1864
J . Phys. Chem. 1987, 91, 1864-1867
roughly with the number of electrons. 3. Several of the stationary points were found to be saddle points (one imaginary frequency) or double saddles (two imaginary frequencies). In several instances the nature of the stationary point changed with basis set or with the inclusion of electron correlation, another consequence of the general flatness of these potential surfaces. 4. In all cases the inclusion of electron correlation corrections leads to an increase in the association energies. This effect is small for Li--(H)z--Li but increases markedly as the number of electrons in the complex increases. The correlation effects are additive to within *2 kcal mol-' for the most stable isomers when the following values are used: Li, 1 kcal mol-'; Be, 5 kcal mol-'; B, 11 kcal mol-'. The correlation contribution is greater when the number of bridging hydrogens is greater. Although correlation
does not alter the relative isomers stabilities, it does lead to significant binding in HBe- -(H)3--Be which is unbound at HF/631G**//HF/6-31G*. 5 . Dimerization reactions of LiH and of BeHz proceed without activation with the HF/3-21G theoretical model. The formation of diborane from 2BH3 also proceeds without activation on the MP2/6-3 lG* potential energy surface. These results are expected since no bond-breaking or repulsive interactions are involved.
Acknowledgment. This work was supported by the National Science Foundation under Grant C H E 81-01061-01 and by the Fonds der Chemischen Industrie and was facilitated by an Alexander von Humboldt award to John A. Pople. Registry No. Li2H2, 12435-82-4; LiBeH,, 25282-1 1-5; LiBH4, 16949-15-8; Be,H4, 29860-66-0;BeBH,, 39357-33-0; B,H6, 19287-45-7.
Test of the Hubbard-Onsager Dielectric Friction Theory of Ion Mobility in Nonaqueous Solvents. 1. Ion-Size Effect K. Ibuki and M. Nakahara* Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan (Received: August 19, 1986; In Final Form: October 15, 1986)
The Hubbard-Onsager (HO) dielectric friction theory was tested systematically for mobility of the alkali metal, halide, and tetraalkylammonium ions in four types of polar solvents at 25 O C ; the solvents studied were water, alcohols (methanol, ethanol, and 1-propanol), amides (formamide and N-methylformamide), and dipolar aprotic solvents (acetone and acetonitrile). Instead of the conventional Walden product, we used the residual friction coefficient (A{) defined as the total friction coefficient minus the Stokes friction coefficient for slip. The experimental curve of A{vs. ion radius is V-shaped in all solvents. The HO theory is successful in predicting the size dependence of A{ for the relatively small ions not only in the aprotic solvents but also in the other hydrogen-bonded solvents. However, the negative values of A{ observed for the medium-sized ions in the hydrogen-bonded solvents and the increase in A{ with increasing size of the tetraalkylammonium ions are not explained by the continuum theory.
Introduction For a long time ordinary hydrodynamics has been applied to interpret the dynamic properties of ions in solution. Such a typical example is seen in the study of ion mobility and molecular diffusion where the Stokes-Einstein-Walden (SEW) framework based on the Navier-Stokes (NS) hydrodynamic equation of motion is prevalent. The N S hydrodynamic equation may be applicable to uncharged systems but, in principle, not to ionic systems of our interest. To overcome the fundamental limitations of ordinary hydrodynamics, Hubbard and Onsagerltz (HO) developed the dielectric friction theory and invented the epoch-making electrohydrodynamic equation for migrating ions; H O extended the NS hydrodynamic equation to ionic systems in a self-consistent manner by adapting the Debye dielectric relaxation theory to the flow system involving a charged sphere. The apparent limitations of the SEW framework is often ascribed to the neglect of the liquid structure of solvent rather than to the neglect of the essential effect of the charge on an ion in the continuum model; the effect of the liquid structure on ion dynamics is to be discussed only after the effect of charge is properly taken into account as in the HO theory. In the treatment of ion dynamics in solution, now is the time to replace the classical SEW framework by a modern one due to the HO dielectric friction theory. For this reason an extensive test of the reliability of the H O theory is necessary. Since the experimental test of the HO theory has been confined to aqueous systems in the previous series ~~
(1) Hubbard, J.; Onsager, L. J. Chem. Phys. 1977, 67, 4850. (2) Hubbard, J. J . Chem. Phys. 1978, 68, 1649.
0022-3654/87/2091-1864$01.50/0
of papers,3-" it is extended to nonaqueous systems in the present series. Such a test limited to aqueous systems is insufficient for an understanding of the general feature of the HO theory because of the well-known peculiarities of water. We must test the theory also in nonaqueous systems. The high precision and abundance of measured ionic conductances are of benefit to the stringent test of the general feature of the H O electrohydrodynamic equation of motion which is applicable to ion translation'-" and rotationlZJ3 and electrolyte viscosity.14 It has been clearly shown in a previous paper" that the undue transformation of the limiting ionic conductance ho into the conventional Walden product distorts and obscures the original meaning of Xo. Instead, we have recommended the transformation (3) Takisawa, N.; Osugi,J.; Nakahara, M. J . Phys. Chem. 1981,85,3582. (4) Nakahara, M.; Torok, T.; Takisawa, N.; Osugi, J. J. Chem. Phys. 1982, 76, 5145. (5) Takisawa, N.; Osugi, J.; Nakahara, M. J . Chem. Phys. 1982, 77,4717. ( 6 ) (a) Takisawa, N.; Osugi, J.; Nakahara, M. J . Chem. Phys. 1983, 78, 2591. (b) Nakahara, M.; Takisawa, N.; Osugi, J. In High Pressure in Science and Technology, Homan, C . ; MacCrown, R. K.; WhaIley, E., Eds.; North Holland: New York, 1984; Part 11, p 169. (7) Nakahara, M.; Zenke, M.; Ueno, M.; Shimizu, K. J . Chem. Phys. 1985, 83, 280. (8) Ibuki, K.; Nakahara, M. J. Chem. Phys. 1986, 84, 2776. (9) Ibuki, K.; Nakahara, M. J . Chem. Phys. 1986, 84, 6979. (IO) Nakahara, M.; Ibuki, K. J . Phys. Chem. 1986, 90, 3026. ( 1 1 ) Ibuki, K.; Nakahara, M. J. Phys. Chem. 1986, 90, 6362. (12) Nakahara, M.; Ibuki, K. J . Chem. Phys. 1986, 85, 4654. (13) Felderhof, B. U. Mol. Phys. 1983, 48, 1283. (14) (a) Ibuki, K.; Nakahara, M. J . Chem. Phys. 1986, 85, 7312. (b) Ibuki, K.; Nakahara, M. J. Chem. Phys., in press.
0 1987 American Chemical Society
HO Theory of Ion Mobility
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987
into the so-called residual friction coefficient A{ which is defined as the overall friction coefficient minus the Stokes friction coefficient for slip. When the conventional Walden product was used in the early testI5 of the HO dielectric friction theory, it was ambiguous to what extent the modern dielectric friction theory1q2 is superior to the early one due to Zwanzig (Z).’6,’7 Since the superiority of the HO theory was made clear by the use of A{ in our previous ~ o r k ,we ~ ,take ~ A{ in the present test. It has been found in the previous ~ o r k that ~ - A{ ~ is negative for medium-sized ions in aqueous systems. This anomaly indicates that the actual mobility of these ions is larger than the upper limit set by the ordinary hydrodynamic continuum model, and a special structural mechanism (called the passing-through-cavities) has been proposed for the e~planation.~”We examine here whether the anomaly that cannot be explained by any continuum model is specific to water or not. Water is known for its peculiar properties, which are considered to arise from the hydrogen-bonded structure. To understand the water anomalies, we have to study the corresponding properties of solvents other than water and critically compare them. In the first part of the present series, we examine the dependence of the residual friction coefficient on the ion size in a variety of polar nonaqueous solvents. The temperature and pressure effects on A{ will be separately discussed in the following two papers. Theoretical Section The fundamental difference between the modern (HO) and classical (NS) continuum models mentioned above can be understood formally from the following HO electrohydrodynamic equation of motion for the viscous and dielectric fluid: ~ V ’ V= V p - !L2[Eo X (V X P*)
+ Eo(V*P*)]
(2)
This equation leads to the well-known Stokes law in the form (3)
where Cs is the Stokes friction coefficient for the slip boundary condition and R is the crystallographic radius of the io11.’~9’~Thus only the size effect can be handled by the NS hydrodynamic equation, whereas the charge effect as well as the size effect is treated by the HO electroydrodynamic equation in a self-consistent manner. The ionic friction coefficient { is given by the limiting conductance Xo as { = lelF/Xo
As emphasized previously,I0 however, the simple hydrodynamic relation eq 3 does not hold in the ionic system. Thus, the Stokes radius is introduced just as an adjustable parameter to retain the SEW framework.” Although the invalidity of the Walden rule (the constancy of the product Xoq in the denominator on the right side of eq 5) was noticed a long time ago, the product itself still continues in use in almost all papers on ionic conductance, theoretical or experimental. We should have abandoned the Stokes law and the Walden product for the moving ion in solution at the same time for the sake of self-consistency. The trouble mentioned above is not brought about by the use of the residual friction coefficient3J0 defined as
A t = P - Ps
(6)
where lSand {are given by eq 3 and 4, respectively. Since purely viscous friction is taken as a reference in eq 6, A{ reflects mainly the dielectric friction and all the other factors neglected in the primitive hydrodynamic model for lS. As elucidated previously,’0 the residual friction coefficient is much more useful and meaningful than the so far widely used Walden product. The numerical solutions to the linearized form of eq 1 are given in terms of the dimensionless quantity of the form
x = P/&o
(7)
where RH0 denotes the Hubbard-Onsager radius, and x depends only on R and RHO.A kind of solvent parameter RHOis defined as
(1)
where 7 is the solvent viscosity, v the velocity field in the fluid, p the pressure, Eo the 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 on the right side of eq 1 represents the contribution of dielectric friction which is coupled to viscous friction. Full coupling of the two kinds of frictions renders the HO theory superior to ordinary hydrodynamics and the early dielectric friction theory’6s17in the treatment of ion dynamics in solution, as clearly shown e l ~ e w h e r e . * * ~ -When ” ~ ’ ~ the charge on an ion is neglected (Eo = 0), eq 1 reduces to the linearized form of the NS equation given by l p v =vp
1865
(4)
where F is the Faraday constant and e is the ionic charge. If we assume that tSin eq 3 is equal to {in eq 4, we have the “Stokes radius” as (15) Evans, D.F.; Tominaga, T.; Hubbard, J. B.; Wolynes, P. G. J . Phys. Chem. 1979,83,2669. (16) Zwanzig, R. J . Chem. Phys. 1963,38,1603. (17) Zwanzig, R. J . Chem. Phys. 1970,52, 3625. (18) Pauling, L. The Nature o f t h e Chemical Bond; Cornell University: New York, 1960. (19) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1968.
where T , eo, and c, are the dielectric relaxation time and the static and high-frequency dielectric constants, respectively. In order to facilitate the systematic application of the HO theory to a variety of conditions, we have refineds,9the numerical solutions of x and fitted them to a fourth-order polynomial equation instead of tabulation.2 The improved result is expressed within an error of 1% by (9)
where y = 4~ for the slip boundary condition considered here and the coefficients are given as follows a , = -2.7866,
for RIRHo
a2 = 8.6216, a3 = -3.3425, a4 = 0.39550
> 0.3, and
a. = 15.610, a_, = -11.303,
a-2 = -12.056, a-3 = 14.081, a-4 = -3.5834
for RIRHo < 1.6. In the range 0.3 < R/RHo < 1.6, differences in the two expansions are within 1% and one can use both expansions. The HO theoretical value of the residual friction coefficient is obtained by combining eq 6, I , and 9 as
A{ = q(XRH0 - Y R ) = ?RHOCaj(RHO/RY
(10)
which indicates that not the total friction coefficient but the residual friction coefficient due to dielectric friction can be expanded to the fourth power of R H o I Ror RIRHo to a good approximation. Equation 10 involves both dielectric and viscosity properties of solvent, reflecting the coupling between the viscous and dielectric frictions. When the physical properties of the solvent involved in eq 8 are all available, the HO value of the residual ionic friction coefficient A{ can be calculated for the ion characterized by the radius R and the charge e according to eq 10 under any conditions. Results and Discussion Data Sources. First we summarize the data sources used in the present computation. The polar solvents treated here are protic
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Ibuki and Nakahara
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 4
TABLE 1: Solvent Properties and RHOfor Monovalent Ions at 25 O C solvent n/cP €0 TIDS RunlA water 0.890 78.3 5.2 8.2 1.50 0.551 5.6 47.7 3.17 32.6 methanol 1.087 24.3 4.4 170 3.94 ethanol 1-propanol 1.967 20.4 3.9 340 4.21 FA 3.130 109.5 3.0 39.0 1.48 2.23 10 172 NMF 1.65 182.4 0.303 20.6 1.9 3.1 2.13 acetone 3.9 1.93 0.341 36.0 2.0 acetonitrile
,
CI- 13
1
15 0
2
1
3
4
I
I
I
5 12
R d ) Figure 1. Variation of A!: with ion size in water at 25 O C . The broken and full lines indicate the theoretical and experimental results, respectively. The number attached to the open circle denotes the carbon number in the alkyl chains of each tetraalkylammonium ion. The open and solid circles denote the cations and anions, respectively.
\I
\
-9
E
a m
p
RdCTI-
v
L/,
a
3
0
-
10
-3
E a u
- PrOH
0:
P
Figure 3. Variation of A!: with ion size in formamide (FA) and Nmethylformamide (NMF) at 25 OC. The circles and squares indicate the experimental values of A!: and FA and N M F , respectively. The other notations are the same as those in Figure 2.
v
LI,
a
5
CI- 1-
0 0
I
I
I
I
1
2
3
4
R
5
(8)
Figure 2. Variation of A!: with ion size in methanol, ethanol, and 1propanol 25 OC. The circles, triangles, and squares indicate the experimental values of A!: in methanol, ethanol, and 1-propanol, respectively. The numbers on the right top denote the carbon numbers in the alkyl chains of each tetraalkylammonium ion. The open and solid circles denote the cations and anions, respectively.
or hydrogen-bonded solvents, such as water, alcohols (methanol, ethanol, and 1-propanol), formamide (FA), and N-methylformamide (NMF), and aprotic solvents, such as acetone and acetonitrile. The physical properties of the solvents at 25 OC listed in except Table I are taken from the references cited by Evans et for ethanol,* l-propanol,20,21and NMF,22,23and used in the ~
(20) em and T from: Kosi, T.; Arie, E.; Nakamura, M.;Takahashi, H.; Higashi, K . Bull. Chem. SOC.Jpn. 1974, 17,618. (21) v and €0 from: Barthel, J.; Watcher, R.; Gores, H.-J. Mod. Aspects Electrochem. 1979, 13, 1. (22) e- and T from: Bass, S. T.; Nathan, W. I.; Meighan, R. M.; Cole, R. H. J. Phys. Chem. 1964, 68, 509. (23) 7 and eo from: Gopal, R.; Thatnagar, 0. N. J. Phys. Chem. 1966, 70, 3007.
0
0
1
2
3
R
(A)
4
5
Figure 4. Variation of A!: with ion size in acetone and acetonitrile at 25 O C . The upper and lower parts are for acetone and acetonitrile, respectively.
calculation of RHOfrom eq 8. We have taken the experimental values of the limiting ionic conductances from the references cited in the same placeI5 except for those in ethanol,24 l - p r ~ p a n o l , ~ ~ and NMF.23,26 (24) Kay, R. L.; Broadwater, T. L. J. Solution Chem. 1976, 5, 57. (25) Barthel, J.; Watcher, R. Schmeer, G.; Hilbinger, H. J. Solution Chem. 1986, 15, 531.
H O Theory of Ion Mobility
Success of HO Theory. The experimental and theoretical residual friction coefficients, which are given by eq 6 and 10, respectively, are plotted against the ion radius in Figures 1-4 for the alkali metal, halide, and tetraalkylammonium ions in the four types of polar solvents. In these figures we notice the following common features: all the experimental curves are V-shaped, and the HO theoretical values of A{ initially fall rapidly and then asymptotically approach zero (the macroscopic limit). Figures 1-4 show that the HO theory is successful in predicting the size dependence of the residual friction coefficients for the relatively small ions not only in the aprotic solvents but also in the hydrogen-bonded solvents where the effect of liquid structure is often emphasized. It is noted that the left part of the V-shaped curves of A{ is determined by the charge effect and described by the H O theory. This conclusion is consistent with our previous one based on the studies on t e m p e r a t ~ r e , ~ p, ~r e” s ~ u r e , ~solvent ’ isotope,’ and c o s ~ l v e n teffects ~ * ~ in aqueous systems. Thus it is not safe to discuss the effect of the solvent structure on the ion transport process without an ideal reference like the continuum model tested here; for the case of small ions, in particular, we should pay attention to the charge effect. The agreement of the HO theory with experiment is satisfactory for the lithium ion even in a quantitative sense when it is compared with the early dielectric friction theory by Zwanzig; for example, the Z theory predicts the A{value for the Li+ ion in ethanol as 2.3 X 10” P cm, while the HO theory predicts it as 5.9 X P cm, in fair agreement with the experimental value of 8.2 X P cm. The large discrepancy for the Z theory comes from the neglect of the effect of the coupling of the viscous and dielectric frictions on the velocity field. Figure 2 shows the effect on A{ of the length of the alkyl chain in the homologous series of alcohols. For any ion, the observed value of A{ increases with increasing chain length. The observed trend is well described by HO theory; it is caused by an increase in RHOand in 7. Limitations of HO Theory. In Figures 1-4, we find also the limitations of the H O dielectric friction theory as referred to below. First, the cations and anions show a difference in the ion-size dependence of A{ in most solvents; the difference is negligibly small in water as clearly shown in Figure 1. The larger the residual friction coefficient the stronger the ion-solvent interaction. Consequently, Figure 2 suggests that the cations are more reactive than the anions of corresponding size, while Figure 4 suggests that, as often observed, the anions are more reactive in dipolar aprotic solvents. In this respect, N M F and FA resemble the former and the latter cases, respectively. The difference between the cations and anions cannot be explained by the HO theory because the ionsolvent interactions are characterized simply by the magnitude of the charge and the size of the ion. Second, in all the solvents the observed ion-size dependence of A{ for the tetraalkylammonium ions is contrary to what is predicted by the H O theory combined with the bulk properties of the solvent and the slip boundary condition; the slope of the A{ curve is rather steep in the highly polar solvents. Although the reason for this discrepancy is not well understood at present, the (26) Singh, R. D.; Gopal, R. Bull. Chem. Soc. Jpn. 1972, 45, 2088
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 1867 effect of the hydrophobic nature of these ions on the solvent properties may be relevant; the fact that the peculiar behavior is not restricted to aqueous systems is in favor of the more general term “solvophobic” interactiom8 The change of the boundary condition leads to no plausible interpretation; while the use of the stick boundary condition makes the theoretical A{ closer to the observed value for Bu4N+,it makes the observed Acvalue more negative for Me4N+. Third, negative residual friction coefficients are observed for medium-sized ions in the hydrogen-bonded solvents (see Figures 1-3), although the HO theoretical values are always positive irrespective of the ion size. Furthermore, when the alkyl chain of the alcohol molecule becomes longer, the region where A{ is negative expands. Such trends are found also for diffusion of neutral molecules; by computing the residual friction coefficients ({) from the diffusion coefficients (D)27 using the Einstein relation ({ = k T / D ) ,we notice that the Arvalues are negative and become more negative with decreasing size ratio of the solute to the solvent molecules. Hence these anomalies are not ascribed to the charge effect but to the size effect which is beyond the continuum model. The anomalies mentioned above are the most serious limitations of the continuum theory and require a new theory where the molecular nature of the solvent is somehow taken into account. An idea introduced for the explanation is the special transport mechanism called the passing through cavities (PTC).3“ The PTC mechanism is a process where an ion in the hydrogen-bonded framework composed of neighboring solvent molecules can move rapidly over a short range with little friction. The previously mentioned fluctuation& of the framework structure is considered to play an important role in the cooperative or coupled migration of the ion trapped in the cage, as seen in the case of the solid electrolyte.28 Thus the mechanism of solid-state diffusion remains as a memory in the short-range order characteristic of the liquid state even after the long-range order is lost by melting. The results of the present test are very similar to those of our previous study in aqueous systems. This is surprising because water is famous for its peculiarities due to the hydrogen-bonded “structure”. Our conclusions here are as follows: (1) the HO theory works well in both aqueous and nonaqueous solvents for small ions, and (2) the limitations for medium- and large-sized ions are ascribed not to the “unique structure” of aqueous solutions but to a more general molecular nature of the solvents. It is interesting to see the more general aspects of the reliability and limitations of the HO theory by varying such conditions as temperature and pressure in the polar nonaqueous solvents examined in the present work. These important subjects will be discussed in the following papers.
Acknowledgment. The authors are grateful to Professor J. Barthel for informing us of his conductance data before publication. This work is supported by the Research-Grant-in-Aid from the Ministry of Education, Science and Culture (No. 61 134043). (27) (a) Evans, D. F.; Tominaga, T.; Chan, C. J . Solution Chem. 1979, 8, 461. (b) Evans, D. F.; Tominaga, T.; Davis, H. T. J . Chem. Phys. 1981, 74, 1298. (28) (a) Ueda, A.; Kaneko, Y. Butsuri 1985,40,866. (b) Hokazono, M.; Ueda, A.; Hiwatari, Y. S o l i d S f a f eIonics 1984, 13, 151.
