3.1. Complexities of the VRFB-Cell Electrolytes

The electrochemical cell designed for the VRFB technology is comprised of the (a) anode, anolyte (anodic half-cell); (b) cathode, catholyte (cathodic half-cell); and (c) Nafion-117 proton exchange membrane (separator). In the normal operation process, the anolyte (an H₂SO₄ (aq.) matrix of V²⁺ [V (H₂O)₆]²⁺) and catholyte (an H₂SO₄ (aq.) matrix of VO₂⁺ [VO₂ (H₂O)₄]⁺) solutions are pumped into the electrochemical cell from the respective electrolyte storage tanks, and the former is oxidized while the latter is reduced electrochemically into the anodic and cathodic half-cells. Due to the exceptional hydration propensities of each adjacent state vanadium ions in the aqueous electrolyte systems, the anolyte tank stores V (II) fuel in [V (H₂O)₆]²⁺ form (lilac-colored), and the catholyte tank stores V (V) fuel in [VO₂ (H₂O)₄]⁺ (yellow colored). Therefore, both the anodic and cathodic half-cells of the VRFB cell are supplied with the hydrated complexes of the active vanadium ions, viz., [V (H₂O)₆]²⁺ and [VO₂ (H₂O)₄]⁺, respectively, along with the predominant quantity of free H₂O molecules and the H⁺ carried H₂O molecules. These dissimilarly charged vanadium fuels or electrochemically active hydrated vanadium ions undergo spontaneous charge/discharge reactions in the electrochemical cell as mentioned in Scheme 1 (for simplicity, the charge/discharge reactions in both negative and positive half-cells are separated, and the sum of these partial equations gives a net cell reaction of the VRFB cell). The Nafion-117 membrane (separator) is used to balance the diffusion-gradient governed H⁺ ion (or its hydrated forms), H₂O molecule, HSO₄⁻/SO₄²⁻ ion, and hydrated vanadium ion concentrations in the two half-cells so that the net junction potential developed at the electrode–electrolyte and membrane–electrolyte interfaces is nullified to the most extent. At variable SOCs of the battery, the concentration gradients of these diffusing ionic/molecular specimens, mass/charge transfer toward/from the electrodes, the internal working mechanisms of the cell at different interfaces under variable concentrations, the charge–discharge processes under the depletions/depositions/agglomerations/asymmetric distributions of the molecular/ionic specimens throughout the anolyte/catholyte of the anodic/cathodic half-cell, the ionic/molecular blockage of the SO₃⁻ conductive sites of the Nafion-117, and their hindrance to the proton conductivity (Grotthuss and vehicular) procedures of the latter play very crucial roles deterministic to the exceptional performance of the cell and its integrities (battery). In addition to these, the consequences caused by the hydrated moieties of the adjacent state vanadium ions (released through the charge/discharge processes); their populations and abrupt increase/decrease in concentrations at the electrode/electrolyte and electrolyte/membrane interfaces at variable SOCs; their impacts to the smooth flow of the diffusion gradient-controlled ionic/molecular specimens across the electrolyte; their involvement in the unidirectional “water-flooding” and its prevention strategies; their physicochemical affinities toward the proton-conducting SO₃⁻ sites of the Nafion-117, free H₂O molecules, HSO₄⁻-type monovalent ions, and proton-carrying H₃O⁺-type hydrated species; and the entire microdistributions/structural configurations of them in the nanometer range are imperative means of assessing internal complexities of the VRFB cell working electrolytes which in turn directs the optimum active material utilization strategies and the overall energy-storing capacities of the VRFB technology. Toward understanding these ideal prerequisites, the present study employed MD simulation as the most potential theoretical mean whose mathematically produced rigorous datasets and therein stored statistical microphysicochemical properties were accessed directly by the programmatically designed radial distribution function (RDF) tools (g (r)), and the required macroscopic statistical properties lying within the current MD simulation framework were retrieved. The detailed explanations of them are presented part by part in the following sections.

3.2. Microstructures Around Free Water Molecules

The radial distributions and the microstructures around any free water molecules (unbound state to the central Vⁿ⁺ ion but bound each other by the intermolecular hydrogen bond) present in the entire simulation cell (designed to cope with the internal working conditions of VRFB cell) mostly depict the exact molecular network of water system, and its most probable breakage/cleavage of the hydrogen-bonded assembly while incorporating them with the Vⁿ⁺-hydrated complexes, monovalent charged species (H₃O⁺, HSO₄⁻), and the proton exchange SO₃⁻ functional groups appended polymeric Nafion-117 membrane. The exact quantitative structural information related to the same explains the internal distributions of free water molecules in nanometer range which in principle stands as a standalone fact for outlining the proton conduction and ionic diffusion phenomena plus the most potential factors involved in retarding the ionic/molecular flow rates. As shown in Figure 7(a), the RDF g(r) tools derived radial distribution plot of the free water molecules computed over the last 900 ps of the MD run demonstrate the inhomogeneous number density (population) of them even within the radial distance framework of 5.0 Å. The abbreviations mentioned in the inset such as O_w and H_w indicate the central and centrally bound O and H atoms of the reference H₂O molecule (Ref_H₂O), respectively, and the functions $$ and $$ signify the typical radial O−O and O−H distances measured from that Ref_H₂O, respectively. The sharp intense $$ peaks appeared at 1.8 Å and $$ peak at 2.8 Å justify the typical hydrogen bond length and the distance between the two O atoms of any two nearby H₂O molecules, respectively. These RDF values ensure the existence of closed packing microstructures of water in VRFB-cell electrolyte; an identical network to that exists in pure bulk water systems. Moreover, no RDF peak shifting of these typical spectral bands is noticed while comparing with the MD simulation results associated with the simulation systems containing unhydrated Vⁿ⁺ ions as reported by the present author elsewhere [11]. It means that the hydrated moieties of the vanadium ions (in the presence of monovalent charged species), viz., V (H₂O)₆]²⁺ and [V (H₂O)₆]³⁺ (anodic side) and [VO₂ (H₂O)₄]⁺ and [VO (H₂O)₅]²⁺ (cathodic side) do not disrupt the water channel and its hydrogen bond bound assembly, further underscoring the excellent maintenance of proton conduction pathways (Grotthuss and vehicular) in VRFB-cell electrolyte system taking place facilely through the undisruptive hydrogen-bonded network of water assembly. Again, both of these $$ and $$ radial distributions with those two sharp and intense spectral bands are found to be as equal as the typical O−H and O−O distances of any two nearby H₂O molecules present in the absolute aqueous environments with no Nafion-117 threads and monovalent H₃O⁺ and HSO₄⁻ specimens (blank test) [11]. The $$ and $$ diffused type spectral patterns appeared at the radial distances of ∼3.6 Å and ∼4.5 Å, respectively, are simply due to the repetitive and periodic structural organizations of the H₂O molecules in liquid state. The general conclusions we can extract from these quantitative explanations include no significant internal microstructural changes around any of the H₂O molecules present in the aqueous H₂SO₄ matrix of the vanadium electrolytes, no disruption of the hydrogen-bonded network of water assembly (a schematic pathway for efficient proton conductions), and no internal inductions of the Vⁿ⁺-adjacent state hydrated moieties, their electric charges, and their electrostatic interactions in making the VRFB-cell electrolyte systems electrochemically inactive.

3.3. Microstructures Around Hydronium Ions

The quantitative radial distributions and the microstructural configurations of the water molecule around any H₃O⁺ (a proton-carrying H₂O molecule) specimen act as the descriptive parameters for identifying whether the proton hopping proceeded well through the Grotthuss and vehicular dynamics via the hydrogen-bonded water channel of the working electrolyte matrix or not. The movement of the H₃O⁺ ions (proceeding of the proton conduction) across the water network can actually be inspected from the concerned RDF plot, which in fact demonstrates the same by specifying the formation and involvement of its other higher yet less stable hydrated states, viz., Eigen [H₉O₄]⁺ and Zundel [H₅O₂]⁺ states. The RDF-derived information confirms their existence in the electroactive electrolyte systems by quantizing the experimentally/theoretically determined specific interatomic distances and hydrogen bond length values. The concerned RDF plots computed over the last 900 ps of the MD run are shown in Figure 7(b) where the symbols $$ and $$ mentioned in the inset represent radial distribution functions of O atom of H₃O⁺ (O_h) and O atom of water (O_w), and H atom of H₃O⁺ (H_h) and O atom of water (O_w), respectively. The sharp $$ spectral band appeared at 2.5 Å radial distance stands as a quantitative measures of the interoxygen distance of the two hydrated H₂O molecules of the central H⁺; a characteristic length designated to the Zundel state of the proton (Figure 8(a)) (the Zundel state has hydration number (n) = 2, viz., [H₂O------H⁺------OH₂] or [(H₃O) (H₂O)]⁺; where the symbol ------ represents hydrogen bond), and the relatively less intense $$ spectral peaks appeared at 1.2 Å and 1.5 Å radial distances serve as the distinctive yet distinguished hydrogen bond length values of the Zundel and Eigen states of the proton, respectively. Accordingly, the experimentally determined 2.55 Å distance exists between the central O atom of H₃O⁺ and the central O atom of any one hydrated H₂O molecule (out of three) of the Eigen ion, which is also confirmed by the sharp $$ spectral peaks standing at the radial distance r = 2.5 Å (the Eigen ion has a hydration number (n) = 3 with a central H₃O⁺ ion, viz., [(H₃O) (H₂O)₃]⁺). All these RDF-predicted characteristic bands of the Zundel and Eigen ions at distinctive radial distances r are found to be as equal to those measured in their DFT-optimized equilibrium structures reported elsewhere [12, 20]. As per the blank tests conducted by the present author with and without including bare-state Vⁿ⁺ ions, HSO₄⁻, and polymeric (N₇P)₁₀-type Nafion-117 threads [11] in bulk water systems, none of those designated radial distances are found to be deviated significantly, suggesting the absence of predominant roles of the hydrated moieties of the adjacent state Vⁿ⁺ ions deterministic to the proton conduction mechanisms and ionic/molecular flow rates existing in the VRFB-cell electrolyte systems. Furthermore, the characteristic bands of the $$ and $$ radial distribution functions appeared at radial distances r = ∼5.0 Å and ∼3.0 Å regions due to the periodicity of the water molecules hydrated around the central H⁺ and H₃O⁺ ions of the Zundel and Eigen states, respectively; an unique distribution trends yet molecular arrangements of the hydrated proton in bulk water systems or water enriched electrolyte systems whose composition is alike to the present aq. H₂SO₄ matrix-supported VRFB-cell electrolytes. The relatively diffused $$ RDF bands at r = ∼5.0 Å suggests that the considerable exchange of the water molecules used up in protonic hydration takes place between the first and second hydration shells, while the central protons start to diffuse or hop through the hydrogen-bonded water channel ubiquitously. As a whole, all these quantitative interpretations justified H₃O⁺, [(H₃O) (H₂O)]⁺, and [(H₃O) (H₂O)₃]⁺ ionic detections/conformations (Figure 8(a)) in all the four simulation cells designed for the VRFB-electrochemical cell verify the presence of predominant dynamisms of the hydrated protons (water carrying proton) in VRFB electrolyte systems, which in fact stand as the root cause of proving two mainstream Grotthuss and vehicular proton hopping dynamics; the mechanisms that govern the entire proton conduction processes of every aqueous electrolyte-fueled battery technologies and biochemical processes [21, 22].

3.4. Microstructures Around Adjacent State Vanadium Complexes

The nanometer-ranged radial distributions and microstructural configurations of the free water molecules around any hydrated complexes of adjacent Vⁿ⁺ ions (V²⁺, V³⁺, VO²⁺, and VO₂⁺) are quite essential to understand the hydration propensities of the latter and their free H₂O molecules gaining/loosing and/or exchanging affinities in quantitative scales. The structural stabilities of these complexes with neutral monodentate H₂O ligands during the entire annealing procedure of the MD simulation, after the successful ending of the same scheme, and within the parametrizations of the NVE iterations run over the significantly longer timescales can also be inspected and theorized from the MD trajectories linked RDF-based structural interpretations. The confirmation of their undisrupted hydrated moieties in equilibrium conditions even within the extreme parametrizations range of the annealing and NVE procedures ensures us to validate a few potential issues associated with the VRFB cell such as proton conductions, trends, retardations, or facilitations of the ionic/molecular flow rates, hindrance or simplistic of the conductive pathways, size dependencies of the active vanadium ions on their diffusion rates, and “water-flooding” problems of the VRFB cell. Similarly, the RDF-based distributional interpretations and population density analyses of the HSO₄⁻ ions nearby to the hydrated “all-vanadium” species specific to the anodic and cathodic half-cells are very much applicable to justify whether the highly dense HSO₄⁻ species ([HSO₄⁻]:[SO₄²⁻] = 99:1) present in the bench H₂SO₄ matrix-supported vanadium electrolyte show remarkable or negligible affinities toward making anionic type complexes/clusters with the active vanadium ions or not, and to understand the effective roles of such type monovalent ions toward surpassing or restricting the conductivity rates and diffusion propensities of the protonic/ionic/molecular specimens residing into the same working electrolyte. Moreover, the closed chemical affinities of the SO₃⁻ functional groups of the Nafion threads toward the hydrated vanadium moieties and the most or the least probable blockage of them to the former sites (the bulkier sized hydrated motifs screen the former sites) can also be figured out from the concerned MD trajectories linked RDF distribution trends. Toward understanding all of these rigorous matters in question in quantitative scales, the MD-produced massive datasets of every simulation cell accessed over the last 900 ps of the MD run were assessed through the RDF means, and the required interpretations of thus retrieved RDF plots (Figures 8(b), 9, 10) were carried out. For easing the theoretical elucidations presented herewith explicitly, the representative symbols such as $$ (n = +2, +3, +4, +5), $$, and $$ are used in the insets that, respectively, stand for the radial distribution functions of the hydrated vanadium complexes (vⁿ⁺_complex) in respect to the free water molecule (O_w = O atom of water), HSO₄⁻ ions (S_HSO4 = S atom of HSO₄⁻), and SO₃⁻ conductive sites of the Nafion−117 (S_SO3− = S atom of SO₃⁻).

Since the simulation cells designed in this study contain both the free and fixed (solvated molecules of Vⁿ⁺) water molecules separately, the undisrupted hydrated Vⁿ⁺ complexes and the microdistributions of the free water molecules that tend to remain completely far away from the Vⁿ⁺-centered hydrated moieties are clearly seen in the VMD-rendered graphical images (Figure 8(b)) produced over the 900 ps of the MD run (within this simulation timeframe, the annealing procedure was completed with the attainment of the absolute equilibrium descriptors underscored in Section 2, and the NVE simulation was run iteratively with extreme parametrizations). The dispersing propensities and the outbound distributions of the free water molecules from the central Vⁿ⁺ ion of the given hydrated complexes are depicted quantitatively in Figure 9 where the symbols “V-free H₂O” and “V-fixed H₂O” mentioned in the inset abbreviated the interactions between the central Vⁿ⁺ and outbound (mobile) water molecules, and the interactions between the central Vⁿ⁺ and its inbound water molecules (hydrated), respectively. The intense and distinctive $$ spectral bands appeared at the radial distance r in the range of ∼1.8 Å –∼2.5 Å illuminate the unequal inbound affinities of the adjacent states Vⁿ⁺ ions with water molecules, which is however very typical to the intensities of the electric charges carried by them (V²⁺, V³⁺, VO²⁺, and VO₂⁺) and their solvation propensities. These radial distance r ranges are justifiable to the DFT-derived Vⁿ⁺-OH₂ bond length of each hydrated complex in a respective low energy equilibrium state. The present author reported this bond length value elsewhere [5] for the adjacent state Vⁿ⁺-hydrated complexes as 2.4 Å, 2.0 Å, 2.1 Å, and 2.1 Å, respectively, where the trial structures used for each of them were constructed in the graphical interface used with Gaussian and optimized explicitly into the global minima states via the DFT method. The considerable radial distance deviations to that of the DFT-predicted bond lengths are seen in the first two bivalent and trivalent complexes, viz., [V (H₂O)₆]²⁺ and [V (H₂O)₆]³⁺, which are however lying in the satisfactory range if we consider the extent of the polarizabilities induced by these two intensive electric charges and the moderate computing skills of the currently employed force-field databases [23]. In addition to this, the spectral bands standing at the radial distance r ∼ 4.0 Å present pictorially the exact location in which the optimum population density of the free water molecules is dispersed. This range of r explains quantitatively that none of the free water molecules dare to approach the vicinity/bonding regions of any of the hydrated Vⁿ⁺ complexes (Figure 8(b)) unlike the cases of bare-state Vⁿ⁺ ions that were found to show strong affinities toward the same while simulating them together in the presence of Nafion threads, H₃O⁺ ions, and HSO₄⁻ ions as discussed elsewhere [11] by the present author. This reluctant tendency of the mobile water molecules toward going nearer to the hydrated complexes of adjacent state “all-vanadium” ions would be an esteemed fact to declare that every Vⁿ⁺-hydrated complexes employed throughout this study attained their saturated levels of hydrations despite showing some degree of flexibility toward loosing closely inbound water molecule more especially by the [O = V = O]⁺ unit bearing [VO₂ (H₂O)₄]⁺ species, but keeping it in the hydrogen bond bounding region [1]. Again, the radial distance r that lies in between 2.8 Å and 3.5 Å region of the same RDF plot bears no spectral peaks at all or represents a nodal plane into which no number density of free water molecules is residing, ensuring no exchange of any H₂O molecules takes place in between the fixed-water (inbound state) and free water (outbound state) spheres. However, the nonzero region appeared immediately after the completion of the latter sphere (after the radial distance r = ∼5 Å) of the same plot indicates the spatial homogeneously distributed free water molecules, figuring out the internal pictures of the repeatedly aligned mobile H₂O molecules dispersed throughout the electrolyte matrix designed for the VRFB cell.

Alike to the aforementioned discussions, the closed interactive proximity of the HSO₄⁻ ions toward the hydrated complexes of adjacent state vanadium ions can be analyzed from the concerned RDF plot shown in Figure 10(a). The intense spectral bands standing at an abnormally higher radial distance r ≥ 5 Å region while varying the oxidation states of the hydrated vanadium complexes signify the least probability of forming new bonding moieties with the freely yet predominantly available monovalent type HSO₄⁻ ions. Despite their unwilling trends, the appearance of such intense peaks simply measures the approximate radial distance r region wherein the optimum number density of the HSO₄⁻ ions is accumulated. In fact, this specific region lies in the outbound distances in the perspectives of the covalent bonding principle between any two nearby atoms that are required to be departed by the distance shorter than the sum of their Van der Waals atomic radii r_w for new bond formation. For example, if the r_w of the central S atom (1.80 Å) of the HSO₄⁻ ion and the central V atom (1.79 Å) of each of the hydrated complexes are referred, their sum (r_sum = 3.59 Å) does not exceed the radial distance value (r = ≥ 5 Å) speculated from the concerned RDF plot. This quantitative regime guaranties the null chances of developing bonding complexes between the HSO₄⁻ ion and the hydrated vanadium complexes within the present 2M−5M H₂SO₄ concentration frameworks used in preparing vanadium electrolytes. The same is diagrammatically depicted (in the inset of Figure 10(a)) by the VMD graphical image rendered from the MD simulation databases retrieved over the 900 ps. These quantitative justifications are very much useful to conclude the facts that none of the predominantly available monovalent HSO₄⁻ ions residing into the VRFB-cell electrolyte are used up to trap the electrochemically active hydrated vanadium complexes (the fuel materials that store chemical energy convertible into electrical energy). It means none of the “all-vanadium” fuel materials are wasted or electrochemically disabled to undergo complete oxidation/reduction reactions just only due to the excessive presence of HSO₄⁻. However, a slight retardation of the diffusions or flow rates of the protons/ions/molecules throughout the conducting electrolyte medium may be caused by the [HSO₄⁻] thickened viscous matrices. The similar type unopposed roles and negligible impacts of the HSO₄⁻ ions to the vanadium-based working electrolyte were also noticed while considering the bare-state Vⁿ⁺ ions as the electrochemical-active species [11].

Accordingly, the ionic interactions and the microstructural distributions of the hydrated vanadium complexes with dissimilar charged states (electronic charges) in the vicinity of SO₃⁻ conductive sites of the Nafion-117 can be theorized from the concerned MD trajectories derived RDF plot shown in Figure 10(b). Except the case in V (IV)-hydrated state, viz., [VO (H₂O)₅]²⁺ ion, the relatively less intense RDF peaks are seen beyond the radial distance of ∼4 Å region, justifying the low number density of V (II)- and V (V)-hydrated complexes, viz., [V (H₂O)₆]²⁺ and [VO₂ (H₂O)₄]⁺ accumulated in the relatively far away distance from the SO₃⁻ conductive sites of the Nafion-117. However, the case is pronouncedly different in V(IV) complex state: the relatively more intense, obtuse yet wide-range occupied RDF peak is observed at the radial distance ∼2 Å–∼8 Å region, signifying the asymmetric distributions of more number density of [VO (H₂O)₅]²⁺ ion in the both inbound and outbound regions of the SO₃⁻ conductive sites of the Nafion-117. The RDF g(r) spectral trends and the peak distributions for the V (III) complexes are excluded in the same diagram due to the negligible range expansions across the plotting scales. This might be due to the lack of proper databases in the designated force-field packages because of which the effects of the induced-charge polarizabilities created by the +3 unit electric charge of the [V (H₂O)₆]³⁺ ion were unable to be addressed properly. In terms of the covalent bond developing probabilities between the adjacent state hydrated vanadium complexes and the SO₃⁻ sites of Nafion, the aforementioned bonding principle is still applicable. Since the intense RDF peak position for the SO₃⁻ functional group (from the central S atom, r_w = 1.80 Å) in respect to [VO (H₂O)₅]²⁺ and [V (H₂O)₆]²⁺, [VO₂ (H₂O)₄]⁺ (from the central V atom, r_w = 1.79 Å) is appeared at radial distance r = ∼4 Å and r > ∼4 Å regions, respectively, the sum of the Van der Waals radii r_w of these atoms (r_sum = 3.59 Å) never exceeds that value does not satisfy the critical conditions required for the bonding, illuminating that none of the Vⁿ⁺-hydrated complexes can form covalent bond with any of the SO₃⁻ atoms of Nafion-117. It further means none of the SO₃⁻ groups of Nafion are obsessed by any of the Vⁿ⁺ complexes, and therefore, they can exhibit exceptional conductivities and proton exchange rates even in the present complex electrolyte matrix functionalized battery system. Interestingly, this typical g (S−Vⁿ⁺_complex) radial distance r = ∼4 Å specific to the V (IV)-hydrated complex is found to be as consistent as the closest separation distance maintained in between the two nearby groups, viz., a central V atom of [VO (H₂O)₅]²⁺ and a central S atom of SO₃⁻ Nafion (distance = 4.2 Å) as reported earlier by Intan et al. [24]. Therewith, the closest departing distance between the O atom of the former and the S atom of the latter is measured as 1.5 Å in their closely linked motifs developed in the electrolyte system while delivering ionic mobility/conductivity across the electrolyte–membrane interfaces. Nevertheless, this quantitative interpretation reveals the fact that none of the SO₃⁻ conductive sites of the Nafion experience either partial or complete blockage of the adjacent state hydrated vanadium complexes, ensuring that all the conductive sites and polymeric channel of the Nafion-117 remain completely unoccupied and activated by means of which protons/ions/molecules exhibit smooth diffusions and notable dynamism provided that the Nafion has priory attained the optimum water content (λ = 22) in its hydrophilic domains.

3.5. Microstructures Around SO₃⁻ Functional Groups of Nafion

The complete protonic/ionic conduction abilities and the diffusive flow rates assessing propensities of the SO₃⁻ sites of the Nafion-117 are only be materialized if the microdistributions of the free water molecules and the hydrated protons H₃O⁺ nearby to them are traced out thoroughly. Since the SO₃⁻ functional groups are regarded as the most electrochemically active conductive sites of the Nafion bearing a −1 unit electric charge, they chemically prefer to exhibit positively charged H⁺ exchanging abilities with the nearby H₃O⁺ species leaving behind the free H₂O molecules into the polymeric matrix, which in turn accept H⁺, and further participate in a similar type exchanging mechanisms with another nearby SO₃⁻ unit of the same polymeric thread, and so on. In addition to this, the protonic conduction takes place into the Nafion matrix (hydrophilic domain) through the Grotthuss mechanisms as well in which the H⁺ ions are hopped to the free H₂O molecules dispersed into the same, and the continuous processes happen across the networking channels. Either of these schemes is pronouncedly affected by the water content λ of the Nafion, and its water retaining capacities (swelling property of Nafion: a distinguished characteristic feature when the (a) EW_Nafion ≥ 100 and (b) relative humidity RH = 0, λ_Nafion ≅ 1; RH = 100, λ_Nafion ≅ 14 and in a perfectly water enriched wet condition, λ_Nafion ≅ 22) [14] plus the relative abundance of the H₃O⁺ and H₂O-type species nearer to its SO₃⁻ proton exchanging sites. Additionally, none of the SO₃⁻ functional groups of Nafion are found to be made electrochemically inactive by the excessive presence of the Vⁿ⁺-hydrated complexes, and the HSO₄⁻ ions, and neither of the latter specimens plus their asymmetric distributions are found to involve in impeding the conductive potentialities of the former, and more importantly, none of them dare to form the bonding complexes with the SO₃⁻ groups as presented in Section 3.3. Therefore, the only way to inspect the facile conductivity rates of the Nafion membrane and the proton exchange abilities of its SO₃⁻ functional groups is to monitor the number density of H₂O and H₃O⁺ gathered nearby to them either due to their mutual electrostatic attractive forces or in the course of delivering protonic conductions. The theoretical mean that can correlate all these consequences quantitatively via the graphical representations is none other than radial distribution function plots. Herewith, the radial distribution functions for the H₂O g(S − O_w) and H₃O⁺g(S − O_h) species in reference to SO₃⁻ sites of Nafion are presented by varying the adjacent state Vⁿ⁺-hydrated complexes as shown in Figures 11(a) and 11(b), respectively, and interpreted the results associated with them. In order to figure out their closed electrostatic interactions and the optimum nearest distance, the VMD graphical images rendered from the MD simulation datasets produced over the last 900 ps are also displayed in the respective insets. As can be seen in Figure 11(a), the four intense RDF spectral peaks of g(S − O_w) at radial distance r = 3.8 Å approximate the nearest region of H₂O in reference to SO₃⁻ site of Nafion wherein the optimum number density of the former species are gathered. This S − O_w radial distance r is very much shorter relative to the r estimated for SO₃⁻−SO₃⁻ (∼5.0 Å), SO₃⁻−Vⁿ⁺_complex (∼4.5 Å), and SO₃⁻–bare_Vⁿ⁺ (∼4.2 Å) (Figure 12), signifying that sufficient number density of the water molecules can be accommodated in between any two SO₃⁻ conductive sites of the Nafion by means of which the actual proton exchange and proton hopping processes are possible. In fact, the swelling properties of the Nafion and the length of its hydrophilic tunnel that separates any two of its SO₃⁻ groups made spherical micelles are dependable observables to the extent of which it can uptake and hold up freely moved polar solvents such as water into its conductive polymeric matrix. As per the literature analyses, these types of the micelles developed in the polymeric network of Nafion are of 4–6 nm diameter, and their water-filled connectors are of 1 nm width and 5 nm length, and more importantly, the water retention performance of it is a function of temperature [25]. It means the basic facts underlying the significant proton exchange and the ionic/protonic conduction deliveries of the Nafion membrane are none other than its net water retaining abilities for longer timescale which in turn is only furnished by its SO₃⁻ groups as discussed above. If the same diagram (Figure 11(a)) is furthermore referred, the most distinctive yet optimum number density of the H₂O molecules holding g(S − O_w) spectral band is seen in the case for V (V) complex [VO₂ (H₂O)₄]⁺ (this typical RDF peak has more height and width relative to others). This might be due to the least diffusion-gradient governed freely moved water obstructing abilities of the [VO₂ (H₂O)₄]⁺ caused by its smallest ionic radii (mean bond distance d_V=O = 1.6 Å, d_V−O = 2.024 Å) with a distorted octahedral geometry [7, 24], and the least effects of the induced-charge polarizabilities (lie in the addressable range of currently employed force-field packages) caused by its lowermost electric charge unit. Again, the similar analyses of the diagram confer the symmetric, flat, and nonzero regions appeared beyond r ≅ 4.5 Å, further speculating the excessive presence of significant number density of water molecules aligned periodically in an absolute outbound region of the SO₃⁻ groups. This explanation justified one of the most indispensable requirements of the Nafion membrane to exhibit exceptional proton exchange and free water diffusions across its polymeric hydrophilic matrix in both extreme and nonextreme wet conditions.

Alike to these interpretations, the microdistributions and the most possible involvements of the H₃O⁺ ions in exchanging H⁺ with the SO₃⁻ sites of Nafion can be depicted on the basis of g(S − O_h) plot shown in Figure 11(b). The first and second sharp RDF peaks standing at the radial distances r ≅ 3.0 Å and ≅ 3.6 Å explain the stronger yet unequal strength electrostatic interactions, and the nonzero region lying in between ∼4.0 Å and ∼5.5 Å speculates the occurrence of reversible movements of H₃O⁺ that takes place mainly in the time when the proton exchange between the SO₃⁻ and H₃O⁺ groups is proceeded. In general, the approaching number density of the H₃O⁺ species toward the SO₃⁻ sites of the Nafion is extremely affected by the size of the hydrated Vⁿ⁺ complexes that remain there with them in an absolute working electrolyte condition of the VRFB cell. In this case, among the Vⁿ⁺ complexes, the V (V) moiety [VO₂ (H₂O)₄]⁺ is the smallest one in size and, hence, cannot screen fully the approaching number density of H₃O⁺ toward the SO₃⁻ group of Nafion. Therefore, the relatively high abundance of the H₃O⁺ species exposed directly to the SO₃⁻ groups is observed in the system containing [VO₂ (H₂O)₄]⁺ complex. More importantly, the RDF plot predicted radial distance value r = 3.0 Å–3.6 Å between the SO₃⁻ and H₃O⁺ groups is distinctly smaller than the separation gap of SO₃⁻−SO₃⁻ and is noted as the lowest one (Figure 12); the optimum accommodation and the excellent involvement of the H₃O⁺ species in proton exchange processes with the SO₃⁻ sites of Nafion are justifiable. Again, the nonzero H₃O⁺ population density region of the same RDF plot even beyond the 7.0 Å radial distance indicates the presence of a symmetrically distributed hydrogen-bonded network of H₃O⁺ across the polymeric matrix of Nafion, which is in principle a fundamental basis of conferring both the vehicular and/or the Grotthuss proton hopping dynamics. Actually, the first mechanism takes place with the obsolete movement of the H₃O⁺ unit across the hydrogen-bonded water network retained inside the polymeric chain, and the second involves an exchange of H⁺ between the H₃O⁺ and nearby H₂O of the same matrix, either of them are absolutely facilitated by the symmetric population distributions of the both H₃O⁺ and H₂O specimens (Figures 11(a) and 11(b)) in the regions lying beyond the radial distance r = 6.0 Å in respect to SO₃⁻ group of Nafion.