6362
J . Phys. Chem. 1986, 90, 6362-6365
may not be easily resolved a t kHz frequencies, or that at temperatures not very far below Tg,the @-processwould also freeze out on a time scale of one's experiment. Thus the possibility of the Occurrence of molecular motions in >30 wt % water-glucose mixtures at T I Tgis much less than it is in more-glucose-containing mixtures. In the Vogel-Fulcher-Tamman equation, f, = A exp(-B/( T - To)),the values of parameters A, B, and Toare subject to both the method of analysis and the number of data points available for the a n a l y ~ i s . ' ~ The J ~ values given in Table I represent merely an empirical fit. The only significance is in the ratio T,/T, which decreases on addition of 10 wt % water to glucose. On further addition of water, the ratio increases. This means that In f, vs. 1/T plots for >30 wt % water-glucose mixtures would become increasingly more steep at temperatures near TB. The activation energies for the @-process are in the range of 50-64 kJ mol-I. There is a decrease in its value when 10 wt % water is added to glucose and thereafter an increase on further dilution with water. The change is relatively small and their interpretation within our present knowledge of the origin of @process seems less clear. However, this range of values is similar to that observed for @-process in other glassy materials.6 The presence of a @-processin the glassy state has been suggested to indicate localized high-volume, high-entropy regions, in the otherwise rigid matrix of a glass, where molecular orientation and translational motions persist.13 An analysis of Figure 5 shows that at a constant value of T/T,, the rate of @-process progressively decreases as water is added to it. The decrease in the rate of @-process may involve a closer packing of the highvolume, high-entropy regions allowed by the presence of hydrogen-bonded water molecules, which can also hydrogen bond to the hydroxyl groups of glucose. If such packing involved a greater correlation of dipole vectors of the water molecules or the dipolar (16) Johari, G. P.; Whalley, E. Furuduy Symp., Chem. SOC.1972,6,23. (17) Angell, C. A.; Smith, D. L. J . Phys. Chem. 1982,86, 3845.
groups of glucose molecule, the strength of the @-processwould also be increased. The latter effect has been discussed earlier here and is also seen in Table I and Figure 9. The Effect of Plasticization with Water. The effect of water on glucose, as for organic polymers, is to lower the glass transition temperature. The effect in polymers has been interpreted as due to an increased free volume and the consequent loosening of the interchain and/or intrasegmental motions? and in this sense the effect observed in glucose is similar to that in polymers. But the effect of plasticization by water on the @-processin glucose is quite the opposite of that observed in amorphous polymers. External plasticization of most polymers (for a list of such polymers see ref 4 and 18) does not seem to affect the location of the @-relaxation peak in a temperature plane or its frequency in an isothermal spectrum.18 It also does not affect the height of the @-relaxationpeak. Plasticization of glucose causes the @-relaxation peak to shift to higher temperatures as Seen in Figure 3, or to lower frequencies as seen in Figure 5 . It also raises the height of the @-relaxation peak by more than a factor of two (Figure 9). Clearly, the plasticization effects in glucose are dominated by the directionality of the hydrogen bonds between water molecules and the OH groups of a glucose molecule, as discussed in the preceding section. The Static Permittivity. The strength of relaxation, AE = eo - nD2,where eo is the low-frequency permittivity which decreases with increase in temperature in all cases as seen in Figure 2. The product TAt is found to decrease with temperature for anhydrous glucose but to remain approximately constant at 1.0 X lo4 and 1.2 X lo4 K for 10 and 15 wt % water-glucose mixtures. Thus the orientational correlation of the dipole moments in pure glucose decreases on increasing the temperature while little change occurs in their water mixtures in the relatively narrow temperature range of our study. Registry No. Glucose, 50-99-7. ~
(18)
~~
~~
Heijboer, J. In?. J. Polym. Murer. 1977,6, 11, and references therein.
Interpretation of the "Stokes Radius" in Terms of Hubbard-Onsager's Dielectric Frictlon Theory K. Ibuki and M. Nakahara* Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan (Received: April 1 , 1986; In Final Form: July 7, 1986)
The velocity field around the lithium ion in water has been schematically presented according to the solution of the Hubbardansager electrohydrodynamic equation where the coupling of viscous and dielectric frictions is taken into account in the self-consistent manner. The HO velocity field around the ion is found to be much smaller than the Stokes velocity field around the corresponding uncharged sphere as a result of the strong attractive interaction between the ion and solvent dipoles. The often-used concept of the "Stokes radius" and the apparent validity of the Walden rule exceptionally for the lithium ion in water against temperature are successfully explained by means of the HO dielectric friction theory. The failure of the early dielectric friction theory by Zwanzig is ascribed to the incomplete treatment of the coupling of viscous and dielectric frictions on the migrating ion on the basis of systematic comparison of the early and modern dielectric friction theories.
Introduction The purpose of the present study is to promote our understanding of the significance and usefulness of the modem dielectric friction theory developed to an almost final form by Hubbard and Onsager (H0).1*2 At present this kind of work is of great necessity because little proper attention is paid to the modern dielectric friction theory on the experimental side. The conventional pro(1) Hubbard, J.; Onsager, L. J . Chem. Phys. 1977,67, 4850. (2) Hubbard, J. J . Chem. Phys. 1978, 68, 1649.
0022-3654/86/2090-6362$01.50/0
cedure for interpreting the experimental value of the limiting ionic conductance Xo remains in an underdeveloped state basically relying upon the classical hydrodynamic (Navier-Stokes) equation leading to the Stokes-Einstein law or the Walden product as criticized in our previous paper.3 Although the effects of ion size and solvent structure on ion mobility have been discussed with the aid of the classical hydrodynamic equation, this is not reliable because the equation completely neglects the charge effect (3) Nakahara, M.; Ibuki, K. J . Phys. Chem. 1986, 90, 3026.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 23, 1986 6363
Interpretation of Stokes Radius characteristic of the ion. The modern continuum model used in the H O theory takes account of both size and charge effects at the same time and can provide at least a reliable reference that may lead to an actual finding of the specific or anomalous effects due to the liquid structure of the solvent; without a suitable reference, the often-attempted statement of the effect of the solvent structure (say, of water) on ion dynamics is groundless. The fundamental difference between the modern and classical continuum models mentioned above can be understood formally from the following HO electrohydrodynamic equation of motion for the viscous and dielectric fluid 1
?V2v = V p - -[Eo 2
X
(VXP*) + Eo(V-P*)]
(1)
where 11 is the solvent viscosity, v the velocity field in the fluid, p the pressure, Eothe electrical field at v = 0, and P* the polarization deficiency. In this equation P*expresses the departure of the orientational polarization from that at v = 0 due to slow dielectric relaxation of the flowing solvent. The second term represents the contribution of dielectric friction which is coupled to viscous friction in the self-consistent manner; this full coupling renders the HO theory superior to other theories for ion mobility. When the charge on ion is neglected (E, = 0), eq 1 reduces to the linearized form of the Navier-Stokes (NS) equation given by l p v =vp
(2)
which leads to the Stokes law {s = 4 ~ 7 R
(3)
where lS is the Stokes friction coefficient for the slip boundary condition and R is the crystallographic radius of the ion.4 Thus only the size effect can be treated formally by the N S hydrodynamic equation in contrast to the HO electrohydrodynamic equation. The idea of dielectric friction was put forward by Born to incorporate the charge effect into the framework of the spherein-continuum model for ion dynamics; the theoretical advance is reviewed e l ~ e w h e r e . ~Here we mention only the contributions by Zwanzig (Z)6,7and Hubbard and Onsager. The Z7 and HO' theories are quite different with respect to the starting equation of motion; the former and the latter start from the NS hydrodynamic and HO electrohydrodynamic equations, respectively. As a result, although the early dielectric friction theory' has not been very successful,* the HO theory has been confirmed to be successful in particular for small ions like Li+ with limitations exhibited for large univalent ions under various experimental
condition^.^^^-'^ It has been stated from a molecular point of view near the appearance of the HO theory that the continuum dielectric friction theory is appropriate in the limiting case of weak, long-range interactions which are realized for solutions of "large" univalent ions and that the molecular theory reduces to the old solventberg
picture, which is relevant to the Stokes radius and very familiar to chemists, in the case of strong, short-range attractive interactions.16 Moreover, the concept of the Stokes radius appears to be favored by the well-known fact that the Walden product is invariant with temperature exceptionally for the lithium ion in water. Since the Stokes radius is introduced just as an adjustable parameter with the intention of keeping the validity of the classical continuum model, the empirical concept should be tested theoretically by using the modern continuum model; it is interesting to see whether the H O dielectric friction theory is compatible or not with the solventberg picture in the case of the small ion. To solve the problems mentioned above, we have to return to the details of the solution to the linearized form of eq 1. In the present paper, we develop a schematic description of the fluid velocity around the moving ion according to the detailed solut i ~ n . ~The . ' ~ picture of the stationary velocity distribution in the space for the lithium ion is instructive and full of information to understand the difference between the early and modern dielectric friction theories and to clarify the meaning of the Stokes radius within the content of the dynamic continuum model; analogously, our imagination has been stimulated for its application to a large extent by the pictorial description of the electron cloud in the molecule obeying the Schrodinger equation of motion.
Comparison of Stokes and HO Velocity Fields In this section we attempt to visualize the difference between the H O and Stokes fields which are stationary solutions of eq 1 and 2, respectively, and discuss the physical chemical meaning of the difference. The electrohydrodynamic equation of motion eq 1 is valid for slow motion of the ion "relative" to that of fluid around it; very rapid random motions of the bath particles and their frequent collisions with the tagged particle are prerequisite for the validity of the continuum model in the condensed phase. The solution of eq 1 for the steady flow around a fixed charged sphere has been obtained by Hubbard* in the form 1 v = ,V X If(r)(u X r)] L
where u is the given velocity of the solvent at infinite distance from the center of the ion taken as the origin of a coordinate system, andflr) gives the dependence of the radial velocity on the distance r. The function f ( r ) monotonously increases from zero at the surface of the ion to unity at r = m, and depends not only on r but also on the ionic radius R and the Hubbard-Onsager radius RHO.RH0 which characterizes the coupling of viscous and dielectric frictions is defined as
RHO= [e2(eo - e , ) r / ( 1 6 ~ q e ~ ~ ) ] ' / ~
1982, 76, 5145. (11) Takisawa, N.; Osugi, J.; Nakahara, N. J . Chem. Phys. 1982, 77, 4717. (12) Takisawa, N.; Osugi, J.; Nakahara, M. J . Chem. Phys. 1983, 78, 2591. (13) Nakahara, M.; Zenke, M.; Ueno, M.; Shimizu, K. J . Chem. Phys. 1985, 83, 280. (14) Ibuki, K.; Nakahara, M. J . Chem. Phys. 1986,84, 2776. (15) Ibuki, K.; Nakahara, M. J . Chem. Phys. 1986, 84, 6979.
(5)
where e is the ionic charge, eo and em are the static and highfrequency dielectric constants, respectively, and 7 is the dielectric relaxation time of the solvent. When the well-known Stokes field due to eq 2 is expressed in the form of eq 4, f(r) for the uncharged system is given by f(r) = 1 - R / r
(4) Pauling, L. The Nature of rhe Chemical Bond; Cornell University: New York, 1960. (5) Wolynes, P. G. Annu. Reo. Phys. Chem. 1980, 31, 345. (6) Zwanzia, R. J. Chem. Phvs. 1963, 38, 1603. (7) Zwanzig, R. J. Chem. Phys. 1970, 52, 3625. (8) Kay, R. L. Water; Franks, F., Ed.; Plenum: New York, 1973; Vol. 3, Chapter 4 and references cited therein. (9) Takisawa, N.; Osugi, J.; Nakahara, M. J. Phys. Chem. 1981,85,3582. (10) Nakahara, N.; T6rBk, T.; Takisawa, N.; Osugi, J. J. Chem. Phys.
(4)
(for slip)
(6)
We consider here only the slip boundary condition because it is more favorable at the molecular l e ~ e l . ~We * ~notice ' ~ an important difference in the distance dependence off(r) between the Stokes and HO fields; the latter increases much more slowly than the former because of the dependence of f(r) on RH0 involving th,: ionic charge as in eq 5 . This difference arises from the full coupling of viscous and dielectric frictions exerted on the ion. In order to see the charge effect on the velocity field, let us make visible the difference of the Stokes field for the uncharged sphere and the HO field for the charged sphere on the basis of the solution to eq 1.*,14 Dependence of the velocity of the solvent in the ( x y ) plane normal to the direction of the macroscopic fluid flow u is (16) Wolynes, P. G. J . Chem. Phys. 1978, 68,473.
6364 The Journal of Physical Chemistry, Vol. 90, No. 23, 1986
result of the influence of the ion charge. Thus the charged sphere drags a larger amount of solvent than the uncharged one. The same feature can be seen in Figure 1 after the transformation of the coordinate system into the space-fixed one. According to this transformation, the velocities of the ion and the solvent become -u and v' = v - u, respectively. The distance dependence of v' tells us that the translating ion drags a larger amount of solvent in the H O field than in the Stokes field. For this reason, a larger friction is exerted on a charged sphere than on the uncharged one.
Stokes
T R'2.0 Stokes HO
Figure 1. Profile of the HO and Stokes solvent-velocity fields around the Li+ ion in water at 25 OC in the stationary state in which the ion is at rest and the solvent velocity far from the ion is constant. The arrow indicates the fluid velocity at the distance r from the center of the ion for the slip boundary condition; the integers in the vertical series denote r in A. R = 0.6 and 2.0 A are the crystallographic and Stokes radii, respectively, for the Li' ion. The dashed line presents the velocity profile of the Stokes field for the stick boundary condition. The left portion shows the Stokes field around a sphere with the Stokes radius 2 A.
shown for the two cases in the right part of Figure 1 as a function of the distance r from the center of the ion; the flow direction is taken here as the z axis of the Cartesian coordinate system, and the velocity is symmetric about the z axis. The velocity in the xy plane is calculable from the explicit form of f ( r ) as follows:
[
v(r) = u f ( r )
I:
+ -5 -
(7)
The numerical solution used in Figure 1 is obtained for the Li' ion in water at 25 OC (RHO= 1.5 A). It is clear in Figure 1 that there exists a marked difference in the velocity field as a result of the full coupling of dielectric and viscous frictions. The velocity in the vicinity of the ion in the HO field is very small compared with the corresponding one in the Stokes field despite the use of the (large) bulk dielectric constant even in the vicinity. The velocity at the ion surface is almost zero in the HO field despite the slip boundary condition. Although the zero velocity at the ion surface can be attained by combining the stick boundary condition with the Stokes field (the dashed line), the velocity profile in the short range is quite different from that of the HO field. The very small value and its slow increase in the HO field arise from the long-range force exerted by the ion through the dielectric; the characteristic of the HO field would be the reason why the account of dielectric s a t ~ r a t i o n ' ~little ,'~ alters the essential feature of the HO theory for ion mobility. Thus an increase in the effective viscosity near the small ion results from the additional energy dissipation due to the fully coupled dielectric relaxation of the solvent. The right part of Figure 1 is adequate to tell us how and why the Stokes-Einstein-Walden framework is invalid for the ion; note that the friction coefficient for the migrating ion is determined by the whole pattern of the velocity field. It is interesting to compare the present result for translating motion with the previous result19 for rotational motion. The solution of eq 1 applied to rotation of a charged sphere has been obtained by Felderhof,20and the (HOF) velocity field around the rotating ion has been made visible and compared with the Stokes field by the present authors. It has bee.] found that the dielectric friction which is electrohydrodynamically coupled to the viscous friction does act on the rotating ion under the stick boundary condition despite the spherical symmetry, and that the HOF field extends to a much longer range than does the Stokes field as a (17) Hubbard, J. B.; Kayser, R. K.Chem. Phys. 1982, 66, 377. (18) Stiles, P. J.; Hubbard, J. B.; Kayser, R. F. J . Chem. Phys. 1982, 77, 6189.
(19) Nakahara, M.;Ibuki, K. J . Chem. Phys., in press. (20) Felderhof, B. U. Mol. Phys. 1983, 48, 1283.
Ibuki and Nakahara
Stokes Radius in Terms of the HO Field Figure 1 gives us a rational interpretation of the classical Stokes radius in terms of the modern dynamic continuum model. First, let us show how to calculate numerically the Stokes radius based on the HO theory. Hubbard* has provided the friction coefficient for the translating ion in the form = XqRHO
(8)
where x is the dimensionless friction coefficient, a particular function of R and RHO.We have shown e l ~ e w h e r e ' ~that , ' ~ x can be expressed approximately by the fourth-degree polynomial 4
x = 47O/RHo)
+ JZaJ(RHo/RY =l
(9)
On the other hand, the friction coefficient {and the Stokes radius Rs are defined in terms of Xo as { = lelF/Xo
(10)
where F is the Faraday constant and the other quantities are already defined above. Hence, combining eq 8, 10, and 11, we have the HO expression for the Stokes radius as
Rs = X R H O / ~ T
(12)
As seen from eq 9 and 12, the HO theory predicts that Rs varies little when RH0 is constant over a range of temperature, pressure, or solvent. In the case of water, the variation of RH0 with temperature is actually small; RH0 is 1.49 A a t 0 OC and i .5 1 A at 40 "C. This is the reason why the Walden product Xoq or Rs in eq 11 is constant against temperature for the case of the Li+ ion in water. Thus the exceptional success of the classical hydrodynamics (eq 2) for the small ion is just accidental. The observed increase of the Walden product for the Li+ ion in water with increasing pressure is also successfully predicted by the HO theory because RH0 decreases from 1.51 to 1.44 8, when pressure is increased from 1 to 1000 bar. Why are both classical and modern theoretical frameworks are successful exceptionally for the Li+ ion in water? It follows from eq 6 and 7 that in the Stokes field for slip, the velocity at the ion surface in the xy plane is equal to u/2 and independent of the ion size. In the case of the HO field for the Li' ion in water, the distance where the velocity becomes u/2 is about 2.0 A. At distances beyond this, the HO velocity field is almost identical with that in the Stokes field for R = 2.0 A as seen in the lower part of Figure 1. This identity permits us to consider that the Stokes radius means the distance where the solvent velocity gets nearly equal to u/2. The Rs value calculated from eq 12 for the Li+ ion in water is 1.8 A which is close to 2.0 A referred to above. This similarity is in favor of the above consideration. Hence it can be said from the standpoint of the HO theory that the use of the Stokes radius for a small ion like Li+ is allowed by the accidental similarity of the two velocity fields as shown in the lower part of Figure 1. It is to be noted that the epoch-making H O theory can explain the behavior of the lithium ion as shown elsewhere9J0J2J4Jswithout using any adjustable parameter.
Comparison of HO and Z Theories The classical hydrodynamic framework is desired to be replaced as early as possible by the modern electrohydrodynamic one. For this purpose it is indispensable to show how superior the HO theory is to the Z theory. The second version of the Z theory' takes
The Journal of Physical Chemistry, Vol. 90, No. 23, 1986 6365
Interpretation of Stokes Radius
20
0
0
1
2
3
4
R (%,
Figure 2. Plots of observed, HO, and Z values of A{ for univalent ions in water at 25 "C against the ion size. From the left, the circles indicate the Lit, Nat, Kt, Cst, C1-, Br-, I-, Me4Nt, Et4Nt, Pr4Nt, and Bu4Nt
40
Temp. ( ' C )
5
Figure 3. Plots of observed, HO, and 2 values of A{ for the Li' ion in water against temperature.
ions, respectively.
account of the coupling of viscous and dielectric frictions only within the level of the first-order perturbation and the Stokes field is used as the unperturbed velocity; the first version6 takes no account of the coupling. Since the fully coupled HO field is significantly different from the Stokes one as clearly shown in Figure 1, the incomplete treatment of the coupling is the main reason for the failure of the early dielectric friction theory as shown below. Since the early test2' of the HO and Z theories against experiment was made by taking the conventional Walden product, it was obscure which dielectric friction theory was refer able.^ To compare the reliability of the early and modern dielectric friction theories, we transform Xo into { as in eq 10 and take the residual friction coefficient A{ defined as where ls is given by eq 3. Since the hydrodynamic friction coefficient for slip is taken as a reference in eq 13, A{ reflects mainly the dielectric friction and any other factors controlling ion mobility; it is pointed out elsewhere3 that the residual friction coefficient is much more meaningful than the conventional Walden product. Figure 2 shows the size dependence of A{ for monovalent ions in water a t 25 OC;physical properties of water are collected in ref 9 and Xo values are taken from ref 22. As seen in Figure 2, the observed residual friction coefficient decreases with increasing the ion size below about 3 A with the cations and the anions lying on the same curves. Both HO and Z theories can predict the observed trend, but quantitatively, the former is more reliable; the smaller the ion the larger the overestimation of the Z theory. The large dielectric friction predicted for the small ion by the Z theory comes from the unrealistically large velocity around the ion as shown in Figure 1; note that the faster the fluid flow around the ion the more effective the friction due to the kinetic depolarization deficiency caused by the finite dielectric relaxation time. The modern dielectric friction theory explains neither the negative value of A{ for the medium-sized ions nor its rising trend for the larger ions; they are limitations of the continuum model combined with the bulk properties of the ~ o l v e n t . ~ - ' ~ Figure 3 shows the temperature dependence of the residual friction coefficient for the Li+ ion in water; the relevant data are
i -T
LIrrzZA --
0
HO
500
1000
Press ( kg cm-* )
Figure 4. Plots of observed, HO, and Z values of A{ for the Lit ion in water at 25 "C against pressure.
available in ref 9. The observed decrease of the residual friction coefficient with increasing temperature is predicted by both dielectric friction theories. At any temperature, however, the absolute value predicted by the Z theory is larger by more than 1 order of magnitude than that predicted by the HO theory, the experimental value being always between the two theoretical values. The Z theory exhibits much stronger temperature dependence of the residual friction coefficient than does the HO theory or the experimental value, reflecting the incomplete treatment of the electrohydrodynamic coupling in the Z theory. Figure 4 depicts the pressure dependence of the residual friction coefficient for the Li+ ion in water at 25 OC;the relevant data are taken from ref 10. The HO theory predicts the pressure dependence of the residual friction coefficient is negligibly small in agreement with the observation, but the Z theory predicts a large decrease with increasing pressure. With respect to the relative change, however, both HO and Z theories can predict a slight decrease in A{ in conformity with the observation. We can summarize the above comparison of the HO and Z predictions for the lithium ion in water as follows: although the Z theory can correctly predict the trends of the size, temperature, and pressure dependence of A{, it always overestimates these effects to a large extent. This defect can be ascribed to the use of the undesirably large Stokes velocity field around the ion shown in Figure 1. On the other hand, the HO theory successfully predicts the observed values with a slight underestimation. Now it is clear that the full coupling has to be considered in a future molecular theory for ion migration, although the coupling is completely neglected in the molecular theory put forward by Wolynes.I6 The present comparison suggests t h a t the neglect of
(21) Evans, D. F.; Tominaga, T.; Hubbard, J. B.; Wolynes, P. G. J . Phys. Chem. 1979, 83, 2669. (22) Robinson, R. A,; Stokes, R. H. Elecrrolyte Solutions; Butterworths: London, 1968.
the cross correlation between the soft and hard forces cannot be rationalized. Registry No. Li, 7439-93-2.
