez et al. [
62] have already shown that the dependency of the retention as a function of pH is described by the S-shaped curve. The values of retention rates obtained experimentally and by means of the detailed DSPM model were identical, and therefore, in this work, there is no difference in plotting experimental or calculated retentions.
In general, the possible mechanisms for the separation of electrolytes are sieving, electrostatic interactions between the membrane and the ions or between the ions mutually, differences in diffusivity and solubility, or a combination of all those listed [
19,
71]. A high retention for multivalent ions is frequently combined with a moderate retention for monovalent ions. In our study, the pore size of the membrane was large enough to demonstrate that salt retention is only affected by size effects to very little extent. Taking into account the difference between the membrane cut-off (which is equal to 450 Da) and the studied ion radii—which, e.g., for the Pb
2+ ion (the largest of the investigated ions) is equal to 11.9·10
−11 m—the steric effect may not justify the obtained ions retentions. For all experiments, the highest retention was achieved for Cu
2+. At a pH equal to 9, retention reached values above 97%. For the Fe
3+ and Zn
2+ ions, the highest degrees of retention rates was also achieved for a pH = 9, but the values were much lower and equal to 80.3% and 58.8%, respectively. Whereas for the Pb
2+ ion, the highest retention was achieved for pH = 6.9 which was 90.2%. Such values of retention could be related to the differences in diffusivities and electrostatic interactions between ions and membrane. The maximum retention for Cu
2+ may have resulted from the lowest values of diffusion coefficient of all ions and the minimum retention of Zn
2+ from the highest diffusion coefficient of all ions (compare with
Table 3).
For the estimated values of the total volume charge density for each set of ions present in the aqueous solutions, the membrane becomes different in terms of individual charge, or—in other words—the apparent charge densities on and in the membrane are significantly different. That dependence is associated with the nature of the electrolyte in the system and with the specific adsorption on the membrane surface and pore walls. For solutions containing Cu
2+ ions (1st variant in
Table 4), the
Xd varied with pH changes in the range from 37.6 to 890.6 mol/m
3; for the 2nd variant (
Table 4), for solutions containing Fe
3+ ions from −120.9 to −37.0 mol/m
3; for the 3rd variant (
Table 4), containing Zn
2+ ions from −289.4 to −150.9 mol/m
3 and for last variant, which contained Pb
2+ ions, from -245.0 to 105.6 mol/m
3. At first glance, the variation of the
Xd sign is surprising, especially due to the fact that all of the investigated heavy metals were in ion forms. It should also be noted that NO
3− ions were present in all the listed variants. They were present because the investigated cations were introduced into the solution in the form of nitric (V) salt. Moreover, Na
+, OH
- and Cl
−, H
+ ions were present in the aqueous solution, which originate from sodium base and hydrochloric acid, respectively—used for the regulation of pH.
The obtained inflections of the membrane charge density curves for all ions were confirmed; in each system, the minimal value of total volume membrane charge density was in the range of pH 4.5–6.0, which corresponds to the IEP of the studied membrane. For the Cu
2+, Fe
3+, and Pb
2+ ions, the minimal values of
Xd were for pH = 4.6 and for Zn
2+ for pH = 6. The type of membrane material used for the active layer influences the membrane structure, and thereby affects the membrane separation ability, but also has an influence on membrane surface charge, which depends on the material isoelectric point value. The membrane possessed a positive charge during the filtration of separated solutions with pH lower than the IEP value, whereas during the filtration of solutions with a pH higher than IEP, the membrane possessed a negative charge. Therefore, the obtained trend of total volume membrane charge densities is correct. For example, when Cu
2+ ions are present in the system at a pH below IEP, the Cu
2+ ions are repelled and the anions present in the feed solution are bound to the membrane, so that the overall stable charge on the membrane during that separation is negative and the retention level is lower due to the formed negative layer which attracts Cu
2+ cations. In cases when the pH of the feed is higher than that of the IEP, the Cu
2+ ions are attracted, and thus retention increases and the change in overall membrane charge
Xd might reflect the partial surface adsorption of cations. Such behaviour of the membrane at different pH values is explained by the amphoteric behaviour of the TiO
2 active membrane layer reported by Van Gestel et al. [
27], which is schematically visualised in
Figure 8. Unfortunately, the relation between IEP and the inflection point for the obtained curves for all investigated cases—when di(tri)-monovalent salts were studied—does not work properly for mono-monovalent salts. Van Gestel et al. [
27] studied zeta potential measurements as a function of pH for mono-monovalent and mono-divalent salts (Na
2SO
4, CaCl
2). They concluded that mono-monovalent salts can be considered as indifferent electrolytes for the NF membrane, and that the inherent charge is due to the protonation and dissociation of surface hydroxyl groups (IEP = 6). Whereas for mono-divalent salts, that trend was totally different. The sign of the zeta potential is altered with the type of salts and salt concentrations. Those phenomena were explained by the selective adsorption of cations or anions. Depending on the forms of –Ti–OH surface groups, ions are able to form complexes. Increasing values of membrane charge densities may be caused by the selective adsorption and additional ions adsorption; the first stage is complexation and the next is the adsorption of additional ions. Moreover, some ions may be adsorbed by the pore wall and influence the membrane charge, as suggested by Takagi et al. [
72].
It is postulated that the total volume membrane charge density is determined by the sum of the fixed membrane charge density and the number of adsorbed ions. The possible mechanism for the formation of the membrane charge assumes that the ions are partitioned from the bulk solution into the membrane pore under the influence of the Donnan potential. Among the partitioned ions in the membrane pores, either cations or anions are adsorbed selectively by the pore wall. Next, the adsorbed ions are bound on the pore wall and give the electric charge to the membrane. In our opinion, the electric charge given to the membrane, which includes all these phenomena, can be seen as the total volume membrane charge density, as presented in
Figure 8. In view of this, the values of total volume membrane charge density
Xd will always be different depending on the type of solute (electrolyte) which is subjected to the NF process, and hence, on the ion types and the pH values as well. Such dependence was obtained for the investigated solutions. For each studied solution, the
Xd values in pH range varied from 2.0 to 9.0 are different. Therefore, it can be assumed that the mechanism of selective ion adsorption acts according to the membrane sign, which is positive at a low pH and negative at a high pH; cations or anions are adsorbed on the membrane (see
Figure 8), changing the values and, in two cases, the charge of
Xd, which is also visible in
Figure 4,
Figure 5,
Figure 6 and
Figure 7.
Figure 8 shows the possible explanations of the transport of copper ions below and above IEP; however, it also should be considered as a general explanation of ion transport, whether transports of di- or tri-valent ions are studied.
Normally, the membrane became more negative at a higher pH of the feed. It needs to be highlighted that such a trend exists for monovalent salts—for example, NaCl. In this work, asymmetrical salts were considered and the observed trends were similar to those presented by Mazzoni et al. [
73]. Additionally, the
Xd values stated in this work are values of membrane charge density after nanofiltration process stabilization, i.e., in steady state operation. The membrane active layer functional groups (TiO
2) take forms which depend on the pH of the feed solution contacting the membrane surface, therefore obtaining the adequate surface charge. With the advent of such charge, adsorption and charge exclusion occur, leading to stable separation and reaching the estimated
Xd values.
In an acidic environment, metals occur in the form of free ions, and the absence of soluble charged metal hydroxides render the formation of an additional active separation layer on the membrane surface impossible. At low pH values, retentions are always lower than when pH increases. As the pH of the environment increases, so too does the amount of soluble metal hydroxides. Due to electrostatic effect of the separated mixture, i.e., metal hydroxides–membrane interaction, an active filtration layer can form on the membrane surface, and the retention rate increases, for Cu
2+ from 72% (pH = 2) to 97.7% (pH = 9), for Fe
3+ from 70.2% (pH = 2) to 80.3% (pH = 9), for Pb
2+ from 56.8% (pH = 2) to 86.20% (pH = 9), and for Zn
2+ from 36.1% (pH = 2) to 58.5% (pH = 9). The formation of that layer results in an increase in the density of positive charge in the membrane, which causes the cation retention to increase for all of the investigated experimental variants, as is also presented in [
60]; the values of total volume membrane charge density for each variant increase, which is also presented in
Figure 4,
Figure 5,
Figure 6 and
Figure 7.
Divalent ions have an important effect on the surface charge—divalent cation adsorption on the membrane surface reduces its negative charge. On the other hand, when both divalent cations and anions are present in the solution, the effect of the divalent anion is opposite to the effect of the divalent cation [
35]. Therefore, the obtained total volume membrane charge densities can be related to the apparent interactions between ions present in the mixtures. These phenomena can explain the observed different values of
Xd for different ions, because, as mentioned above, for each variant, all ions present in each system were taken into account. For example, for the Cu
2+ variant in the system, ions such as NO
3−, OH
−, Na
+, H
+ were also considered. Therefore, besides Cu
2+ and membrane interactions, all various phenomena associated with those ion–ions interactions (selective absorption, Donnan partitioning) occur, which significantly influence the total volume membrane charge density values which are inherently included. Additionally, changes in the additional ions ratio in the systems influence the pH values of the feeds.
Generally, when the membrane makes contact with the aqueous electrolyte solution, it takes the electric charge according to a few possible mechanisms: functional group dissociation, the adsorption of ions from solution, and the adsorption of polyelectrolytes, ionic surfactant or charged nanoparticles. Such charge has an influence on ions distribution in the solution, in view of the electroneutrality requirements of the separated system [
74]. This charging mechanism can occur on the exterior membrane surface and in the interior pore surface, due to the distribution of ions in the solution to maintain the electroneutrality of the system [
75]. The membrane has the internal and surface charge density. Surface charge may be assigned to constant membrane charge (intrinsic), which is generated when the membrane is soaked in the electrolyte. This is caused in view of the acid/base dissociation or ionization of other functional groups, or ions adsorption on the membrane surface from the solution. Therefore, in this study, the overall membrane charge was considered, which presents the total volume membrane charge density created during the NF separation.
In order to enable a comparison of the obtained data with the literature data, the effective membrane charge density was rearranged to the surface charge density according to Equation (19), with the assumption that membrane surface charge is uniformly distributed on the entire intergranular volume between cylindrical pores [
39]:
where
σ is the surface charge density [C/m
2],
rp is the pore radius [m], and
F is the Faraday constant [C/mol]. The values of the total volume membrane charge densities after conversion to surface charge densities
σ [C/m
2] are presented in
Table 5. These values are in good qualitative agreement with the values presented in [
76].
3.3. Determination of Corellation of the Total Volume Membrane Charge Density
In order to determine the correlation which would provide at least limited re-use of the obtained estimation results of the presented modelling, the correlations of the estimated total volume membrane charge densities were obtained. In the trial-and-error search of the feasible form of a correlation relating
Xd and pH, including the Newton’s and Lagrange’s interpolating polynomial methods, Equation (20) was finally proposed:
where
a,
b,
c,
d,
e,
f are the coefficients, the values of which are presented in
Table 6 as first set of parameters.
The parameters of correlation were regressed with use of the least squares method, and the so-defined objective function reached values between 0.50 for Pb2+ and 611.44 for Zn2+. The presented form of Equation (20) gives the first view of how function Xd = f(pH) might be shaped, and through which values it can progress.
In this study, the measure of model compatibility with empirical data was based on the variance of random component method. The starting point was model residuals. The assessment of the random component variance, the so-called remainder variance, is expressed by Equation (21):
where
Xd,i(pH) is the total volume membrane charge density determined experimentally [mol/m
3],
is the total volume membrane charge density calculated with regression model [mol/m
3],
n is the number of observations, and
m is the number of estimated model parameters.
The root of the remainder variance is the standard deviation of the residues
Se (also known as the estimation standard error). This value indicates the average difference between the observed values of the explanatory variable and theoretical values. As seen in
Figure 9a, the obtained correlations converge well with the computationally obtained results of
Xd. In
Figure 9, the horizontal thin lines mark the range of IEP, whereas the horizontal bold line marks the value of the IEP of the TiO
2 membrane. As mentioned earlier, the obtained inflection of the membrane charge density curves for all ions is confirmed, and in each system the minimal value of total volume membrane charge density was in the range of pH 4.5–6.0. The shapes of the obtained correlation functions are in good agreement, and the inflection points of each ion are generally close to the limits of IEP, except for solutions containing Fe
3+.
After the analysis of the first set of parameters reported in
Table 6, it was proposed to unify the parameters present in the denominator of Equation (19) for divalent cations and perform the parameter optimization. The results of those optimizations are reported as the second set of parameters in
Table 6. Although the second set of parameters exhibit higher values of
Se in comparison to the first set, they are still in good quantitative agreement (see
Figure 9b).