7394
J. Phys. Chem. B 2001, 105, 7394-7398
Transport Coefficients of an Inorganic Brownian Particle in Solution: The Tungstosilicate Anion T. Olynyk, M. Jardat,* D. Krulic, and P. Turq Laboratoire Liquides Ioniques et Interfaces Charge´ es, case courrier 51, UniVersite´ P. et M. Curie, 4 place Jussieu, F-75252 Paris Cedex 05, France ReceiVed: October 18, 2000; In Final Form: April 2, 2001
Self-diffusion coefficients of the tungstosilicate anion SiW12O404- have been determined in an HClO4 medium of various concentrations by voltammetric measurements at a glassy carbon electrode. The results have been extrapolated to zero ionic strength using an analytical transport theory (mean spherical approximation (MSA) transport calculations). The electrical conductivity of aqueous sodium tungstosilicate solutions with and without added salt has been then calculated using both the MSA transport method and Brownian dynamics simulations. Calculations have been compared to our experimental data. It has been shown that the extrapolation at infinite dilution of the conductivity of sodium tungstosilicate solutions gives a value for the self-diffusion coefficient of the polyion consistent with the voltammetric determination. Moreover, the sodium tungstosilicate electrolyte exhibits a behavior similar to a simple electrolyte in aqueous solution, whereas the addition of a salt yields an ionic association phenomenon.
I. Introduction Continuous solvent models of solutions are currently considered as “primitive” and reasonable only for large solute particles such as colloids, polyelectrolytes, or macromolecules. However, it has long been known1 that the Brownian motion is a dynamical limit for any object of large mass and size with respect to solvent molecules for long observation times. If those entities are charged, they undergo not only the diffusional Brownian motion but also an electric force in the presence of an external electric field. Here, relatively small but heavy species are considered. The interest of studying small Brownian particles (brownons) is that they have well-defined size, molecular weight, and charge, contrary to large objects whose structural properties are usually averaged values of distributions seldom monodisperse. Charged particles with their counterions in solution coexist possibly with an added salt, which strongly influences electrostatic interactions. Thus, the dynamical properties of brownons will depend not only on their charge, mass, size, and concentration but also on the relative value of their concentration with respect to that of a possible added salt. There are few real examples of small Brownian particles. The structural2 and dynamical3-5 properties of cryptate salts of different charges in water have been already investigated. The cryptate is an organic complex of roughly spherical shape, made of an alkaline cation included in a cryptand6,7 molecule. The radius of this ion is of the order of 0.6 nm. In the present paper, an inorganic negatively charged Brownian particle is studied, the tungstosilicate anion SiW12O404-. Its size is similar to that of the cryptate ion but its charge is higher (-4 with respect to +1 or +2). The tungstosilicate ion, which contains numerous oxygen atoms and two other elements in positive oxidation states, is an example of heteropolyanions. For more than a century, the number of publications concerning heteropolyelectrolytes has been in constant expansion. * Corresponding author, E-mail:
[email protected], Fax number: 33 144273834.
Whereas the structural, spectroscopic, and redox properties of heteropolyanions as well as their role in homogeneous catalysis have been widely studied,8 their transport properties in aqueous solution have attracted less interest. The SiW12O404- ion has a Keggin structure, which involves WO6 octahedrons sharing their edges to form 3-octahedron groups linked to one another by corner sharing. The center of this assemblage is occupied by the silicium atom with a tetrahedral coordination.9 This compound is heavy, rigid, and roughly of spherical shape. Tungstosilicates are chemically unstable in nonacidic solutions and may then require the addition of a slightly acidic supporting electrolyte. Aqueous solutions of tungstosilicate ions are also easily obtained from solid tungstosilicic acid (H4SiW12O40), which ensures a pH sufficiently low to stabilize the complex. The purpose of this paper is to study the dynamical properties of tungstosilicate ions in aqueous solution. We have determined the diffusion coefficient of this species by voltammetry in the presence of a large excess of supporting electrolyte, which suppresses the migration of the electroactive species. The selfdiffusion coefficient at infinite dilution is indeed the key parameter in the analysis of other transport properties in the framework of continuous solvent models. MSA (mean spherical approximation) transport calculations, which combine Onsager’s continuity equations with MSA equilibrium correlation functions, do use, as input parameters, the self-diffusion coefficients at infinite dilution of ions. Such calculations have allowed, for example, the calculation of transport coefficients of simple electrolyte solutions (1-1, 1-2, and 2-2 electrolytes) which have been found in good agreement with the experimental data. Self-diffusion coefficients10,11 and the conductivity of two simple ionic species12 and of ionic mixtures,13,14 in the 0-1 molar concentration range have been predicted. MSA transport theory has the advantage to be an analytical treatment of equilibrium15,16,17 and of transport properties of electrolyte solutions, giving asymptotically Debye-Onsager limiting laws in the low concentration range. A version of such a theoretical approach (MSA tracer diffusion) has been used in this paper to extrapolate
10.1021/jp003833k CCC: $20.00 © 2001 American Chemical Society Published on Web 07/14/2001
Transport Coefficients of Tungstosilicate Anion the self-diffusion coefficient of the tungstosilicate ion at zero ionic strength from data obtained by voltammetric measurements. To the best of our knowledge, it is the first experimental determination of this value. The self-consistency and adequacy of models and experiments have been then checked by measuring the electrical conductivity of tungstosilicate solutions of various concentrations (3 × 10-3 to 5 × 10-2 mol dm-3) in the presence of increasing quantities of added salt (sodium sulfate). Again, MSA transport equations have been used to analyze experimental data. For large concentrations and highly dissymmetric electrolytes, such as those encountered in the present work, it has been necessary to check the validity of the approximate MSA expressions. This has been done by comparison with numerical simulations, which give an exact numerical solution of the dynamical continuous solvent problem. An efficient Brownian dynamics (BD) simulation method has been recently introduced18 in order to study the dynamical properties of electrolytes in aqueous solution. This method has allowed the calculation of transport coefficients of simple electrolyte solutions,18 electrolyte mixtures,19 and cryptate solutions,5 the latter being asymmetric in size. It has been shown in the case of simple electrolytes (1-1)5,18,19 that hydrodynamic interactions must be taken into account to agree with experiments. In the present work, Brownian dynamics simulations are performed in order to calculate the electrical conductivity of sodium tungstosilicate solutions in the presence of sodium sulfate. The paper is organized as follows. In section II, we describe the experiments. The essential points of the transport theory and of the simulation procedure are presented in section III. Finally, section IV deals with results and discussion. II. Experimental Section For any experiment, fresh aqueous solutions prepared from tungstosilicic acid hydrate (Aldrich) in ultrapure water of 18 MΩ‚cm resistivity were used. All chemicals were of analytical grade purity. The concentration of the tungstosilicic acid was determined by acid-base titration. Voltammetry. The electrochemical behavior of tungstosilicate species has been already investigated.20-22 The reduction process of the SiW12O404- ion was studied by cyclic voltammetry in acidic solutions.22 Attention was focused on the first of the different observed reduction waves. In HClO4 medium, up to 3 mol dm-3, this wave corresponds to a monoelectronic reversible reaction.22 We have determined the self-diffusion coefficient of the tungstosilicate ion by square wave voltammetry (SWV), at a glassy carbon electrode, using the wave corresponding to the first oxydo-reduction stage. This wave behaves, effectively, as a simple reversible wave without any apparent complication and presents a well-developed foot in the anodic direction. With a mercury drop electrode, we observed an erratic behavior under the same conditions. SWV allows a good separation between two consecutive waves and a good rejection of background currents. In addition, the degree of reversibility of the electrochemical reaction can be easily checked from the half-wave width. Experiments were performed with a PAR 283 potentiostat interfaced with a PAR model 616 electrode. The working electrode was a PAR glassy carbon disk electrode of surface S ) 12.6 mm2. An Ag/AgCl NaCl saturated reference electrode was used (measured potential -52 mV/calomel saturated electrode). The counter electrode was a platinum wire. Solutions were deaerated with argon and thermostated at 25 ( 0.1 °C.
J. Phys. Chem. B, Vol. 105, No. 31, 2001 7395
Figure 1. Square wave voltammetry of the tungstosilicate ion (5 × 10-4 mol dm-3) in HClO4 (0.10 mol dm-3).
TABLE 1: Self-Diffusion Coefficients of the Tungstosilicate Ion in Aqueous Solution Obtained by SWV Measurements, at Various HClO4 Concentrations Cac Cac/mol dm- 3 D/10- 10 m2 s-1
0.11 4.6
0.14 4.6
0.10 4.5
0.20 4.5
Cac/mol dm-3 D/10- 10 m2 s-1
0.40 4.9
0.60 4.8
0.80 4.8
1.00 4.9
0.30 4.7
Before each experiment, the glassy carbon working electrode was cleaned with a polishing cloth, “Buehler microcloth”, and impregnated with “Buehler” diamond suspension of 0.1 µ granulosity. After this treatment, the electrode was abundantly rinsed with ultrapure water. It can be considered that the SWV waveform results from the superimposition of impulses of amplitude ∆E and of duration τ on the steps, of height ∆Es, of a potential staircase.23 The experiments were performed with ∆Es ) -2 mV, ∆E ) -50 mV, and τ ) 40 ms. For these values, the half-wave width of a monoelectronic reversible wave, at 25 °C, is 99 mV24 and the peak height, ∆ip, is given by23
|∆ip| ) 0.555FSc* xD/πτ
(1)
where F is Faraday’s constant, c* the bulk concentration of the electroactive species, and D its diffusion coefficient. We have found that, for concentrations of HClO4 between 0.1 and 1 mol dm-3, the peak height is proportional to the bulk concentration in the range 5 × 10-5 to 10-3 mol dm-3. For all of these waves, the half-wave width was 99 ( 3 mV. The diffusion coefficients were fitted using eq 1 from experiments performed with concentrations of tungstosilicate ion close to 5 × 10-4 mol dm-3 in HClO4 medium of various concentrations. For this concentration of the tungstosilicate ion, the foot of the anodic part of the wave is horizontal and the individual current densities do not exceed 1.2 µA mm-2. The wave presented in Figure 1 was obtained with c* ) 5.0 × 10-4 mol dm-3 in HClO4 0.10 mol dm-3. For each concentration of HClO4, at least five measurements were performed and the mean value of ∆ip was taken into account. The scattering of the measurements was of the order of 1.5%, which corresponds to an uncertainty of 3% on the self-diffusion coefficient. In Table 1, the values of the diffusion coefficients determined in HClO4 [0.1-1 mol dm-3] are presented. Conductivity. Four groups of aqueous solutions were prepared. The first one contained sodium tungstosilicate, and other
7396 J. Phys. Chem. B, Vol. 105, No. 31, 2001
Olynyk et al.
TABLE 2: Experimental Molar Conductivity of Na4(SiW12O40)(Na2SO4)n in Aqueous Solution at 298 Ka n
C/mol dm-3
0 0 0 0 1 1 1 1
3.54 × 1.77 × 10-2 7.09 × 10-3 3.54 × 10-3 5.05 × 10-2 2.53 × 10-2 1.26 × 10-2 6.31 × 10-3
a
10-2
κmes/mS cm-1 10.34 5.55 2.46 1.32 20.59 11.53 6.28 3.38
Λ/S cm2 mol-1 285 311 344 368 407 439 479 515
n
C/mol dm-3
κmes/mS cm-1
Λ/S cm2 mol-1
2 2 2 2 3 3 3 3
5.05 × 2.53 × 10-2 1.26 × 10-2 6.31 × 10-3 5.05 × 10-2 2.53 × 10-2 1.26 × 10-2 6.31 × 10-3
26.37 14.88 8.15 4.40 31.67 17.98 10.01 5.45
521 586 642 693 626 711 791 860
10-2
C is the concentration of the electrolyte, κmes the measured conductivity, and Λ the molar conductivity deduced from κmes.
groups contained both sodium tungstosilicate and sodium sulfate electrolytes. Sodium hydroxyde was added to mixtures of tungstosilicic and sulfuric acids under pH control to prepare sodium tungstosilicate and sodium sulfate electrolytes. The pH of the solutions was always less than 4.5. The formula of the dissolved ionic compound was finally Na4SiW12O40(Na2SO4)n with n in the range 0-3. For each value of n, several solutions of this electrolyte mixture were then obtained by dilution, in the range of 5 × 10-3 to 5 × 10-2 mol dm-3. The conductivity cell (Inforlab Chimie) had platinized electrodes. The cell constant was determined from a KCl solution with conductivity equal to 12.880 mS cm-1 at 25 °C. The cell was immersed in a water bath, the temperature being maintained constant at 25.0 ( 0.1 °C. Solution resistances R were measured with a Wayne Kerr bridge (model 6425) at four frequencies f: 10, 5, 2, and 1 kHz and extrapolated to infinite frequency by linear regression of R ) f(1/f). The ionic conductivity of hydronium ions was subtracted in order to obtain the conductivity κ of the “pure” electrolyte Na4(SiW12O40)(Na2SO4)n in aqueous solution. This correction was always less than 2% of the measured value. The molar conductivities Λ ) κ/C of the electrolyte, with C being its molar concentration, i.e., the SiW12O404- concentration in any case, are given in Table 2. III. Theoretical Section Transport Theory in the Mean Spherical Approximation. Continuous solvent models are based on the assumption that the solvent is a continuum characterized by its dielectric constant and its viscosity. In this case, the dominant forces that determine the deviations of transport processes in electrolyte solutions from the ideal behavior (i.e., without any interactions between ions) are assumed to be hydrodynamic interactions and electrostatic relaxation forces.25 When external forces are applied on the solution, the different particles do not respond in the same manner and have thus different drift velocities. Hydrodynamic forces (friction forces), mediated by the solvent, tend to equalize these velocities. This effect is called the electrophoretic effect. Moreover, when the ionic equilibrium distribution in an electrolyte solution is perturbed by an external force, electrostatic forces appear, which tend to restore the local electroneutrality. This effect is called the relaxation effect. Recently, several transport phenomena have been described in the framework of a linear response theory in which Onsager’s continuity equations are combined with the MSA equilibrium correlation functions, using the Green’s response functions.10,12,13,26-28 In the present study, this transport theory has been used to calculate the self-diffusion coefficient of a tracer, here the tungstosilicate ion, in a concentrated electrolyte solution, here HClO4 medium. Three ionic species were then considered in this calculation: SiW12O404-, H+, and ClO4-. Another version of this theory has allowed us to compute the electrical conductivity of aqueous solutions containing three different ions (Na+, SiW12O404-, and SO42-).
In the case of the tracer diffusion in a concentrated electrolyte solution, the relaxation effect mainly causes departures from the ideal behavior. The self-diffusion coefficient of the tracer, denoted by D1, is then given by10,11
(
D1 ) D01 1 +
)
δk1 k1
(2)
where δk1/k1 takes into account the relaxation effect at the first order. The detailed expression of this correction can be found in ref 11. To calculate the electrical conductivity of an ionic solution, both relaxation and electrophoretic effects must be taken into account. An expression of the conductivity as a function of these corrections at the first order has been proposed by Durand-Vidal et al.13 in the case where the solution contains three different ions:
κ)
e2NA kBT
3
o rel ciD0i z2i (1 + δVhyd ∑ i /Vi )(1 + δki /ki) i)1
(3)
where NA is the Avogadro number, kB the Boltzmann constant, e the elementary charge and T the temperature. Index i is referred to ion i of charge zi with ci its molar concentration, and Di0 its self-diffusion coefficient at infinite dilution. δVhyd is the hydroi dynamic correction on the velocity of species i and Vio its infinite dilution velocity. δkrel i is the projection of the relaxation force acting on species i along the electric field axis and ki the external force acting on i (due to the external electric field). o rel The detailed expressions of corrections δVhyd i /Vi , and δki /ki can be found in ref 13. Brownian Dynamics Simulation. Transport coefficients of electrolyte solutions may be also computed from Brownian dynamics simulations in the framework of the continuous solvent model.18 Four systems were simulated at 298 K, which correspond to the solutions studied by conductivity: Na4(SiW12O40)(Na2SO4)n, with n varying from 0 to 3, the electrolyte concentration being in any case 0.025 mol dm-3. The RotnePrager tensor31 has been used to calculate hydrodynamic interactions. The diffusion matrix has been calculated with the same approximations as those used previously.18 The solventaveraged interaction potential between ions Vij is modeled by pairwise repulsive soft-core interactions32 and the Coulomb part:
(
)
Aije2 ai + aj 1 Vij(r) ) 4π0 12(ai + aj) r
12
+
qiqj 4π0rr
(4)
Here ai is the radius of the ith ion, 0 the dielectric constant of the vacuum, r the relative dielectric constant of the pure solvent, taken equal to 78.54, and Aij is an adjustable parameter. A similar interaction potential has been already used to simulate the dynamical properties of simple electrolyte solutions from Brownian dynamics,18,19 and it has led to results in good
Transport Coefficients of Tungstosilicate Anion
J. Phys. Chem. B, Vol. 105, No. 31, 2001 7397
TABLE 3: Radii of Ions and Self-diffusion Coefficients at Infinite Dilution Used in the Numerical Simulations r/nm D°/10-9 m2 s-1
Na+
SO42-
SiW12O404-
0.130 1.333
0.258 1.065
0.525 0.51
agreement with experiments. The coefficients Aij have been chosen in order that the minimum of the potential between ions of opposite charges corresponds to a distance that is the sum of their radii. The same expression for the parameters Aij has been taken for any ionic pairs: Aij ) zizj/r. Radii of ions and selfdiffusion coefficients at infinite dilution are given in Table 3. The self-diffusion coefficient of the tungstosilicate ion extrapolated at zero ionic strength was deduced from the voltammetric experiments as it is explained in the following part. In each simulation, 64 tungstosilicate ions and appropriate numbers of other species were placed in a cubic box with periodic conditions. To compute the soft core interactions we used a spherical cutoff of half a box length, applying the minimum image convention. Coulomb interactions were computed using the Ewald summation technique33,34 with the conducting boundary condition. The simulation time step ∆t was chosen such that the acceptance ratio remained higher than 70%, which led to values between 0.12 and 0.18 ps. Dynamical properties were calculated by averaging over five successive trajectories, each containing 214 or 216 time steps according to the total number of particles. It must be noted that the computation took several days.
Figure 2. Self-diffusion coefficient of the tungstosilicate ion in perchloric acid solution at 298 K as a function of the square root of the perchloric acid concentration Cac.
IV. Results and Discussion Extrapolation of the Self-Diffusion Coefficient of the Tungstosilicate Ion at Zero Ionic Strength. The self-diffusion coefficient at infinite dilution of the tungstosilicate ion, D0SiW12O404was fitted from the experimental results given in Table 1 with the aid of eq 2, which gives the self-diffusion coefficient D1 of a tracer in a concentrated electrolyte solution. In this calculation, the values taken for radii of H+ and ClO4- ions were the ionic radii, respectively, 0.138 and 0.240 nm. The self-diffusion coefficients of hydronium and perchlorate ions at infinite dilution were taken from their individual limit ionic conductivity:35 D0H+ ) 9.309 × 10-9 m2 s-1and D0ClO4- ) 1.792 × 10-9 m2 s-1. The value taken for the tungstosilicate radius was the crystallographic value which was also used in the simulations (see Table 3). Figure 2 presents the self-diffusion coefficients as functions of the square root of the perchloric acid concentration. Both experimental data and results of the MSA transport calculations are given. The self-diffusion coefficient of the tungstosilicate ion extrapolated at zero ionic strength by the theoretical method is
D0SWV(SiW12O404-) ) (0.51 ( 0.02) × 10-9 m2 s- 1 It should be noticed that this estimated error on D° concerns only the experiments. The value obtained from the Stokes relation D° ) kBT/6πηr with r being the crystallographic radius 0 (SiW12O404-) ) 0.47 × 10-9 of the tungstosilicate ion is DStokes 2 1 m s , which is very close to the previous value. Electrical Conductivity. The numerical resolution of eq 3 has allowed us to calculate the electrical conductivity of the four groups of solutions studied in the Experimental Section. Brownian dynamics simulations with and without hydrodynamic interactions have been also performed at a given tungstosilicate concentration for several quantities of added salt. Figure 3 presents the ratio between the molar conductivity Λ of the electrolyte, Na4(SiW12O40)(Na2SO4)n, in aqueous
Figure 3. Molar conductivity of Na4(SiW12O40)(Na2SO4)n aqueous solutions at 298 K obtained by Brownian dynamics, by experiments and by MSA transport calculations for n varying from 0 to 3.
solution and its limiting value Λ0, obtained by experiments, by Brownian dynamics, and by MSA transport calculations. The tungstosilicate concentration was 0.025 mol dm-3 in any case, and n varied between 0 and 3. Simulation results are in good agreement with MSA transport calculations. The BD simulation including HI was already checked for 1-1 electrolytes in solution:5,18,19 it is shown here that this method also allows the description of electrolytes that are disymmetrical in charge and in size. Figure 4 gives the molar conductivity as a function of the square root of the electrolyte concentration. Both experimental data and results of the MSA transport calculations are reported. In any case, the parameters of the calculations were the same as those involved in the interaction potential described in section III.B. In particular, the self-diffusion coefficient at infinite dilution of the tungstosilicate ion was the one deduced from voltammetric experiments as described in the latter section. All of the parameters used in simulations as well as in MSA transport calculations were thus issued from experiments. As it can be observed in Figure 4, the agreement between experiment and theoretical calculations is better in the case where n ) 0, which corresponds to a sodium tungstosilicate aqueous solution without added salt. In this case, the value of
7398 J. Phys. Chem. B, Vol. 105, No. 31, 2001
Olynyk et al. framework of the Brownian dynamics simulation, whereas MSA transport calculations are practically restricted to charged hard sphere mixtures with three species at the most. The Brownian dynamics with hydrodynamic interactions describes properly the conductivity of electrolyte solutions and appears then to be the appropriate method to study such a system. Nevertheless, as it has been noted in this paper, this simulation method requires at the moment long computer times and needs to be improved in order to more easily treat large numbers of particles. Work on this point is in progress. Acknowledgment. We gratefully thank S. Durand-Vidal and O. Bernard who allowed the use of their MSA transport calculation programs and C. Treiner for his help concerning conductivity measurements. References and Notes
Figure 4. Molar conductivity of Na4(SiW12O40)(Na2SO4)n aqueous solutions at 298 K as a function of the electrolyte concentration for n varying from 0 to 3.
the self-diffusion coefficient at infinite dilution that would be obtained by extrapolation of the experimental data using MSA transport theory is D0conductivity(SiW12O404-) ) 0.47 × 10-9 m2 s- 1. This result is consistent with the value deduced from voltammetric measurements (D0SWV(SiW12O404-) ) 0.51 × 10-9 m2 s- 1). It should be noted that the interaction potential used here is similar to the one already chosen to simulate the transport coefficients of simple electrolyte solutions.18,19 This means that the sodium tungstosilicate electrolyte behaves roughly like a simple electrolyte although it is dissymmetric in size and in charge. However, when sodium sulfate is added to this aqueous solution (n ) 1 to 3), calculations overestimate the experimental molar conductivity, all the more as n is increasing. Conductivity is decreased by a partial association between ions of opposite charges, especially for electrolytes dissymmetric in charge with a large charge-surface ratio, such as Na+ and SO42-. The difference between the experimental conductivities and the calculated ones in the presence of added salt can thus be explained by this phenomenon, as the potentials used in MSA transport calculations and in simulations do not take into account this association. Conclusion In the present work an experimental and theoretical investigation of the transport properties of the tungstosilicate ion in aqueous solutions has been exposed. The self-diffusion coefficient at infinite dilution of this polyion was deduced from voltammetric measurements. The knowledge of this transport coefficient was essential to implement theoretical tools such as MSA transport calculations and Brownian dynamics simulations. The electrical conductivity of sodium tungstosilicate aqueous solutions with and without added salt was measured and compared to calculated values. The extrapolation at infinite dilution of the conductivity of sodium tungstosilicate solutions gives a value of the tungstosilicate self-diffusion coefficient consistent with the voltammetric determination. Moreover, it was shown that the sodium tungstosilicate electrolyte in water exhibits a behavior similar to a simple electrolyte, whereas the addition of a salt seems to yield an ionic association phenomena. The apprehension of the intimate interactions of particles in solution needs the development of a more subtle potential, taking into account specific details (for example adding short-ranged attractive terms32) and, possibly, the consideration of more than three species. This is feasible without restrictions in the
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