THE SPECIFIC INTERACTION OF TWO IONS IN A STRONG AQUEOUS ELECTROLYTE BY S. LEVINE AND H. E. WRIGLEY Dept. of Mathematics, University of Manchester Received 1Ofh July, 1957 A method of calculating the interaction of two univalent ions in water at small sepa- rations is developed. The region of water surrounding a typical ion is divided into two parts, a co-ordination or hydration shell and the rest of the water. The water beyond the co-ordination shells is treated as a continuous dielectric medium. Assuming a given configuration of molecules in contact with the ions, the induced dipole moment in each hydration shell is obtained by introducing the reaction field. The interaction energy is determined by first evaluating the corresponding energy of the two ions with their co- ordination shells in V ~ C U O and then adding on the work of bringing up the dielectric.The case of non-overlapping co-ordination shells is considered and also that of smaller sepa- rations at which the overlapping is not yet sufficient to necessitate the removal of any molecule from these shells. Deviations from the usual Coulomb energy with the macroscopic value of the dielectric constant are obtained at small ionic separations. The correction to this classical energy takes the form c4 c6 c7 @+@+@+. . .9 where R is the separation and the C’s are constants. Numerical computations have been carried out for potassium fluoride, on the assumption that the K and F ions have the same radius and polarizability. The correction is found to be positive for both like- charged and oppositely-charged ion-pairs and is about 8 % at a separation of 6-8 A. The free energy of interaction of two ions at a distance apart in an aqueous medium may be written as (1) where e and e’ are their charges, E: is the dielectric constant of the pure water and W(R) is a correction term which becomes significant at small separations.I t is the purpose of this paper to develop a method of determining W(R). The region of water surrounding an ion is divided into two parts, a “ co-ordination ” or hydration shell and the rest of the water, which is treated as a continuous dielectric medium. (For brevity, an ion and its co-ordination shell will be called an ion complex.) This approach is preferable to one in which the whole of the water is regarded as a continuum with variable dielectric properties, since the validity of macroscopic theory within the ion complex is doubtful.Since an accurate determination of W(R) presents considerable difficulties, particularly at very small X, in the present paper we shall choose a simplified model which, however, illus- trates the main features of our general approach. In particular, dielectric satura- tion and electrostriction are neglected in the medium surrounding the ion complexes and the values of R are such that either the hydration shells do not overlap or overlapping is possible without the removal of a co-ordination molecule from its shell. V(R) = ee’/& + W(R), POTENTIAL DISTRIBUTION DUE TO ION COMPLEXES For a uni-univalent electrolyte there are three cases: case I, two positive ions, case 11, two oppositely charged ions, case 111, two negative ions.It is 4344 SPECIFIC INTERACTION OF TWO IONS sufficient to examine cases I and IT. For simplicity, it is assumed that in all cases the two ions have the same radius and polarizability and that the configuration of water molecules in the two hydration shells can differ only in the sense in which the dipoles are pointing. The water dipoles of an ion complex are assumed to point either away from a positive ion or towards a negative ion when R is infinite ; this simplifies the mathematics. It is quite possible to extend the cal- culations to more precise models and orientations of the molecules in the co- ordination shells,l-4 but this will not be attempted here. Let 01 be the centre of the ion carry- ing the (positive) charge e and 0 2 that of the other ion with charge e in case I and - e in case IT, where e is the electronic charge.The distance 0 1 0 2 = R and P is any point in the vicinity such that 01P = r l , 02P = r2 and 81 and 02 are the angles between 0 1 0 2 and 01P and 02P respectively (fig. 1). If P lies in the continuous medium the electrostatic potential $0 at P satisfies Laplace's equation and may be expanded in spherical harmonics, namely, P + e ' _ c FIG. 1. retaining only three terms. Throughout this paper, we use the convention that if alternative plus and minus signs appear in an equation, the upper sign refers to case I and the lower sign to case 11. In case I the normal derivative of $0 at the median plane between the two ions is zero and in case 11, $0 = 0 on the plane.For a specified configuration of the water molecules in the hydration shells, $0 will depend on the azimuthal angle about the line 0102. In a more elaborate treatment the energy of interaction would be determined for each configuration of the co-ordination molecules and then the appropriate average over all such con- figurations can be interpreted as W(R). However, it is expected that the depend- ence on the azimuthal angle will be small and so as a first approximation this will be neglected. By making use of the expansion, valid for I rl/R I < 1 5 (Po (COS 62) = 1, P1 (cos 02) = cos &), the following convenient form for 40 at the boundary of the ion complex 01, which is assumed to be a sphere of radius a, is obtained :S .LEVINE AND H. E. WRIGLEY 45 The potential #i inside the co-ordination shell of ion 01, for values of r1 greater than the distance of any charge inside the shell from the origin 01 can be similarly expressed as +i = Bo + - + B1r1 + - cos 61 + B2r12 + - P2 (cos 8,) + . . . . (5) rl ( :2) :3) We need to determine the seven constants in the formulae (4) and (5). EVALUATION OF p AND 4 The constants p and q represent the dipole and quadrupole moments of the charge distribution in the ion complex 01. Suppose that there are four water molecules in each hydration shell forming a tetrahedral configuration with the dipoles pointing towards a negative ion and away from a positive ion. Then at R = CO both dipole and quadrupole moments vanish, i.e.p = q = 0. How- ever, at small R the reaction field6 that one ion complex and the surrounding dielectric medium produce inside the other ion complex will induce dipole and quadrupole moments. Inside the ion complex 0 1 , the potential from which this reaction field is derived is (6) +R = ~~r~ cos el + ~ ~ r ~ 2 ~ ~ (cos 8,) + . . . . If co-ordinate axes x, y, z with centre at 0 1 and the z axis along 0 1 0 2 are intro- duced, then (6) reads +R = BIZ + 92(2z2 - x2 - y2). (7) The dipole moment p is determined as follows. Let the centre 01 of a given water dipole in the ion complex have Cartesian co-ordinates x, y, x and spherical polar co-ordinates r, + rw, 81, $, where ri and rw are the radii of the ion and of a water molecule respectively. Then x = (Is + I.,) sin 81 cos 4, y = (ri + rw) sin 61 sin 4, z = (ri+ rw) cos 61.(8) At R = co the dipole lies on the radial line pointing away from and thus 81 and $ define its direction, where 81 is the angle between the dipole and the line 0 1 0 2 . When the dipole is subjected to the reaction field at finite R, the angle 4 is unaltered since 3$~/34 = 0 but 81 becomes 81 -- /3 say, where it may be assumed that p is small. Thus if pw is the dipole vector its components are (9) where pw is the dipole moment of a water molecule. As a first approximation the interaction between adjacent water molecules inside a hydration shell may be neglected compared with that between the ion and a hydration molecule. Then, making use of (7)-(9), the energy associated with the field acting on the given water dipole is readily calculated to be pw sin (81 - cos $, pw sin (8, - p) sin 4, pw cos (6, - p>, + B2~.w(ri + rw)(2 cos 61 cos (01 - /3) - sin 61 sin (81 - /$I, (10) where the first term on the right represents the interaction of the dipole with the ion.The angle p is now determined by requiring that w be a minimum. The condition dw/dp = 0 reads46 SPECIFIC INTERACTION OF TWO IONS The dipole moment in the direction 0 1 0 2 produced by this change in orientation of the water dipole is given by p,(cos (81-/3)-cos el> = pw sin 81 (tanp + . . .) + pw cos 191(--+ tan2p + . . .). (12) The expression (11) for tan P is substituted into (12) and the resulting form for (12) is developed as a power series in B1 and B2. The sum of (12) over the four molecules of the ion complex is obtained as follows.We may suppose that the four water molecules of 0 1 are situated at alternate corners of a cube (fig. 2). FIG. 2. It is now convenient to introduce co-ordinate axes XI, y’, z’ which are attached to the ion complex such that the point 01 is the origin and the cube edges are parallel to the axes. Further let I, rn, n be the direction cosines of the line 0 1 0 2 relative to these axes. If ,E denotes summation over the four molecules then it is readily seen that Zsin281 = 4 - Z c o s 2 8 1 = 8/3 since z cos2 el = 4t(z + rn + n)2 + (I - rn - n)2 + ( - I - m + n)2 + (- I + nz - 4 2 1 = 4/3. Similarly, C sin 81 sin 281 = 2.Z (cos 81 - cos3 81) = 0. It follows that those terms in (12) which are linear in B1 and B2 contribute to the induced dipole moment in the direction 0 1 0 2 an amount Other such linear terms are due to the polarizabilities 01, and ai of a water molecule and ion respectively.The z-component of the induced dipole of the water molecule at x, y , z is Summing over the four molecules and noting that Zcos 81 = 0, we obtain the contribution - 4cc,B1. The corresponding induced dipole moment of the ion is - ~$1. Adding up the three terms, the induced dipole moment due to the homogeneous part of the reaction field is - (010 + 4a, + ai)B1 =- aB1 where 01 may be described as the polarizability of the ion complex. We may expect B1 > 0 in case I and B1 < 0 in case 11. It is noteworthy that cc is independent of the orientation of the tetrahedral configuration of the ion complex.The sum over the four molecules of the higher terms in (12) depend in general on the orientation of the tetrahedral arrangement, but this will be ignored. A convenient orientation is that for which the cube in fig. 2, has its edges parallel - *BlpwC(ri + rw121eI =- EOBl, say. - cc,(B1 + 2B2z) =- ccc,(B1 + 2B2(ri + rw) cos e,>.S . LEVINE AND H . E. WRIGLEY 47 to the x , y , z axes. (This means that in fig. 2 the cube is rotated until the z' axis coincides with the line 0 1 0 2 . ) Then it may be verified that the contribution to the induced dipole moment from the quadratic terms in B1 and B2 on the right- hand side of (12) is simply - ' ~ ' ~ w [ ( ~ i + rw)s/e2]B1B2 =- aB1B2, say. Thus the induced dipole moment is p =- CiBl - 0B1B2 + . .., (1 3) neglecting higher terms. The determination of q is considerably more difficult and we shall only estimate its value. As with (T, the dependence of q on the orientation of the tetrahedral arrangement will be ignorcd, and it is sufficient to retain the first term on the right of (6). Suppose that the orientation of the ion complex is such that the axes x, y, z, which we denote here by X I , x2, x3, are the principal axes associated with the tensor ec say, which defines the quadrupole moment and, in addition, let ell = 132. Then making use of the usual formula for the potential due to a quadrupole,sS 7 it may be shown that q = e33 - ell. (14) For simplicity we follow Bernal and Fowler 1 and assume that each water molecule consists of two charges & e at a distance d = p,/e apart, with the negative charge - c at the centre of the molecule.A particular orientation of the ion complex 0 1 which satisfies the required conditions on eQ is that already introduced above to evaluate G. Then the contribution to q from the rotation of the water dipoles (assumed to be rigid), when subjected to the field ] B1 I is where ,f3 is obtained from (1 1) by putting B2 = 0 and sin 81 = 4 2 / 3 . This result still applies if the cube is rotated about 0 1 0 2 , since the potential distribution re- mained independent of the azimuthal angle +. There will be a contribution to q from the induced charge distributions in the water molecules and ion. To determine this the quadrupole moments produced by a homogeneous field are required for a water molecule and an ion. Since these do not seem to be known we shall make a rough estimate of this polariza- bility term with the aid of the following simple model.It is assumed that inside a water molecule of the ion complex 0 1 , a negative charge - r],e, where 7, is of order one, is displaced a distance s from its centre in the direction 0 1 0 2 , leaving behind a charge Twe. Then s > 0 in case I and s < 0 in case 11. The contribution to e33 is i 2~wes(ri + rw) cos w - Twes2, where o is the angle between 0 1 0 2 and the radial line from the point 01 to the centre of the molecule. Since Zr cos w = 0, summation over the four molecules yields the contribution to 4. The corresponding contribution from the ion is written as - 4ui2B12/qe. The other components of the tensor eij are unaffected by this polarizability effect.Thus, adding together the three terms above 4 = X W , where The second term on the right of (16) is independent of the orientation of the tetra- hedral configuration provided the water molecules are assumed to be isotropic. For simplicity, we shall choose yw = ~i = 1. Since both terms in (16) are probably48 SPECIFIC INTERACTION OF TWO IONS rough estimates and are of the same order of magnitude, the value of x may be in considerable error. Furthermore the form (2) for #O implies that in case I1 x is replaced by - x for a negative ion. Fortunately in the present calculations, x is found to be so small that it may be neglected as a first approximation. APPLICATION OF BOUNDARY CONDITIONS The five remaining constants in the expressions for #o and #i are obtained by applying the boundary conditions at ~1 = a, namely, On equating the coefficients of cos 81 and P2 (cos 81) in the usual manner we have the four relations : (21) The remaining relation which determines Bo is not required.It is convenient to solve these equations by a method of successive approximations as follows. Eqn. (18) and (19), yield A1 and B1 in terms of A2, B2 and B1B2, namely, ~ X B I 2ea 6Ala 3A2 12A2a 2B2a--=~ &-&--- a4 [ cR3 R4 a4 *TI a where, introducing t = a/R and u = a/a3, Good approximations sides of (22) and (23) c = 1- u - (1 + 2U)/€, D = 1 - u + (1/2€)(1 -t 2u) ct3, (26) to A1 and B1 are given by the first term on the right-hand respectively. Substituting these values for A1 and B1 into (20) and (21), we can solve for A2 and B2 and so obtain (27) (28) A2 =& 2 e 3 [ ( 1 -;)(I - 3 5 3 ) ”s 45 z2] X’t , 3 EF where x’ = eX/a6, a dimensionless quantity, and G = 1 - - 1 F 2t(2 + 4).andS. LEVINE AND H. E. WRIGLEY 49 These values for B1, A2 and B2 can then be inserted into the right-hand sides of (22) and (23) and this yields better approximations for A1 and B1. Clearly this iteration process can be repeated to yield any degree of accuracy but this was not found necessary. The terms in the square brackets in (27) and (28) which are proportional to X I were found to be of the order of 10-6 and 10-5 respectively and therefore have been neglected. EVALUATION OF INTERACTION ENERGY The energy V(R) may be obtained by carrying out the following steps.(i) It is imagined that a single ion complex with its water molecules in the configuration corresponding to intinite R is situated in vacuu. The internal energy of interaction of this complex is not required. The dielectric medium is now brought up from infinity and surrounds the complex, under the condition that the charge distribution inside the complex is held fixed. Then the work of introducing the dielectric is 6 9 7 where V is the volume of the dielectric, Eo is the electric field in the vacuum before it is filled by the dielectric and E is the corresponding field in the dielectric medium after this operation. Since the dipole and quadrupole moments of the (undis- torted) charge distribution in the ion complex are zero and we are ignoring the higher multipoles, Eo .E = e 2 / ~ 4 , where r is the radial distance from the centre of the ion complex. Thus (30) reads The corresponding energy of the two ion complexes at separation R is now evaluated in three stages as follows. (ii) Consider the two ion complexes at infinite separation in vacuo with their charge distributions in the state corresponding to this position. This distribution must now be changed to that actually prevailing at the given finite separation R. The energy expended in rotating a water molecule of the complex 01, through an (ri + rw)2 Expanding the expression (1 1) for substituting into (32), the sum of (32) -- - 'we [+ tan2 p +. . .I. (ri + rwI2 tan p into a power series in B1 and B2 and then (32) over the four molecules of one complex is + e (33) p w h f r w ) 4 B 2 2 t....Only the coefficient NO is independent of the orientation of the ion complex and we have again chosen the particular orientation used previously. The work required to polarize the water molecules and ions is next found. If X is the electric field on a water molecule in complex 0 1 , due to its ion, the work of polarizing this water molecule is &@&(x - V#R) (x - V#R) - *awx2, where X is the magnitude of the vector X. If x , y , z denotes the position of the centre of the molecule relative to 0 1 as centre, then X has components and the components of V { ! R can be determined from (7). Summing over the four molecules and adding on the corresponding work of polarizing the ion itself, we obtain (e/(ri + rwI3>(x9 Y , 23 W," = 2aw{B12 + 2B22(ri + ~ w ) 2 > + 4CtiB12.(34)50 SPECIFIC INTERACTION OF TWO IONS The work of displacing the charge inside the two ion complexes is then 2w1 = 2(W1' + W,") = aB12 + yB22 + . . ., where y = 8cc,(ri + r d 2 + (2pw/c)(ri + r d 4 . It is observed that the leading term in this expression is the work of polarizing the ion complexes in a homogeneous field of intensity I B1 I. The resulting charge distribution is held fixed for the remaining two operations. (iii) The two ion complexes are moved in vacuu from infinite distance apart to separation R and the work required is evaluated. This is (35) e2 2ep 2p2 2eq R R2 R3 2 W 2 = & - & - & - & R j + . . . . (iv) The dielectric is again introduced to fill the vacuum surrounding the two complexes, stationed at distance R apart.The expression (30) yields the work of bringing up the dielectric. Here E =- V$O and if YO is the potential function f + . ., (37) then Eo =- VYo. Since both #O and Yo satisfy Laplace's equation, this work may be expressed as applying Green's theorem. Here S is the surface bounding the volume V of the dielectric and 3/3n denotes differentiation along the normal to S drawn outwards from V. It is convenient to replace V by the infinite half-volume bounded by the median plane and the sphere r1 = a and take twice the value obtained for the energy (38). On the median plane 3$0/3n = 3Yo/3n = 0 in case 1: and $0 = YO = 0 in case I1 and so there is no contribution. The integral over the surface at infinity (e.g.an infinite hemisphere) also vanishes. Either surface integral in (38) may be used. To calculate the first integral over the sphere rl = a we substitute the expansion (4) for #*, evaluated at rl = a and the corresponding expansion Then, neglecting the terms in q since these are very small, 2 +'j(*,R2+;;z ea A1 f- 2Ala *-) 2 R3 R4 ea2 3Ala2 R4 + ~ d ~ ~ ~ ) ( + ~ = t ~ ) + . A2 . .I. (40) R5 +3 f-f- '( E R ~S . LEVINE AND H . E. WRIGLEY 51 After some algebraic manipulation, the energy of interaction can finally be expressed as w?) = 2Wl + w2 + w3 - WO) f 1 + b3t3 -1 b5t5 3- b6t6 -I- . . . , (41) ER 1 where b3, 05 and b6 are slowly varying functions of t = a/R, given by 3F b6 ='F -[ 1 (1 -+) ( 1 + 5u - g) - 2 3 3 0 C 9 whereH=- 1 -- 4 1 +- f- 0 "( r) - ( iE) 2 0 6 ' Here the dimensionless quantities y' = y/a5 and (T' = oela6 have been introduced, We have ignored higher powers of t since these would also require a consideration of higher terms in the expansions (2) and (5).The values of b3, b5 and b6 are shown in the table for a KF solution. The K and F ions are assumed to have the same radius ri = 1.33 A and the same polar- izability tci = 0.93 X 10-24 cm3 (the mean of their actual polarizabilities). Choos- ing Y, = 1-38& 01, = 1.68 x 1 0 - 2 4 ~ ~ 3 and p = 1*84D, it is found that cc = 15.3 x 10-24 cm3. An upper estimate of a is given by a = ri + 2r, = 4-09 A, when u = a/a3 = 0.223. If a tetrahedral structure is assumed for the water surrounding an ion, the radius of the sphere touching the nearest molecules which are not in contact with the ion is 3.09 A and this gives a lower estimate of a when u = 0.518. The distance R = 6.18 A is approximately equal to the closest distance of approach of the two ions without causing the disturbance of the tetra- hedral structure inside each shell.The effective dielectric constant of the ion complex is readily seen to be 2 = 1 + 474(4na3/3) = 1 + 3u. Thus 7 =1-66 for the upper estimate and 2-56 for the lower estimate of a. The fist value of C seems too small since it is expected that when dielectric saturation of the co- ordination molecules is complete 2 = d, where n is the refractive index of the ion complex. Of course, our value of a can only be approximate since a crude model has been chosen for the co-ordination shell. We have computed b3, b5 and b6 for the two values of u and for t = 3 (contact of the ion complexes) 6 and +. It is seen that the correction to the Coulomb energy is positive in both cases I and 11, i.e.the correction behaves as an additional repulsion. It is instructive to consider the case where u = 0 when we have two point charges at the centres of spherical cavities. The corresponding values of b3, b5 and bg are given in the table and again there is repulsion. Now there is a well-known theorem in electrostatics that if the dielectric constant of the medium in an electric field is reduced but the source of the field is unchanged then the energy is increased. Hence the removal of the water molecules from the co- ordination shells to produce the spherical cavities must lead to an increase in energy and this may be regarded as the origin of the positive correction. The coefficient b3 becomes negative for u > 1 (when E' > 4) and this shows that in order to obtain a negative correction the value of the polarizability a must be appreciably greater than that estimated here.TABLE 1 ul h) case I I1 0 (a’ = y’ = 0) I 0.223 I1 (a = 4.09A) I I1 (a = 3.09A) 0,518 65 b6 b3t3 4- b5t5 f bgt6 b3 5 a 3 t t 3 t 0.583 0.527 0.501 0.744 0.673 0,656 -0.373 - 0.345 - 0.332 0.090 0.032 0.008 a419 *458 ,481 -582 -635 -650 -29 1 -311 -322 -075 so30 -008 -615 -534 -497 ~750 *677 660 -a827 - ‘779 - -756 a087 *U31 so08 0350 a438 -469 ,585 -639 ,655 *680 -729 .73& -073 -030 -008 -697 -551 ‘487 a775 a697 -679 - 2.007 - 1.915 - 1.870 *080 -029 -008 -288 -382 -437 a600 ~656 *673 1-713 1.791 1.833 -081 so30 so08 When a = 3.09& x’ = 1.31 x 10-3, U’ = 0-385, 7’ = 0.515. When a = 4-09 A, x’ = 2.44 x 10-4, U’ = 0.072, 7’ = 0.127. 1 Bernal and Fowler, J . Chem. Physics, 1933, 1, 515. 2 Eley and Evans, Trans. Furuduy SOC., 1938, 34, 1093 ; Everett and Coulson, Trans. Furuduy SOC., 1940,36,633 ; Verwey, Rec. trav. chim., 1941,60,887 ; 1942, 61, 127. 3Lennard-Jones and Pople, Proc. Roy. SOC. A, 1951, 205, 155. Pople, Proc. Roy. SOC. A, 1951, 205, 163. Duncan and Pople, Trans. Furuduy SOC., 1953, 49, 217. 4 Campbell, J. Chern. Physics, 1952, 20, 141 1. 5 Hobson, Spherical and Ellipsoidul Harmonics (Cambridge University Press, 193 l), 6 Boettcher, Theory of Electric PoZurization (Elsevier, Amsterdam, 1952). 7 Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). p. 140.