J. Phys. Chem. 1983,87,5071-5083
5071
FEATURE ARTICLE Hydration and Mobility of Ions in Solution R. W. Impey, Division of Chemistry, National Research Council of Canada, Ottawa K1A ORB, Canada
P. A. Madden,+ Theoretical Chemistry Department, University Chemical Laboratory, Cambridge CB2 lEW, England
and I. R. McDonald* Department of Physical Chemistry, University of Cambridge, Cambridge CB2 IEP, England (Received: March 16, 1983, In Final Form: August 4, 1983)
Molecular dynamics calculations are reported for the systems [M+],, and [X-I,,, with M = Li, Na, K and X = F, C1. The results for the static structure, including coordination numbers, ion-water geometries, and radial distribution functions, are in very good agreement with those obtained from neutron diffraction experiments and the diffusion coefficients of the ions show the expected increase with ion size. However, problems caused by the fact that the molecular dynamics system contains only one ion make it more profitable to discuss the dynamics in terms of the motion of the water molecules in the first coordination shell of the ion. A definition of residence time in the coordination shell is proposed and leads to values for the hydration numbers which are consistent with estimates derived from electrochemical measurements. The ionic mobilities are analyzed in terms of a theoretical model involving a nonuniform viscosity. The latter may be characterized either by the reorientation times of water molecules in the first coordination sphere or by their residence times. In either case, good agreement is obtained with the experimentally determined Walden products.
1. Introduction
The way in which the structure of bulk water is modified by the introduction of ions and the extent to which this determines the dynamical properties of the solution are classic problems of electrochemistry.'Y2 Our present understanding of the changes which occur is strongly influenced by the work of Frank and G ~ r n e y and ~ - ~may be summarized as follows. The water surrounding the ion is considered to be divided into three regions. In the innermost (or primary) region, A, the water molecules are strongly oriented by the ion and tend to be carried by it as the ion moves through the solution. Next, there is a secondary region, B, in which the molecules are only weakly oriented by the ion, though the structure is broken down to some extent by comparison with bulk water. Finally, in the outermost region, C, far from the ion, the structure is the same as that of bulk water. Depending on the relative importance of regions A and B, ions may be classified as structure makers or structure breakers. Small ions are structure makers; the ions in region A are strongly bound to the ion and it is appropriate to think in terms of a well-defined ion-water complex. Larger ions are structure breakers; their influence is felt mainly in their disruption of the hydrogen-bond network characteristic of bulk water. This explains the high mobilities of the larger alkali and halide ions. However, very large ions may be structure makers, since hydrogen bonding between water molecules can be enhanced at the surface of the ion; these are the so-called "hydrophobic" structure makers, Present address: T h e Royal Signals and Radar Establishment, Malvern, Worcestershire WR14 3PS, England.
of which the tetraalkylammonium ions are well-known examples. The picture that we have sketched is the familiar one of a hydrated ion; the number of molecules which contribute to the hydration is called the hydration number of the ion. Much experimental effort has been devoted to determining the hydration numbers of different ions. The traditional methods are largely indirect: a value of the hydration number is inferred from measurements of properties such as mobility, compressibilit,y, dielectric constant, etc. The experimental situation up to 1969 has been reviewed by Hinton and A m h 6 The most striking feature of the data that they compile is the fact that different experimental techniques can lead to vastly different estimates of the hydration number. In the case of Na+, for example, values ranging from 1 to 71 have been reported,2 and such a spread is by no means exceptional. A number of factors contribute to this uncertainty, including the difficulty of partitioning the hydration number for a salt into separate contributions from the cation and anion. However, the main reason for the wide spread in results is undoubtedly the fact that different types of measurements are sensitive to ion-solvent correlations over dif(1) Robinson, R. A.; Stokes, R. H. "Electrolyte Solutions"; Butterworths: London, 1955. (2) Bockris, J. O'M.; Reddy, A. K. N. "Modern Electrorhemistry"; Plenum Press: New York, 1973. (3) Frank, H. S.; Evans, M. J. Chern. Phys. 1945,13, 507. (4) Gurney, R. W. "Ionic Processes in Solution"; McGraw-Hill: New York, 1953. (5) Frank, H.S.; Wen, W. Y. Discuss. Faraday SOC.1957, 24, 133. (6) Hinton, J. R.; Amis, E. S. Chern. Reu. 1967, 67, 367.
0022-3654/83/2087-5071$01.50/00 1983 American Chemical Society
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ferent ranges of both distance and time; in theoretical treatments of the problem, this question is often given inadequate attention. For the concept of hydration number to be useful, a definition a t the microscopic level is required. Following Bockris and Reddy? we shall interpret the hydration number of the ion as being the number of water molecules which are bound to the ion for times sufficiently long that they participate in the diffusive motion of the ion. Such a quantity could appropriately be termed a dynamic hydration number. In a simple theory, the ion and its hydration shell may be treated as a single kinetic entity. One of the main objects of the present paper is to propose and examine the usefulness of a dynamic hydration number defined in terms of the residence time of water molecules in the region close to the ion. It is also possible to speak of a coordination number, equal to the mean number of water molecules in the first coordination shell of the ion. The definition of the latter is itself arbitrary, but once a convention has been chosen the coordination number is unambiguously specified. The terms coordination and hydration are sometimes used interchangeably, but for our purposes it is precisely the difference between them which is of interest. The coordination number is determined solely by the static structure of the solution and is measurable, in principle, by X-ray and neutron diffraction methods. The concept of hydration number is a wider one, since it involves the adoption of a suitable time scale. In recent years, our knowledge of the microscopic structure and dynamics of aqueous solutions has grown substantially. There are several reasons for this progress. First, the technique of neutron diffraction has been improved to the point where it is possible to determine both ion-solvent and ion-ion c~rrelations.~-llThere have also been considerable developments in spectroscopic techniques applied to the study of solution^.^^^^ Second, much is now known about ion-water interactions from ab initio potential energy calculation^.^^-^^ The availability of realistic models of the water-water and ion-water interactions has encouraged the application of the Monte Carlo method to the study of ions both in bulk water18J9and in clusters of water molecules.20-22 In addition, the group at Mainz have carried out a number of molecular dynamics calculation^^^-^^ on alkali halides in water, based largely (7) Soper, A. K.; Neilson, G. W.; Enderhy, J. E. J. Phys. C 1977, IO, 1793. (8) Enderhy, J. E.; Nielson, G. W. Adu. Phys. 1980, 29, 323. (9) Enderby, J. E.; Neilson, G. W. Rep. Prog. Phys. 1981, 44, 593. (10) Newsome, J. R.; Neilson, G. W.; Enderby, J. E. J . Phys. C 1980, 13, L923. (11) Newsome, J. R. Ph.D. Thesis, University of Bristol, Bristol, England, 1981. (12) Franks, F., Ed.; “Water: A Comprehensive Treatise”; Plenum Press: New York, 1973; Vol. 3. (13) Clementi, E.; Popkie, H. J . Chem. Phys. 1972, 57, 1077. (14) Kistenmacher, H.; Popkie, H.; Clementi, E. J. Chem. Phys. 1973, 58, 5627. (15) Kistenmacher, H.; Popkie, H.; Clementi, E. J . Chern. Phys. 1973, 58, 5627. (16) Kistenmacher, H.; Popkie, H.; Clementi, E. J . Chem. Phys. 1973, 58, 5842. (17) Schuster, P.; Jakubutz, W.; Marius, W. Top. Curr. Chem. 1975, 60, 1. (18) Beveridge, D. L.; Mezei, M.; Swaminathan, S.; Harrison, S. W. “Computer Modeling of Matter”; Lykos, P. G., Ed.; American Chemical Society: Washington, DC, 1978. (19) Mezei, M.; Beveridge, D. L. J. Chem. Phys. 1981, 74, 6902. (20) Clementi, E.; Barsotti, R.; Fromm, J.; Watta, R. 0. Theor. Chim. Acta 196, 43, 101. (21) Clementi, E.; Barsotti, R. Chem. Phys. Lett. 1980, 59, 21. (22) Engstrom, S.; Jonsson, B. Mol. Phys. 1981, 43, 1235.
on empirical potential energy functions. Both types of simulations are capable of yielding information on the properties of aqueous solutions which goes far beyond what can be learned experimentally. Finally, there has been significant progress in both the equilibrium and nonequilibrium statistical mechanics of ionic solutions. Work in these areas has been reviewed by Andemen,%Friedman and Dale,29and W ~ l y n e s . ~ ~ The purpose of the present paper is to report the results of a series of molecular dynamics calculations of aqueous solutions containing monatomic ions at effectively infinite dilution. The specific systems considered are [M+],,, with M = Li, Na, K, and [X-I,,, with X = F, C1. The outline of the paper is as follows. In the next section we give some details of the simulations and of the potential energy functions used to describe the water-water and ion-water interactions; in section 3, we discuss the structure of the solutions in terms of partial radial distribution functions, coordination numbers, and functions relevant to neutron diffraction experiments; sections 4 and 5 are concerned with the translational motion of the ions, the former with the self-diffusion coefficients and the latter with velocity autocorrelation functions; in section 6 we discuss the related concepts of residence time and hydration number and suggest definitions of each in terms of quantities which are easily measurable in the simulations; in section 7 , the same definitions of residence time and hydration number are used to build a simple theoretical model which accounts satisfactorily for the measured mobilities of the ions; and in section 8, we summarize the main results of our work.
2. Molecular Dynamics Calculations For the most part, the technical details of the molecular dynamics calculations were the same as in our earlier work on pure water.31 The primitive molecular dynamics cube contained one ion and either 64 or 125 water molecules, and the usual periodic boundary conditions were imposed. The water molecules were treated as rigid objects and the equations of motion were solved by the method of constraints developed by Ryckaert et al.;32 the standard finite difference algorithm33was used for both the water molecules and the ion, with an integration time step of 0.005 ps. Approximately 2000 time steps were allowed for equilibration, apd static and dynamic properties of the system were calculated by averaging, typically, over a further 6000 time steps, covering a total real time of order 30 ps. All electrostatic interactions (ion-solvent and solventsolvent) were evaluated by the Ewald method. To the extent that no account was taken of any counterion, the calculations correspond to conditions of infinite dilution, though the use of the periodic boundary condition means that this interpretation is not strictly correct. The (23) Heinzinger, K.; Vogel, P. G. Z. Naturforsch. A 1976, 31, 463. (24) Vogel, P. C.; Heinzinger, K. 2. Naturforsch. A 1976, 31, 476. (25) Pilinkis, G.; Riede, W. 0.;Beinzinger, K. Z. Naturforsch. A 1977, 32, 1137. (26) Bopp, P.; Dietz, W.; Heinzinger, K. Z. Naturforsch. A 1979, 34, 1424. (27) Szasz, G. I.; Heinzinger, K.; Reide, W. 0. Ber. Bunsenges. Phys. Chem. 1981,85, 1056. (28) Andersen, H. C. “Modern Aspects of Electrochemistry”;Conway, B. E., Bockris, J. O’M.; Eds.; Plenum Press: New York, 1975. (29) Friedman, H. L.; Dale, W. D. T. “Statistical Mechanics, Part A: Equilibrium Techniques”; Berne, B. J., Ed.; Plenum Press: New York, 1977 _.
(30) Wolynes, P. G. Annu. Rev. Phys. Chern. 1980, 31, 345. (31) Impey, R. W.; Madden, P. A,; McDonald, I. R. Mol. Phys. 1982,
4fi - - , 512 - - -.
(32) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J. Comput. Phys. 1977, 23, 327.
(33) Verlet, L. Phys. Reu. 1967, 159, 98.
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(b)
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W
Figure 1. Possible structures of the ion-water complex for (a) a cation and (b) an anion.
0
1
2
3
4
5
6
7
8
RIA
density of the system was taken to be that of pure water a t the temperature of interest. The interactions between water molecules were modeled by the analytical potential energy function of Matsuoka, Clementi, and Yoshimine (MCY), obtained by fitting to the results of ab initio quantum-mechanical calculations on the water dimer.34i35 A large number of potential models have been used in the simulation of liquid water, the best known of which is the ST2 potential of Stillinger and Rahman.36 Our choice of the MCY potential was dictated in part by a desire to maintain consistency with our previous work on pure water, but more importantly by the fact that the model is known to yield a fair description of many of the static and dynamic properties of real ~ a t e r . ~ lIn 8 ~the MCY model, the short-range interactions between molecules are described by atom-atom potentials of the Born-Mayer type, and the charge distribution of the monomer is represented by three fractional point charges corresponding to a dipole moment of 2.19 D (the measured gas-phase value is 1.85 D). To describe the interactions between ions and water molecules, we have used the potentials of Kistenmacher et a1.16 (their "simple" models), which are also based on ab initio calculations a t the dimer level. The short-range ion-molecule forces are again described by Born-Mayer type potentials, but the charge distribution of the water molecules is now modeled by six fractional charges with dipole moments ranging from 2.34 D (for the case of K+) to 2.51 D (for Li+). The fact that the water molecule presents a different charge distribution to an ion than to another water molecule is physically reasonable; an enhancement of the effective dipole moment is necessary in the ion-water case if account is to be taken of the polarizing effect of the ion. Of the previously reported simulations of aqueous solutions, the work of greatest relevance to our own is that of Beveridge and c o - w ~ r k e r s . ' ~ JThese ~ authors have studied the same systems and have used the same potentials, but, because their interest has been focused on static (34)Matsuoka, 0.; Clementi, E.; Yoshimine, M. J. Chem. Phys. 1976, 64, 1351. (35) Lie, G. C.; Clementi, E.; Yoshimine, M. J. Chem. Phys. 1976,64,
2314. (36)Stillinger, F. H.;Rahman, A. J. Chem. Phys. 1978, 68, 666.
Figure 2. Radial distribution function g l o ( R )for Li+ as a function of temperature.
TABLE I: Positions of Peaks in the Radial Distribution F u n c t i o n s a ion
T/K
Li'
278
Na'
K' F' C1-
282 274 278 287
R(1-O)/A 1.98 (1.95 2.29 2.76 2.67 3.29 (3.34
t
?-
0.02)
0.02)
R(1-H,)/A 2.57 (2.55 2.95 3.35 1.73 2.35 (2.25
i
R(I-%)/ A
0.02) 3.07 3.73
t
0.02)
a Quantities in parentheses are e x p e r i m e n t a l results" for a 3.57 m s o l u t i o n of LiC1.
properties, all their calculations to date have been made by the Monte Carlo method. 3. Structure The structure of the water molecules about an ion I is described by the two partial radial distribution functions gIo(R) and gI,(R) (I = M, X). The local order to which the forms of the calculated functions correspond is conveniently discussed in terms of the two idealized models of the ion-water complex which are pictured in Figure 1; these are the same models used by Enderby, Neilson, and co-workers7-11in analyzing the results of neutron diffraction experiments on a number of electrolyte solutions. In the context of the suggested structures, the quantities of principal interest are the angle 0, which measures the deviation from planarity of the M+-H20 complex, and the angle +, which measures the deviation from linearity of the X--H-0 bond in the X--H20 complex. Partial Radial Distribution Functions. The functions gIo(R) and gIH(R) for each of the five ions are plotted in Figures 2-9. The trends with ion size are those to be expected on intuitive grounds, and the positions of the peaks are broadly consistent with the structures of Figure 1,as we shall see below. The separations at which the main peaks occur are summarized in Table I; for Li+ and C1-, there is good agreement with the peak positions obtained in the neutron scattering experiments'O on a 3.57 m solution of LiCl.
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25, 1983
Li+
0
I
I
I
I
I
I
i
1
2
3
4
5
6
7
8
R /A
Flgure 5. Radial distribution functions for K+ at 274 K. Plain curve:
gIo(R) for a system of 64 water molecules. Curve with triangles: gIo(R)for a system of 125 water molecules. The circles are the Monte Carlo results of Mezei and Beveridge” for gIc(R). 4.0
I
I
I
I
I
I
I
1
2
3
4 R/A
5
6
7
3.5 3.0
2.5
-lY
(9
-I
0
1
2
3
4
5
6
2.0 1.5
7
R /A Figure 3. Radial distrlbutlon function gIH(R)for Li+ as a function of temperature.
,
I
I
I
I
I
O
1
I
-0.5 -OI .o
1
0
Flgure 6. Radial distribution function gIH(R) for K+ at 274 K. Plain curve: results for a system of 64 water molecules. Curve with triangles: results for a system of 125 water molecules.
7
6
-a
.
5
- 4 (3
3 2 I
0 -I
I
0
I
I
I
I
I
I
i
1
2
3
4
5
6
7
0
R/A
Flgure 4. Radial distribution functions for Na+ at 282 K. Plain curve: glo(R). Curve with triangles: gl$?). The circles are the Monte Carlo results of Mezei and Beveridge’ for glc(R).
From the figures, we see that the results for the positive ions are characterized by a very pronounced main peak in gMo(R). The height of this peak decreases in the series Li+ Na+ K+, as does that of the much weaker second peak. For both Li+ and Na+, the function falls almost to zero between the first two peaks; for K+, the minimum is
- -
-I
I
0
I
1
I
I
I
I
I
I
2
3
4 R /A
5
6
7
I
8
Flgure 7. Radial distribution functions for F- at 278 K. Plain curve: glo(R). Curve with triangles: gIH(R). The arrows mark the expected peak positions for a linear F - 4 - 0 bond. The circles are the Monte Carlo results of Mezei and Beveridge’’ for gIc(R).
The Journal of Physical Chemistry, Vol, 87, No. 25, 1983 5075
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TABLE 11: C o o r d i n a t i o n Numbersa A
ion
T/K
Li' Na'
278 282 274 278 28 7
K+ F-
-a 0
c1-
2.0
%on
5.3 ( 5 . 5 6.0 7.5 5.8 7.2 (5.9
i;
0.3)
i;
0.2)
Quantities in parentheses are e x p e r i m e n t a l results" for a 3 . 5 7 rn s o l u t i o n of LiC1. a
1.5
1
I .o
,
I
I
I
5
6
7
-0.5
- 1.00
1
2
3
4
%
R/A
Flgure 8. Radial distribution functions for CI- at 287 K. Plain curve: g,,(R) for a system of 64 water molecules. Curve with triangles: g,,(R) for a system of 125 water molecules. The circles are the Monte Carlo results of Mezei and Beveridge'' for g,(R).
J
0.0 '
-
-0.5-
I
I
1
I
I
I
I
shallower and, in the results for the 125 water molecule system, there is a hint of a third peak. In the same way, the first peak in &H(R) becomes weaker in the order of increasing ion size. In each case, there are two well-resolved and widely separated peaks. The results of Figures 2 and 3 show that the structure around the Li+ ion is only slightly relaxed as the result of an increase in temperature of order 100 K. The fact that the structure is so welldefined and apparently so persistent (see below) is cause for slight concern in one respect: the results may be sensitive to the choice of initial configuration. However, by taking different starting conditions, we have found that in molecular dynamics runs of the length reported here the final results are independent of the way in which the initial arrangement of water molecules is chosen. Consequently, the final state achieved can be regarded as the unique equilibrium one. In the case of the negative ions, the main peaks in gxo(R) and gxH(R) are of roughly equal height. The two peaks seen in gxH(R) are closely spaced and correspond to the two nonequivalent hydrogen atoms H1, H2 of Figure Ib;
in the case of the positive ions, the second peak in gXH(R) derives from a different, more distant shell of water molecules. Again, the degree of structure seen in the distribution functions decreases with ion size. Comparison urith Monte Carlo. The results of the present work should agree within the limits of statistical error with the Monte Carlo results of Mezei and Beveridge,19since the same potential energy functions were used in both cases. The comparison is slightly complicated by the fact that Mezei and Beveridge plot the function gIc(R) rather than gIo(R),where C denotes the center of mass of the water molecule. The only ion for which we have calculated this function is Li', and there the agreement is good. In Figures 4, 5, 7, and 8, we compare the present results for gIo(R)with the Monte Carlo results for gI,(R). For Na+ and F-, the agreement is satisfactory, and the small differences which exist are explicable in terms of the different temperatures in the two sets of calculations and of the fact that the center of mass is not coincident with the oxygen atom. For K+, the agreement is only barely acceptable, and for C1- it is frankly poor. The Monte Carlo calculations were made for larger systems (215 water molecules), and to that extent are likely to be more reliable, but the molecular dynamics results do not show any significant size effects between 64 and 125 molecule systems. A possible explanation of the differences is that they are associated with the use, on the one hand, of an Ewald sum (molecular dynamics) and, on the other, of a truncated Coulombic potential (Monte Carlo) to describe the electrostatic interaction between the ion and the water molecules. It is known37that use of a truncated Coulombic potential leads to systematic errors in the case of fully ionic systems, including molten salts and plasmas, but little is known about the merits of different schemes for treating mixed systems of the type under discussion here. CoordinationNumbers. The separation R = Rs at which the function gIo(R)has its first minimum may be used to define the first coordination shell of the ion, and integration of R2gIo(R)up to this point yields a value for the coordination number nion.For the smaller ions (Li+,Na+, F-),the latter is a well-defined quantity, since the integral plateaus as a function of its upper limit and the precise choice of outer radius of the coordination shell is unimportant. For these ions, the local coordination is roughly octahedral. In the case of K+ and C1-, the coordination numbers are larger, but they also have a less clear-cut physical significance, since glO(R)has a much shallower first minimum. For a sharply defined coordination shell, integration of g I H ( Rup ) to its first minimum should yield a coordination number equal to 2n,on. This is approximately true for the smaller ions, but for K+ and C1- there are deviations which suggest that there is some penetration by molecules belonging to the second shell of neighbors. The calculated coordination numbers are gathered together in Table 11. The only neutron diffraction results (37) Ceperley, D., Ed. "The Problem of Long-Range Forces in the Computer Simulation of Condensed Media"; NRCC Proceedings No. 9: Berkeley, CA, 1980.
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TABLE 111: C o n f o r m a t i o n of t h e I o n W a t e r Complexa
ion
TIK
Lit Na'
278 282 274 278 287
K'
F-
c1-
e Idea 40 (40
i
10) 14
z v) 0.00
a
\ c
with which useful comparison can be made are those for the 3.57 m solution of LiCl studied by Newsome et al.,1° which are also quoted in the table. Agreement with the present results is excellent for Li+, but less good for C1-. The calculations of Mezei and Beveridge19and the majority of the calculations of the Mainz g r o ~ p also ~ ~ yield - ~ ~coordination numbers for C1- of around 7-8, in agreement with our renlts. On the other hand, Monte Carlo calculations2I on a cluster of water molecules surrounding a Clion have yielded a coordination number of 4.3. Conformation of the Ion-Water Complex. From the characteristic distances listed in Table I and the bond lengths of the water molecule, the average values of the angles 8 and $ can be deduced. The results are listed in Table 111. For the cations, it is clear that on average there is a significant deviation from planarity. The value obtained for Li+ is in excellent agreement with the experimental resultlo for 3.57 m LiCl, though the comparison is complicated by the fact that the experiments show that 0 is concentration dependent. It is not obvious to us that any deep physical meaning can be attached to the calculated values of 8, particularly when there is no apparent trend with ion size (see Table 111). The width of the main peak in gIo(R)makes it improbable that any one structure is dominant. However, the results do show that there is no marked tendency to adopt the cation-water geometry appropriate to the dimer. Ab initio results17 agree in predicting that the minimumenergy configuration is planar, with CZusymmetry, i.e., 8 = 0. In the case of the anions, the situation is different. For both F- and C1-, the values of R(X-0) and R(X-HI) are consistent with a nearly linear X--H1-O bond, which is the geometry favored at the dimer level. This is in agreement with the neutron scattering resultslO for C1-. The values of $ quoted in Table I11 appear to contradict this statement, but shifts of only 0.01 A in the positions of the peaks in gxo(R) and gXH(R)would be sufficient to reduce the calculated angles to zero. In contrast to the results for the positive ions, the experiments show that the geometry of the anion-water complex is almost independent of concentration. In Figures 7 and 9, the arrows denote the positions a t which the peaks would occur if the X--H1-O bond were strictly linear; the positions of the second peaks are very sensitive to small deviations from this ideal geometry. Neutron- Weighted Distribution Functions. The neutron scattering pattern for an alkali halide-water system is the sum of 10 partial structure factors Sap(Iz),where Iz is the wavenumber; each of these is the Fourier transform of a partial radial distribution function gap(R)(a,0 = M, X, H , 0). In practice, because of the large incoherent scattering cross section of the proton, experiments are made on deuterated samples. The complexity of the scattering pattern is such that it can be unraveled to yield information on ion-water correlations only with the aid of some model of the structure. However, if the experiment is repeated following isotopic substitution of one of
.
0
0.031
20 36
1 7 (0) Quantities in parentheses are e x p e r i m e n t a l results" f o r a 3 . 5 7 m solution of LiC1.
::::h 0
i Idea
Impey et al.
Y v)
4
I ;+
.
-0.01 *
::::~1~1 , , , I
I
-0.04
0
I
2
3
7
5 , 6
4
8
9
IO
K/L Figure 10. First-order difference scattering function S,(k) for Li'. Curve: molecular dynamics results at 278 K. Points: experimental results of Newsome et al.'O~ll for a 3.57 m solution of LiCI.
0.03I
I
C
I
I
I
I
I
I
I
I
r
1
0.02 v)
E
0.01
U
-a: m \
3
0.00.
Y
-O.O1 -0.02L
0
I
1
... . .*
I
2
I
3
I
4 R /A
I
5
6
7
8
Figure 11. Fourier transform G,(R) of the first-order difference scattering function for Li'. Curve: molecular dynamics results at 278 K. Points: experimental results of Newsome et al.'O,'l for a 3.57 m solution of LiCI.
the ions and the difference between the two scattering patterns is taken, the contribution from water-water correlations is removed. For example, isotopic substitution of a cation leads to the so-called first-order difference scattering function SM(K),with a Fourier transform GM(R) given by GM(R) = A& @ ,,!)
4-
DgMM(R) -k E (1)
BgMD(R) -k CgM,(R)
with a similar expression for Gx(R). The coefficients A, B, C, and D are determined by the scattering lengths of the nuclei and thg concentration of the solution, and E is chosen to make GM(R) vanish as R a. Since C and D will, in general, be much smaller than A and B, the functions GM(R)and Gx(R) provide a direct measure of the distribution of oxygen and deuterium atoms around the ions. The differencing technique was developed by Enderby, Neilson, and co-workers7-11and both the method and its applications have recently been r e v i e ~ e d . ~The ,~ present results may again be compared with the experimental resultslO#llfor a 3.57 m solution of LiC1. The comparison is shown in Figures 10-13. Overall, the
-
~
The Journal of Physical Chemistry, Vol. 87, No. 25, 1983 5077
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TABLE IV: Self-Diffusion Coefficients
-
...
0.01 -
v)
z a
\
Y
-0.03L!L-J -0.04
I
2
3
4
5
8
7
6
9
IO
K /A-‘ Figure 12. First-order difference scattering function S , ( k ) for CI-. Curve: molecular dynamics results at 287 K. Points: experimental results of Newsome et al.’O~”for a 3.57 rn solution in LiCI.
L t
0.01 2
a
1
rl-
0.02
-a:
-0.01
1 c
-0.02_i
-0.03
~
0
calcd
exptla
Li’
278
0.3 1.8
0.60 3.10 0.91
282 274 278 287
c1-
a
1.0 2.8
0.9 2.3
1.10 0.86 1.60
From ref 1.
perimental facts concerning the dynamics of electrolyte solutions is the observation that the ionic diffusion coefficient does not decrease monotonically with increasing crystallographic radius of the i0n.l At the molecular level, this behavior can be rationalized in Stokes-Einstein terms by supposing that the ion and its hydration shell move as a rigid unit, i.e., a “solventberg”, with an effective radius which is larger than that of the bare ion. Other explanations have been advanced, based on continuum-type theories. Recently, W01ynes~~ has developed a statistical mechanical theory in which the molecular and continuum pictures appear as limiting cases of the same basic approach. In a molecular dynamics calculation, the self-diffusion coefficient can in principle be calculated by monitoring the mean square displacement of the ion and expoliting the well-known relation d 6D = lim -(IR(t) - R(0)I2) t-m dt
-:0.00 U
T/K
Na’ K’ F-
-0.01
-0.050
ion
368
;; -0.02
c?
D/(lO-’ cm2S K I )
-
f q p p... j
I-
2
-
1
2
3
4
5
6
7
8
R/A
Figure 13. Fourier transform G,(R) of the first-order difference scattering function for Cl-. Curve: molecular dynamics resuits at 287 K. Points: experimental results of Newsome et al.’O,ll for a 3.57 m solution of LiCI.
agreement with experiment is very good, particularly in the real space functions. The agreement for Li+ would be much poorer if comparison were made with experimental results for a more concentrated (9.95 m) solution, showing the importance of the counterion in such conditions. It should be noted that the experimental functions include a small contribution from ion-ion correlations, whereas in the theoretical results C = D = 0. 4. Self-Diffusion Coefficients In much of what follows, we shall be concerned with different aspects of the mobility of ions in solution, particularly with the role played by hydration. The ionic mobility is conveniently discussed in terms of the selfdiffusion coefficient D; in the limit of infinite dilution, D is related to the equivalent ionic conductivity Xo by the Nernst equati0n.l For many liquids, including water, the self-diffusion coefficient satisfies the Stokes-Einstein relation D = ~,T/(~TTu) (2) where a is the effective radius of the molecule and 7 is the shear viscosity. However, one of the most striking ex-
(3)
where R(t) denotes the coordinates of the ion a t time t . For completeness, the results obtained in this way are listed in Table IV, together with the experimental results deduced from conductivity measuremental Unfortunately, the molecular dynamics results are of no more than semiquantitative significance. Because the system contains only one ion, there are large statistical uncertainties in the calculations, and the values quoted in the table may be in error by as much as 50%. For this reason, we have also attempted a calculation of the ionic diffusion coefficient by the perturbation-and-difference, linear response technique of Ciccotti and J a c ~ c c i . ~A~t first , ~ ~sight, this is a problem which is well suited to study in this way. In practice, we have found that a systematic response can be measured only for times up to about 0.3 ps, which is too short for the purpose in hand. We believe that the failure of the method is the consequence of a recurrence time effect, involving the propagation through the periodic system of the disturbance linked to the initial perturbation. Taking as the speed of propagation the calculated31speed of sound in MCY water (3040 m s-l), the recurrence time for a system of 64 water molecules would indeed be in the range 0.3-0.4 ps, independent of the mass and size of the ion. Because of the uncertainties in the molecular dynamics results, we do not think it worthwhile to attempt a detailed comparison with the measured diffusion coefficients. To make contact with experimental results on the mobilities of the ions we prefer an indirect approach based on the behavior of the water molecules in the region close to the ion; we shall return to this question in section 7. However, the qualitative behavior appears to be correct; in particular, there is a clear positive correlation between the size of the (38) Ciccotti, G.; Jacucci, G. Phys. Reu. Lett. 1975, 35, 789. (39) Ciccotti, G.; Jacucci, G.; McDonald, I. R. J . Stat. Phys. 1979,21, 1.
5078
The Journal of Physical Chemistty, Voi. 87,
No. 25, 1983
Impey et ai.
i I -
‘..-_ - - --
i
I
0.2
0
02
04 t Ips
Figure 14. Velocity autocorrelation functions for Li+ at 278 K and Na+ at 282 K. Full curves: Z,,(t). Broken curves: Z,,Xf). Curves with dots: A ( t ) . All functions have been normalized to unity at t = 0.
t /ps
04
Flgure 15. Velocity autocorrelation functions for K+ at 274 K and bulk for H,O. Broken water at 286 K. Full curves: Z,(t) for K+ and Z,(t) curves: Z,,,,,(f). All functions have been normalized to unity at t = A
U.
I
ion and the calculated diffusion coefficient, in good accord with what is observed experimentally. The low value obtained for Li+ is consistent with the molecular dynamics calculations of the Mainz This could well be a problem associated with the time scale of the molecular dynamics calculations, since over the length of the run the Li+ ion diffuses (at 278 K) with an essentially intact coordination shell (see section 6). 5. Velocity Autocorrelation Functions
Calculation of the self-diffusion coefficient is useful for the link that it provides between theory and experiment and for the way that it highlights the role played by hydration in determining the mobility of the ion. However, it provides no direct information on the process of selfdiffusion. To obtain some insight into the microscopic dynamics of the ion, we have calculated the velocity autocorrelation function, defined as
0
I i
c 1-
= (v(t).v(o))/(Iv(o)12)
(4) where V(t) is the velocity of the ion a t time t . The selfthrough the relation diffusion coefficient is related to Zion@) zion(t)
0
where m is the mass of the ion, but this route to the calculation of D is not independent of the one discussed in section 4 and suffers from the same problem of poor statistics. The results are plotted in Figures 14-16. An interesting feature in the results for the larger ions, i.e., K+, F-, and C1-, is the fact that there are two distinct frequencies making a contribution to Zio,(t); for Li+ and Na+, on the other hand, the autocorrelation function has the appearance of a damped oscillation characterized by only a single, high frequency. It is natural to ascribe the
I
I
02
0.4 t /ps
Flgure 16. Velocity autocorrelation functions for F- at 278 K and CIat 287 K. Full curves: Z,&). Broken curves: Z,,,,,(t).
high-frequency motion to a “rattling” of the ion in a cage of solvent molecules. To see whether or not this interpretation is justified, we have computed in addition to Zion(t)the time autocorrelation for the center of mass
The Journal of Physical Chemistry, Vol. 87, No. 25, 1983
Feature Article
velocity of the set of molecules forming the first coordination shell of the ion. To be precise, we have computed the function Zshell(t)
= (vS(t)'vS(0))/ (lvs(o)12)
(6)
where (7)
In this definition, V,(t) is the center of mass velocity of a water molecule labeled j and the sum on j runs over those molecules which lie within the first coordination shell, symbolized by S ( t ) . Thus j E S ( t ) if IR,(t) - R(t)l < Rs, where R,(t) denotes the coordinates of molecule j (for simplicity we have chosen R,(t) to be the position of the oxygen atom), R(t) denotes the coordinates of the ion, and Rs defines the outer radius of the first coordination shell. In practice, as in section 3, we have taken Rs to be the position of the first minimum in the distribution function glo(R). As we shall see in the next section, the identity of the water molecules in the first coordination shell does not change significantly over the time scale of interest here (roughly 0.5 ps). The third curve which is plotted in the figure relating to Li+ and Na+ is the normalized difference of Zi,,(t) and z s h e l l ( t ) , i.e., the quantity A(t)/A(O), where A(t) =
([v(t)- Vs(t)l.[v(o) - Vs(0)l)
(8)
This is a function which describes the motion of the ion relative to its first coordination shell. For the ions in question, it is rapidly oscillatory, the characteristic frequency being the same one that dominates the form of Zion(t). Since, at room temperature, the coordination complex of Li+ persists for times of order 20 ps (see section 6 below), the oscillatory motion of the ion relative to the coordination shell makes no contribution to the self-diffusion coefficient and in this case D may be viewed as the time integral of z s h e u ( t ) . The long negative tail appearing in the latter shows that the motion of the coordination shell is strongly hindered. The shape of the function is reminiscent of that obtained by a hydrodynamic calculation for the motion of a sphere in a compressible solvent.40 The description just given of the functions Z,,,(t) and &he,( t ) in the case of Li+ is appropriate also for Na'. For K+, however, there is a similarity between Zi,,(t) and Zsheu(t)which is suggestive of an ion moving in a continuum. In a true continuum, Zshell(t) would simply be the velocity autocorrelation function of a sphere of solvent of uniform composition which a t all times is centered exactly on the position of the ion. Under these circumstances, the functions Zi,,(t) and Zghe]l(t)would be identical. However, the analogy must not be pursued too far, since the detailed behavior of zi,,(t)and Z s h e l l ( t ) in the case of K+ is not suggestive of a continuum-like behavior; the oscillations which are seen are too rapid to be attributable to hydrodynamic effects and must instead be due to some specific restoring force of a microscopic nature. A hint as t o the possible origin of this force may be obtained by considering the results for the negative ions, together with the functions analogous to zi,,(t)(i.e., zbd(t)) and z s h e U ( t ) , for the case of bulk water. (In calculating z s h e n ( t ) for bulk water, an arbitrarily chosen water molecule plays the role of the ion.) These results are displayed in Figures 15 and 16. They show that the function Zshell(t)for F- and C1- and that for K+ are all very similar to the corresponding function for bulk water. In particular, the frequency of the rapid oscillation is roughly the same in all cases. This (40) Zwanzig, R.; Bixon, M. Phys. Reu. A 1970, 2, 2005.
5079
suggests that for these three ions the oscillations in Z s h e u ( t ) are not directly attributable to correlations between the water molecules and the ion, but arise instead from a coupling of the motion of water molecules in the first coordination shell to that of molecules in the bulk, the effect being similar to what is found in pure water. For Li+ and Na+, different considerations apply. In the case of Li+, there is no well-defined oscillation in Zshe]l(t). For Na+, there is an oscillation, but it occurs a t a significantly higher frequency than in the curves for F,C1-, and K+. The frequency is also close to that seen in Zion(t), which is not the case for F- and C1-. That there are some differences between the Li+ and Na+ results and those for F- and C1- is not unexpected, given the differences in structure around the ions. For Li+ and Na+, the water molecules in the first coordination shell are tightly bound with the oxygen atoms all pointing inward. Examination of the running coordination numbers shows that for these ions there is no penetration of hydrogen atoms on water molecules in the bulk into the region of the first peak in gIH(R). Thus, there are no hydrogen atoms from the bulk at sufficiently near distances to engage in hydrogen bonds with the tightly coordinated oxygen atoms, and the hydrogen-bond network is accordingly disrupted. No such argument applies in the case of the anions. Even for the strongly coordinated F ion, there is an overlap of the first peak in g&) and the second peak in gIH(R),so that both hydrogen and oxygen atoms are exposed to interactions with water molecules in the bulk. For K+, the binding of the coordinated water molecules is much weaker than for Li+ and Na+, and the peaks in the radial distribution functions are correspondingly broader. The extent of hydrogen bonding between water molecules in the first and second shells will therefore be considerably greater than for the smaller cations. 6. Residence Times and Hydration Numbers The structure of the different solutions was discussed in section 3 in terms of coordination numbers. As we pointed out in the introduction, the coordination number is defined solely in terms of static quantities. The concept of hydration is a more complex one, which we shall identify with that of persisting coordination. However, this interpretation becomes complete only when a choice of time scale is made. This time scale must be related in some way to the residence time of water molecules in the first coordination shell of the ion. To obtain a precise definition of residence time, we introduce a function P,(t,t,;t*). This is a property of the water molecule j and is equal either to 0 or to 1. It takes the value 1 if molecule j lies within the first coordination shell of the ion at both time steps t , and t t,, and in the interim does not leave the coordination shell for any continuous period longer than t*. Under all other circumstances, it takes the value 0. We may now define an averaged quantity nion(t),characteristic of the ion, by the expression
+
1 Nt ni,,(t) = - C E:P,(t,,t;t*) Nt n=l 1
(9)
It follows immediately from this definition that nio,(0) nion,the coordination number introduced in section 3, whereas nlon(t)measures the number of molecules which lie initially within the first coordination shell and are still there after a time t has elapsed. The parameter t* is introduced to take account of molecules which leave the first coordination shell only temporarily and return to it without ever having properly entered the bulk. The use
5080
The Journal of Physical Chemistry, Vol. 87, No. 25, 1983
TABLE V : Residence Times and Hydration Numbers nhvd
ion Li’ Na K’
+
FCIa
T/K 278 368 282 214 278 281
~ ~ i ~ ~ calcd / p s
33.3 6.0 9.9 4.8 20.3 4.5
ref 2a
4.6
5*1
3.8 2.9 4.6 2.6
4 + 1 3+1 4_+1 2 + 1
See text.
of the parameter t* is only a crude device for dealing with this situation, but it avoids the necessity for any elaborate bookkeeping. For reasons which will become apparent shortly, we have set t* equal to 2 ps. The function nion(t)turns out to have the same general form for all the ions that we have studied. Except at short times, it decays in an accurately exponential fashion with a characteristic time T ~ , Le., ~ ~nion(t) , nionexp(-t/TSion) for large t. The quantity rSion is a correlation time for the persistence of the first coordination shell around the ion; it provides a simple definition of the residence time of water molecules in that shell. The results obtained for rSion are listed in Table V. As would be expected, there is a strong negative correlation with ion size, the residence times decreasing in the order Li+ Na+ K+ for the positive ions and as F C1- for the anions. In the case of Li+, the only ion for which we have made the necessary calculations, there is also a strong dependence on temperature. It is clear from the results that over a temperature range of 90 K the coordination shell of the Li+ ion becomes markedly less stable. This feature is not apparent from studies of the structure alone (see Figures 2 and 3). The other point to note about the results for Li+ is that at the lowest temperature studied the residence time is comparable with the length of the molecular dynamics run used in its determination. Residence times have been measured by NMR for ions with unpaired spins, but studies of monovalent ions (Li+ and F-) have not so far yielded numerical estimates for the quantities of interest here.41 For divalent ions, the resis or longer. dence times are found to be of order The same calculation can be made for bulk water, yielding a function nb,lk(t) which decays with a characteristic time pb,&. For this purpose, the coordination shell is defined by the position of the first minimum in the oxygen-oxygen distribution function goo(R);we recalP1 that for MCY water at 286 K, the first minimum in goo(R) occurs at 3.34 and the corresponding coordination number is nbulk = 4.8. Under the same state conditions, the calculated residence time is TSbulk = 4.5 ps. If in the definition of Pj(t,t,;t*) we set t* = 0, we obtain a function iiion(t)which again decays exponentially but with a characteristic time which is obviously shorter than T ~ In ~the ~case~ of Li+ . and Na+, for which T~~~~is considerably longer than 2 ps, the difference between the two characteristic times is only small; the same would be true of the divalent ions which have been studied by NMR. The choice of t* = 2 ps in the definition of nion(t) was dictated by the fact that this is roughly the value obtained for the characteristic time ?‘bulk; for pure water at 286 K we find that +bulk = 1.8 ps. The quantity TSion bears a close relation to the residence time discussed by Hertz.41 For pure water, Hertz gives an estimate of 8 ps for the residence time, calculated as the
-
- -
-
(41) Hertz, H. G. “Water: A Comprehensive Treatise”;Franks, F., Ed.; Plenum Press: New York, 1973; Vol. 3.
Impey et al.
time required for a water molecule to diffuse over one molecular diameter, Le., from the first to the second peak in the oxygen-oxygen distribution function. Our definition of TSbulk is more closely linked to the time required to diffuse over a molecular radius, i.e., from the first peak to the first minimum in the distribution function. Accordingly, we should expect our value for fSbulk to be about one-quarter of the estimate given by which indeed is true. The physical significance of is that of a characteristic time over which the water molecules in the first coordination shell exchange identity with molecules in the bulk and hence for any correlation between the ion and a particular water molecule to be lost. It is interesting to note that in the case of pure water, the characteristic time TSbulk is approximately equal to the mean of the three correlation times describing the reorientation of the principal axes of the water molecule (3.1, 5.7, and 3.7 ps, at 286 K). This is consistent with our earlier suggestion31 that the breakup of the first coordination shell is initiated by molecular reorientation rather than translation. With the significance of rSlonand TSbulk in mind, we choose to define the dynamic hydration number nhyd of an ion in aqueous solution as the mean number of water molecules which remain within the first coordination shell over the period of time in which the coordination shell in bulk water is renewed. Because of the exponential character of the functions nio,(t) and nbulk(t),this number is given by nhyd = nion eXp(-TSbulk/TSion) (10) Values of nhyd obtained in this way are listed in Table V. They show a relatively wider variation, from 2.6 (for C1-) to 4.6 (for F- and Li+), than do the coordination numbers discussed earlier. The dynamical aspect of the hydration number concept has played an important role in recent work of Enderby and c o - w o r k e r ~ .These ~ ~ ~ authors have shown how quasielastic neutron scattering techniques may be used to study the hydration of an ion. Hydrated water molecules are identified as those which translate with the ionic diffusion coefficient on the time scale of the neutron experiment (30-100 ps). They may therefore be detected only if the residence time exceeds the neutron time scale, which is true for divalent ions. Our estimate of the residence time for Li+ is consistent with the upper bound established in the neutron work. For those ions for which the hydrated water is sufficiently tightly held that it may be detected in the neutron experiment, the hydration and coordination numbers are the same. This would also be true of the hydration number defined above, since in such a case rSion>> @bulk. The hydration number measured in a neutron experiment and the definition offered here have as a common feature the fact that they provide estimates of the number of water molecules which are bound to an ion on a time scale relevant to the microscopic dynamics. Since our values are computed from the ratio of two number, we may hope that inadequacies in the potentials and other defects in the simulation will have been minimized and that the numbers obtained will therefore be realistic estimates of the hydration numbers in real aqueous solutions. As we have seen in the introduction, the experimental situation is confused, since different types of experiments do not measure the same quantity. In Table 2.20 of ref 2, Bockris and Reddy have compiled a list of the “most probable” values of what they term the primary hydration number, a quantity which is closely related to the hydration number defined by eq 10. Their results for the ions of interest here
The Journal of Physical Chemistry, Vol. 87, No.
Feature Article
25, 1983 5081
TABLE VI: Walden Products ion
TIK
RIIa
RcIA
Li'
27 8 282 274 218 28 7
0.6
3.3 3.5
Na'
K
F-
c1a
0.9 1.4 1.2 1.8
3.9 3.5 4.3
rs,onI 4mk 7.0 2.6 1.2 3.7 1.1
h , q / ( n - ' g c m SK')
R
rR ion/ bulk
7.4 2.2
1.1 4.5
1.0
eq 1 8
eq 19
exptla
0.62 0.74 0.77 0.53 0.65
0.51
0.35 0.45 0.65 0.48 0.67
0.67 0.77 0.45
0.65
F r o m ref 1.
are reproduced in Table V. The agreement with the present work is very good, but this may be partly fortuitous, since Bockris and Reddy2 rely to a large extent on thermodynamic measurements. In view of the sharp distinction that we have drawn between a static coordination number and a dynamic hydration number, it would be more appropriate to limit the comparison to experimental estimates based solely on the measurement of ionic mobilities. The relevant data from ref 2 (Tables 2.19 and 2.20) are 3.5-7 for Li+, 2-4 for Na', and 0.9 for C1-. The agreement with our results is now less striking, but it should be emphasized that there is no reason to suppose that the quantity defined by eq 10 is identical with that measured by any particular experimental method.
7. Hydration Numbers and Ionic Mobilities We want now to combine the ideas of previous sections in a way which makes an explicit connection between the hydration number of an ion and its mobility. Though this does require the introduction of a theoretical model, it allows the calcualtion of the self-diffusion coefficient of the ion in a manner which avoids the statistical problems outlined in section 4, since the quantities which appear are properties of the surrounding water molecules rather than of the ion itself. The use of hydration numbers to interrelate ionic mobilities appears already in the solventberg picture of the diffusion process. For strongly hydrated ions, including Li+, the ion is viewed as diffusing with an intact coordination shell. This is a physically appealing picture which provides a simple explanation of the low mobilities of such ions. For weakly hydrated species, the model is less satisfactory. It is now necessary to consider the ion as diffusing in company with a small (fewer than nion)number of attached water molecules and it is difficult to reconcile this with a fully molecular interpretation in which each molecule in the coordination shell must play an equivalent role. We therefore prefer to abandon the notion of a rigid solventberg in favor of an approach which takes explicit account of the lifetime of the ion-water complex. Indeed, to do otherwise would scarcely be consistent with our earlier arguments in which residence time and hydration number appear as closely related concepts. One possibility is to treat the coordination shell of the ion as rigid for times t < T~~~~and as free water a t longer times. The velocity of the diffusing ion may be written42 as the solution of a Langevin equation with memory effects, i.e. V(t)=
- 1 t d 7 E(t - 7) V(7) + X(t)
radius of the sphere may be taken equal to Rs for t < Pion and equal to that of the bare ion, R I e.g., for times longer than this. With these approximations, an expression for D may be derived in terms of R s and RI. Though the model is a plausible one, the results of the calculation are unsatisfactory, since in each case the diffusion coefficient turns out to be that appropriate to the fully hydrated ion of radius Rs. Mathematically, the source of the problem is the fact that the relaxation time of the hydrodynamic friction is much smaller than Pion. The result is therefore dominated by the friction on the hydrated ion. Physically, the weakness of the model is the fact that 7Sionis assumed to characterize the lifetime of a rigid object consisting of the ion and its complete coordination shell. In a more realistic approach, account must be taken of the fluidity of the coordination shell; this may be achieved by supposing that the water in the coordination shell has a viscosity which is different from that of bulk water. Since the residence time is much longer than the relaxation time of the hydrodynamic friction, we may neglect the exchange of molecules between the first coordination shell and the remainder of the solution. The model that we propose is therefore one of a bare ion, of radius RI, diffusing in a medium of nonuniform viscosity. Following a suggestion of W o l y n e ~we , ~ write ~ the friction coefficient 5 = k,T/D as
T o determine R I ,we suppose that the first peak in the cation-oxygen distribution function occurs at RI + Ro and the first peak in the anion-hydrogen distribution function occurs a t R I + R H ,where Ro and R H are respectively the oxygen and hydrogen radii. We set Ro = 1.4 A and R H = 0.5 A and determine R I from the distribution functions shown in Figures 2-9. The resulting values of RI are listed in Table VI and are in satisfactory agreement with traditional valued for the crystallographic radii of these ions. The distance-dependent viscosity is modeled in the following way. For R larger than a characteristic distance Rc, we set q(R)equal to 7, the shear viscosity of bulk water. For RI < R < Rc, we suppose that the viscosity has a different but constant value, $; this is to be determined from the dynamical properties of water molecules lying within the first coordination shell of the ion. With this simple approximation, eq 12 becomes
(11)
where X ( t ) is the random force acting on the ion and [ ( t ) is a generalized friction coefficient. T o represent 4(t) we use the known result for the hydrodynamic friction on an oscillating sphere immersed in a viscous medium.43 The
To choose Rc, we note from Figures 2-9 that the outermost atom in the first coordination shell of both cations and anions is always a hydrogen atom. The most probable distance of this atom from the ion is given, for cations, by
(42) Hansen, J. P.; McDonald, I. R. 'Theory of Simple Liquids"; Academic Press: London, 1976.
(43) Landau, L. D.; Lifshitz, E. M. "Fluid Mechanics"; Pergamon Press: Oxford, 1963.
5082
The Journal of Physical Chemistry, Val. 87, No. 25, 1983
the position of the first peak in g&) and, for anions, by the position of the second peak. We therefore set Rc equal to the sum of the most probable outer distance from the ion and the hydrogen atom radius RH. If p = ps, eq 13 reduces to the Stokes friction coefficient with slip boundary conditions for a bare ion of radius R,; if p / q s = 0, it becomes the slip coefficient for a rigid coordination complex of radius Rc. The true situation is intermediate between these limits, corresponding to 0 < p / p s < 1. The value of p / p s appropriate to a particular solution may be obtained from the calculated reorientation times of water molecules in the first coordination shell, using the rotational version44of the Stokes-Einstein relation, i.e. = 8ppSa3/kBT
(14) where a is the effective radius of a water molecule. Similarly, for water molecules in the bulk, for which the Stokes-Einstein relation is obeyed quite well,45we may write 7Rbulk = 8 i ~ p a ~ / k ~ T (15) 7'ion
Impey et al.
As we have already noted, eq 18 may be regarded as a scheme for interpolating between the slip limits for the bare and fully hydrated ions. However, stick boundary conditions would be more appropriate in the fully hydrated case, since the hydrated complex experiences tangential as well as radial forces due to molecules in the As a physically more appealing alternative to eq 18, we could write Xop in the form
which interpolates between bare slip and fully hydrated stick limits. This brings the calculated values of the Walden product into even better agreement with experiment, as shown by the results given in Table VI. Eq 19 was arrived a t by making a particular choice of R dependence for the nonuniform viscosity appearing in Wolynes' e x p r e ~ s i o neq , ~ ~12. However, a relation of the form
These two results may be used to eliminate p s from eq 13, leaving
In practice, we have chosen 7Rionand TRbulk to be the reorientation times of the molecular dipole axis. At least for the positive ions, this is a quantity which is more intimately related to the breakup of the coordination sphere than, e.g., the reorientation time of the proton-proton vector. The two characteristic times were calculated from the rate of decay47a t long time of the dipole correlation function of water molecules in the first coordination shell of the ion (for 7Rion)and in the bulk (for 7'bulk). Experimental data on the mobilities of ions are often presented in the form of the Walden product Xoq, where X, is the ionic equivalent conductivity. The reason for this is that ?A,, is much less strongly dependent on temperature than Xo itself. In the present case, by focusing on the Walden product, we avoid the necessity for introducing a numerical value for the viscosity of bulk water. The Walden product is related46 to the friction coefficient through the expression A017 = F e / ( E / d (17) where F is$he Faraday and e is the elementary charge. Thus xop =
&( 7F))
- -+ '1
7 ion
l--
(18)
7 ion
where A is a constant. Values of the quantities entering eq 18 are listed in Table VI, where comparison is also made between the calculated Walden products and the experimental values corresponding to infinite dilution. Given the simplicity of the underlying physical model and the absence of any adjustable parameters, the agreement with experiment is very satisfactory. In particular, the trend with ion size is correctly reproduced. (44) Masters, A. J.; Madden, P. A. J . Chem. Phys. 1981, 74, 2470. (45) Krynicki, K.; Green, C. D.; Sawyer, D. W. Faraday Discuss. Chem. Soc. 1978, 66, 199. (46) Kay, R. L. "Water: A Comprehensive Treatise"; Franks, F., Ed.; Plenum Press: New York, 1973; Vol. 3. (47) Impey, R. W. Ph.D. Thesis, University of Cambridge, Cambridge, England 1982.
could be justified on general grounds as a method for interpolating between limits appropriate to the bare ion and the fully hydrated complex. The parameter v is a measure of the extent to which the dynamics of water molecules in the first coordination shell differ from those of molecules in the bulk, and the natural form in which to express it is as a ratio of correlation times. From this point of view there are arguments for choosing not the reorientation times appearing in eq 19, but the residence times introduced in section 6, setting v = 7Sbulk/TSion. Values of this ratio are listed in Table VI; it is gratifying to see that they are very close to the values of v based on reorientation times and their use in eq 20 would lead to similarly good predictions for the Walden product. The theory as we have implemented it is applicable only to weakly hydrated species. For Ni2+,it is known from inelastic neutron scattering experiment^^^ that hydration effects extend to the second shell of water molecules and there is a tendency in this direction for Li+. The criterion that we have used to determine Rc is inappropriate for such species. Indeed, the rather poor agreement with experiment in the case of Li+ could be greatly improved by increasing Rc, and such a modification of the approach would not be unreasonable given the well-defined nature of the second coordination shell for this ion. However, in view of the other approximations involved, we do not believe that any additional insight would be gained by treating Rc as an adjustable parameter. The main lesson to be drawn is that the ratio &&/ 7Slon may be used both in the definition of a hydration number and to characterize the dynamics of water molecules situated close to the ion. It thereby provides a link between the structure (coordination number) and dynamics (ionic mobility) of the solution in a manner which avoids certain of the difficulties inherent in the solventberg approach. 8. Conclusions The results of our work may be summarized as follows. First, computer simulations requiring only modest computing facilities are able to give quantitative information (48) Masters, A. J.; Keyes, T.; Madden, P. A. J. Chem. Phys. 1981, 75, 485. (49) Hewish, N. A.; Enderby, J. E.; Howells, W. S. J . Phys. C 1983, 16, 1777.
J. Phys. Chem. 1983, 87,5083-5090
on the statics and dynamics of dilute aqueous solutions of weakly hydrated, spherical ions. The information obtained is of a type which is useful for the interpretation of experimental data, and the comparison with experiment shows that currently available potentials provide a satisfactory description on the ion-water interactions in solution. There are some statistical problems connected with the fact that the ions are present only in low concentration, but these are confined to studies of the ion dynamics, and there are few difficulties associated with the calculation of either the structure around the ion or the dynamics of the water molecules in the region close to the ion. Secondly, the calculated structural properties are in very good agreement with the experimentally determined radial distribution functions. The results may therefore be used with some confidence in predicting coordination numbers and ion-water geometries for those systems for which experimental measurements are so far lacking. Thirdly, the ionic self-diffusion coefficients, though not measurable with high accuracy, reproduce the tendency to increase with ion size which is a special feature of ionic solutions. However, the mobilities of the ions are more profitably analyzed by an indirect approach based on the behavior of the ions in the first coordination shell. Fourthly, the behavior of the velocity autocorrelation functions of the ions and of their coordination shell suggests that the solventberg model provides a satisfactory picture of the dynamics only for ions such as Li+. Fifthly, a definition
5083
of hydration number in terms of a time scale defined by the relative rates of breakup of the local order in the solution and in pure water leads to results which are in reasonable accord with estimates based on electrochemical measurements. Finally, we have incorporated the results of the simulations into a theoretical model which relates the ionic mobility to the drag experienced by a sphere moving in a fluid of nonuniform viscosity. The fluid viscosity in the vicinity of the ion may be characterized either by the reorientation correlation times of water molecules in the region or by their residence times in the first coordination shell. In both cases, good agreement is found with ionic mobility data. An attractive feature of the model is the fact that it provides a natural link between the structure (coordination number) and dynamics (ionic mobility) of the solution, the concept of residence time (and hence of hydration number) playing the key, intermediary role. Acknowledgment. We are grateful for the financial support provided by the SERC through the award of a Research Studentship (to R.W.I.) and an Advanced Fellowship (to P.A.M.). We thank Dr. G. W. Neilson and Dr. J. R. Newsome for supplying us with their neutron diffraction data in tabular form, and Dr. J. N. Agar for a critical reading of the manuscript. Registry No. HzO, 7732-18-5; Li, 7439-93-2; Na, 7440-23-5; K, 7440-09-7; F-,16984-48-8; C1-, 16887-00-6.
ARTICLES Solvated Phenol Studied by Supersonic Jet Spectroscopy Akira Oikawa, Haruo Abe, Naohiko Mikami, and Mitsuo Ito” Department of Chemistry, Faculty of Science, Tohoku University, Sendai 980, Japan (Received:May 10, 1983)
Fluorescence excitation spectra, dispersed fluorescence spectra, and mass-selected multiphoton ionization (MPI) spectra have been observed for the complexes formed between phenol and various solvents prepared in a supersonic free jet. The spectra of the 1:l (phenol, solvent) and 1:2 complexes have been identified from dependence of the fluorescence excitation spectra upon the pressure of solvent or He and from the mass-selected MPI spectra. It was found that all the solvents studied (water, methanol, ethanol, dioxane, and benzene) induce red shifts of the phenol absorption by formation of the 1:l hydrogen-bonded complex. The second solvent molecule which interacts with the 1:l complex results in red or blue shift of the absorption of the 1:1 complex by the 1:2 complex formation depending on the nature of the interaction. Intermolecular vibrations of the 1:l and 1:2 complexes were also obtained for both the ground and excited states.
Introduction “Solvation” is probably the most important concept which is widely used in all aspects of chemistry. However, solvation is the most difficult subject to explore on the molecular level. The answers to naive questions such as how is a solute molecule surrounded by solvent molecules, how many of the solvent molecules are called solvating molecules, what are the structure and dynamics of the solvated molecule, and so on are still vague. These fundamental questions would be solved if we could pursue
experimentally the processes by which an isolated solute molecule is subsequently surrounded by individual solvent molecules. Recently, the supersonic expansion technique was proved to be very useful to prepare isolated ultracold molecules in the gas phase1 and it can be applied to explore solvation processes on the molecular level. From the above viewpoint, weakly bound intermolecular compounds such (1) See, for example, D. (1980).
H.Levy, Annu. ReL. Ph>s. Chem., 31,
0022-3654/83/2087-5083$01.50/00 1983 American Chemical Society
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