3.2. Energy of Adsorption of OH−, HCO3−, and H3O+ Ions on the Air/Water Interface
We used Equations (11) and (12) to calculate the adsorption energies of the OH
−, HCO
3− and H
3O
+ ions on the air/water interface. The ions with bare radius below 1.35 Å are kosmotropic [
43]. Exclusion makes the acetic ion, whose shape is more complex [
43]. The kosmotropic ions undergo stronger image repulsion by the air/water interface than the chaotropic ions. The reason is their stronger electrostatic field is in close proximity, making their hydration shells tougher than the ones of the chaotropic ions [
50]. Therefore, OH
− and H
3O
+ are kosmotropic, while HCO
3− is chaotropic ion.
The adsorption of a given ion on the air/water interface causes two opposite energetic effects:(i) gain of energy
E1 (see
Figure 2) via the adsorption of the ions by overcoming the image repulsion force; this process is endothermic; (ii) loss of energy
E2 (see
Figure 2) due to displacement of an ensembles of surface water molecules by the ions into the bulk of water; this process is exothermic. The adsorption of the ions on the air/water interface always displaces surfaces water molecules into the bulk (see
Figure 2). The total change of energy
Eads from these two processes determines if the adsorption is favorable energetically or not. Such an approach was missing in the literature. The authors traditionally studied the dispersion interaction of the different ions with the air/water interface, thus establishing their segregation at close proximity of the phase boundary [
2,
3,
51,
52,
53,
54,
55,
56,
57,
58,
59,
60]. As far as the image force between the ion and air/water interface is always repulsive, they arrived at the conclusion that all the ions are repelled by the air/water interface but at different degrees—some ions are more repelled than other ions, thus causing their segregation at the very surface. These results were well paired with the experimental increase of the surface tension at large salt concentration. In addition, they were confirmed by the more precise molecular dynamic (MD) simulations of salt solutions close to the air/water interface [
61,
62,
63,
64]. All these literature studies cannot explain the Ray–Jones effect; neither they can explain the negative origin of the air/water interface. The reason is that they studied salt solutions with high concentration (
c > 0.5 mol/L). Our approach is traditional as well but more general. We are not interested in the segregation of the different ions close to the interphase boundary. The calculation of their adsorption energy
Eads despite being either positive or negative is more practical and definite.
To make our approach clearer, we processed Equations (11) and (12), thus obtaining separate expressions for the energies
E1 and
E2, which are shown in
Figure 2. Hence, for kosmotropic ions we obtained the following relations:
For chaotropic ions, we obtained the following relations:
We wish everyone could be able to repeat and reproduce our calculations. For this reason, we provide the whole information needed for these calculations. The data are presented in CGS system, which we used in our calculation. We present our final output data in SI system.
Table 2 presents the parameters of the water, which we used in our theoretical study.
The ionic parameters of OH
−, HCO
3−, and H
3O
+ ions are presented in
Table 3. One can see that the value of
Nw is not integer number and even can be smaller than one. This looks strange at first glance, but it does not suffer of lack of logics because after the integration, the water is observed as a structureless continuum, whose surface energy is larger than its bulk energy.
Table 4 presents the London constants
Liw and
Lww, the dimensionless gain of energy of transfer of one ion from the bulk to the air/water interface
E1/
kBT, the dimensionless loss of energy of transfer of
Nw surface water molecule to the bulk
E2/
kBT, the dimensionless total energy of adsorption of one ion to the air/water interface (
E1 −
E2)/
kBT, and the adsorption energy
EAds of 1 mol ions on the air/water interface for OH
−, HCO
3− and H
3O
+ ions.
Shown in
Figure 3 are the energies of adsorption of OH
−, H
3O
+, and HCO
3− ions on the air/water interface. One can see in
Table 4 and
Figure 3 that the transfer of ion from the bulk to the air/water interface always gains energy (
E1/
kBT > 0), i.e., the ion overcomes the image repulsion of the air/water interface. Moreover, one can see that the transfer of
Nw surface water molecules into the bulk always losses energy (
E2/
kBT > 0), i.e., these water molecules interact with significantly larger amount of other water molecules when they are located in the bulk, compared to their state on the air/water interface. In this way they decrease their energy. One can see in
Table 4 as well that for the case of OH
− and HCO
3− ions the loss of energy is larger than the gain of energy. Hence, the total change of energy is negative, i.e., the system decreases its energy when they adsorb on the air/water interface. Therefore, their adsorption on the air/water interface is energetically favorable. On the contrary, the gain of energy is larger than the loss of energy for the case of H
3O
+ ions. Hence, their adsorption on the air/water interface is energetically unfavorable. This means that there is an attraction force between the OH
− and HCO
3− ions and the air/water interface, while the H
3O
+ ions exhibit repulsion force by the same surface. One can see as well the values of their adsorption energies per mol ions. These values are significantly smaller than the adsorption energies of the surfactants. For example, the adsorption energy of sodium dodesyl sulfate (SDS), which is common well known surfactant, is
EAds = 44.12 kJ/mol [
67]. The adsorption energies are heat effects of the adsorption of these ions. Therefore, these values represent the changes of the entalpies Δ
H of the ions during their adsorption, i.e., there is a negative change of the entalpy (Δ
H < 0) for case of the OH
− and HCO
3− ions and positive change of the entalpy (Δ
H > 0) for the case of H
3O
+ ions. Nevertheless, the negative change of the entalpy (Δ
H < 0) is a required but not sufficient condition for spontaneous process. The positive change of the entalpy (Δ
H > 0) is required but not sufficient condition for forced process. The change of the entropy is important to know as well.
3.3. Entropy of Adsorption of OH−, HCO3−, and H3O+ Ions on the Air/Water Interface
The entropy of adsorption of ion from the bulk onto the air/water interface is calculated via Equations (15)–(17).
The change of the entropy of transfer of
Nw surface water molecules into the bulk is negligible as far as the water molecules do not move, but only vibrate due to the hydrogen bond network. For this reason, we only calculate the entropy of transfer of the ions from the bulk onto the air/water interface. Shown in
Table 5 are the values of change of the entropy of adsorption of OH
−, HCO
3−, and H
3O
+ ions and the parameters needed for their calculation. One can see that the ions decrease their entropy values if they adsorb on the air/water interface.
Once we have the enthalpy and entropy values of the adsorption of the OH−, HCO3−, and H3O+ ions calculated, we can calculate the related changes of the free Gibbs energy by means of Equation (18).
Shown in
Figure 4 is the change of the Gibs free energy during the adsorption of OH
−, H
3O
+, and HCO
3− ions on the air/water interface.
Table 6 shows the values of the changes of the free Gibbs energy of adsorption of OH
−, HCO
3−, and H
3O
+ ions on the air/water interface. One can see in both
Figure 4 and
Table 6 that the OH
− and HCO
3− ions decrease their free Gibbs energy if they adsorb on the air/water interface. Hence, their adsorption is a spontaneous process. On the contrary, the H
3O
+ ions increase their free Gibbs energy if they adsorb on the air/water interface. They can adsorb on the air/water interface only if they are forced to adsorb. We must recognize that these calculations are valid for ions either in deionized (DI) water or in diluted salt solutions, in which the Ray–Jones effect [
5,
6,
68,
69] is in power (up to 0.002 mol/L). At larger salt concentrations, the adsorption of ions on the air/water interface becomes energetically unfavorable due to an increase of the image repulsion force. The ions located in the bulk interact not only with the water molecules but also with the other ions located in the bulk as well. In DI water or diluted salt solution, the image repulsion is not so strong yet. This allows the adsorption of ions on the air/water interface. Therefore, if we follow the general thermodynamic logics and the values of Δ
G in
Table 6, we arrive at the conclusion that the OH
−, HCO
3− ions adsorb on the air/water interface, while the H
3O
+ ions are repelled by the latter. Thus, an electrical double layer (EDL) is formed at the very bubbles–negatively charged adsorption layer of OH
−, HCO
3− ions and positively charged diffuse layer of H
3O
+ ions.
We should note here that the zeta potential of gas bubbles in water, measured by the different authors, varies within some reasonable limits [
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21] depending on the method of measurement, but all of them agree on the negative value of the zeta potential of the micro-(or nano) bubbles in fresh DI water and its salt solution until the isoelectric point. Our theoretical analysis on the ion adsorption on the air/water interface accounts for the dispersion intermolecular forces and is valid for bubbles with every size. Therefore, the OH
− and HCO
3− shall adsorb on the surface of the bubbles, while the H
3O
+ will be repelled by the bubble despite its size.
3.4. The Law of Parsimony and the Negative Charge of the Bubbles
The negative charge of the bubbles is a 100-year-old scientific problem. A number of explanations were suggested in the literature to solve it. They are complex, and some of them contradict to each other. How to arrive at the right explanation? We search the help of the “Occam’s Razor” philosophical principle known as “The Law of Parsimony”.
The basic principle was pronounced by the Greek philosopher Aristotle (384–322 BC) (“The more limited, if adequate, is always preferable”) and by the Roman mathematician and astronomer Claudius Ptolemy (100–170 AC) (“We consider it a good principle to explain the phenomena by the simplest hypothesis possible”). The English mathematician and physicist Isaac Newton (1643–1727) stated “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances”. Nevertheless, it is assumed that the roots of this principle in its present form lie in the works of the English Franciscan friar William of Occam (1287–1347) [
70] who formulated “Never posit pluralities without necessity” (“Numquam ponenda est pluralitas sine necessitate”). The Franciscan scholastic theological philosopher John Punch (1603–1661) further formulated “Entities are not to be multiplied without necessity” (“Entia non sunt multiplicanda praeter necessitate”). The Scottish metaphysical philosopher William Hamilton (1788–1856) called this maximum in 1852 as “Occam’s razor” or “The Law of Parsimony”. This principle was reconsidered in the light of the contemporary scientific knowledge and technology by the American philosopher Jonathan Schaffer in 2014 [
71] who pointed out that the “razor” is too blunt measure for the ontological economy, failing to distinguish fundamental from derivative entities. Hence, he replaced the “razor” with a “laser”, which is focused specifically on fundamental entities and commands: “Do not multiply fundamental entities without necessity!” Therefore, Schaffer had in mind that “Occam’s razor” should be applied mostly for fundamental entities, having intrinsic nature. Hence, we formulated as a fundamental problem the negative charge of the bubbles in water. We scrutinize the experimental findings on this problem and the related different explanations in the search for the simples one. To be more specific, we present in
Figure 5 the scheme of the experimental findings and their explanations in the literature and the present study as well.
As shown in
Figure 5, the basic entity in form of scientific puzzle consists of experimental knowledge that (i) “the bubbles have negative zeta potential depending on the pH and the ionic strength values”, (ii) “the nonionic surfactants practically do not affect the zeta potential while the ionic surfactants do”, and (iii) “pairs of bubbles from CO
2 in contact do not coalesce for a long time on the contrary of pairs of bubbles from other gases”. Evidently, these findings have one common origin causing negative charge of the bubbles. The question is “what makes them negatively charged?” The right explanation should be a fundamental entity (simple, sufficient, and definite explanation), which should be cause of our basic entity, consisted of its three above mentioned statements. In addition, the right explanation should not introduce new puzzles. Hence, our basic entity should be undoubtfully derivative from the fundamental entity (the right explanation). We will apply Occam’s razor to the explanations from the literature and our own as well.
Reference [
28] suggests a hypothesis that even the purest water contains anionic surface active traces. They developed theoretical model based on this hypothesis, explaining the Ray–Jones effect [
6,
69] and the negative charge of the bubbles. Nevertheless, their model contradicts the experimental fact that the nonionic surfactants do not affect the zeta potential of the bubbles. Hence, this explanation has two flaws: (i) the hypothesis needs a proof, and (ii) the theoretical model only partially explains our basic entity (statements (i), (ii), (iii)). Therefore, Occam’s razor cuts this hypothesis and its related consequences.
Another explanation suggested by Reference [
36] consists of charge transfer from the bulk toward the surface water molecules due to topologically defected hydrogen bond network. This explanation is supported by quantum-chemical calculations. Therefore, we accept that such phenomenon is possible. But this explanation opens the door to another puzzle—in which way the topological defects of the hydrogen bond network are related with the pH and the ionic strength values and the presence of carbon dioxide (CO
2) in water? If we accept this explanation, we accept a new puzzle. Hence, Occam’s razor cuts the multiplication of the entities as not justified in this particular case.
Another well accepted explanation supported by References [
22,
23,
24,
72] suggests a hypothesis that the autolysis of the surface water is different from the autolysis of the bulk water. They developed a theory based on this hypothesis resulting in positive adsorption of both the OH
− and H
3O
+ ions on the air/water interface. Afterwards, they suggested another hypothesis based on their theory that the adsorption of OH
− ions is larger than the adsorption of H
3O
+ ions, thus resulting in total negative charge of the bubbles. Overall, this explanation suggests two hypotheses, which need proofs. Occam’s razor cuts these hypotheses.
Another explanation [
26,
27] suggested a theoretical model pairing the surface tension and the zeta potential, taking into account all the possible ions into the water with variable pH. The model operates with the energy of interaction of the ions with the air/water interface and the specific length for each ion, at which the concentration gradient of the ion is developed. These two parameters are to be adjusted by means of the experimental value of the zeta potential versus pH. Thus developed, the model results in positive adsorption of the ions charging the air/water interface despite being either H
3O
+ ions charging the bubbles positively at low pH or OH
− ions charging the bubbles negatively at higher pH values. In addition, to calculate zeta potential values equal to the experimental ones, the model imposes high values of the energy of adsorption of the ions close to these ones of the weakest surfactants and huge subsurface concentration. Otherwise, the predicted zeta potential values are significantly lower than the experimental ones. Thus explained, the physical reason for the adsorption of the ions on the air/water interface remains unclear. In other words, this explanation supports the hypothesis that the OH
− ions adsorb on the bubbles at moderate and high pH values, and the H
3O
+ ions adsorb on the bubbles at low pH values. Unfortunately, the adjustment of theoretical energy of interaction and specific ionic length to experimental data of the zeta potential is not a proof of the hypothesis, because the physical reason for this adsorption remains obscure. Occam’s razor cuts this hypothesis.
Other explanations [
9,
37,
38] based on completely experimental studies suggest the hypothesis that both HCO
3− and OH
− ions adsorb on the air/water interface thus charging it negatively. When the pH value is increased, both the HCO
3− and OH
− ions increase their concentrations, and this correlates with larger in absolute value negative zeta (or surface) potential. On the contrary, when the pH value is decreased, both the HCO
3− and OH
− ions decrease their concentrations, and this correlates with smaller in absolute value negative zeta (or surface) potential, isolectric point (IEP) in the pH range of 3 to 4 and positive zeta potential at pH < 3. These experimental observations support the hypothesis of adsorption of the ions on the air/water interface, but unfortunately, they are not equal to a proof of the hypothesis. Occam’s razor cuts this hypothesis.
The majority of molecular dynamic simulations (MDS) of the air/water interface [
29,
30,
31,
32,
33] have shown that the OH
− ions should not be adsorbed at the air/water interface, but H
3O
+ ions should, thus charging the bubbles positively. Few authors argued that their MDS analysis [
34,
35] shows the opposite. The results of the MD simulations contradict the experimental findings. Moreover, some of the MD simulations contradict other MD simulations. Occam’s razor cuts these explanations due their contradiction to the experimental data.
We suggest in the present work a thermodynamic approach to solve the puzzle. It is well known that the spontaneous processes decrease the free Gibbs energy of a given system [
48]. On the contrary, the forced processes increase the free Gibbs energy of the system. We calculate the change of the free Gibbs energy of adsorption of HCO
3−, OH
−, and H
3O
+ ions and found out that the HCO
3− and OH
− ions spontaneously adsorb, while the H
3O
+ ions adsorb only when being forced on the air/water interface. We set the limit of validity of the exploited theory up to ionic strength values of about 0.002 mol/L. A possible flaw of this explanation is that the theory always predicts a negative value of the zeta potential although being variable, while at pH < 3 the experiment shows positive values of the zeta potential. Under such conditions, the ionic strength has values of the of order 0.002 mol/L and larger, which violates the validity of the theory. Therefore, we set the validity of the theory in the pH range of 3 < pH < 11 and ionic strength up to 0.002 mol/L. Another possible flaw of this theory is that the energies of adsorption of the ions are close to the thermal energy
kBT, which should result in very low values of the zeta (or surface) potential if the latter is calculated by means of the classical Poisson–Boltzmann equation. From Occam’s razor’s viewpoint, we suggest a fundamental entity causing the basic entity with its three statements and with no hypotheses left. We state that the HCO
3− and OH
− ions adsorb on the bubbles, while the H
3O
+ ions are repelled by them at ionic strength less than 0.002 mol/L. Keeping in mind the age and the toughness of the puzzle, we leave the door open for other fundamental explanations.