**6. Residence time of the solvent molecules in the coordination shells**

An important feature of ions in solutions is a persistence of the coordination shell, because a dynamics of an exchange process may determine a reaction rate. A residence time of the solvent molecules in the coordination shells of the ions has been calculated from a time correlation function R(t), proposed previously (Impeay et al., 1983). The time correlation function is defined as follows:

$$R(t) = \frac{\sum\_{i=1}^{N\alpha} \sum\_{j=1}^{N\_\beta} \kappa\_{ij}(t) \cdot \kappa\_{ij}(t + \Delta t)}{N\_\alpha \cdot N\_\beta} \tag{7}$$

where N and N denote the number of ions and the solvent molecules in the sphere of the radius defined by the first minimum of the gionO(r) function, respectively. ij(t) is the step function; ij(t)=1, if the solvent molecule *j* is in the ion shell and ij(t)=0 otherwise. The calculations of the R(t) functions were performed for at least 500 randomly chosen initial configurations. The calculations were done for several time intervals t within the range 0.01-0.5 ps.

The solvent molecules could leave the ion shell for a period, which was shorter than t, otherwise they were neglected in further calculations. Variations of t and the solvent composition do not influence the character of R(t) functions. For all simulated systems the time correlation functions for Mg2+ and Ca2+ ions decrease rapidly in the period shorter than

of the angles between three oxygens of the solvent molecules in the Na+ and Mg2+ shells shows two peaks at 60 and 90o, respectively. This confirms the octahedral arrangement of

The Ca2+ shell, which contains more molecules, does not show any symmetry. As seen from Figure 7 in aqueous solution the distribution of O-Ca2+-O angles exhibits two peaks, around 67o and 135o, respectively. The former angle is close to the value, which can be expected for tetrahedral or hexahedral symmetry, but the latter angle cannot be correlated with any of the polyhedra. This means that the Ca2+ shell is irregular. When the coordination number of Ca2+ decreases with the increasing methanol content, the most probable O-Ca2+-O angle increases. In methanolic solution the distribution of angles shows two peaks around 75o or 145o, respectively. This means that the cation shell remains irregular. The distribution of the O-O-O angles confirms a lack of symmetry of the Ca2+ shell, because in all studied solutions

The O-Cl- -O angles, computed for the solutions of NaCl, MgCl2 and CaCl2, are very similar.

that the coordination shell does not show any symmetrical arrangement. A lack of the symmetry of the Cl- shell causes that in aqueous solution the distribution of the O-O-O angles is almost uniform, except a small peak at about 54o. It is worthy to notice that such peak is believed to be a distinctive feature of the tetrahedral arrangement of pure water (Gallanger & Sharp, 2003). This means that in the coordination shell of Cl- the water

An important feature of ions in solutions is a persistence of the coordination shell, because a dynamics of an exchange process may determine a reaction rate. A residence time of the solvent molecules in the coordination shells of the ions has been calculated from a time correlation function R(t), proposed previously (Impeay et al., 1983). The time correlation

() ( )

*t tt*

*ij ij*

 

*N N*

 

where N and N denote the number of ions and the solvent molecules in the sphere of the radius defined by the first minimum of the gionO(r) function, respectively. ij(t) is the step function; ij(t)=1, if the solvent molecule *j* is in the ion shell and ij(t)=0 otherwise. The calculations of the R(t) functions were performed for at least 500 randomly chosen initial configurations. The calculations were done for several time intervals t within the range

The solvent molecules could leave the ion shell for a period, which was shorter than t, otherwise they were neglected in further calculations. Variations of t and the solvent composition do not influence the character of R(t) functions. For all simulated systems the time correlation functions for Mg2+ and Ca2+ ions decrease rapidly in the period shorter than

**6. Residence time of the solvent molecules in the coordination shells** 

1 1

*N N*

 

*i j*

( )

*R t*


(7)

the coordination shells.

these angles are either 55o or 107o.

structure partially remains.

function is defined as follows:

0.01-0.5 ps.

As seen from Figure 7 the distribution of the O-Cl-

1 ps, afterward, independently of the time interval, they reach a constant value close to 0.95. This means that about 95% of the solvent molecules do not leave the coordination shells of the cations during the whole simulation time. The coordination shells of the divalent cations are very stable, with the lifetime remarkably exceeding 150 ps, and being independent of the solvent composition. The long lifetime of the primary hydration shells has been reported previously for Ca2+, about 700 ps, and Mg2+ , about 422 ps (Konesham et al., 1998). The long residence time of the solvent molecules has been expected, because the hydrodynamic radii of both cations noticeably exceed the ion radii in crystal (Hawlicka, 1995). This means that the cations move with their coordination shells together, because the ion filed controls the translations of all nearest neighbours.

The R(t) functions for the Na+ and Cl ions decrease monotonously and they can be fitted to a second-order exponential decay:

$$R(t) = A\_1 \exp(-\frac{t}{\tau\_1}) + A\_2 \exp(-\frac{t}{\tau\_2}) \tag{8}$$

The first term describes an escape of the solvent molecules located close to the border of the coordination shell, whereas the second term concerns the persistence of the shell. Parameters A1 and A2 reflect fractions of the solvent molecules involved in both processes. The first process is rather fast and its characteristic time 1 is shorter than 1 ps. The residence time 2 of the solvent molecules in the Cl shell and the methanol molecules in the Na+ shell increase with the time interval t. Such dependence it is shown in Figure 8.

Fig. 8. Influence of the time interval t. on the residence time 2 of the methanol molecules in the Cl- shell.

As seen the residence time reaches the constant value when t is not shorter than 0.2ps. Thus the 2 values discussed below were computed for t=0.2 ps. This means that the solvent molecules leaving the ion shell for the time longer than 0.2 ps were neglected in further calculations.

In aqueous solution the lifetime of the coordination shell of Na+ is long, more than 170 ps, but in methanolic solution this time is much shorter, about 45 ps. Therefore is not surprising that the lifetime of the Na+ shell decreases when the methanol content increases.

MD Simulation of the Ion Solvation in Methanol-Water Mixtures 417

In all studied solutions the H-bond numbers are the same, <nHB>w=3.1. Thus this influence is slight and only in CaCl2 solutions it extends beyond the first coordination shells of the

Differences between the electrolyte solutions appear when the H-bonds of the molecules in first coordination shell of the cations are compared. The Mg2+ and Na2+ ions are sixcoordinated and the angular distributions show, that all water molecules are properly oriented to form two H-bonds as H-donors. Though the average number of H-bonds per the water molecule in the Na+ shell is two, [<nHB>w]Na+=2, a detailed analysis shows that about 65% of the water molecules in the Na+ shell form 2 H-bonds, whereas the reminder of them has either one, about 15%, or three, about 20%, H-bonded neighbours. Most of the water molecules in the Mg2+ shell, about 70%, have two H-bonded neighbours, but 30% of the molecules form only one H-bond. Though both cations coordinate six water molecules the radius of Mg2+ shell is smaller, about 0.27 nm, than that of the Na+ shell, about 0.32 nm. Thus the Mg2+ shell must be more compact and there is probably not enough space for Hbonded neighbours of all water molecules. The water molecules in the Ca2+ shell have also less H-bonded neighbours. Most of them, about 80%, has only one H-bonded neighbour and only 20%of the molecules form two H-bonds with their neighbours in the second shell. The radii of Ca2+ and Na+ ions in crystal, 0.096 and 0.102 nm (Marcus & Hefter, 2004) and their shells, 0.34 and 0.32 nm respectively (see Table 3), are similar. The first shell of Ca2+ consists however of 10 water molecules. Though most of them are oriented properly to have the Hbonded neighbours the shell is compact and only 20% of the molecules have enough space

In mixed solvent the number of H-bonds per water molecule in the shells of Na+ and Mg2+ ions remains unchanged. Such behaviour might be expected, because neither the coordination number nor the orientation of the molecules depends on the solvent composition. In methanol-water mixtures the water molecules from the first shells of Na+ and Mg2+ prefer the methanol molecules as H-bonded neighbours in the next sphere. Such preference can be understood, because the H-bond between the H-donor water molecule and the H-acceptor methanol molecule is energetically favourable (Palinkas et al. 1991).

As seen from Tables 3 and 4 the number of the molecules in the first and second shells of Ca2+ decrease with the increasing methanol content. The radii of both shells are, however, independent of the mixture composition. Thus the first shell becomes less compact. This improves the orientation of the water molecules as H-donors. In consequence all water molecules have two H-bonded neighbours. However they have water molecules as the Hbonded neighbours despite the unfavourable energy of H-bond between two water molecules. The methanol molecules appear in the first shells of Ca2+ in methanol rich solvents, when there is a lack of water to form the coordination shell. Their antidipole

The chloride ion is H-bond acceptor and in aqueous solution the water molecules form almost linear H-bond with Cl- shell. About 80% of the water molecules coordinated by the anion form three H-bonds with the neighbours in the bulk solvent. In mixed solvent the methanol molecules replace the water molecules in the anion shell. The molecules form the

therefore they are H-acceptors and have only one H-bonded

ions (Owczarek et al., 2007).

for two H-bonded neighbours in the second shell.

orientation causes that they have only one H-bonded neighbour.

linear H-bond with Cl-

neighbour in the bulk solvent.

The exchange of the water molecules between the Cl- shell and the bulk is fast. Though the size and composition of the anion shell in solutions of CaCl2, MgCl2 and NaCl are the same, the persistence of the shells is different. In solutions of CaCl2 and MgCl2 the anion shells are more flexible than those in NaCl solutions. In solutions of the alkali earth chlorides about 85% of the water molecules stay in the primary shell of the anion less than 5 ps. This means that the water residence time is shorter than the characteristic time of the water translations, about 6 ps (Hawlicka & Switla-Wojcik, 2000). This explains why the hydrodynamic radius of Cl in aqueous solution is like its radius in crystal (Hawlicka, 1986). In NaCl solution the lifetime of the anion shell is longer. About 75% of the water molecules stay in the shell about 19 ps.

The residence time of the methanol molecules in the Cl- shell is longer than that of the water molecules. In methanolic solutions of CaCl2 and MgCl2 the lifetime of the Cl shell exceeds 50 ps. Shorter lifetime, about 25 ps, has been found for the methanolic solution of NaCl. However even this shortest residence time, 25 ps, exceeds significantly the characteristic time of methanol translations, about 9 ps (Hawlicka & Switla-Wojcik, 2000). This explains why the hydrodynamic radius of Cl- in net methanol is greater than the radius in crystal (Hawlicka, 1986).
