Magnesium can be reversibly deposited from ethereal solutions of Grignard salts of the RMgX type aryl groups, and Cl, Br), and complexes of the type B; Br; R, or aryl groups, and These complexes can be considered as interaction products between bases and Lewis acids. The use of such complexes in ether solvents enables us to obtain solutions of reasonable ionic conductivity and high anodic stability, which can be suitable for rechargeable Mg battery systems. In this paper we report on the study of variety of complexes, where B, Sb, P, As, Fe, and Ta; Br, and F; and ethyl, phenyl, and benzyl (Bu, Et, Ph, and Bz, respectively) in several solvents, including tetrahydrofuran (THF), 2Me-THF, 1-3 dioxolane, diethyl ether, and polyethers from the "glyme" family, including dimethoxyethane (glyme), and (tetraglyme), as electrolyte solutions for rechargeable magnesium batteries. It was found that complexes (R, Bu and in THF or glymes constitute the best results in terms of the width of the electrochemical window from which magnesium can be deposited reversibly. These solutions were found to be suitable for use in rechargeable magnesium batteries. A variety of electrochemical and spectroscopic studies showed that these solutions have a complicated structure, which is discussed in this paper. It is also clear from this work that Mg deposition-dissolution processes in these solutions are far from being simple reactions of redox couple. The conditions for optimized Mg deposition-dissolution processes are discussed herein. © 2001 The Electrochemical Society. All rights reserved.
The success of R&D of nonaqueous lithium batteries that emerged as a practical reality more than 20 years ago 1 focused attention on other active metals such as calcium 2 and magnesium 3 as possible anodes in high-energy-density, nonaqueous batteries. While calcium cannot be deposited electrochemically from nonaqueous electrolyte solutions, 4 it has been known for decades that Mg can be deposited reversibly from ethereal solutions of Grignard reagents, namely, RMgX compounds, aryl groups, and groups such as Cl and Br. 5 6 7
Hence, it seems that the development of rechargeable magnesium batteries can be a realistic goal. However, due to the low anodic stability of the RMgX/ether solutions, it was clear that a first step in R&D of secondary Mg batteries should be the development of new electrolytic solutions of higher anodic stability, in which Mg can be reversibly, electrochemically deposited. About 10 years ago, Gregory et al. 8 reported on a solution (Ph and Bu are phenyl and butyl groups) in which magnesium can be deposited reversibly, which showed improved anodic stability as compared with Grignard salt solutions (1.9 V compared with 1-1.5 V vs. Mg reference electrode 8 ). These authors also suggested that Al, As, and P may replace the boron atom. However, it was not demonstrated nor reported as to its feasibility.
It was also reported recently that amidomagnesium halides in ether solvents may serve as promising salts, whose anodic stability is better than that of classical Grignard salt solutions, and in which Mg electrodes behave reversibly. 9 It is important to note at this stage that in contrast to lithium, magnesium cannot be reversibly plated from simple ionic salts such as etc., in aprotic solvents. Despite the vast amount of effort devoted to this issue for over half the century and some controversial publications, all the credible evidence for electrochemical magnesium deposition is involved with complex organomagnesium solutions in ethers. It is believed that the reason for the inability to plate or dissolve magnesium electrochemically from simple, inorganic/aprotic solvent solutions is a consequence of a buildup of efficient, compact passivation layers on the interface between the magnesium electrodes and the solution. Since magnesium is a divalent ion, these surface films form a real passivation layer, in contrary to the wrongly named "passivation" layers that formed on the monovalent lithium electrode. In the former case, the magnesium ions are located at a specific site in the lattice; thus, their mobility is close to zero, while in the latter case the small lithium ions possess a relatively high mobility in the solid lattice, thus enabling the passing of an ionic current through these interfaces. Electrochemical impedance spectroscopy (EIS) experiments performed on a variety of magnesium solutions support this thesis. 10 A few months ago, we reported on a new rechargeable magnesium battery system, 10 comprised of an Mg metal anode -Chevrel phase Mg insertion cathode, and an ethereal electrolyte solution. 10
We also report on the development of electrolyte solutions for rechargeable magnesium batteries. We studied a large variety of complexes of the types, where Al, Sb, Ta, Fe, As, and P; Br, and F, and Et, Ph, and Bz groups, in several ether solvents including tetrahydrofuran (THF), 2Me-THF, 1-3 dioxolane (DN), dimethoxyethane (DME), diglyme ], tetraglyme [], and diethylether (DEE). We also tested triethylamine as a solvent. Major tools for the study included voltammetry, chronopotentiometry, proton and NMR, and single-crystal X-ray diffraction (XRD).
The aim of this study was to discover electrolyte solutions, from which magnesium can be reversibly deposited and dissolved, and which are characterized by wide electrochemical windows (i.e., their oxidation potentials are as high as possible). We investigated the conditions for high efficiency of Mg deposition-dissolution processes, and for high anodic stability of the above-described solutions.
Experimental
All the preparations for the various electrochemical studies were carried out under highly pure argon atmosphere in VAC, Inc., glove boxes. Basic experimental aspects related to the study of nonaqueous magnesium electrochemistry was already described. 11 12 Highly pure inhibitor-free THF, 2Me-THF, and DN were obtained from Tomiyama, Inc., or Merck KGaA and could be used as received. Tetraglyme, diglyme, and glyme were obtained from Aldrich and distilled from their solutions with benzophenone containing lithium chips (blue solutions) in a homemade glove box under vacuum. Hexane from Aldrich was dried under molecular sieves. DEE (99.99%) and triethylamine (TEA, 99.99%) from Aldrich were used as received. was obtained from Aldrich as 1 M solution in heptane and was used as received after filtration. and were synthesized from the Grignard reagents by addition of dioxane. 13 from Aldrich was purified by sublimation at 150°C under vacuum. was obtained from Aldrich as 1 M solution in heptane and was used as received. All other acids were used as received.
The electrolytes were synthesized by mixing proper amounts of solution and or solution in the desired ratios. The mixing immediately formed white precipitants. The mixtures were stirred for 48 h. The heptane was then completely evaporated, and a proper amount of THF was added to form 0.25 M solutions of the complex with the general formulas In several experiments, THF was replaced by other solvents such as glyme, diglyme, tetraglyme, 2Me-THF, and several mixtures of THF with DEE, TEA, and DN. In several other experiments, the was replaced by different acids using the same synthetic procedure as described previously.
The electrochemical measurements were performed using an EG&G potentiostat model 273 or the Solartron model 1286 electrochemical interface (run by a 586 PC driven by the Corrware software from Scribner). All the voltammetric measurements were carried out in three-electrode cells in a parallel configuration. The working electrode was a polished gold plate; the counter and reference electrodes were a Mg foil and strip, respectively. All the electrolysis measurements were carried out in a three-neck cell in which the necks and the electrodes were separated by thin glass sinters. Pt served as the working electrode (WE), a magnesium strip as the reference electrode (RE) and magnesium or Pt foil as the counter electrode (CE). A measured quantity of solution from each compartment was withdrawn after the electrolysis and dried under vacuum to get a white solid that was dissolved in water. These solutions were analyzed for aluminum and magnesium by inductively coupled plasma (ICP) spectroscopy and for chlorides by potentiometric titration (analytical services by Aminolab, Inc., Israel).
The nuclear magnetic resonance (NMR) measurements were performed on Bruker, Inc., instruments (DPX-300 and DMX-600).
The XRD measurements were carried out using a Nonius KapaCCD diffractometer (0.7 deg φ and ω scans) using Mo Kα radiation. The analyzed crystal, precipitated from a solution, was embedded in a drop of viscous oil to prevent its exposure to air and moisture to which this material may be highly sensitive. It was then instantly cooled to 110 K in order to minimize a structural disorder of the THF ligands and large-amplitude thermal displacements of the lighter C atoms.
Results and Discussion
Figure 1 taken from Ref. 10 presents the typical voltammograms of THF solutions of Grignard salt, organoboron magnesium complex, and organochloro aluminate complex [with the formal formulae: BuMgCl, and ] with noble metal working electrodes (e.g., Au, Pt). This figure reflects the reversible magnesium deposition-dissolution processes that take place in the above-described systems, as well as the different anodic stability of the solutions, in the following order: In parallel studies it was found that reversible Mg deposition in all the solutions occurs in the absence of stable passivating surface films but are accompanied by complicated adsorption phenomena. 14 15
As described in the experimental section, the magnesium aluminates and borates, as well as other possible complexes of the type, B, As, P, Sb, Ta, and Fe, etc., Br, and F, and or aryl groups, can be expected to be products of a reaction between an base and an Lewis acid. We therefore examined a large variety of combinations of bases with Lewis acids in ethereal solutions in terms of the possibility of obtaining reversible Mg deposition and better anodic stability of the solutions. In general, cyclic voltammetry was used for these evaluations. In several cases, the cycling efficiency of Mg deposition-dissolution processes during prolonged cycling tests was also measured by chronopotentiometry.
Table I summarizes data from voltammetric measurements of THF solutions containing different acid-base combinations of different ratios. The typical cycling efficiency of Mg deposition-dissolution cycles and the electrolyte decomposition potentials (defined as the potential in which ) is provided for each combination in this table. The cycling efficiency data was calculated based on the ratio between the integration of the magnesium deposition peak and the integration of magnesium dissolution peak in the cyclic voltammograms (CVs). It should be noted that in addition to the acid-base combination mentioned in Table I, we also studied THF solutions of as a Lewis base, together with the following acids: and With none of these solutions could reversible Mg deposition processes be observed.
Table I.
A summary of Mg cycling efficiency and oxidation potentials of THF solutions containing different combinations of Lewis bases and Lewis acids of the and types, respectively, at different ratios as indicated. | ||||
---|---|---|---|---|
Lewis base | Lewis acid | Acid-baseratio | Mg cyclingefficiency | Electrolytedecompositionpotential |
1-2.00 | 95 | 2.10 | ||
1-1.75 | 95 | 2.05 | ||
1-1.50 | 97 | 2.00 | ||
1-1.25 | 94 | 1.90 | ||
1-1.00 | 96 | 1.80 | ||
1-0.75 | 95 | 1.65 | ||
1-2.00 | 92 | 2.25 | ||
1-2.00 | 80 | 2.08 | ||
1-2.00 | 88 | 2.15 | ||
1-2.00 | 75 | 2.40 | ||
1-1.75 | 74 | 2.30 | ||
1-1.50 | 74 | 2.25 | ||
1-1.25 | 83 | 2.15 | ||
1-1.00 | 86 | 2.10 | ||
1-0.75 | 92 | 2.00 | ||
1-1.50 | 86 | 1.77 | ||
1-1.00 | 68 | 1.60 | ||
1-0.66 | 91 | 1.40 | ||
1-0.50 | 93 | 1.30 | ||
1-1.00 | 80 | 1.20 | ||
1-0.50 | 93 | 1.75 | ||
1-0.20 | 71 | 1.50 |
Figures 2 3 4 5 show typical CVs, measured with solutions containing the following Lewis acids: and respectively, at different base-acid ratios, from which the data in Table I were extracted. As seen in Table I, the highest electrolyte decomposition potential was measured with solutions at the 1:2 stoichiometric ratio. However, the Mg cycling efficiency measured with this solution is rather low. As the acid/base ratio of these solutions decreases, the cycling efficiency increases, but the electrolyte decomposition potential decreases as well. A similar trend was also observed with solutions. As seen in Table I, the best performance in terms of high Mg cycling was obtained with the solutions containing the combinations. In addition, at acid/base the electrochemical window of these solutions on gold electrodes is wider than 2 V, which also makes these solutions highly promising for use in rechargeable magnesium battery systems. Therefore, we concentrated on the study of complex solutions.
Table II summarizes the effect of a cosolvent on the Mg cycling efficiency and electrolyte decomposition potentials of solutions. Several typical CVs related to different solvent combinations (from which the data in Table II has been extracted) are presented in Fig. 6. From this table, it is clear that the nature of the solvent plays a key role in the electrochemical behavior of these solutions. In general, the presence of solvents such as DN and 2Me-THF has a detrimental effect on the Mg cycling efficiency, while the presence of linear ethers such as DEE or tetraglyme does not adversely affect the Mg cycling efficiency. As seen in Table II, the behavior of the THF-tetraglyme solutions is very promising. High Mg cycling efficiency was measured with them, while the electrolyte decomposition potential remains quite high vs. Mg RE). These results are very important in light of the high boiling point and the very low volatility of tetraglyme, properties that are important for battery electrolyte solutions, from the standpoint of safety.
Table II.
A summary of the effect of cosolvent on the magnesium cycling efficiency and oxidation potentials of complex solutions. | |||
---|---|---|---|
Co solvent | Percent byvolume(%) | Mgcyclingefficiency (%) | Electrolytedecompositionpotential |
0 | 95 | 2.10 | |
Dioxolane | 10 | 78 | 1.60 |
Dioxolane | 30 | 85 | 1.50 |
Diethyl ether | 10 | 83 | 2.15 |
Diethyl ether | 30 | 94 | 1.57 |
Triethylamine | 10 | 99 | 1.62 |
Triethylamine | 20 | 97 | 1.44 |
Triethylamine | 30 | 89 | 1.79 |
2Me-THF | 10 | 86 | 2.10 |
2Me-THF | 30 | 88 | 2.30 |
2Me-THF | 50 | 72 | 2.33 |
2Me-THF | 90 | 55 | 2.00 |
Tetraglyme | 10 | 74 | 2.24 |
Tetraglyme | 20 | 92 | 1.73 |
Tetraglyme | 30 | 95 | 1.55 |
Tetraglyme | 50 | 97 | 2.2 |
Tetraglyme | 100 | 90 | 2.2 |
These results focus attention on THF and tetraglyme solutions containing complexes as very promising electrolytic solutions for rechargeable magnesium batteries, as was also confirmed by galvanostatic experiments with these solutions: magnesium was deposited on gold electrodes followed by repeated electrochemical dissolution-deposition (constant current) of 20-25% of the initially deposited magnesium. After 80-100 cycles, the residual magnesium was electrochemically dissolved, and from the charge involved in this last process, the loss per cycle, and hence, the average cycling efficiency of the Mg, were calculated, and reached values close to 100% for both THF and tetraglyme solutions. These results are in line with parallel studies that suggested that there are no irreversible surface reactions between the magnesium deposits and the solution in THF and tetraglyme solutions of the solutions. 14 15 However, the results reported herein clearly demonstrate that Mg deposition-dissolution processes in these solutions are very complicated and are influenced by both the nature of the Lewis acid-base present and the nature of the solvent molecules as well. In an attempt to analyze the structure of the complexes, single crystals were precipitated from the THF solutions of this complex with an acid/base ratio close to 2. The crystals were obtained either by precipitation at low temperatures or by addition of nonpolar solvents (e.g., hexane). Both crystallization methods yield identical single crystals with the structure presented in Fig. 7.
However, it was surprising to discover that the crystals that can be precipitated from solutions are not the electroactive species in these solutions, as THF solutions containing the material whose structure is presented in Fig. 7 do not show reversible magnesium deposition-dissolution as the solution from which they were prepared. In contrast, Mg could be deposited reversibly from the residual solution remaining after the precipitation of the single crystal.
Scheme 1 summarizes the conditions for obtaining reversible Mg deposition-dissolution processes in based solutions. This scheme describes a series of experiments in which solid species were precipitated from the solutions and redissolved in THF, followed by electrochemical measurements. After each precipitation the residual solution was also tested, as seen in this scheme. It appears that addition of to solutions that show poor performance considerably improves their ability to reversibly, electrochemically deposit magnesium. These experiments clearly demonstrate that there is no direct correlation between the precipitated salts and the electroactive species in solutions that lead to the highly reversible Mg deposition-dissolution processes observed. We speculate that the true structure of the system in solution phase is better described as a series of equilibria, in the same manner as the most advanced studies showed for the components of Grignard reagent solutions. The Mg core, stabilized by THF molecules, may be bound to both halide or alkyl groups, counterbalanced by anions, whose structure may also be a complex series of equilibrium. In fact, the structure of both the cationic and the anionic species in these solutions may be determined ad hoc, depending on the reactant acidbase ratio, temperature, total concentration of the acid and base, and the nature of the solvent molecules. However, upon crystallization into solid substance, either at low temperatures or by the addition of
Scheme 1.a nonpolar solvent, they form a single, solid structure (Fig. 7) that is probably the most stable one at oversaturation concentration. Precipitation of each crystals from solutions of acid/base ratio equal to two, leaves Lewis acid molecules in the solution phase. They probably interact with residual in the solutions, thus forming the electroactive solutions as indicated in Scheme 1. The increase in the concentration of the organoaluminate at the expense of the due to the precipitation of the chloromagnesium aluminate complex (Fig. 7), decreases the electrolyte decomposition potential of the residual solution, as expected (Scheme 1). It is therefore clear that as the electron withdrawing power of the acid is higher (i.e., high chlorine content), it stabilizes the complex salts against electrochemical oxidation. This property of the Lewis acid is reflected by the high electrolyte decomposition potential of the solutions containing as the acid (Table I and Fig. 3). The mechanism of the anodic stabilization of the complexes by the Lewis acids could be further studied by NMR analysis of predeuterated THF (d8) solutions of a and or at different ratios. From the NMR measurements, it is clear that our starting materials (as received from Aldrich) contain both -butyl and sec-butyl groups, in a ca. 1:0.7 ratio. This was confirmed by 2D-NMR techniques ( correlation and correlation). The chemical shifts of the alkyl moieties are as follows. -butyl: δ −0.59, 1.57, 1.29, and 0.90; δ 7.98, 33.83, 32.29, and 14.40. sec-butyl: δ 1.28, −0.30, 1.59, 1.67, and 0.99; δ 22.50, 20.63, 34.11, and 17.40.
We prepared solutions comprised of THF d8 and at different ratios, and their NMR spectra were measured. The peaks of the hydrogens of the ethyl groups, as well as the peaks of the hydrogens on carbons 2-4 of the butyl groups, were quite similar to those separately measured with the THF or the THF solutions. However, the NMR peaks of the hydrogens on the carbons bound to magnesium (either of the primary or the secondary Bu group) are shifted to a lower field as the acid/base ratio is higher. This correlates with the increase in the electrolyte decomposition potential (high ) as the acid/base ratio in these solutions is higher. These results are summarized in Table III and are demonstrated in Fig. 8. Hence, Table III and Fig. 8 show the linear correlation between the increase in the electrolyte decomposition potentials and the chemical shift of the hydrogen on the carbons bound to the magnesium in as the acid/base ratio increases. We can conclude that what limits the anodic stability of these solutions is the oxidation of the R-Mg bond. Hence, the presence of the Lewis acid leads to electron-withdrawal interactions, namely, electrons from the electron-rich R-Mg bond are attracted by the Lewis acid, and thus, oxidation of the R-Mg bond becomes harder as the concentration of the Lewis acid in solution is higher.
Table III.
The dependence of the chemical shift in the NMR spectra of the hydrogens bonded to the carbon of the Bu groups which are bonded to the Mg atom, and the solution oxidation potential as the function of the or ratios. | |||
---|---|---|---|
Acid-base ratio | (ppm) | (ppm) | Electrolytedecompositionpotential |
0 | 1.41 | ||
1 | 0.1 | 2.05 | |
1.5 | 0.15 | 2.1 | |
2 | 0.3 | 0.09 | 2.25 |
0.35 | 0.25 | 2.55 |
We were able to obtain more information about the complicated structure of the complexes in THF solutions from electrolysis of solutions in three-compartment cells. Such experiments can offer information about the transference numbers of the ionic species in the solution. 16 These transference numbers can be calculated from the change in the solution components concentration during bulk electrolysis. Since the accurate structure of the electroactive species in the complex salt solutions is unknown, we chose to follow the changes in the concentration of the individual elements, Mg, Al, and Cl, which constitute the possible species in the solutions. In a typical experiment we passed an amount of charge that is equal to 0.4 electrons per salt molecule, as defined by the simplistic equation. The electrolysis was carried out in a fine glass cell that was separated into a three-component cell (cathodic, middle, and anodic part). At the end of electrolysis, 4 mL of solution was withdrawn from each compartment, the THF was evaporated, and the solid residue was reacted with 4 mL of water and analyzed for its Cl, Mg, and Al concentration.
Table IV summarizes the data from three experiments. As was shown in similar systems, accurate values for transference numbers could not be calculated from these measurements. In contrary to simple ionic solutions, in which precise values can be obtained, dynamic equilibrium systems composed of complex ions can be studied qualitatively only. Despite these difficulties, a few conclusions could be extracted from these measurements. In the experi-ments labeled 1 and 2, magnesium served as cathode as well as anode. In these experiments the magnesium is electrochemically dissolved in the anode compartment with simultaneous magnesium deposition in the cathode compartment. These electrochemical reactions are accompanied by mass transfer of ions for charge compensation. In experiment no. 3, platinum mesh served as the anode; thus, the oxidation reaction was not magnesium dissolution. During this experiment evaluation of gas was observed in the anode compartment. The following conclusions can be drawn from these experiments. In all cases, despite the consumption of magnesium ion by metal plating, the magnesium ion concentration increased in the catholyte, as compared with its content in the solution of the middle compartment. This means that the migrating cation contains more than one ion of Mg, based on the assumption that the transference number of the cation is less than 1. As indicated in Table IV, concentration is reduced in the cathode's compartment during electrolysis. Based on this observation we can conclude that the migrating cation is probably not which is the cation species of the solid precipitated from these solutions (Fig. 7). The concentration decreases upon electrolysis in the catholyte while it increases in the anolyte. This means that aluminum is contained in the moving anionic species, which migrates toward the anode side. Upon electrolysis, the Mg concentration decreases in the anode solution, meaning that the Mg ions migrate as dimers or oligomers toward the cathode side.
Table IV.
A summary of analysis of electrolysis experiments with solutions. [Concentrations of the various elements in the catholyte, anolyte, and in the middle chamber (C, A, M, respectively) in several experiments as indicated]. | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Experiment | Electrodes | Cl (g/L) | Mg (g/L) | Al (g/L) | |||||||
Cathode | Anode | C | M | A | C | M | A | C | M | A | |
1 | Mg | Mg | 31 | 36 | 31 | 5.7 | 4.9 | 4.1 | 8.1 | 12 | 12 |
2 | Mg | Mg | 31 | 35 | 21 | 5.5 | 4.7 | 2.8 | 7.8 | 10 | 8.8 |
3 | Mg | Pt | 30 | 32 | 32 | 6.3 | 4.4 | 1.0 | 8.2 | 12 | 16 |
Conclusion
Interactions between Lewis bases of the type and Lewis acid of the type, aryl, B, and mostly Cl, in ethereal solutions, may form complex solutions of relatively high anodic stability, high ionic conductivity (several mS at room temperature), and from which magnesium can be reversibly deposited. From a wide variety of combinations, THF and tetraglyme solutions of mixtures of and at acid/base ratios close to 2, exhibit especially promising properties in terms of anodic stability and magnesium deposition-dissolution cycling efficiency close to 100%. Hence, these solutions are very suitable for use in rechargeable Mg batteries. At low temperatures or upon addition of nonpolar solvents to these solutions, they precipitate as a single solid product that was clearly identified as (6THF)⋅ However, this solid product does not reflect the structure of the electrolytes in solutions. In fact, the electrolyte's structure in solutions may be indefinable, and the stoichiometry of both the cations and the anions in solutions may be determined ad hoc, depending on the acid-base ratio, the nature of the other molecules, and the temperature. Based on electrolysis experiments, we can assume that the active cation includes more than one Mg ion, and hence, may have the following general structure: ⋅ROR, while the anion probably has the general structure of It appears that the presence of R group as part of the cation structure is crucial for obtaining reversible magnesium deposition (Scheme 1). The anodic stability of these systems is determined by the stabilization of the R-Mg bonds by the Lewis acid in solution. We assume that at sufficiently high potentials, electrons can be readily withdrawn from the carbon-magnesium bonds, and radical species can thus be formed. Hence, the high electron affinity of the Lewis acid probably leads to the existence of strong interactions between the Lewis acids in solution and the R-Mg bonds in the Lewis bases that inhibit their easy oxidation. Therefore, an increase in the Lewis acid concentration enhances the solution's anodic stability. However, strong Lewis acids mean a high chlorine content at the expense of R groups (in the cation species), which are important for the reversibility of the Mg deposition process. Thereby, the optimization of these solutions for use in rechargeable Mg batteries is based on a compromise between the strength and concentration of the Lewis acid needed for the anodic stabilization and the high content of R groups in the di- or polymagnesium cations required for obtaining reversible Mg deposition. It therefore appears that combinations of and close to a 2:1 ratio bring these solutions to an optimum in terms of the considerations given.
Acknowledgment
Partial support for this work was obtained from the BMBF, the German Ministry of Science, in the framework of the DIP program for Collaboration between Israeli and German Scientists, and by the Israeli Ministry of Science and Technology, in the framework of the Infrastructure Research Program.
Bar Ilan University assisted in meeting the publication costs of this article.