7932
J . Phys. Chem. 1989, 93,1932-1942
"Mg2+ and %I- Quadrupolar Relaxation in Aqueous MgCI, Solutions at 25 OC. 1. Limiting Behavior for Infinite Dilution R. P. W. J . Struis,+ J. de Bleijser, and J. C. Leyte* Gorlaeus Laboratories, Department of Physical and Macromolecular Chemistry, University of Leiden, P. 0. Box 9502, 2300 R A Leiden, The Netherlands [Received: October 18, 1988; In Final Form: May 31, 1989)
The nuclear magnetic relaxation rates of 35Cl-and 25Mg2+at infinite dilution in water are reported with increased accuracy. The quadrupolar relaxation due to coupling with the solvent dipoles is described by use of a model in which both the structure and the internal dynamics of the hydration sphere may be specified. Dynamic correlation in the hydration sphere is treated and its effect is evaluated for simple cases. Correlated internal reorientation of the hydration water molecules of Mg2+ is indicated. The theory is applied to mono-, bi-, and trivalent cations.
Introduction
For some time the study of the quadrupolar relaxation of strong electrolyte ions dissolved in solution has been of interest for both the experimentalists and the theoreticians.' The basic theory of the relaxation of a quadrupolar nucleus by fluctuating electric field gradients has been discussed by A b r a g a n 2 The main objective of the present paper is to interpret the limiting behavior of the chloride and magnesium relaxation rates i n infinitely diluted, aqueous MgCI, solutions at 25 "C. In a forthcoming study the magnesium and chloride relaxation in finite concentrated MgCI, solutions will be a n a l y ~ e d . ~ The 2sMg2+and 35Cl-nuclei both have spins exceeding For these ions the magnetic relaxation rates are dominated by the interaction of the nuclear quadrupole with the electric field gradients at the nucleus. For the interpretation of the experimentally determined relaxation rates, several theories have been developed in order to estimate the strength and the fluctuation in time of the electric field gradients at the relaxing nucleus. A molecular interpretation of the behavior of ions in polar solvents resulted in a variety of theories ranging from a continuum approach4 to a purely microscopic treat~nent.~Among these theories, the electrostatic theory independently developed by Hertzb8 and Valiev and co-workers9 proved to be quite successful. Hertz's electrostatic theory will also be applied in the present study. According to this theory it is assumed that in the limit of infinite dilution the main contribution to the ionic quadrupolar interaction stems from the fluctuating electric field gradients arising from the reorienting water dipoles. The electrostatic theory is semimacroscopic because the theory incorporates the dynamical and structural properties of water molecules in the near surroundings of the studied ionic nucleus with a continuum-like correction to account for the contribution arising from the water molecules located further away. For the latter contribution the polarization factor is introduced. Often the polarization factor derived on the basis of a continuum theory by Cohen and Reiflo is used. Recently, some criticism was directed at the applicability of the continuum polarization factor in Hertz's electrostatic t h e o r ~ . ~ J I In the present study, the uncertainty associated with the appropriate value of the polarization factor may be circumvented by interpreting the 35CI-, )'CI-, 79Br-, 81Br-,and 1271- quadrupolar relaxation rate at infinite dilution. Reexamination of the ionic relaxation was stimulated by advances in experiments on MgCI2solutions, such as 2H, I7O, IH, and IH-l7O NMR,12J3quasi-elastic neutron scattering,I4 28Mg2+ and 36Cl-tracer diffusion,I5 neutron diffraction experiments (in NiC12/D20),'6 and X-ray" studies, and by theoretical studies, e.g., molecular orbitalI8 and MD.I9 Combining the results obtained from the various studies, rather detailed, ion-specific information can be obtained on the dynamical and structural properties of the water molecules in the near surroundings of the Present address: Physikalisch-Chemisches Institut der Universitat Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland.
0022-3654189 f 2093-1932$01 S O f 0
relaxing nucleus. Often, these properties are referred to as the hydration model. In the present study this is illustrated by the interpretation of the chloride and the magnesium relaxation data. As is supported by the results of the above-mentioned studies, it is concluded that the chloride ions exert a moderate influence on the dynamical behavior of the water molecules and the relaxation rate may therefore be interpreted in terms of the so-called fully random distribution (FRD) hydration model. Contrary to the electrostatic interpretation available in the literature,*O in the present study the recently determined value of the reorientational correlation time in pure water ( T = 25 "C), viz., T ~ "= 1.94 f 0.12 ps, will be applied.12 It will be shown that the present interpretation is in accord with the experimentally determined 3sCl- relaxation rate in the limit of infinite dilution. However, the magnesium ions exert an appreciable influence on both the dynamical and structural properties of the neighboring water molecules. This may be illustrated by the fact that the water molecules reside for a rather long time (microseconds) within the first hydration layer.21 The magnesium hydration water reorients anisotropically, and the water dipoles have nonradial orientations.I2J3 Weingartner and Hertz22 analyzed the ion-water
( I ) N M R of Newly Accessible Nuclei; Laszlo, P., Ed.; Academic: New York, 1983; Vol. 1 and 2. (2) Abragam, A. Principles of Nuclear Magnetism; Clarendon: Oxford, 1961. (3) Struis, R. P. W. J.; de Bleijser, J.; Leyte, J. C. J . Phys. Chem., following paper in this issue. (4) Hynes, J. T.; Wolynes, P. G.J . Chem. Phys. 1981, 75, 395. (5) Versmold, H. Mol. Phys. 1986, 57, 201. ( 6 ) Hertz, H. G.Z . Elektrochem. 1961, 65, 20. ( 7 ) Hertz, H. G.Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 531. (8) Hertz, H. G.Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 688. (9) See, e.g., references cited in ref 5. ( I O ) Cohen, M. H.; Reif, F. Solid State Physics; Academic: New York, 1957; Vol. 5, 321. (1 I ) Engstrom, S.; Jonsson, B.; Jonsson, B.J . Magn. Reson. 1982,50, 1. (12) Struis, R. P. W. J.; de Bleijser, J.; Leyte, J. C. J . Phys. Chem. 1987, 91, 1639. ( I 3) Struis, R. P. W. J.; de Bleijser, J.; Leyte, J. C. J . Phys. Chem. 1987, 91, 6309. (14) Hewish, N. A.; Enderby, J. E.; Howells, W. S. J . Phys. C 1983, 16, 1177.
(15) Harris, K. R.; Hertz, H. G.; Mills, R. J . Phys. 1978, 75, 391. (16) Enderby, J. E.; Cummings, S.; Herdman, G. J.; Neilson, G. W.; Salmon, P. S.; Skipper, N. J . Chem. Phys. 1987, 91, 5851. (17) Pilinkas, G.; Radnai, T.; Dietz, W.; S d s z , Gy. I.; Heinzinger, K. Z. Naturforsch. 1982, 37a, 1049. (18) Ortega-Blake, I.; Novaro, 0.;Les, A,; Rybak, S. J . Chem. Phys. 1982, 76. 5405. (19) Dietz, W.; Riede, W . 0.;Heinzinger, K. Z . Naturforsch. 1982,37a, 1038. (20) Hertz, H. G.;Holz, M.; Klute, R.; Stalidis, G.; Versmold, H. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 24. (21) Neeley, J. W.; Connick, R. E. J . Am. Chem. SOC.1970, 92, 3476.
0 1989 American Chemical Society
zsMg2+and 3sCI- Quadrupolar Relaxation in MgC12. 1
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 7933
contribution for ions with structured hydration spheres. Internal mobility within the sphere was introduced by allowing rotation of the water molecules around their ion-oxygen axes. The model was not given a specific name. In the present paper essentially the same model is used, and to indicate the connection with previous work (Le., the N O S and FOS models; see Theoretical Section), it will be indicated with DOS (dynamically oriented solvation). The present treatment of the DOS model is, however, different from earlier work and so are the conclusions drawn from the results. Cross-correlation functions for the water molecules in the hydration sphere are explicitly formulated, and they are evaluated in simple extreme cases. It is found that the role of the solvent dipole component perpendicular to the ion-oxygen axis is sufficient to explain the ionic relaxation rate in contrast to previous emphasis on the radial component. It will be shown that with the modified theory reasonable agreement can be obtained between the theoretically estimated and the experimentally determined magnesium relaxation rates in the limit of infinite dilution. Furthermore, it will be demonstrated that the present interpretation is also applicable for other cations, such as 7Li+, 9Be2+ 43Ca2+,87Sr2+ , 135Ba2+, 137Ba2+ 69Ga3+, and 27A13+, for which it is reasonable to expect that the hydration water spheres are structured albeit to a varying degree. Finally, it has been notedz3 that for some of these cations the application of the less appropriate FRD model, introducing typically pure water parameter values, quantitatively leads to rather satisfying results as well. Therefore, the connection between the DOS and the FRD hydration models will be analyzed.
Theoretical Section General Theory. In the extreme narrowing limit the quadrupolar relaxation rate RQ ( = l / T Q , l= 1/TQ,2)is given by2
Here, I is the spin of the relaxing nucleus, Q is the nuclear quadrupolar moment, e is the charge of the proton, h is Dirac’s constant, and J(0) is the spectral density function at zero frequency given in eq 2
J ( 0 ) = 2 $ - ( V;)(O) 0
V;)(t)*)dt
with m = 0, f l , or f 2 . V’!!)(t) are defined here as components in the laboratory system of the field gradient tensor at the position of the relaxing nucleus. The electric field gradient at the position of the studied nucleus will be referred to as the local field gradient. The integrand is the time correlation function of the electric field gradient. In the electrostatic theory it is assumed that in a solution of a strong electrolyte the relaxation mechanism arises from purely electrostatic origin consisting of a contribution to the field gradient a t the studied nucleus due to the surrounding water dipoles, d , and an additional contribution from the point charges of the surrounding ions, A. Neglecting cross-correlation, the observed relaxation rate is8
F Q=~FQ(d + A)
(3) Here FQ is a constant that comprises the nuclear properties of interest; d and A are respectively the so-called ion-water and ion-ion contributions to the local field gradient. The constant FQ is 2 7 ( 2 I + 3 ) eQ(1 - -ym) FQ = l O P ( 2 I - 1 ) l h
I
2
+
where p is the electrical dipole moment of the water molecule, C, is the water concentration in particles per cubic centimeter, r, is the reorientational correlation time of the water molecules, and ro is the distance of closest approach of the water dipoles and the relaxing nucleus. Throughout this study, for ro the sum of the ionic and water molecular (1.4 A) radius will be taken. In eq 5 , the polarization factor P is introduced to account for the many-body cross-correlation contributions to the spectral density function J(0). The polarization factor will be discussed in more detail in section A under Results and Discussion. In view of the rather moderate influence of the chloride ions on the dynamical and structural properties of the neighboring water molecules, in infinitely diluted MgC12 solutions it may be assumed that the correlation time, 7,, will be of the order of the pure water value. Whenever the pure water value is used, this will be indicated by the index O , e.g., 7,O and Cwo. However, for small and highly charged cations, like Mgz+, the application of the FRD model, as discussed above, is of course less appropriate. The results of the experimental and theoretical studies on MgClz and NiClz solution^^^-^^ clearly show that the dynamical and structural behavior of the six water molecules within the first cationic hydration layer differs significantly from that of water molecules located beyond this hydration layer. For the latter class of water molecules the behavior may be expected to be comparable with the one found in the pure solvent. Therefore, their ion-water contribution may still be estimated with the FRD hydration model, eq 5 , in which the pure water values of 7, and C, will be maintained, but with the distance of closest approach, ro, replaced by the radius of the second coordination sphere, b.29 In the following discussion this contribution will be denoted as d6RD(b). In addition to this ion-water contribution, contributions may also arise from water dipoles that are located within the magnesium hexaaquo units. It will be assumed that each of the hydration water molecules has a sharp radial ion-water distribution function, with ro the average closest distance of approach between the ion and the water dipole. Noticing that in a point dipole approximation the water dipole coincides with the position of the
(4)
For the constant FQ it is noted that besides the molecular parameters defined previously in connection with eq l , in the electrostatic theory the Sternheimer or antishielding factor, ym, (22) Weingartner, H.; Hertz, H. G. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 1204. (23) Lindman, B.; Forsbn, S.; Lilja, H. Chem. Scr. 1977, 11, 91.
is introduced to take into account the contribution to the electric field gradient due to deformations in the charge distribution of the electric cloud surrounding the relaxing nucleus under It is noted that in eq 4 the term ( 1 - -ym) differs from the term normally used in the electrostatic theory, viz., ( 1 ym). This alteration has been introduced here to obtain consistency with the -ym values given in the l i t e r a t ~ r e . ~ ~In- ~the ~ limit of infinite dilution the ion-ion contribution, A, being proportional to the electrolyte concentration, becomes negligibly smalls8 Therefore, further discussion can be entirely focused on the ion-water contribution, d . The Ion-Water Contribution in the Limit of Infinite Dilution. Depending on the model of hydration, in certain cases explicit formulas can be obtained for the ion-water contribution, d. A well-known example is the fully random distribution (FRD) model in which uniform random distribution of the centers of mass and random water dipole orientation throughout the solution is assumed.6 The 3sCl-relaxation rate in infinitely diluted MgCIz solutions is most successfully interpreted in terms of this FRD model.20 For the ion-water contribution, Hertz obtained
(24) Sternheimer, R. M. Z . Naturforsch. 1986, 410, 24. (25) Sternheimer, R. M. Phys. Rev. 1966, 146, 140. (26) Lucken, E. A. C. Nuclear Quadrupole Coupling Constants; Academic: London, 1969. (27) Sen, K. D.; Narasimhan, P. T. Advances in Nuclear Quadrupole Resonance; Smith, Ed.; Heyden: London, 1974; Vol. 1 . (28) Schmidt, P. C.; Sen, K. D.; Das, T. P.; Weiss, A. Phys. Rev. B 1980, 22, 4167. (29) Helm, L.; Hertz, H. G. Z . Phys. Chem. 1981, 127, 23.
7934 The Journal of Physical Chemistry, Vol. 93, No. 23, 19'89 oxygen nucleus, more specifically this hydration model implies a sharp radial ion-oxygen distribution function within the first hydration layer. This assumption is meant here when referring to a "distinct" hydration model or layer. In the literature various distinct hydration models have been considered. These models and corresponding ion-water contributions are discussed most conveniently if one starts with the result obtained for a more general distinct hydration model. In the present study we will consider n,, nonradially oriented water dipoles, all located within a distinct first hydration layer. The field gradient fluctuations in time are thought to stem from two random diffusion processes. These are the isotropic overall reorientation of the hydration complex and, within the complex, an internal rotation of each water dipole around its ion-oxygen axis. Furthermore, it will be assumed that for each of the water molecules the angle between the water dipole moment vector and the ion-oxygen axis is fixed and equal to 6. The total hydration model, that is, the above-described distinct hydration model in combination with the FRD hydration model for the bulk water molecules, will be indicated with DOS (dynamically oriented solvation). The ion-water contribution, d p s , is given by eq 6. Its derivation is presented in the Appendix.
Here, 5 denotes the ratio ( r O v / qof) the two correlation times rov and q, which characterize respectively the overall and the internal diffusion processes. In order to avoid confusion it is noted that the definition of 5 in this study differs slightly from the one used in ref 12. The parameter gQ takes into account the effect of the radial ion-water (oxygen) pair distribution.22 Dealing with distinct 1. AR and AT are dimenhydration models one requires gQ sionless quenching parameters for which explicit formulas are given in the Appendix. AR and AT account for the quenching of the total electric field gradient due to correlations between the orientations of the water molecular dipole moment vectors within the hydration layer. A R and AT refer to the radial and tangential components, respectively. It is noted that if a very sharp radial distribution applies and the hydration layer has an octahedral symmetry, then the effective field gradient contribution arising from the radial components of the n, dipole moment vectors 0. For A T it is noted that, quenches completely, because A, besides the octahedral symmetry and sharp radial ion-water distribution functions, additional orientational correlation effects between the tangential components of the n, water dipole moment vectors are required to obtain the limiting value AT 0. On the other hand, if tangential correlation effects are completely absent, or, with complete correlation effects under specific relative orientations of the n, dipole moment vectors, one may obtain AT =
-
Struis et al. specifically, the system considered was that of one point charge and one point dipole in a spherical cavity surrounded by a dielectric medium with dielectric constant t representing the rest of the water. From a simulation with 50 water molecules these authors concluded that the polarization effect described above is negligible. With the distinction in mind concerning the polarization factor, some of the current distinct hydration models and corresponding ion-water contributions discussed in the literature can be derived from eq 6 and will be briefly reviewed here. The fully oriented solvation (FOS) model7 considers a distinct first hydration layer with radially orientated water dipoles, Le., /3 = 0. From eq 6 one obtains (7) Another model is the nonoriented solvation (NOS) in which it is assumed that within a distinct first hydration layer the orientations of both the ion-water and the dipole moment vectors are randomly distributed, e.g., A R = AT = 1. In view of these assumptions it may be clear that in this case it is meaningless to consider an internal diffusion process around a well-defined axis. Hence, in eq 6 one may take 5 0, and here the correlation time T~~ does not refer to the simple overall diffusion process of the hydrated ion but to a random internal reorientation, possibly in combination with an overall reorientation. The ion-water contribution, d y , is then easily obtained when in eq 6 one averages over all angles B, assuming an uniform distribution of the dipole orientations. The result is given in eq 8. Analogously,
-
,!OS
=
5P2nsTov + d&RD(b)
9rO8
Weingartner and Hertz22obtained dGos from their general model by taking 5 = 3.8. For the interpretation of the magnesium relaxation rate, in the present study a distinct hydration model will be applied that is intermediate between respectively the FOS and N O S models in order to obtain consistency with the characteristic dynamical and structural properties within the magnesium hexaaquo units. In this intermediate model, hereafter referred to as the TAN model, one considers a distinct hydration layer that is octahedrally symmetric with respect to the centers of mass and in which the n, (=6) dipole moment vectors have nonradial orientations, characterized by a constant tilt angle p. A radial contribution may contribute to dDoSthrough fluctuations in the ion-dipole distances. These fluctuations will be very rapid, however, with frequencies corresponding to the far-IR region and characteristic times of the order s. A second source for a radial contribution arises from fluctuations in the value of 0 for the individual hydration water molecules, again with characteristic times in the s. These effects may be expected to be small: From, order of 1. e.g., diffraction studiesI6 in NiCl,/D20 at room temperature, one One may note that in eq 6 the water-water correlation effects may roughly estimate the fluctuation in ro. For example, in the within the first hydration layer have been taken into account by 1.46 m NiCI2/D20 solution one has ro = rNi4 = 2.07 A, with, the parameters AR and A T and be ond the first hydration layer by the polarization factor P (in d[ 2D ( b ) ) .It is important to note at worst, a half-width at half-height of 0.12 A. The relative fluctuation in ro of ca. 6% leads to a contribution of ca. 60% of that, in the derivation of the ion-water contribution due to the s this results in a water dipoles within a distinct hydration layer, in the l i t e r a t ~ r e ~ , ~ ~p 2 / r o 8 ,and with a correlation time of moderate rate effect S(1.5 X 10-3)p2n,/r,8,or less, because in eq the polarization factor, P, is introduced to take into account 8 one has T~~ = 4 X 10-l' s. We will therefore concentrate on water-water correlation effects of the hydration water molecules the tangential components of the hydration dipoles and take AR with the water molecules located in outer hydration layers. This results in a factor P2 in the net ion-water contribution in q ~ e s t i o n . ~ = 0. For this case the ion-water contribution will be referred to as d v N ,and the result is given in eq 9. In the present derivation this correction has not been applied because the above-mentioned correlation effect and corresponding ion-water contribution is likely to be small, because of the r-4 dependence of the dipolar electric field gradient and the weak dynamic coupling of hydration and bulk water molecules. It is Experimental Section interesting to note that the expected unimportance of this correlation is well supported by the Monte Carlo study of Engstrom For the determination of the 2sMg2+and 3sCl- relaxation rates et al.'I These authors derived an expression for the polarization in MgCI2 solutions, several different stock solutions were prepared here defined as the quotient of the main component factor, PDImLE, on the basis of weight, with distilled and deionized water, and of the electric field gradient tensor at the position of one point MgCI2.6H2O. Distilled water was deionized and filtered with a charge for a system with and without a polarizable medium. More Milli-Q water purification system (Millipore Corp.). The specific
-
-
25Mg2+and 35C1- Quadrupolar Relaxation in MgC12. 1 TABLE I: =Mg2+ and "CI- Longitudinal, R , , and Transverse, R,, Relaxation Rates as a Function of MgCI, Concentration at 25 "C 25MgZt 35~1m(MgC1A mol/ke; H,O R,, s-I R2, S-' R,, S-' R2, S-' 0.0240 4.6 f 0.5 4.5 f 0.5 0.0484 4.1 f 0.4 4.4 f 0.4 0.0718 4.3 f 0.3 4.3 f 0.1 0.0972 4.2 f 0.1 4.2 f 0.2 0.246 4.56 f 0.06 4.55 f 0.04 34.5 f 0.2 35.9 f 0.4 0.488 4.79 f 0.03 4.85 f 0.10 39.8 f 0.2 41.4 f 1.1 0.985 5.56 f 0.04 5.65 f 0.11 51.9 f 0.9 53.9 f 1.7 I .49 6.68 f 0.04 6.7 f 0.08 64.8 f 1.0 71.7 f 2.3 2.19 8.42 f 0.08 8.43 f 0.04 93.6 f 1.5 99.2 f 4.5 m+O
4.16 f 0.03 s-I"
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 7935
1001
I
L'
x R ;
29.2 f 0.6 s-I"
n
"Obtained by linear extrapolation of the RI and R2 data determined in MgCIz solutions with concentrations 50.985 m, as discussed in the Experimental Section.
conductance of the water did not exceed 1 X lod cm-' Q-I. The analytical reagent MgC12.6H20 was obtained from Baker. The magnesium concentrations in the stock solutions were checked by complexometric titrations in EDTA. The other solutions were prepared by mixing weighed amounts of different stock solutions or by dilution with distilled and deionized water. It is noted that some of these solutions (m = mol of MgC12/kg of water L 0.246) have also been used in the 2H and I7O N M R study,I3 and in order to improve the accuracy of the experimentally determined 2H and 170relaxation rates, the concentrations have been increased slightly above the natural abundances; that is, the respective mole fractions were less than 0.7% (I7O,I8O) and 1.7% (2H). The N M R tubes (Wilmad 10 mm) were heated in an EDTA solution, heated in a N a H C 0 3 solution, and stored for several days filled with distilled and deionized water. The longitudinal 2sMg2+and 35Cl- relaxation rates ( R Q , ~= 1/Tq,1) were measured by the alternating phase inversion recovery method.30 For all RQ,l measurements, 100 data points were collected and fitted to a single exponential by a nonlinear leastsquares procedure based on the Marquardt-Levenberg algorithm.31 For 2sMg2+,the transverse relaxation rates, R , , = ~ / T Q Jwere , measured by using a Carr-Purcell-Gill-Meiboom (CPGM) sequence.2 For 3sCI-, both the spin-echo2 and the CPGM sequence were applied. The measurements have been performed on a modified S X P spectrometer (Bruker), equipped with a 6.3-T superconducting magnet (Oxford Instruments). The temperature was maintained at 25 f 0.5 OC by fluid thermostating using Fluorinert FC-43 (3M Co.). The relaxation rates, as a function of the molal MgC12 concentration up to 2.19 m, are presented in Table I and shown in Figure 1. The relaxation rates and accuracies denoted in Table I are the average values and the standard deviations calculated with the results obtained from several relaxation rate determinations spread over a period of time. In all cases it was observed that the extreme narrowing limit, Rq = RQ,l = RQ,2, applies and that the free induction decay falls off exponentially with the time. Inspection of the data shown in Table I shows that, for most of the MgC12 solutions studied, the relative accuracy of R2 is less than the one obtained for the corresponding R l value. It is noticed that especially the spin echo and CPGM and, to a lesser extent, the inversion recovery sequences are rather sensitive to the optimal setting of the pulse lengths. The nuclei studied here have a moderate sensitivity; therefore many data accumulations are required to obtain an acceptable experimental setting. Of course the realization of an acceptable setting becomes increasingly difficult for a decreasing amount of the nucleus studied. This is well illustrated by the accuracies obtained in the magnesium relaxation rates, as a function of the MgCI2 concentration. Es(30) Demco, D. E.; van Hecke, P.; Waugh, J. S . J . Magn. Reson. 1974, 16, 467. (31) Nash, J. C. Compact Numerical Methods; Adam Hilger: Bristol, 1979.
1
I 3 2 m Figure 1. Longitudinal (0)and transverse ( X ) relaxation rate in s-' of 2sMg2+(RMg)and 35CI(Ra) as a function of the molal MgCIz concentration (m)at 25 "C. 0
1
pecially for 2sMg2+,it is noted that the determinations of R M ~ J and R$gZ become rather time-consuming due to the experimental condition that the repetition delay time between successive data accumulations must be 5/RMg,I or larger. For example, in the 0.024 m MgClz solution a single determination of R M g , ~with , a relative accuracy typically on the order of 15-30%, took more than 45 h, in which each of the 100 data points was obtained by 800 accumulations. For the chloride relaxation data it is noted that, within the concentration range studied, &l,2 tends to be somewhat ( f 4 % ) larger than the corresponding Rcl,I value. However, in view of the moderate accuracy reached in Ra,2, this apparent trend is thought to be not physically significant. Usually, the magnesium and chloride relaxation rates at infinite dilution are estimated by the means of linear extrapolation of the data obtained in moderately concentrated solutions. However, one may argue the validity of the linear extrapolation approach of, e.g., the magnesium relaxation data, in view of the c1/2 dependences of the quadrupolar relaxation rates of monovalent cations, as observed by Sacco et aLs2in diluted aqueous solutions of NaClO,, NaI, NaBr, LiBr (in D20), RbC1, RbBr, and RbI at 25 OC. Inspection of their relaxation data, as a function of the electrolyte concentration, shows that the c1/2dependence may become detectable for concentrations lower than ca. 0.2% = 0.25 m). Therefore, longitudinal and transverse magnesium relaxation rates have been determined in MgCI2 solutions down to 0.024 m. Visual inspection of Figure 1 clearly shows that, within experimental accuracy, the magnesium relaxation rate changes linearly with the electrolyte concentration. The dependence expected from the results of Sacco et al. could not be detected. Hence, the 25Mg2+and 35C1-relaxation rates in the limit of infinite dilution have been estimated by the means of linear extrapolation of the longitudinal and transverse relaxation data obtained in solutions equal to and smaller than 0.985 m. The extrapolated values were determined by a weighted linear least-squares procedure. The data were weighted with the reciprocal value of the squared uncertainty in the relaxation rate. The results are respectively RM,(m-O) = 4.16 f 0.03 s-l and R,(m-O) = 29.2 i 0.6 s-l. These results agree well with the values reported in the literature; Le., at 25 O C one has RMg(m+O)= 4.1 f 0.3 s-I 29 and R,(m-O) = 28.0 f 1.4 One may note that the accuracy of the experimentally determined relaxation data in the limit of infinite dilution has improved considerably. (32) Sacco, A,; Holz, M.; Hertz, H. G. J . Magn. Reson. 1985, 65, 82. (33) Holz, M.; Weingartner, H. J . Magn. Reson. 1977, 27, 153.
7936 The Journal of Physical Chemistry, Vol. 93, No. 23, 1989
Struis et al.
e,
TABLE II: Parameten Used To Estimate the Polarization Factor, P,from the Quadrupolar Relaxation Rate of =CI-, "Cl-, nBr-, *'Br-, and InI- at Infinite Dilution and 25 O C in Water and the FRD Hydration Model, Eq 5" (1 - rdb
nucleus 35~137~1-
79Br*IBr1271-
I 312 312 312 312 512
Q,37
b
-0.0802 -0.0632 0.332 0.282 0.785
used 69.8 69.8 167 167 300
lowest 50.3 50.3 100 100 139
highest 86.9 86.9 21 1 21 1 398
rion,8, 1.81 1.81 1.96 1.96 2.20
PQW, s-I 29.2c 17d 1500~ 1050~ 46001
~,(n?-+O)/iWo
0.9 0.9 0.6 0.6 0.3
P 0.40 0.39 0.40 0.39 0.56
Superscript numbers are references. (1 - y-) used from ref 20; (1 - y-) lowest and (1 - y-) highest are respectively the lowest and highest (1 - y-) values cited in ref 25-28. cThis work. dFrom Figure 1 of ref 40; T = 28 " C .
Results and Discussion 35Cl-and 25Mg" Relaxation Rates as a Function of MgCI, Concentration at 25 "C. A . The 3 5 C tRelaxation Rate in Infinitely Diluted MgCl, Solutions. X-ray,I7 MD,3'36 and neutron diffractionI6 studies show that the radial chloride-water and the water-water distribution functions in the near surroundings of the chloride ion are not very pronounced. The residence time of the water molecules near to the chloride ions is of the order of picosecond^,^^ which is comparable with the reorientational correlation time of the water molecules in pure water.I2 Therefore, a distinct first hydration layer is not assumed here and the fully random distribution model seems most appropriate to interpret the chloride relaxation rate at infinitely diluted MgCI2 solutions. For Rcl(m-O) one has
For the water molecular particle concentration, C,, the pure water value will be used, Le., C," = 3.346 X water molecules/cm3. For the other parameters in eq 10, the following values are used:2o I = 3/2, P = I f 2 , Q = -8.02 X lo-, b,37 e = 4.80 X 1O-Io esu, h esu, = 1.055 X ergs, I.( = p(gaseous water) = 1.82 X ro = rcI rw i= 1.81 1.40 = 3.21 A, and 7- = -68.8.28 As is shown by, e.g., IH N M R experiments in aqueous electrolyte solutions,38 the influence of the CI- ions on the water dynamics . estimate will also be is moderate, Le., T,(m+O) i= 0 . 9 ~ ~ 'This used in the present interpretation. A refinement of the water molecular dynamics into a rapid librational motion (o(O.1 ps)) and a less rapid diffusional motion (e.g., rotational diffusion, u (ps)), as is apparent from the M D simulation results of the quadrupole relaxation of Li+, Na+, and C1- in dilute aqueous solution,34 has not been introduced here, because the effective correlation t i m e , I ' ~with, ~ ~ e.g., the data given in ref 34, shows that the slow diffusional motion induces the major part (590%) to the relaxation rate. It is important to note that, with respect to the interpretation given in the literature,20 in the present interpretation two major alterations have been introduced: (1) , ~present study Instead of the usually applied value of 2.5 p ~in the the recently determined value of 7," = 1.94 f 0.12 ps (25 O C ) I 2 will be used. (2) For the Sternheimer factor, ym,the result obtained in a more recent calculation of Schmidt et a1.,28Le., ym = -68.6, will be used. In earlier N M R s t u d i e the ~ ~ value ~ ~ ~ of -56.6 has often been used. Introducing the parameter values in eq 10, one obtains Rcl(m-+O) = 46.3 s-l. Taking T ~ O= 2.5 ps and 7- = -56.6, in the literature the calculated value of ca. 40 s-I is ~ b t a i n e d . ~Although .~~ the latter result agrees rather well with the presently derived result of 46.3 s-I, it is noted that the present interpretation is more in line with the presently available information on the water dynamics and the Sternheimer factor.
+
+
(34) EngstrBm, S.; Jonsson, B.; Impey, R. W. J. Chem. Phys. 1984, 80, 5481. ( 3 5 ) Impey, R. W.;Madden, P. A.; McDonald, I. R. J. Phys. Chem. 1983, 87, 5071. ( 3 6 ) Bounds, D. G. Mol. Phys. 1985, 54, 1335. (37) Lindman, B.; ForsBn, S. Chlorine, Bromine and Iodine N M R ; Springer: Heidelberg, 1976. (38) Endom, L.; Hertz, H. G.; Thiil, B.; Zeidler, M. D. Ber. Bunsen-Ges. Phys. Chem. 1967, 71, 1008.
The presently calculated result of 46.3 s-I agrees reasonably well with the experimentally determined value of 29.2 S-I. The agreement between the calculated and the experimentally derived chloride relaxation rate could be improved by taking, e.g., the Sternheimer factor normally used in the literature. For example, with = -56.6 one obtains Rcl(m-O) = 31.5 S-I. However, this quantitative agreement may be fortuitous due to, e.g., the uncertainty associated with the appropriate value of the Sternheimer factor, ym,and the polarization factor, P. In aqueous electrolyte solutions containing chloride ions the appropriate value of the Sternheimer factor is not known. As was noticed by Reimarsson et al.,39a reasonable value may be expected to lie in the rather large range of reported calculated free ion values between -49.3 and -84.9. This range of values can be found, i.e., in ref 25-28 and other results cited therein. Another source of uncertainty is given by the polarization factor, P. The polarization factor enters the electrostatic theory in the calculation of the two-particle cross-correlation functions of the local field gradient which contribute to the spectral density function at zero frequency, J(0). Here, (P2- 1) is introduced as a constant of proportionality between the self-correlation function for a given water molecule and the sum of two-particle cross-correlation functions of this molecule over the entire system. For the continuum approximation of the polarization factor, the result derived on the basis of a continuum theory by Cohen and Reif,Io eq 11, is normally used: 2t 3 PCR
=
+
where t is the static dielectric constant of the dielectric medium. Taking in eq 11 t on the order of 80, the value of PCRreaches near its limiting value of Usually the value of 0.5 is taken. In the literature some criticism has been directed on the applicability of PCRin the electrostatic theory.4J1 It has been noted, e.g., that (1) Cohen and Reif considered the case of the local electric field, c.q., gradient arising from an external ionic nucleus, while in the electrostatic theory, especially in the limit of infinitely diluted electrolyte solutions, one considers a different case; that is, the local field gradient arises from a discrete water dipole. (2) The application of PCRin diluted electrolyte solutions in water may be argued against because PcR has originally been derived for quadrupole effects in solids, where usually one deals with significantly smaller t values than the values one encounters in the present studied system, e.g., e = 80. Therefore, it may be interesting to estimate the value of the polarization factor, P, from the experimentally determined quadrupolar relaxation rates of 35Cl-,37Cl-,79Br-,81Br-, and 1271at infinite dilution. For these anions the FRD hydration model seems most a p p r ~ p r i a t e . The ~ polarization factor, P, can be obtained by introducing the known parameter values in eq 10. In Table I1 the relevant parameter values are given. For (1 - y-) the results obtained in a more recent study of Schmidt et are used. As is indicated in Table 11, the differences between the reand applied (1 - ym)values are on the order of 30%. (39) Reimarsson, P.; Wennerstrom, H.; Engstrom, S.; Lindman, B. J . Phys. Chem. 1977,81, 789. (40) Weingartner, H.; Miiller, C.; Hertz, H. G. J . Chem. SOC.,Faraday Trans. 1 1979, 75, 2112.
25Mg2+and 35CI- Quadrupolar Relaxation in MgCI2. 1 7
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 7937 TABLE 111: Correlation Time T, and Tilt Angle fl as a Function of the MK12 Concentration at 25 O C "
H20
m(MgC12), mol& 0.246 0.488 0.985 1.49 2.19 2.78 3.47 3.98 4.49 4.99 5.39 5.49
Figure 2. Orientation of the cation-oxygen axis (CO) relative to the water molecular frame (x,y,z). The water molecule lies in the xz plane. The orientation is characterized by the Eulerian angles a and 8.
For the correlation time ~ ~ ( m - 0relative ) to the pure water result, rW0 = 1.94 ps, it is noted that for Br- and I- the result, as obtained from 'H N M R experiment^,^^ depends on the number of hydration water molecules assumed. In the present interpretation the result obtained taking the number of hydration water molecules equal to six will be used. For P the average result is P = 0.43 f 0.08. It is noted here that the results obtained in MDM and MC"*41studies of monovalent ions, such as Na+, Li+, and CI-, in water also suggest roughly this magnitude for P. These studies show that the main contribution to J ( 0 ) arise from water-water correlation effects in the near neighborhood, e.g., within two water molecular layers of the ions in question. From the results shown in Figure 2 of ref 11 one can estimate that the effective field gradient fluctuation, ((P$2))2), caused by the 15 nearest water molecules, relative to the sum of the separate contributions, is of the order of 0.29 f 0.14 (average result over Li+, Na+, and C1-). Interpreting this value as a rough estimate of P2, one obtains P = 0.54 f 0.13. M D simulation study on quadrupole relaxation of the same ions in dilute aqueous solution with 64 water molecules supports the estimation given here:34 Using the results given in Table I1 of ref 34, one obtains as an average result P = 0.56 f 0.18. Also, in a more recent M D simulation study on NQR of xenon- 131 in water it was concluded that the electric field gradient fluctuations are mainly caused by the molecules in the first hydration shell and for the polarization factor the value P = 0.47 has been obtained.42 Therefore, it may be concluded that P = 0.43 f 0.08 (average result of results shown in Table 11) is a satisfactory estimate for the water-water correlation effects in (in)finitely diluted electrolyte solutions provided that the ions under study exert a moderate influence on the dynamical and structural properties of the neighboring water molecules. This result will be used in the analysis of the Mg2+ relaxation rate in the next section. B. The 2 5 ~ $ + Relaxation Rate at Infinitely Diluted MgC12 Solutions. As was mentioned in the Theoretical Section, for the magnesium hydration water a distinct hydration model may be introduced, provided that within the first hydration layer the radial ion-oxygen distribution functions are rather sharp. This assumption is supported by the NiZ+-O distribution functions as obtained experimentally by neutron diffraction experiments in (various concentrated) NiCI2/Dz0 solutions16and from the results obtained in MDI9 and X-ray17 studies in MgCI2 solutions. For example, from the neutron diffraction experiments one obtains a very narrow distribution of the Ni-0 distance around r,,, = 2.07 A. (See also the discussion given in connection with eq 9.) In former 2sMg N M R s t ~ d i e s ~it~was q ~assumed ~ * ~ ~ that within the first magnesium hydration layer the water dipoles are radially oriented, and hence, the FOS model seemed most appropriate here. However, the results of the experimental and theoretical studies in strong electrolyte solutions mentioned in the Introduction ne(41) Engstrom, S.;Jonsson, B. Mol. Phys. 1981, 43, 1235. (42) Schnitker, J.; Geiger, A. Z.Phys. Chem. 1987, 155, 29. (43) Simeral, L.; Maciel, G. E. J . Phys. Chem. 1976, 80, 552
8, deg 29.7 29.0 27.6 27.4 28.0 28.4 28.3 28.2 27.6 27.5 27.5 27.5
PS f 0.86 f 0.44 f 0.19 f 0.12 f 0.09 f 0.08 f 0.06 f 0.04 f 0.02 f 0.02 f 0.01 f 0.01
7 i 3
f 1.7
3.22 2.98 2.55 2.50 2.68 2.83 2.80 2.82 2.19 2.96 3.07 3.17
f 1.0 f 0.9 f 0.7 f 0.7
f 0.8 f 0.7 f 0.7 f 0.7 f 0.8 f 0.8 f 0.8
"6 and 7i are taken from ref 13. Uncertainties are estimated standard deviations as explained in the text. 3.5
n
5; 3.0
2.5
2.0
-
0
2
6 m Figure 3. Correlation time ri (in ps) as a function of the molal MgCI2 concentration m at 25 "C. The drawn line represents the linear relation: q ( m ) = (2.39 f 0.01) (0.12 f 0.02)m. L
+
cessitate a reappraisal of the applicability of the FOS model in this case and the assumptions made regarding the dynamical and structural properties of the magnesium hydration water. This will be discussed below in detail. From the results obtained from the ZHand 170N M R studies in MgCI2 solution^,^^*^^ and supported by the results obtained in neutron diffraction studies in several electrolyte solutions containing bivalent cations,16 it has been concluded that within the magnesium hydration units the water dipoles are not radially oriented. More specifically, these studies show that the water dipole points away from the cationic nucleus, its orientation is not collinear with the cation-oxygen axis, and the latter axis is confined to the water bisectrix plane. Characterizing the orientation of the cation-oxygen axis relative to the water molecule by the Eulerian angles a and @ (see Figure 2), within the water bisectrix plane one has a = 90". Regarding the dynamical behavior of the water molecules in the MgC1, solution, in the above-mentioned 2 H and I7O N M R study in MgCI2 solutions it was shown that at infinite dilution the overall reorientation of the hydration units as a whole takes place in a time 7 , ( m 4 ) = 36 f 4 ps ( T = 25 "C). Apart from this overall diffusion process each hydration water dipole may be considered to reorient around the cation-oxygen axis. This internal diffusion is characterized by the correlation time 7i. For the tilt angle, @, and the correlation time, q,the results obtained in the mentioned N M R study13 will be extrapolated in order to obtain the values at infinite dilution. These results are presented in Table 111. For the estimation of the standard deviations in respectively 7i and /3 two contributions have been taken into account: The first is the error in the experimental data necessary to calculate these two parameters (see the evaluation procedure in section 2E of ref 13). A relative error of 2% has been assumed in Rel, (x = 0, D) and D 2 ( 0 ) / D 2 ( m )respectively. , One may note that the uncertainty in 7i and fi increases for decreasing MgCI2concentrations. In addition, an uncertainty for respectively q and @ has been
7938 The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 calculated that corresponds with an uncertainty in the overall correlation time 7," of ca. 11%. For m 5 4 the uncertainties in and 7i are mainly determined by the latter contribution. Inspecting these results, one may conclude that the tilt angle, p, is concentration independent, e&, p = 28'. In Figure 3 the correlation time Ti(m)is shown as a function of the molal MgC1, concentration. Apart from the results obtained for 0.246 and 0.488 m, one may conclude that T~changes linearly with increasing electrolyte concentration from 2.4 ps, (m 0) to 3.1 ps ( m = 5.49 m ) . The results obtained at 0.246 and 0.488 m may be ignored here because, as is indicated in Figure 3, for these lower concentrations the estimated accuracy in the T~ becomes increasingly less. For the sake of completeness, it is noted that the results obtained in MgCI2 solutions beyond 5.49 m deviate from the linear concentration dependence introduced above, but these results hardly affect the extrapolated q(m-0) value. In view of the above discussed dynamical and structural properties of the hydration water molecules it is clear that the ion-water contribution is more adequately interpreted in terms of the TAN hydration model, eq 9. In addition to this contribution, also contributions may arise from water molecules located beyond the first hydration layer. Neutron diffraction studiesI6 in NiCl,/D,O and quasi-elastic neutron scattering experiments14 in MgC12/H20 indicate a moderate influence respectively on the structural and dynamical properties of water molecules residing beyond the cationic hydration layer. For these water molecules the FRD model may still be maintained (eq 5) with C w ( m + O )= Cwo= 3.346 X IO2, water molecules/cm3, with T , ( ~ + O ) = 7,' = 1.94 ps, and by replacing the distance ro by the closest ion-dipole distance for water molecules located beyond the magnesium hydration layers, b. From a geometrical point of view the distance b may be estimated with29 b = ro/3l/, + (4rW2 - (2/3)r02]1/2 (12)
-
+
where ro = pion r,. With r M g= 0.66 A and rw = 1.40 A one obtains b = 3.43 A. Contributions arising from an exchange of water molecules between the hydration and the bulk water phases can be ignored here because the residence time of water within the first magnesium hydration layer is of the order of microseconds.21 The total ion-water contribution to the magnesium relaxation rate at infinite dilution is given in eq 13.
Struis et al. between water molecules that are localized within the magnesium hydration units. More specifically, AT refers to the tangential components of the dipole moment vectors in question. Provided that the value for the polarization factor, P, is known, one may calculate AT from eq 14 and the experimentally determined result, RMg( m - 4 ) = 4.16 s-'. Although the value of P is not precisely known, from eq 14 it can be seen that AT will range from ca. 0.05 1 (if P = 0) to 0.038 (if P = 1). However, as is indicated by the results in neutron diffractionI6 and in quasi-elastic neutron scattering experiments,14 it may be assumed that the influence of the magnesium ions on the water molecules located beyond the first hydration layer is as limited as the moderate influence of the C1-, Br-, or I- ions on neighboring water molecules. Therefore, it may be expected that, in the interpretation of these anions and the 25Mg2+relaxation rate, P refers to comparable water-water correlation effects, and hence in eq 14 the result obtained in section A, that is, P = 0.43 f 0.08, may be used. Introducing this estimate for P in eq 14, one obtains AT = 0.049 f 0.006. It is noted here that the estimated uncertainty in AT mainly stems from the uncertainty in rov(m+O), e.g., T,,(m-O) = 36 f 4 ps. (The uncertainty in y- is not included in this estimate.) It is concluded that the main contribution to the 25Mg2+relaxation rate probably stems from the nonradially oriented components of the respective hydration water dipoles. In particular the internal diffusion process 0 ) determines the time with si(m-0) = T,' > T ~ ca. , 3 times larger than T ~ O . (See Table V.) It may be concluded that, for the bivalent ions, the TAN model does not yield a better agreement of calculated and experimental rates if compared to the FRD model. The T A N model is, however, preferable in view of its consistency with known
+
7940 The Journal of Physical Chemistry, Vol. 93, No. 23, 1989
Struis et al.
features of the structural properties of these hydration spheres of these ions.
Conclusions The solvent contribution to the quadrupolar relaxation rates in aqueous solution may be described by the DOS model, which includes structural and dynamical aspects of the hydration sphere. From the interpretation of the quadrupolar relaxation rates of 35Cl-, 37Cl-, 79Br-, *'Br-, and 1271- at infinite dilution in water, it is concluded that P = 0.43 f 0.08 is a satisfactory estimate for water-water correlation effects in (in)finitely dilute electrolyte solutions provided that the ion under study exerts a moderate influence on the dynamical and structural properties on the neighboring water molecules. For ions with a structured first hydration layer, especially for small and highly charged cations, it is indicated that the dipole components perpendicular to the ion-dipole axies are responsible for the nuclear relaxation. The reorientation of the transverse dipole components of the different water molecules in the hydration sphere is found to be correlated. Appendix In this appendix the ion-water contribution to the quadrupolar relaxation rate, Rws, will be derived for the following distinct hydration model. %his general model considers n,, nonradially oriented water dipoles, all located within a distinct first hydration layer around the ionic nucleus under study. Each point dipole represents the charge distribution of a hydration water molecule. The point dipole will be located at the position of the oxygen nucleus and the ion-oxygen distance is thought to be constant in time and equal to r,. As was mentioned in the Introduction, both autocorrelation and crawcorrelation functions of the local electric field gradient (EFG) contribute to the spectral density function at zero frequency, J ( 0 ) . Here, the local field gradient is the gradient at the position of the relaxing ionic nucleus. For the calculation of an autocorrelation function, a single water dipole, u, is considered; for a two-particle cross-correlation function one considers the correlation with a secnd dipole, v. The electric field gradient fluctuation in time is thought to stem from two random and mutually uncorrelated diffusion processes: the isotropic overall diffusion of the hydration unit and, within the unit, an internal rotation of each dipole around the ion-oxygen axis in question. The angle between the dipole moment vector and the internal rotation axis is fixed and equal to p. Furthermore, it will be assumed that the hydration unit is rigid; the orientations of the n, ion-oxygen vectors relative to each other do not change in time. In the extreme narrowing limit, the quadrupolar relaxation rate, Rgos, is given in eq A l . RSoS = C(&T~;Q) J(0) (AI) Here, the constant C(&y,;Q) comprises some of the molecular properties of interest as given in eq A2.
lz'
/
'ion
'VI
Figure 4. Location and orientation of dipole u with respect to the principal axes of its SAS frame, (x',y',z?. The ionic nucleus is located at the origin.
with m E (O,fl,*2). In eq A4 denotes the summation over v excluding the contribution for v = u. "V$)(O)and, respectively, "!!)(t) denote the mth spherical tensor components of the second-rank local EFG tensor, F2),caused by the dipole moment vector u, at time t = 0, and the dipole moment vector v, at time t. The asterisk indicates the complex conjugate. In eq A4 an autocorrelation term is obtained whenever u equals u; otherwise a cross-correlation term results. The spherical tensor components are defined in the laboratory frame but are explicitly known in the molecular frame, hereafter referred to as the simple axis system (SAS) frame. For each of the n, dipole moment vectors a SAS frame can be found. The characteristic property of a SAS frame is the following: In the SAS frame the ionic nucleus is located at the origin of the principal axes system, (x';y';z?,and the point dipole moment in question lies on one of the principal axes, here denoted as z', at a distance ro from the origin. The orientation of the dipole moment vector is confined to the z'x' plane. As an example, in Figure 4 the SAS frame of dipole u is shown. One may note that the principal axis, z', coincides with the symmetry axis of the internal diffusion tensor; hence, as is indicated in Figure 4, the azimuthal angle equals the tilt angle 6. Of present interest are the components of the EFG tensor " ~ , Z k ) as defined in eq A5, in terms of the rectangular coordinates x',
y', and z'. Here, 4J(Fo) is the electric potential at the position of the ionic nucleus caused by dipole u. In the SAS frame, the irreducible components of the EFG tensor are given respectively in eq A6-A8. The results on the respective right-hand sides of
The spectral density function at zero frequency, J(O), is given in eq A3 J ( 0 ) = 2JmGg)(t) 0 dt
('43)
where G t ) ( t )is the correlation function of the local electric field gradient. Restricting the calculation to the sum of autocorrelation and two-particle cross-correlation functions, for @ ( t ) one may write
eq A6, A7, and A8 are the irreducible components in terms of the azimuthal angle, p, the distance of closest approach, ro, and the electrical dipole moment of dipole u, p. The following discussion will concern a general term of the correlation function G:)(t), that is, the term e;'" as () defined , before in eq A4: c y ( t ) = (("vy(O)){"vc,2)(t)*))
(A9) The spherical EFG tensor components defined in the laboratory fixed frame, viz., "F$)(O) and uV'!!)(t), are most conveniently
25Mg2+and 3sCI- Quadrupolar Relaxation in MgCI2. 1
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 7941
"2)
TABLE VI: Time-Dependent and -Independent Transformations and Corresponding Transformation Angles, Q,between the Relevant Principal Axis Systems for Dipoles u and Y dipole u laboratory
T
QW)
I
overall diff of u (11 int diff axis of u )
T QW 1
internal diff of u
T
dipole u laboratory
T
QW I
overall diff of u
T 1
overall diff of u
T
QW
SAS of u
internal diff of u
1
T
uV2)= C uV$2)Df,2)(Rg)Dj?(RuJ D$:A(Ry)e-ih@3u('413) jhsy
function, P;"(teq ),A9, and assuming that the overall and the internal diffusion are independent processes, for P;'(t),one may write CXt)=
C C Ai$DJ:l(Quu)(Df,y(R$(O))D$:X(RW)))
j,ha k,q
(qI(QY(O))
uv2).
uv2),
X
q z m w ) ) ('414)
with A!;: = ( ' ~ l Z ) e - i h ~ ~ Y ) ( u ~ i q In 2 ) *eq e ~A14 ~ ~ 3$;l u )and . 4; may and therefore will arbitrarily be chosen equal to zero. The term concerning the overall diffusion can be calculated by (D$:A(QY(o)) Di:z(QY(t))) =
J ]dRY(o)
dQ';(t) G,,{QY(o);Qf(t),tt X P{QP(o)lqA(QY(o))Dk:z(nf(t)) ('415)
SAS of u
obtained from the irreducible EFG tensor components defined in the SAS frame in question by making use of the transformation properties of the Wigner rotation matrices of second order, P2)(R). In Table VI the time-dependent and -independent transformations and corresponding transformation angles, R, between the relevant principal axis systems are shown for dipoles u and v . In this table, the term "overall diff of U" has the following meaning. As the overall reorientation is assumed to be isotropic, the principal axis system for this process may be chosen arbitrarily for one value of t . It is chosen to coincide with the internal principal axis system for v at t = 0. Therefore at t = t , R:(t) involves only a rotation around the actual z axis. For dipole u a transformation R, has to be inserted to obtain the internal axis system of u ("overall diff of u") as modified by the overall movement only. The transformations are described extensively here to prevent confusion. In previous work,22instead of the SAS system with its origin in the ionic nucleus, a principal axis system (PAS) was used with the dipole on the z axis and in the origin. In PAS the field gradient was calculated in a point on the continuation of the dipole. Rotation of PAS to a system with z parallel to the ion-dipole direction still yields the gradient in the same physical location on the continuation of the dipole. This point does not, however, coincide with the ionic nucleus if the dipolar direction is not radial from the ion. In eq A10 and A1 1 the irreducible tensor components of O f 1 2 ) and which in eq A9 are defined in the laboratory fixed frame, are written in terms of the irreducible tensor components defined in the SAS frames of dipole u, viz., and the SAS frame of dipole u, viz.,
(A12)
k-q
QUU
QU,
1
= cuq2)D$:i(Ry)D$:,,(Rp)e-i9@30
Here, P{Ry(O)]= 1/8a2 is the probability density for an arbitrarily set of Eulerian angles Ry at time t = 0, and Go,is the conditional probability function for the three-dimensional isotropic overall diffusion.s5
Go,= CCL p,r
+
2L 1 D$' (Ry (0)) Dg)(RY(t ) ) d L + l)Dmr 8r2
(A 16)
where Do, is the diffusion constant that characterizes the overall diffusion process. Introducing eq A16 into eq A15, and applying the orthogonality properties of the Wigner rotation matrices, for eq A15 one obtains
(o$:A(Ry(o)) D f X ( R Y ( t ) ) ) = (1 / 5 ) 6 k ~ e " ~ (A17) ~~ Introducing this result into eq A14, for P;'((t)one obtains Cy(t)= (1 /
5 P m '
C A.Z:"$~(n,,)(D~y(ng(o))
D$:X(Qy(t)))
J,h,k.q
(A18) Now a more complicated task is to calculate the term in P;"((t) which concerns the internal diffusion processes of the dipoles in question, viz.
(a2)(fi;(o)) Dgj'(Qu,(t)))
('419)
As was mentioned in connection with the discussion of Table VI, the set of Eulerian angles R2 only consists of a phase angle a, viz., 0;= {a;(O);O;O} and RY2)(f) = {a:(t);O;O]. Hence, from eq A19 only nonzero results can be expected if h = j' and q = k. Therefore, C;'((t), eq A18, can be simplified to
Pi"(?) = (1 / 5)e"D0vfCA~;u,D~~i(Ru") ( e-ilha~Y(0)-ka~'(r)~) (A20) h,k
Here Dg',(R) denotes the m,nth component of the Wigner rotation matrices of second order,s3 and R denotes the set of Eulerian transformation angles.s4 For R = {a;&y},Dg',(R) equals e-ima ti:,)"(@)
e+?.
As mentioned before, one of the principal axes of the SAS frame coincides with the symmetry axis of the internal diffusion tensor. Hence, the set of Eulerian angles R3 only consists of a phase angle, &, viz., R;l = {#;O;O] and Ry = {+f;;O;O]. Therefore, for eq A10 and AI 1 only nonzero results can be obtained if q = w and h = s. The simplified equations are given in eq A12 and A13, respectively. Introducing eq A12 and A13 in the correlation (53) Brink, D. M.; Satchler, G. R. Angular Momentum; Oxford University Press: London, 1962. (54). Hobson, E. W.The Theory of Spherical and Ellipsoidal Harmonics; Cambridge University Press: Cambridge, 1931.
The calculation of the autocorrelation contribution (u = u ) from eq A20 can be done in a straightforward manner and may be discussed first. For the autocorrelation term, PkU,in eq A18 the set of Eulerian angles Ruu equals {O;O;O). Hence, only nonzero '; is h = k. Taking h = k , u = u, terms may be obtained for C and introducing D$?.(R,,) = 1, from eq A20 one obtains ci'J(t) = (1 / ~ ) e " ~ ~ ~ ~ ~ A ~ ~ ( e - i k ~ a z ' ( o )(A21) -az'(f)~) k
The term between brackets can be calculated as given in eq A22 ( e-ikl~zY(0)-azY(f)l)=
1
day(0) dag( t ) flag(0) ] GinJay(0 );cry( t );t)e-ik~a2"(o)-a~u(f)~ ('422)
( 5 5 ) Versmold, H. Z. Naturforsch. 1970, ZSa, 367.
1942
The Journal of Physical Chemistry, Vol. 93, No. 23, 1989
where P(aq(0)) = 1 / 2 is~ the probability density for an arbitrarily value of the phase angle a; at time t = 0, and Gin,is the conditional probability function for the one-dimensional internal diffusion, for which one may use the Green’s function given in eq A2355 1 E G,,, = 2T J = - +-
eiJluz”(o)-~z’(~)l-J20,r
(A23)
where Di is the diffusion coefficient characterizing the internal diffusion process. Applying the orthogonality properties of the Wigner rotation matrices, for eq A22 one obtains ( e-ikla2Y(0)-a2Y(i))) = e-kzD,i (‘424)
Struis et al. pending on the relative orientations of the radial components of the respective hydration water dipoles, one has 0 IAR I0. AR = 0 is obtained if the hydration unit is rigid and is octahedrally symmetric with respect to the relative orientations mentioned above. It is noted that eq A28 has also been derived by Hertz,7 who introduced an additional parameter, A, to take into account the width of the angular distribution in the hydration layer. For AT the result depends on the correlation assumed between the orientation of the tangential components of the respective water dipoles. The general expression for AT is
And, finally, for the autocorrelation contribution, C y ( t ) ,one obtains
P;’((=t) (1 /5)e”Dw‘xAz:fe-kzDii
(A251
k
For the cross-correlation contribution, the term in eq A 18 concerns the internal diffusion processes of both the dipoles, u and u. Generally, the appropriate conditional probability function is unknown. Before discussing some of these cases, for the sake of clarity it may be helpful to resume the derivation of the quadrupolar relaxation rate, The steps are summarized in eq A26a-A26e:
Estimating d y D , the ion-water contribution due to the water dipoles located beyond the ionic hydration layer as discussed in the Theoretical Section, finally one obtains
RF.
Analogous with the treatment of the effects of radial ionsolvent pair distribution by Weingartner and Hertz,22 in eq A31 the function gQhas been introduced. For the present interest it is noted that if for a distinct hydration layer one assumes that the radial ion-solvent pair distribution function is very sharp, one requires gQ 1. Only under selected conditions can explicit results for AT be obtained. Some of these cases will be discussed below. That is the case that the internal diffusion process of dipoles u and u are not mutually correlated and the case of complete correlation, e.g., a;(t) = ha;(t). For AT the results obtained in the cases of no correlation and complete correlation are summarized in eq A32-A34:
-
no correlation: correlation:
AT
= 1
(‘432)
(A26e) Performing the integral in eq A26e, and introducing the constant FQas has been defined in eq 4 of the Theoretical Section, one obtains
where rOv= 1/60,,, ri = 1/6Di, and f denotes the ratio r,,/ri. In eq A27, AR and AT are quenching parameters that account for the correlation effects within the ionic hydration layer between the radial and between the tangential components of the respective hydration water dipoles. The parameter AR is given in eq A28.
AR = 1 + C’dh$(@u,)
,
Here, d@(PU,)denotes the function (3 cos2 (P,,,)
-
1}/2. De-
where dit](Pu,) = cos2 (&) - (1 - cos (Puu)l/2and di:!] = -cos2 + (1 + cos(Pu,)1/2. In practice, the measure of correlation between the respective dipole orientations within the hydration unit is not known. The results shown in eq A33 and A34 may serve as a guide to the range of values one can expect for AT. Assuming that the relative orientation of the n, dipole moment vectors is such that for all dipole moment pairs considered one may set the argument of the cosine functions in eq A33 and A34 equal to zero, within a rigid, octahedrally symmetric ion hexaaquo unit one obtains from eq A33 AT = -1 and from eq A34 AT = 0. Depending on the symmetry, the hydration number, the respective dipole orientations, and the measure of correlation involved, in principle one may also obtain absolute values for A R and AT that are larger than 1 .
a”)
Registry No. MgCI2, 7786-30-3; 35CI-, 32997-85-6; 25Mg2+,6665006-4.
