joumal of MOLECULAR
LIQUIDS E L S EV I E R
Journal of Molecular Liquids 78 (1998) 7-18
Solvation of ions in binary solvents - experimental and MD simulation studies Ewa Hawlicka and Dorota Swiatla-Wojcik Institute of Applied Radiation Chemistry, Technical University, Zeromskiego 116, 90-924 Lodz, Poland I. I N T R O D U C T I O N In binary solvent compositions solvation shells of ions and a bulk solvent might differ. This phenomenon is called preferential solvation and, in general, is expected to be due to stronger interactions between the ions and one of the solvent components. One should however remember that the nature of the phenomenon is still under discussion and that the term preferential solvation has been introduced to stress non-linear variations of solution properties with the composition of binary solvents. Various experimental techniques have been employed to investigate preferential solvation but the lack o f theory means that conclusions are inconsistent, and even the ionic solvation in alcohol/water mixtures is still under question. NaC1 solutions in methanol/water mixtures serve as a good example o f these contradictory conclusions [1-3]. Only X-ray and neutron diffraction experiments are expected to give a direct insight into a structure and composition of the ion solvation shells. Unfortunately neutron diffraction experiments on electrolyte solutions in binary mixtures have not been reported and the X-ray technique does not provide decisive results for the methanol-water mixtures. Structures of the ion shells in methanolic and aqueous solutions are very similar [4,5] and the reason probably is that "the traditional method of analysis of average geometrical models by a multi parameter fit to the experimental X-ray structure is not sensitive enough for eventual small changes" . MD simulation can help to investigate preferential solvation of ions in methanol-water mixtures. This method gives an opportunity to reproduce various solution properties. Agreement of the computed properties with the experimental ones enables more direct insight into the solution by making separate computations for the subsystems: the solvation shells of ions and the bulk solvent. Thus MD simulation can be a substitute for a theory leading to better understanding o f the nature of the ion solvation. © 1998 Elsevier Science B.V. All rights reserved. 2. P A I R P O T E N T I A L S Several potentials have been proposed to describe the interactions between water and methanol , but only a few of these assume the molecules to be non-rigid bodies [8-11]. It is known  that flexible models reproduce dynamic properties better. This is important for our purpose as dynamic properties of electrolyte solutions supply information concerning the solvation o f ions. Our models assume three force centers for water and methanol; located on oxygen and two hydrogens and on oxygen, hydroxyl hydrogen and methyl group, treated as a pseudo-atom, respectively. The flexible models permit internal vibrations and therefore a total pair potential consists & t w o parts representing the intra- and inter-molecular interactions: 0167-7322/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved
PII SO167-7322(98) 00079-8
V(ro~o,Pi) : Vt.... (pt)+Vmter(rafl)
V .... (p,)+q~qP
The intermolecular potential Vi,t~(r~) is separated into Coulomb and non-Coulomb parts. The partial charges, located on the force sites, are listed in Table 1. The non-Coulomb parts of BJH potentials for water [ 10] and PHH for methanol [ 11] have been reported previously. The intramolecular terms are expressed as the power series of the internal coordinates pi for 'stretch' and 'bend', including three body interactions:
(2) Definitions o f the internal coordinates according Carney et al.  -and parametrs o f the gas phase equilibrium geometry of methanol and water, used to compute the coordinates, are listed in Table 1..The parameters of the intramolecular potentials for both solvent components have been published previously [ 10,11 ]. Table 1. P a r t i a l c h a r g e s , parameters of the gas phase equilibrium geometry and definitions  of the i n t e r n a l coordinates for methanol and water Methanol
0.35 e 0.25 e
0.9451 A 1.425 A 108.53 °
0.9572 A 104.52 °
charges oxygen hydroxyl hydrogen methyl group bond lengths OH OCH3
angles internal coordinates
(ton - r o l l "q ) I r o n (roc- roc"q )/roe 0~OH
- 0UCOH eq
(rom - rou"q )/rom (rom - ronm )/rouz ( X H O l l - C L H O H eq
The pair potentials for all the interactions involving the ions are expressed by the following equation:
q~qB V(r,~) :
+ _~-r~ + B~ exp(-C~r~)
The parameters of the ion-methanol and ion-water interactions have been reported previously [13,14], and the potential energies of the complexes between ions and solvent components are shown in Fig. 1. The binding energies have been computed for the coplanar arrangement with the antidipole and dipole orientation for cation and anion, respectively. As seen the pair potentials Na÷-water and Na'-methanol are very similar, although the energy minimum o f the complex involving water is about 2% lower than that for methanol.
Interactions of CI" with methanol and water are also similar. The minimum for methanol is slightly deeper and is found at slightly shorter anion-oxygen distance. 50
Na*-H zO~ . - . . . .
0 i ~ ' S'
Fig. 1. The pair pot,mtials for interactions between cation (left), az~on (right) and solvent component
3. DETAILS OF SIMULATION The simulation was performed for NaC! solutions in pure water and methanol-water mixtures. For comparison the binary solvents were also simulated. In all cases the simulation runs were done for the standard NVE ensembles; the periodic cubes contained 400 molecules o f solvent, 8 cations, and 8 anions. The size of the periodic box was calculated from the experimental densities at 25°C. The parameters of simulated systems are summarized in Table 1. Temperatures shown in the table have been averaged over the simulation runs o f the equilibrated systems. Table 2. Parameters d the simulated NaCI Methanol mole fraction x~ 0.0 0.1 0.5 0.9
Concentration of NaC! mol-dm-3 1.11
1.03 0.80 0.65
density at 298 K ~G-m"3 1.11410 1.0165 0.9122 0.8170
Box length A
23.046 23.782 26.713 29.568
299 :l: 4 297 + 5 290+5 297 "1-6
The starting configurations were obtained by random displacement of solvent molecules and ions in the cube. The time step was 0.25 fs and after about 7 ps and 15 ps of equilibration for the solvents and NaC! solutions respectively the simulation of each system was extended over 50 ps.
10 4. RESULTS 4.1. S t r u c t u r e o f s o l u t i o n s
In order to gain insight into the solvation shells of ions partial radial distribution functions have been computed, making a distinction between oxygen and hydroxyl hydrogen atoms belonging to the molecules of methanol and water. Thus the structure of the ion solvation shell in the methanol-water mixture is described via five partial radial distribution functions: ionoxygens of methanol OM and water Ow, ion-hydroxyl hydrogens of methanol HM and water Hw and ion-methyl group (carbon). As an example of the partial radial distribution functions the cation-oxygen, gN~oM(r) and gN~ow(r), and anion-oxygen, gc~oM(r) and gctow(r), functions obtained for an equimolar methanol-water mixture are shown in Fig. 2. 15-
c~ion-oxyg~l diaance, A
anion-oxygen dislance, A
Fig. 2. Partial radial distribution functions for cation (left) and anion (right) in equimolar mixture. Dotted and full lines correspond to water and methanol oxygens, respectively.
The gNaoM(r) and gNaow(r) functions are very similar within the first peak but only the gN,ow(r) exhibits astructure beyond the first solvation shell. Sharp first peaks are centered at 2.34_+ 0.02 A and 2.27+0.02 A, respectively. The gNaoM(r) peak is noticeably higher, suggesting that there are more methanol rather than water molecules in the first solvation shells. For the anion this effect is more remarkable. Only the gc~oM(r) function exhibits a wellpronounced peak centered at 3.25_+ 0.02 A, indicating that there is mostly methanol in the first solvation shells. The positions of the first maximum Rma~and the first minimum Rm~. of the g~l~(r) functions are summarized in Table 3. In order to show the effect of the cosolvent on the partial radial distribution functions the ratio of the peak heights in the binary solvent g~f~(Rm,~)m~and in net components g~(Rma~),¢t, water and methanol, respectively, are also presented in Table 3. The running coordination numbers nj(R~m) were calculated by integration of the partial radial distribution functions up to R ~ : Rmm
G (Rm,o) = 4Jrpj Ig, o~Oj(r) r2dr
where pj denotes the number density of either methanol or water oxygens. These were used to compute the methanol mole fraction in the solvation XM~h~"from the following relationship:
II x~" -
n . (R~,°) r/M (Rmi n ) + nw (Rmi n )
The ratio o f the methanol mole fractions in the shell and in the solution, xMShell/x M , can be a clear measure o f the preferential solvation o f ions. Table 3. Characteristic parameters of the ion-oxygen radial distribution functions obtained for the mixed solvents: R~i, and Rm~, distances (in A) where the function g,,l~(r) shows its maximum and minimum, respectively; the ratio of the peak heigths in the mixtures g~a(l~,)mi, and in net solvent components, g,,~(R~,,) ,,t, coordination numbers nj(R,,L,), the methanol mole fraction in the solvation shell xM'h'n and the ratio
In all the mixtures studied the positions of the gN,oM(r) and gN,ow(r) peaks are very close to each other, although the distance between Na + and the water oxygen Ow is slightly shorter. It agrees with the experimental relationship. The distances reported for aqueous and methanolic solutions are 2.30+ 0.05 A  and 2.38+ 0.05 A , respectively. The position ofgNaoM(r) and gsaow(r) peaks is independent of solvent composition. It is worth stressing that the radial distribution functions for the ions and the methyl groups exhibit very broad peaks. Thus it becomes clear why X-ray experiments cannot distinguish between the molecules o f methanol and water in the first solvation shells. As can be seen from Table 3 the g~,oM(r) peaks in mixed solvents are lower than in the methanolic solution, whereas the influence of methanol addition on the gyaow(r) peak is unexpected The peak height increases despite the decrease of water concentration although this does not reflect an excess of the water content in the solvation shells o f Na'. The coordination numbers show that in water-rich and equimolar mixtures the cations are
12 preferentially solvated by methanol. Only in methanol-rich solvent has a weak preferential hydration o f N a + been found. An effect o f the mixture composition on anion solvation is striking. Only in water-rich solvent both functions, g o o ~ r ) and goow(r), show well pronounced first peaks. Their positions, at about 3.25i~0.05 and 3.37iO.05 A, for methanol and water respectively agree with the values 3.26i0.1A and 3.24i-0.1A, obtained from the neutron diffraction data for net solvent components . The g o o ~ r ) peak decreases with increasing xM but its position remains unaffected. The goow peak vanishes with increasing methanol concentration. Although only in methanol rich solvent the water molecules do not enter the first solvation shells of anions it is worth noticing that in these mixtures the excess of methanol content is less remarkable as compared with other mixtures. 4.2, Interaction energies Although preferential solvation of ions is expected as a result o f stronger interactions o f the ion with one o f the solvent components it cannot be the reason for preferential solvation of the Na" and Cl" ions by methanol. As seen from Fig. 1 the potential energies of the complexes Na%H20 and Na+-CH3OH, as well as CI'-CH3OH and CI'-H20, are very close. These energies were computed for an artificial system consisting o f one ion and one solvent molecule. Therefore the question appears whether interactions between the solvent molecules may affect the potential energy o f the ion-solvent interactions. Accordingly the average potential energies o f ion-solvent interactions have been computed for all simulated solutions as a function o f the ion-oxygen distance. The average potential energies of the solvent molecules in the field of ions in an equimolar mixture are shown in Fig. 3. Similar dependencies have been found for all simulated systems.
Fig. 3. Average potential enevgiel of methanol (full) and water (dotted) in the field of cation (left) and anion (right) in equimolar methanol-water mixture
As seen from Fig. 3 the average potential energy of methanol < V ~ ( r ) > , and water , in the field o f cations is the same within the statistical error. The depths of their minima are very close to those shown in Fig. 1. Similar relationships have been found in all mixtures studied and confirm a supposition that the preferential solvation of Na + is not due to
13 the stronger interactions with methanol molecules. The location o f the minima in < V ~ ( 0 > and coincides with the positions o f the gmou(r) and gmow(r) peaks. Average potential energies o f methanol and water in the field o f anions and are also similar. Positions o f their minima coincide with the first peaks o f the goou(r) and goow(r) functions. As compared with the potential energies o f the anion-solvent molecule complexes shown in Fig. 1 the difference between and is noticeable. Such behaviour might be expected because the energies o f the complexes have been computed for the dipole orientation o f the solvent molecule. This unfavorable orientation occurs because the anion forms an almost linear H-bond with the solvent molecules. Contrary to water, methanol can form only one H-bond with CI', thus its orientation in the vicinity o f an anion is more restricted and the minimum o f is deeper by about 15 %. Important sources o f our knowledge about the ion-solvent interactions are data on enthalpies o f ion solvation and on ion transfer from aqueous solution into other solvents. Extrathermodynami¢ assumptions are necessary to separate experimental enthalpies into ionic contributions, whereas MD simulation can gain direct insight into the interaction between the ion and the solvent components. However, direct computation o f the ion solvation energy is impossible, because the solvation enthalpy (energy) depends on the potential energy o f the ionsolvent interactions, and on the energy o f the solvent reorganization AEss. This energy reflects changes in the interactions between the solvent molecules induced by the salt addition. A separation o f the AEss into its ion contributions also needs artificial assumptions. Accordingly we have calculated the solvation energies o f the salt E~, from the following equation: =
= y. E,, +
where EELs denotes the sum o f the potential energies resulting from the interactions o f the ions and solvent components. The solvent reorganization energy AEss represents the difference between the potential energy o f the solvent interactions in the presence o f the salt (Ess)N,cl and in the mixed solvent (Ess),o~,. The energy o f NaC! solvation F_~, the solvent reorganization AEss, and the energy o f the NaCI transfer AEu~ which is the difference o f the solvation energy in mixed solvent (F~l)~x and in aqueous solution (E~),q, are listed in Table 4. Also summarized are the potential energies o f the solvent-solvent interactions in the electrolyte solution (Ess)~,o and in the pure solvent (Ess),~. Table 4. Energies of NaCI mlvatioa E~, transfer AEw., solvent reorganization AEss, i o n solvent interactions
xM 0.0 0.1 0.5 0.9
in NaCI solution
(Ess)~ kJ mol-' 41.2 -40.8 -37.4 °32.5
(Ess)~o kJ tool"I - 28.3 -27.2 -25.5 -21.0
a n d i n p u r e s o l v e n t (F-,ss),,h.
Es~s kJ tool~ - 470 - 510 - 470 - 340
Ea~ kJ moll - 310 - 350 - 320 - 230
aEss Id mol" 12.9±0.2 13.6±0.2 11.9±0.3 11.5±0.2
F~ k J" mol -l
-780 + 10 -840+10 -785 + 10 -560 + 20
-64+3 -11 + l +220 + 15
A direct comparison of the energies presented in Table 4 with the experimental results is difficult because the measured values are usually extrapolated to infinite dilution to avoid the ion-ion interactions. In our simulations the salt concentrations ranged from a moderate concentration 1.11 molal NaC1 in water up to 0.653 molal NaCl in the methanol rich-mixture. The hydration energy of NaCl resulting from MD is comparable with the experimental data (-770 kJ/mol , -784 kJ/mol ) despite the significant difference in the NaCI concentration. For XM=0.1 the solvation energy reaches the lowest value which can suggest the strongest solvation of ions; here E~o~increases with increasing xM. Unfortunately, there are no data reported for solvation enthalpies in water-deficient mixtures. The solvation enthalpy measured in the methanolic solution is -787.3 kJ/mol . The noticeably higher value of Esol in methanol-rich solvent (by about 25%), can be attributed to the significant difference in NaCI concentration as the simulated system is very close to that of a saturated solution.
4.3. Dynamic properties Self-diffusion experiments are important for studies of ionic solvation, because the selfdiffusion coefficients can be determined independently for all components of the solutions In MD simulations the self-diffusion coefficients have been derived from the center of mass velocity autocorrelation functions via the Green-Kubo relationship: t
D= l i m ~ < v(o). ~(t)>dr t ~
The average of the velocity autocorrelation functions has been calculated as follows: C~(t)::
X r X ~ Z Z v ~ ( t , ) . v j ( t i +t) t I /=1
where N,, denotes the number of particles, NT the number of time averages and vj(ti) the velocity of the particle ct at time ti. The normalized velocity autocorrelation functions for the ions, methanol and water molecules in an equimolar mixture are shown in Fig. 4.
.~ 04 C)
; time, ps
Fig. 4. Normalized velocity autocorrelation functions of solvent components (left) [methanol (full), w a t e r (dotted)] and ions (right) [ Na + (full), CI- (dotted)] in an equimolar mixture.
For all simulated solutions the dependences of Cvv(t) are similar. As a compromise between good statistics and long correlation time, the time limit of 2.5 ps was assumed for the calculation o f the self-diffusion coefficients. The self-diffusion coefficients of methanol DM, water Dw, and the ions sodium DNa and chloride Dcl, are listed in Table 5. All values have been extrapolated to 298 K for better comparison with experimental data [ 17,18]. Table 5. Self-diffusion coefficients of methanol, DM, water Dw, sodium DN, and chloride Dcl ions in methanolwater mixtures at 298 K DM- l0 s, cm~s ~ Dw- l0 s, cm2sa Dr¢~'l0 s, cmZs-1 Dcr l0 s, cm2s-I XM MD Exp MD Exp MD ExpI181 MD Exp 0.0 1.5 :l=0.2 2.09 1.2 =£-0.1 1.42 2.0 =1.-0.2 1.98 0.1 1.0 + 0.2 1.04 1.2 _+0.2 1.48 0.7 + 0.2 1.04 1.0 _+0.2 1.30 0.5 0.9 + 0.2 0.95 1.0 + 0.2 0.96 0.7 5:0.3 0.85 0.9:1_-0.2 0.90 0.9 1.8 5:0.2 1.65 1.5 + 0.2 1.20 0.7+ 0.3 1.03 0.9 _+0.2 1.12
The calculated Dr, values are in good agreement with the experimental ones and the difference between them does not exceed 10%. For water the difference between the experimental and computed values is more noticeable, reaching 30%. One should notice however that the D-values resulting from MD simulation correctly reproduce the influence of the solvent composition. It is well known that the self-diffusion of water, alcohols and other compounds forming Hbonds does not result from motions of single molecules but it does from collective motions of diffusion units . The effect of electrolyte on D-values of the solvent components is expected to be complex. On the one hand the molecules belonging to the solvation shells of ions may be noticeably retarded, but on the other the electrolyte may destroy the H-bond network o f the solvent, making the motion of the solvent molecules more independent. In order to characterize the influence of NaCI on the self-diffusion of the solvent components the ratios, Fi, of the D-values in the NaCI solution and the corresponding mixed solvent have been computed. The calculated Fi data are in excellent agreement with the experimental data (see Table 6). Thus the Fi ratios calculated for the molecules constituting solvation shells o f ions could reliably reflect the ion field effect on the solvent mobility. The D-coefficients for the solvent molecules in the solvation shells have been compared with the corresponding D-values in mixed solvents and the Fi ratios are listed in Table 5. Table 6. Ratios of the self-diffusion coefficients of methanol FM and water Fw computed from MD simulation for all molecules of the solvent component (total), for the molecules belonging to the solvation shells of cations (Na shell) and anions (CI shell) and obtained from the experiments (Exp) . xM 0.0 0.1 0.5 0.9
0.77 0.79 0.77
0.60 0.45 0.45
0.54 0.53 0.50
0.79 0.82 0.77
total 0.95 0.78 0.79 0.64
Na shell 0.5 0.4 0.4 0.4
CI shell 0.99 0.94
Exp 0.94 0.85 0.83 0.50
As can be seen the methanol molecules in the vicinity of ions become less mobile. A similar effect can be observed for water molecules in the field of cations, whereas the mobility of
16 water molecules in the solvation shells of anions remains unchanged. In such a case it becomes obvious why ion movement methods lead to a zero value for the hydration number o f CI" . The spectral densities of the hindered translations of all components have been calculated by Fourier tranfformation of the normalized center of mass velocity autocorrelation functions: S"
Jtc°) = - - ~ Jo < - ~ > cos(2avc°Odt
where c and mj are the velocity of light and the massof the j-th component, respectively. In aqueous solution the main peak of the spectrum of hindered translations of Na ÷ appears at about 170 cm "~ with a satellite peak at about 290 cm"~. Because of strong cation-water interactions, none of these frequencies can be assigned to the motions of bare cations. In methanol-deficient solvent and in equimolar mixture these peaks become less pronounced but in methanol-rich solvent two peaks are found at about 200 cm"~ and 270 cm "t. Their positions are very close to those reported for the methanolic solution of NaCI [ 13]. An effect o f the solvent composition on the hindered translations of anions is different. In aqueous solutions there are two peaks at about 36 cm"~ and 140 crn-~, respectively. The low frequency peak can be attributed to the motion of bare anions, whereas the interaction o f Cl" with its neighbourhood leads to the higher frequency peak. It is worth noticing that the low frequency peak is higher. A small addition of methanol results in a blue shift of the low frequency peak and a red shiR of the higher frequency one. Moreover, the heights o f the peaks become similar in size. Three similar peaks at about 50 cm-~, 110 cm~ and 130 cm ~ are found for the equimolar mixture. In methanol-rich solvent there are only two peaks at about 40 cm~ and 125 cm "~. Their positions are very close to those reported for the methanolic solutions of NaCI  and MgCl2 . In all mixed solvents the low frequency peak is lower, indicating that the motion of bare anions is of less importance. It confirms the conclusions drawn from the self-diffusion calculations. The velocity antocorrelation functions Cw~(t) have been computed for the a-sites of water (ct= O, H, H) and methanol (ct= O, H, CH3). The Fourier transformation o f C ~ ( t ) yields the partial spectral density S~(t0) of the intramolecular vibrations. A summation over all these densities S~(to) 3
s ~ , (co) = ~ s" (co)
gives the total power spectrum of methanol and water. The flexible models, BJH and PHIL of methanol and water respectively correctly reproduce frequencies of the basic modes of pure liquids [ 10,11 ]. Thus an effect o f the ion field on the molecular vibrations can be investigated. The NaCI influence is particularly evident for the OH stretching frequencies of both solvent components. In order to show changes of the OH stretching vibrations due to the presence of NaCI the fiequency shifts Ac~t, have been calculated as the difference o f the (r0ce)tmm frequency o f methanol and water in NaCI solutions and in the corresponding mixed solvent . Similar shifts have been computed for three subsytems of the solutions: the solvation shells of cations (¢~m)s.a=a, anions ( ~ ) c ~ , and the bulk solvent (C0OH)~k: the results appear in Table 7.
17 Table 7. F r e q u e n c y s h i f t s o f O H streW'_b|Q s
c m t , f o r m e l i t a a o l s a d w a t e r i n rite N a C I
I~ee~u~ Metb__sno/ xM O.O 0.1 0.5 0.9
teal - 43 - 15 -20
N a dufll
- 7 - 18 -19
- 108 - 71 -69
- 42 - 7 +6
N s s/re/!
- 15 - 12 - 7 -20
- 4 - 3 0 -12
-89 o 103 - 140 -144
sltell -27 -10
The addition o f NaCI shifts the OH frequencies of methanol and water to lower wavenumbers. In the presence o f NaCl the H-bond network o f the solvent is partially destroyed. Thus the red shift is due to a strengthening o f the H-bonds. Despite the almost linear hydrogen bonds between the anion and the solvent molecules the main red-shift contribution o f the OH frequencies results from the interaction with cations. The red shift o f the OH water frequency increases with increasing methanol concentration and demonstrates
the cooperative effect deduced from IR experiments : the stronger the H-bonds which exist in the mixture the stronger is the effect o f N a ÷ observed. 5. C O N C L U S I O N S MD simulations were performed to investigate preferential solvation o f ions in methanol/water mixtures. Although the models employed for the solvent molecules and the pair potentials are simple the physical properties o f the solutions derived from MD simulations, radial distribution functions, energies o f the NaC! solvafion, self-diffusion coefficients o f all components, power spectra, agree reasonably with reported experimental data. Thus we can take advantage o f MD simulation and compute physical properties for the subsystems: ion solvation shells and the bulk. Our results o f MD simulations indicate that preferential solvation o f ions is not due to the ion-solvent interactions but is a consequence of the incompatibility o f the structures of net water and net methanol. Both ions and methanol molecules cannot fit into the tetrahedral structure o f water. They are forced to form "common aggregates"- the preferentially solvated ions. Acknowledgements
The authors would like to thank the Computer Center of the Technical University in Lodz for computational facilities and the Polish Committee for Scientific Research for partial support by Grant No. 3 T09 010 10. REFERENCES 1. A. K. Covington, K.E. Newman and T. H. Lilley, J. Chem. Soc. Faraday I, 69 (1973) 973. 2. M. Holtz, H. Weingartner and H. G. Hertz, J. Chem. Soc. Faraday I, 73 (1977) 71. 3. E. I-Iawlicka, Z. Naturforsch. 41 a (1986) 939. 4. G. W. Neilson and J. E. Enderby, Annu. Rep. Prog Chem. C, 76 (1979) 185. 5. M. Yamagami, H. Wakita and T. Yamaguchi, J. Chem Phys., 103 (1995) 8174. 6 . . T. Radnai, I. Bako and G. Palinkas, ACH, Models in Chemistry, 132 (1995) 159.
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