Mass density in hydration shells of ions

ELSEVIER

Physica B 245 (1998) 34--44

Mass density in hydration shells of ions I. D a n i e l e w i c z - F e r c h m i n

a, A . R . F e r c h m i n b'*

aInstitute of Physics, A. Mickiewicz UniversiO,, ul. Grunwaldzka 6, PL-60-780 Poznan, Poland b lnstitute o f Molecular Physics, Polish Academy o f Sciences, ul. Smoluchowskiego 17, PL-60-179 Poznah, Poland

Received 7 May 1997; received in revised form 8 September 1997

Abstract

The densities of the first hydration shells of 43 different ions are calculated on the basis of an electrostatic and thermodynamic approach applying the concept of a local electrostriction pressure. The plot of the local density in the hydration shells, as a function of the reduced (divided by square root of valence) distances from the centres of ions, presents a common smooth line even for ions with different valences. :f~; 1998 Elsevier Science B.V. All rights reserved. PACS: 61.20.Qg; 77.65.-j; 77.22.Ch. Keywords: Density; Electrostriction; Dielectric saturation; Electrolyte; Hydration

1. Introduction

There is ever growing experimental evidence for one or two hydration shells formed around tri-, di-, and monovalent ions ([-1, 2] and references therein). The ions are sources of extremely high electric fields, up to about 1011 V/m at the distances of the first hydration shells. With this in mind, an approach has been developed, based on an extension of the theory of nonlinear effects in dielectrics in high fields [-3], and applied to calculate the state of electric polarization of water about ions as a function of the radial distance from their centres [4]. In Ref. [4] the major concern was in the degree of radial order of the dipole moments of water tool-

* Corresponding author.

ecules. Very recently, the problem of the density of water placed in electric fields of strengths exceeding 10 l° V/m has been stressed in relation with X-ray diffraction experiments [5]. Although the experiments concerned double layers at charged electrodes, the authors noted that fields of comparable strengths can be encountered in the neighbourhoods of ions in water solution. It has been demonstrated [6] that the experimental results of Ref. [5] can be quantitatively explained by considering the electrostriction [3, 7, 6] effects derived based on the conditions of thermodynamic equilibrium. The latter has been expressed in the form of the balance of chemical potentials per water molecule. The change ffw in the chemical potential ~ is due to the work done by the electric field during the reorientation of the dipole moments of H e O molecules. To maintain the equilibrium, the pressure in the system

0921-4526/98/$19.00 '.i 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 4 9 9 - 7

35

I. Danielewicz-Ferchmin, A.R. Ferchmin / Physica B 245 (1998) 34-44

grows with applied field. This pressure gives rise to an electrostriction [3] work L related to the change in the chemical potential (L. Equilibrium required mutual compensation of those changes in the chemical potential, hence -- ~w = (L.

The results are compared with diffraction experiments and in part also with the data derived by computer simulations. The question how our methods are applicable to diluted electrolytes is also discussed.

(1)

The same reasoning can be applied to the problem of density in aqueous environment of electrical charges of ions. Since the water molecules are in a Coulomb field, the phenomenon of electrostriction is a natural link between the presence of charges on the ions and the density of the molecules about them. Experimentally, the density distribution around ions can be found from neutron diffraction isotopic substitution data providing the twopoint correlation functions or radial distribution functions (RDF). In this paper, based on the same electrostatic and thermodynamic approach as in Ref. [6], we shall calculate the local density of water in the hydration shells. Expressed as a function of the reduced (i.e., divided by the square root of valence) radius of the hydration shell, it reveals a characteristic regular behaviour (Fig. 4). The values of water density about many ions are calculated and given in Table 1. In order to compare the results of our calculations with data available in literature, we expressed the former in terms of coordination numbers h [1, 2, 8]. The h values are obtained from our calculated local density data with the help of the volumes of hydration shells given by Hahn I-9] (cf. Fig. 3 and Table 1 below). The present work stands at the crossroads of several domains of investigation. It concerns the properties of electrolytes. It applies the radii of the hydration shells derived from X-ray and neutron diffraction experiments as input data. It applies the theory of nonlinear dielectric phenomena in extremely strong fields to find the permittivity, a quantity bearing information on the degree of orientation of the dipoles of water molecules within the shells about ions. There follows the idea that the knowledge of the field and permittivity within a shell is sufficient to find its density via the effect of electrostriction, provided that the density of water under high pressure is known. Thus, we perform a thermodynamic calculation of the electrostriction.

2. Method of calculation: electrostatic and thermodynamic approach

Consider a constant volume V comprising N i water molecules localized at an ensemble of wellseparated ions (infinite dilution limit) and denote N ° the number of molecules in a portion of water of the same volume V far from the ions (at infinity). The superscripts i and o mark quantities inside and outside the electric field, respectively. The volumes per molecule are vi and v°, respectively, with the condition V N i v i = N°v °. Let us start with a calculation of the work W done by the electric field of an ion on a water molecule transferred from infinity to a point at a reduced distance x = rlZI- 1/2 from the centre of the ion; r denotes the distance from the centre of the ion, Z its valence. The strength E of the field is =

E-

Zq

_

4X88or 2

q

(2)

4~88oX 2 ,

where eo is permittivity of vacuum and q is the elementary charge. The relation between the permittivity e and the electric field strength is, according to the Onsager local field model [3], expressed as (cf. Refs. [4, 6]): (8 - -

1/2)

x ~ N b - (cos 0),

(3)

where (cos0)=tanh

( 2(e +an 2 / 2 )) ,

(4)

where (cos 0) is the mean cosine I-3] of the angle 0 between the dipole moment of the molecule and the radial electric field, and a -

#q(n 2 + 2) 8rtk TEo

and

b-

4x/~(n 2 + 2) 3Vq

36

I. Dunielewicz-Ferchmin,

A.R. Ferchmin / Physica B 245 (1998) 34-44

where p is the dipole moment of a water molecule, n the refraction index. Throughout this paper, temperature T and number density N/V are taken at ambient conditions. The hyperbolic function tanh results from statistical mechanical calculations, considering that the inversion of the dipole moment from p to - p, due to the motion of protons, is more energetically favourable than its rotation, since at room temperature the hydrogen bond energy is one order of magnitude larger than kT [3,4]. The work W done by the electric field is -dy,

where y = cr

Note that from definition (Eq. (6)) 0 is a function of x, which in turn depends on E (Eq. (3)). The derivatives @a/&) and (&/aN) are obtained from Eq. (3):

a0

0

FE.

=

(5)

where

and

CJ=&.

at;

Let us introduce

(-)aN

the notation:

(13)

C-

where The work W done leads to a change in the value of the free energy of water:

axm2 “9- ____ ’ - E i- n2/2

From the definition

Substituting Eqs. (12) and (13) into Eq. (1 l), one obtains the change in the chemical potential iW. In Fig. la, iW is plotted as a function of the reduced distance from the centre of an ion. The striction work L is calculated, within the same approximation as in Ref. [6], by integrating the area under the isotherm [lo, 111 V = V(P):

v

’ =

of the chemical

potential

i

c-j i?F

aN

T,C,,o

one obtains

the change

in [ due to the work W:

P

L=

iw=

(10)

where c’O= V/N”. Henceforth, we shall drop the subscripts T and V, since constant volume and isothermal conditions are assumed throughout this paper. From Eq. (10) we obtain

V(P) dP. s p”

(14)

Above, the lower limit of the integral P” denotes the pressure in water far from the ion. It is equal to the applied pressure. The upper limit of the integral in the Eq. (14) is equal to the value of the local electrostriction pressure P’ in water at the ion. The change in the chemical potential cr. due to this work is

(11) where

CL

=&

j+;V(P)dP

= [Lu(P)dP.

(15)

L Danielewicz-Ferchmin, A.R. Ferchmin/Physica B 245 (1998) 34-44

60i/ 60 j4o o>

)

sure/7. Thus, close to the ion density and pressure bear local values different from those in bulk water far therefrom. The pressure found in this way is equal to the upper limit of the integral in Eq. (16), p i = /7. The work ~L done by the pressure/7 compensates the decrease in energy ~w due to the work of reorientation of the dipoles. With the knowledge of the pressure one finds the relative water density d (cf. Fig. 3 in Ref. [6]):

20

uo

1.5

2.5

0

x,(A) (a)

40 80 120

P (GPo) (b)

Fig. 1. (a) Variation in the chemical potential of water molecules ~w per volume v°, due to the work done by the electric field, as a function of the reduced distance x l from the centre of an ion. (b) Variation in the chemical potential of water molecules ~L per volume v°, due to the work done by the pressure in field-free conditions, as a function of this pressure P.

Taking into account Eqs. (11) and (15) one can rewrite Eq. (1) in the following form:

~o

~Y

~

N ~

~

vo

oV(P) dP'

(16)

or in a shorter form: -

Cw -

~L -

U°

pi

d . . ~)1. . pO .

0.5

37

(17)

U°

where v° is the volume per molecule in absence of the field. In Fig. lb the right-hand side of Eq. (16), i.e., ~L/V°, has been plotted as a function of pressure p = pi _ the upper limit of the integral in Eq. (16). The value of this integral is equal to the area under the isotherm found using the data of Refs. [10, 11]. Eq. (17) means that at the equilibrium at a given distance from a particular ion the ordinates in Fig. la and Fig. lb must be equal. This enables one to find from Fig. 1 the value of the local pressure at a reduced distance x from the centre of an ion. With varying distance from the charge, hence with varying the electric field, maintaining the equilibrium (Eq. (17)) requires a variation in specific volume and pressure. The latter is the electrostriction pres-

(18)

The values of the local density pm at the reduced radii of the first hydration shells of ions are given in Table 1 column 4. The radii of the first hydration shells of ions are taken from Refs. [1, 8]. In Fig. 2a-Fig. 2d the relative density of water d in the hydration shells is plotted as a function of the reduced radius of the first shell xt. One can notice that in the plot of density versus the reduced radii of the first shells, the densities of shells about ions of different valence lie on a single smooth curve (Fig. 2a). Literature often presents coordination numbers, i.e., numbers of water molecules in the first shells of ions. To find them from our density data we need the volumes of the first shells. They can be found in Ref. [9]. F r o m Fig. 4b therein the values vu in cm 3 per mole of the ions (as a function of the radius r) have been read out, expressed as Vl in ~3 per cation, and re-drawn as a function of the reduced radius Xl of the first shell, in our Fig. 3. The volumes vl given in Table 1 follow from the parabolic fits to the data given in Ref. [9], as shown in Fig. 3. The coordination numbers h (numbers of water molecules in the first shells) have been calculated according to the formula:

h - pllvl

(19)

M ' where M = 3 x 10 .23 g is the mass of a water molecule, and p~ is the water density in the field of the ion within its first shell. The calculated values of the coordination numbers h (calc) are given in Table 1, column 6. In the above formulas there appears the radius of the hydration shell which for the particular ions is taken from diffraction experiments [1, 2, 8], cf.

1. Danielewicz-Ferchmin, A.R. Ferchmin / Physica B 245 (1998) 34-44

38

2.7 -AI

-X

-x d 2,5 - ~

x ion .3+ o ion 2 +

d

A ion 1+ x

o ion 1 -

2,0

2.4

ixGa Cr × Fe xx Rh

k

x

In x TI Tm Ybx) DxYlruSm Lu × Nd ErTbXX Pr

2.1

1.5 0

1.0 1.0

,,,, I i t I L , , , I L , [ A I , a~, a I ,--~-~I

1.5

2.0

2.5

3.0

(a}

....

,,,,

1.1

1.2

1.3

2.5

1.4

1.5

~(A)

(b)

XI(A)

~¢l~×~,L,°

1.8

3.5 1.2

~LI

Be o

d 1.1

2.0

F'e MnOC d

Sn Hg

1.5

Na,~Ag

1.0

K

Rb TI Cs

ClBrl

Ca o

Sro 1.0 1.0

I t i f I I J I I I J I I I [ I I I J

t,2

1.4

(c)

1.6

1,8

0.9

2.0

x,(a)

Itliilllllllllll .9

2.4

2.9 xl(A)3.4

(d)

Fig. 2. (a) Density of water d within the first hydration shell as a function of the reduced distance Xl from the centre of an ion. Data for several negative monovalent ions are included (cf. Fig. 2d). Note that density data for ions of different valence Z follow a smooth curve indicating the universal character of the variable x,. (b) Part of Fig. 2a with trivalent cations, extended to show details. (c) Part of Fig. 2a with divalent cations, extended to show details. (d) Part of Fig. 2a with monovalent cations and anions, extended to show details. Note that due to the too low Coulomb fields no noticeable variation of water density is seen about monovalent ions, except for Li +.

Table 1. The coordination number h is obtained from the density calculated in the framework of the present theory and, in addition, the volume of the first shell given in Ref. [-9]. The values of h thus obtained are compared with literature data ([1,2, 8], and references therein) in Table 1.

3. An overall picture of water about ions

Fig. 4 shows an overall picture of the properties of water emerging from the present calculations. A striking feature in the behaviour of d -1 (cf. Eq. (18)) and
of an ion, followed by a steep variation range starting at a certain value of x. The values of relative volume, d-~, expressed as a function of distance x, in the range 1 A < x < 3.75 A, follow a common smooth line for all ions investigated in this work. It is characterized by a plateau, d-~ = 1, at larger x, preceded by a steep rise with growing x to reach the plateau. At the plateau no distinction can be made between the relative volume of water in the electric field of an ion and far from the ion (bulk water). The steep variation of d -1 and
0

+

o

+

+

~

+

~

~

+

~

~

~

~

~

+

-

~

+

-

~

+

~

~

+

~

+

~

0

+

....

0

+

~ -

~

+

o

0

+

+

+

~ +

~ + +

~ + +

~"

~'~_~ : : r ~ -'-~

~ ~

E

~-~

~ ~ ~.=~

o

E'

I

~D ',0

b

1. Danielewicz-Ferchmin, A.R. Ferchmin / Physica B 245 (1998) 34 44

40 Table I (Continued)

xl (,~)

vx (,~3)

pi (g/cm 3)

Co 2 + Mg 2 +

1.48 1.50

120 120

1.8 1.8

Fe 2 *

1.50

120

Zn 2+ Mn 2+

1.51 1.555

Sn 2 + Cd 2 * Ca 2+

Hg 2 + Sr 2 +

h (calc.)

h (ref.)

Method

References

8 8

7.3 7.4

1.8

8

7.4

125 130

1.8 1.75

7.2 6

7.4 7.7

1.56 1.62 1.70

130 140 160

1.7 1.7 1.35

5.9 4.5 3.2

7.7 8 7.1

1.70 1.87 2.2

160 190

1.35 1.2

3.2 1.4

7.1 8.5

5-6.3 6~8 6 5.1-6 6 6 6 5 2.4.3.8 6.9 8.8 6 8 6.4-10 9.2 6 8, 7.3 15 _+ 1

X X MD X ND X X ND X X X ND MD X X ND

[8] E8] [8] [8] [22] [1, 8] [12] [12] [8] [8] [8] [12, 23] [8] [8, 12] [1, 24] [25]

Monovalent cations Li + 1.95

I lO

1.2

0.9

4.4

4 5.5; 6 6 4 5.3

X ND MD MC

Na *

160

1.0

0.11

5.3

5--6

X

4.9-6 6 7 4.3 6 2 4 3.7-4.1 6; 4 3.9-8 6.6,7.6 5.1 7.5 5.6 3 8 7 7.9 4

ND MD MC X ND X ND MD MC X X MD X

[12] [12, 26] E26, 83 [26] [123 [26] [8] [26] [8] [12, 26] [8, 12] [26, 8] [8, 26] [26] [27, 28] [12, 8] [8] [12]

2.40

/-/(GPa)

Ag ÷

2.40

160

1.0

0.11

5.3

K+

2.70

180

1.0

0.06

6.1

Rb + Cs +

2.89 3.15

190 205

1.0 1.0

0.04 0.03

6.5 6.8

T1 +

3.20

205

1.0

0.027

6.9

deduced from the neutron diffraction isotopic substitution experiments [2], although it contrasts with that drawn earlier on intuitive grounds by Gutmann et al. (cf. Fig. 12 in Ref. [29]) where water density smoothly and slowly varied with distance from a cation.

3.1. Partition of the hydration shells The data given in Table 1 can be divided into three distinct sets corresponding to three areas in Fig. 4.

The reduced radii of the first shells of most diand trivalent cations fall into the shaded area A. As seen in Fig. 4, water in the first shells (area A) has the relative volume d-~ < 0.6. It means that water density therein is markedly higher than in the bulk water. The orientation of the HzO dipoles is characterized by a deep dielectric saturation, (cos 0) = 1 (~, < 4, cf. Tables 1 and 2 in Ref. [4]). This is characteristic of the hydration shells of the majority of di- and trivalent ions for which xl ~< 1.62 A (those ions whose reduced radii of the first shells are lower than that of C d 2 + ion). The second

1. Danielewicz-Ferchmin, A.R. Ferchmin /Physica B 245 (1998) 34 44

v,(A

X Jr-

24O

41

Ion 3+ v,=217.246-395.652xl+248.064xl 2

C)

ion 2+ v,=79.8839-101.321x~+86.6585xl 2

A

Ion 1+ vl=-279.165+274.845x~-38.4365xl 2

A Cs

200

,o

EuTb~

/

/

Dy M

/ /

/Felda~ OCd

CyNi

~i~ L'

@

8O

1.0

1.5

2.0

2.5

3.0

x,(A) Fig. 3. Volumes v~ of the hollow spherical hydration shells of cations as a function of the mean reduced radii x~. Least square fit curves are plotted for trivalent, divalent and monovalent ions separately. The equations of the corresponding curves are written in the plot. The data are taken from Ref. [9].

shells have been observed about all of them. The lower the reduced radius of the first shell xl the stronger the ion is hydrated. The reduced radii of the second shells of these cations fall into the shaded area B. The properties of their second shells are shown in the area B in Fig. 4: d-1 = 1, as in the bulk water with no applied field, and 0.4 < (cos 0} < 0.9 (50 < e < 76). The first shells of monovalent cations, except Li +, and the first shells of anions: F - , C I - , B r - and I - (formed at reduced distances x > 2.7 .A, cf. Table 2 in Ref. [4]) fall also into the shaded area B. The relative density of water in their shells is d = 1 (cf. Fig. 2d and Fig. 4). The reduced radii of the sole shells of the Ca z +, Hg 2+, Sr e+, and Li + ions fall into the range between the areas A and B in Fig. 4. Water in their shells has a relative volume d - 1 between 0.6 and 1. In this range, 0.9 < (cos 0} < 1 (7.5 < ~ < 50, cf. Tables 1 and 2 in Ref. [4]), which is classified as

a shallow dielectric saturation. The behaviour of water about the Ca 2+, Hg 2+, Sr 2+, and Li + ions is different from that sketched above for either A or B ranges.

3.2. Some specific examples 3.2.1. 0 3+ In Ref. [2] the strongly hydrating character of Cr 2 + ion seen in experiments is discussed. It can be inferred from our calculations (Table 2) that the first shell of Cr 2 + ion is in the state of a deep dielectric saturation and its density is very high. In its second shell the degree of the orientation order of the dipole moments is still high, but the density is equal to that of the bulk water far from the ion. F r o m the calculated data presented in Table 1 it can be inferred that the hydration of A13 +, G a 3 +,

42

L Danielewkz-Ferchmin, A.R. Ferchmin / Phvsica B 245 (1998) 34 44

those in the shell of Fe 2 + ion. Hence for the shells of Fe 3 + and Fe z + ions our theoretical approach provides results essentially consistent with what can be deduced from experiment [2]. 3.2.3.

0.5

1.s

2.s

3.s

×(~,)

Fig. 4. Reciprocal relative water density d t {solid line) and mean cosine (cos 0) of the angle 0 between electric field and the dipole moment of H2O molecule (dashed line, after Ref. [4]), as a function of the reduced radius x. Reduced radii of the first hydration shells of most of divalent (Me 2 +) and trivalent (Me 3 +) cations fall into the range of the shaded area A. The reduced radii of the second shells of most Me 2 * and Me 3 + cations and of the first shells of monovalent ions fall into the range of the shaded area B.

Table 2 Hydration characteristics of Cr 2+ Shells First

Second

1.15 2.5 1 2.45

2.45 54 0.85 1

3.2.4. Reduced radius (x, A) Permittivity (~:j Mean cosine ((cos 0)t Density (p~, g/cm31

Fe a+, Rh 3+ and Be 2+ ions can be comparable to that of Cr 2 + ion. 3.2.2. F e 2+ a n d F e 3+

In Fig. 10 in Ref. [2] the first order difference functions (A Gve) for Fe 3 + and Fe 2 + are shown (see Ref. [2] for the meaning of A G @ . The deep first minimum in AGve~,~ at about 3.2 A is known to be indicative of the long lifetime [16, 22, 30] of the Fe 3 + (OHz), complexes. In the A Gve2+ function that minimum is more shallow indicating that Fe z + ion is less strongly hydrated than Fe 3+ ion. In the terms applied here, the reduced radius of the first shell about Fe 3 + ion is smaller than that of the first shell about Fe z + ion (Table 1) and hence the water density and the degree of orientation ordering of dipoles in the shell of Fe 3 + ion are higher than

Cb/2+ a n d

Z n 2+

In Table 1 the ions are ordered according to the reduced radii of the first hydration shells. If, instead, they were ordered according to the reduced radii x of the ions themselves, Cu 2+ ion would be situated after Ni 2 + ion (xyi2~ = 0.69 A, Xcu~'~ = 0.72 ,~). This is the only characteristic of Cu 2 + ion that can be considered as peculiar within our approach. More subtle effects pointed out in Ref. [2] associated with C u 2 + ion cannot be accounted for within our approach due to the approximations made. Also, due to our assumption of infinite dilution, the phenomena related to the cation-anion complexation [2], observed in the case o f Z n 2 + ion, are outside the scope of the present work. In the following, specific properties of ions from the intermediate range between the shaded areas A and B (Fig. 4) are discussed. Ca 2 +

For C a 2 + ions the position of the first shell is well defined, but however the second shell has not been found, and the coordination numbers deduced from different experiments vary over a wide range (cf. Table 1). Moreover, the coordination of Ca 2+ ion reveals strong concentration effects [12, 23]. 3.2.5. Li +

The Li + ion happens to be called 'weakly hydrated' in literature [2], despite the well defined position of its first shell (cf. Refs. in Table 1). 3.2.6. S r 2+

X-ray diffraction measurements have led to the values of the radii of the first shells about S r 2 + ions lying between the areas A and B in Fig. 4, and some uncertain indications of the formation of second shells have been reported [1]. Neutron diffraction isotopic substitution experiments [25] indicate the presence of a highly disordered first hydration shell a b o u t S r 2 + ion at a reduced radius larger than that deduced from X-ray diffraction. It would be interesting to find if the anomalous hydration of Sr 2 +

L Danielewicz-Ferchmin, A.R. Ferchmin / Physica B 245 (1998) 34-44

ion at ambient conditions has anything in common with the anomalies observed about Ni 2+ ion at nonambient conditions [-30, 31]. This will hopefully be the subject of another work.

43

From the values of densities within the first hydration shells given in Table l, the coordination numbers can be calculated at a cost of assuming additionally the models proposed by other outhors (e.g., Refs. [9, 32]) for the volumes of the shells. The coordination numbers h(calc) within the first hydration shells about ions have been calculated (Eq. (19)) and given in Table 1 together with their values deduced from X-ray (X) and neutron diffraction (ND) experiments, as well as molecular dynamics (MD) and Monte Carlo (MC) calculations by various authors. The values of h(calc) seem to exceed slightly those deduced from experiment. In this context note that the volume values [-9] of the first coordination shells affect systematically the calculated h(calc) numbers (Eq. (19)). The credibility of Hahn's model is extensively discussed in Ref. [9]. An additional estimate of the uncertainty of the volume values of the first hydration shells can be obtained by comparison of data provided by different authors. Let us take as an example the Ni 2 + ion, which has been the subject of many investigations (e,g., Ref. [,21] and references therein). From Table 5 in Ref. [32] one obtains the value vl of the volume of the first hydration shell of Ni 2 + ion v~ = 130 ~3. This value exceeds that calculated in Ref. [9] cf. Table 1 in this paper. For other ions the volume values given by Barta et al. [32] exceed those of Hahn [-9]. Another factor that can affect both the water density and h(calc) is the approximate replacement of the electrostriction pressure by the field-free pressure, on calculating the striction work L, as discussed in detail in Ref. [6].

(MD) ([-1, 2] and references therein). Some MD results found in literature are given in Table 1. Among them only two provide coordination number values closer to experiment than our ones, the rest mostly do not differ in accuracy. The fundamental problem in the simulation of a realistic system seems to rely upon finding an adequate description of the forces between particles. Many models have been proposed with various expressions for the total intermolecular potential energy; adjustable parameters were not scarce in this domain [1]. We shall discuss calculation of average quantities as applied to the present case of an aqueous electrolyte. Strong electric fields due to the charges of the ions decay rapidly with the radial distance into water due to the Coulomb law. As a consequence, the molecules within the first hydration shells are in the fields of considerably higher strength than those outside them. In this way the first hydration shells about ions are distinguished by the field from the whole electrolyte. Taken together, the first shells form a subsystem of molecules in the same physical conditions, although they do not have a common macroscopic boundary. As mentioned above (cf. Eq. (1)), the chemical potentials of various parts of the system in equilibrium, including the set of the first hydration shells, the sets of the further more or less well defined shells, and the rest in zero field, are equal. The subsystem of all first shells forms a macroscopically large ensemble of molecules in the same physical conditions and one is allowed to perform on them the corresponding statistical averages leading to values of their thermodynamic parameters. The same applies to the second shells, if appropriate. Although dispersed in space, the set of hydration shells can be treated in much the same way as a layer of molecular thickness in the theory of electrolytes at an electrode [33, 6]. Thus, one can freely derive thermodynamic quantities concerning the sets of hydration shells dispersed in space, but otherwise macroscopic, by statistical methods, as done throughout this paper.

5. Discussion

6. Conclusion

One possible theoretical approach to the aqueous environment of ions is the molecular dynamics

Densities of the first hydration shells of 43 trivalent, divalent, and monovalent ions have been

4. Coordination numbers

44

I. Danielewicz-Ferchmin, A.R. Ferchmin / Physica B 245 (1998) 34 44

calculated. The results have been t a b u l a t e d a n d plotted as a function of the reduced radial distance x from the centre of an ion, which behaves like a universal variable for all valence values Z. It means that density is a s m o o t h function of the reduced distance irrespectively of the valence. The local density in a shell is due to the local electrostriction pressure /7 b e a r i n g values 4-5 orders of m a g n i t u d e higher t h a n the a m b i e n t pressure. It has been found that water in the vicinity of ions forms a very i n h o m o g e n e o u s system: passing from the first to the second h y d r a t i o n shell of di- and trivalent cations one e n c o u n t e r s j u m p s in density a n d average o r i e n t a t i o n of water molecules. This forms the essential finding of the present theory a n d is n o t inconsistent with what can be deduced from the n e u t r o n diffraction isotopic s u b s t i t u t i o n experiments on a variety of ions [12]. In addition, with the help of the values of the volumes of the hyd r a t i o n shells given by H a h n [9], a c o m p a r i s o n is performed between the calculated c o o r d i n a t i o n n u m b e r s h(calc) of the cations a n d those h(ref) taken from the literature of X-ray diffraction, neutron diffraction, molecular d y n a m i c s a n d M o n t e Carlo simulations. A reasonable general agreement is found, but n o t without a slight systematic deviation. The classification of hydrated ions illustrated in Fig. 4 m a y serve, i.a., to discern between the ions behaving in a s t a n d a r d way a n d those which do not. The latter are, as discussed in Section 3, such ions with first shells of reduced radii between the ranges A a n d B in Fig. 4: Ca 2 +, H g 2 +, Sr 2 +, a n d Li+. Their peculiar b e h a v i o u r k n o w n from experim e n t [1, 2] h a p p e n s to be related to a single parameter: the reduced radius of the first h y d r a t i o n shell.

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