J. Phys. Chem. 1995, 99, 12647-12654
12647
Ionic Association of Na+-CI-, Na+-Na+, and Cl--CI- in Methanol: Mean Force Potentials and Friction Kernels G. SesC and E. Guhdia Departament de Fisica i Enginyeria Nuclear, Campus Nord. Mbdul B4, Universitat Politkcnica de Catalunya, c/ Sor Eulhlia d'Anzizu s/n, 08034 Barcelona, Spain
J. A. Padr6* Departament de Fisica Fonamental, Universitat de Barcelona, Avda. Diagonal 647, 08028 Barcelona, Spain Received: October 20, 1994; In Final Form: May 31, 1995@
Solvent-averaged potentials for Na+-Cl-, Na+-Na+, and Cl--CI- in methanol have been evaluated using the constrained molecular dynamics simulation technique. Solvent contributions to the total force on the solute pairs have been analyzed carefully in all of the systems. In order to get a deeper insight into the influence of the solvent on the ion pairs, the most probable solvent distributions at the relevant points of the calculated mean force potentials have been studied. Our mean force potentials show important discrepancies with the ones obtained using the RISM (reference interaction site model) approximation. Friction kemels for the relative dynamics of the ion pairs have also been evaluated. The passage across the barrier existing in the Na+-Cl- mean-force potential has been analyzed in light of Kramers and Grote-Hynes theories. It tums out that the association-dissociation process takes place in the polarization caging regime. A comparison between our results and the ones corresponding to the same ion pairs in water has been done.
1. Introduction Methanol is a very common nonaqueous solvent that plays an important role in a great number of chemical processes as a principal reaction media. A lot of experimental work has already been published on it,' but computer simulations have been mainly oriented toward the study of water solutions. Methanol molecules have a dipolar moment that is slightly smaller than the one of water molecules, but the macroscopic behavior of both liquids is quite different. For instance, the dielectric constant is 78 for water and 32 for methanol. Moreover, there are big differences between the hydrogenbonded structures made up by both solvents. Then, a detailed information of microscopic properties of methanol solutions would be highly desirable. Computer simulation methods have been used to study static and dynamical properties on pure Simulations of single ions in m e t h a n ~ l ~and - ~ dilute solutions* have already been reported. There is also a molecular dynamics study of NaCl in methan01,~but a detailed analysis of the behavior of pairs of ions in methanol is still lacking. The mean force potentials (W(r))between ions in solution play a key role in the study of properties of liquid solutions. Hirata and Levyroperformed the first calculations of W(r)for ions in methanol. They assumed a three site model for methanol, and they used the extended RISM integral equation approximation. Computer simulation techniques, as the constrained molecular dynamics (MD)," are also very useful in the calculation of W(r). Constrained MD is based on the calculation of the mean force acting on the solute particles due to solvent molecules, while the solute particles are kept at fixed separations. It has given satisfactory results in the calculation of W(r)in aqueous solution^.'^.'^ We have applied constrained MD to the calculation of W(r)between Na+ and C1-, Na+ and @Abstractpublished in Advance ACS Abstracts, July 15, 1995.
Na+, and C1- and C1- in methanol and to the study of the dynamics of these ion pairs. The present paper is organized as follows. A description of the methodology and the systems under study is given in section 2. In section 3 the results on ion-ion mean forces and mean force potentials obtained for all the simulated systems are presented. The ion pair friction kemels are analyzed in section 4 . Section 5 is devoted to the ion pair interconversion process for the Na+-Cl- system. The final section gives a view of some concluding remarks.
2. Methodology and Description of the Systems The mean force potentials for Na+-Cl-, Cl--Cl-, and Na+Na+ in methanol have been evaluated. In each case, the systems under consideration were made up by 2 ions and 214 methanol molecules located in a cubic box with periodic boundary conditions. The size of the cubic box ( L = 24.4 A) was chosen to give a solvent density of 0.7866 g/cm3, and the temperature of the system was 298 K. We assumed for the methanol molecules a rigid model with three interaction sites representing the methyl group, the oxygen atom, and the hydrogen atom in the hydroxyl group. The hydrogen-oxygen and the oxygen-methyl distances were 0.945 and 1.430 A, and the bonding angle was 108.5". Then, each methanol molecule has a dipole moment of 2.22 D. The interactions within the methanol molecules have been evaluated using the OPLS (optimized potential for liquid simulations) potential by JorgensenI4 which refers to the following mathematical expression,
where A? = 4 ~ i a i 'C? ~ , = 4~iuf',and qi is the charge assigned to the i site. This potential describes correctly many thermo-
0022-365419512099-12647$09.00/0 0 1995 American Chemical Society
12648 J. Phys. Chem., Vol. 99, No. 33, 1995
TABLE 1: Interaction Potential Parameters MeOWion particle E (kcaymol) u (A) MeOH Me 0.207 3.775
c1Na+
0 Ho
0.170
3.071
0
0
0.1 17 1.600
4.420 1.900
SesC et al.
q (e) 0.265 -0.700 0.435 -1 1
dynamical and structural properties of liquid methano12q4and represents a significant improvement with respect to the TIPS (transferable intermolecular potentials) m ~ d e l . ~ Both . ’ ~ OPLS and TIPS models have the same functional form but slightly different parameters. The interactions involving the ions have been calculated using a potential form which is analogous to (l), and the Lennard-Jones parameters had been derived from ab initio calculations of single ions in water.I5 As it is costumary practice, the transferability has been assumed. In the reference interaction site model (RISM) calculations of ref 10, the TIPS potential model has been considered. For all the potentials, the Lennard-Jones type part has been truncated at half the box length and the Ewald summation has been used to calculate the Coulombic part. The potential parameters are gathered in Table 1. More realistic models including solvent flexibility or polarization could have been used, but in these cases the computational cost of the simulations would increase significantly, and some studies for ion pairs in waterI3si6indicate that similar results are obtained for W(r) if such aspects are taken into account. The equations of motion have been solved at every time step for each interaction site using the leap-frog Verlet integration algorithmi7 with a time step of 2.5 fs. The SHAKEIS method has been used to maintain the geometry of the solvent molecules. The constrained MD methodology has been used in the calculation of the ion-ion mean force potentials, which are defined as W(r) = - ~ B Tln(g(r)),I9 where g(r) is the solutesolute radial distribution function. The method requires the MD simulation of a system, where a constraint is added so that the ion-ion separation is kept fixed by means of the SHAKE technique. If a system is composed by two ions (A,B) and N solvent molecules, the force (AF(t;r))exerted by the solvent molecules along the ion-ion internuclear axis can be expressed as
where F A S ( ~ and ; ~ ) FBs(t;r) are the total forces on the solute particles due to the solvent molecules, mA and mB are the masses of the solute particles, ,u = mAmB/(mA mB) is the reduced mass, and F is the unit vector along the AB direction. This quantity is computed at each time step, and it is averaged over the whole simulation. If Fd(r) is the direct ion-ion force, the total mean force between ions may be written as
+
F(r) = F J r )
+ AF(r)
(3) where AF(r) (AF(t;r)). Then, the mean force potential can be calculated by integrating
W(r)= W(ro)- J:F(r) d r
(4)
W(r0)was chosen so that the calculated mean force potentials should coincide with the macroscopic Coulomb potential at long distances. We took ro = 8 8, for the potentials involving Na+ and ro = 9 8, for the Cl--Cl- potential, as the decay of the latter is much slower. The experimental value of the methanol dielectric constant (6 = 32) was assumed.
TABLE 2: Error Estimates in the Methanol Contribution to the Ion-Ion Mean Force in Units of kBT/A = 150 PS
SRUN
= 75 PS
ion pair
r(A)
s(ps)
ffAF
EAF
OAF
EAF
CI--ClNa+-Na+
5.4 3.6 4.6 6.0 2.6 3.4 4.6
0.23 4.00 0.50 0.25 4.00 0.70 1.25
6.97 10.75 10.56 10.90 6.59 8.60 9.07
0.39 2.48 0.86 0.63 1.86 0.83 1.17
7.06 10.11 10.36 10.88 8.29 8.58 9.12
0.28 1.66 0.60 0.45 1.35 0.59 0.83
Na+-CI-
ZRUN
For each like ion pair 27 simulations were performed to cover
an interionic separation ranging from 2.8 to 8.0 8, in the NafNa+ system and from 3.8 to 9.0 8, in the Cl--Cl- case, with increments of 0.2 8,. For the Na+-Cl- pair, 28 simulations have been erformed and the interionic distance interval from 2.4 to 8.0 has been explored. In order to minimize possible hysteresis effects, several initial configurations have been generated. In each case, after an initial equilibration period of 25 ps, the ion-ion separation was increased or decreased to cover the whole range of distances. For each value of r, we carried out a further equilibration period of 10 ps which was then followed by a production period ranging from 75 to 150 PS. The statistical errors on AF(r) and on W(r) have been evaluated according to the procedure described in ref 20. In Table 2 we have gathered the values for the statistical inefficiency, that is, for the time interval necessary to obtain statistically independent samples of AF(r), the values for the standard deviation of AF(r),and its estimated statistical errors at several interionic separations and for different simulation lengths. According to these results, we can assess that it is not necessary to perform simulations longer than 75 ps in most of the cases, as this does not result in a significant decrease of statistical uncertainties. The estimated errors for the mean force potentials are lower than ~ B for T the Na+-Cl- and Na+-Na+ pairs and than 0 . 4 k ~ Tfor the Cl--Cl- pair. In addition, we have calculated some functions which provide an instrument for the analysis of the ion pair relative dynamics. It is well-known that the function kernels for the relative dynamics of ion pairs in a solvent are related to the stochastic forces through the fluctuation-dissipation theorem
8:
If it is assumed that the interionic distance is maintained at a fixed value (rigid bond approximation), one can prove that21.22 R(t) = AF(t;r)- AF(r) (6) This equation may be applied provided that there is a clear separation of time scales between translation and reorientational motion of the ion pair. This condition is fulfilled by the sort of systems considered in this work.22 We have used (5) and (6) to evaluate the friction kernels in our constrained MD simulations at different interionic distances. 3. Ion-Ion Mean Forces and Mean Force Potentials
The mean force on each ion (F(r))is displayed in Figure 1 together with the direct (Fd(r))and the solvent (AF(r))contributions for the Haf-C1-, Cl--Cl-, and Na+-Na+ pairs. The contribution of the solvent (AF(r))has the opposite sign than the direct force between the ions in all the systems. This is consistent with the tendency of polar solvents to form stable complexes in the case of like ion pairs and to dissociate the ions with opposite sign.”-’3 The balance between F d ( r ) and
Na+-Cl-, Na+-Na+, and Cl--Cl70 1
J. Phys. Chem., Vol. 99, No. 33, 1995 12649
in Methanol
\
1
I
CI'-CI'
\ \ \ \
30
It is interesting to analyze the relevance of the molecular structure of the solvent by making a comparison between our results and the continuous model predictions. In this approximation, we can assume that the direct interionic force has a short range term (FSR(r)),which comes from a Lennard-Jones potential in our systems (see eq l), and a Coulombic part,
where ql and q2 are the ionic charges and r is the interionic distance. In the presence of a solvent, the Coulombic part is modified by taking into account the dielectric constant E , and the mean force becomes
-30
Then, the contribution of the solvent is the difference between eq8and7 AF(r) = ( 1 / ~- 1
-
L W
LL
-30 -50
I I I
-
/
/
/' '
/
I
/ \ /
I
50
/ -
/ /
-
I
I
I
I
I
Na*-Na+
\ \ \
I
3
5
1 L
W
LL
) ~ (9) r The numerical values given by (9) for every pair of ions are represented in Figure 1. They agree with the results of our simulations at interionic distances greater than 7 A. From this, two statements follow. First, our model is consistent with the long distance behavior commonly accepted. Second, our systems are big enough to cover the interionic distances at which the solvent does not behave as a continuum. In Figure 2 we have plotted the contribution of the solvent to the ion-ion mean force, having methanol or as a solvent. At short interionic distances, the molecular structure of the solvent is crucial and there are important discrepancies between the influence of both solvents onto the solute particles. At longer distances, the results obtained for both solvents are quite similar. It should be noted that the contribution of a solvent to AF(r) is given in this case by the ( 1 / ~- 1) factor (see eq 9) and, despite the very different dielectric constants of water and methanol, differences in this factor are smaller than 2%, which goes beyond the precision of our results. The mean force potentials have been obtained by integration of F(r) according to eq 4. In Figure 3, the W(r)of Na+-Cl-, Na+-Na+, and Cl--Cl- in methanol are displayed, together with the ones obtained for the same pairs of ions in water. Whereas we have oscillations in the potentials corresponding to Na+-Cl- and Na+-Na+, the W(r)for Cl--Cl- in methanol decreases monotonically. The result for Cl--Cl- is qualitatively different from the one obtained having water as the solvent, where a shallow minimum appears at 5.4 kl* In order to analyze the origin of these differences, we have calculated the mean number density of methyl groups and oxygen and hydrogen atoms in a plane which contains the interionic axes. Figure 4 contains the density plots for the interionic distance of 5.4 A. Although methanol does not form particular stable structures around the Cl--Cl- pair, the most probable orientation of its molecules can be deduced from the density curves of Figure 4 and we have displayed it in Figure 5. It is apparent that the methanol molecules are located around the pair of ions with their hydrogen atoms oriented toward C1- because of the strong electrostatic interaction between these two particles. The methyl group is also positively charged, but every hydrogen has been assigned a charge almost twice the charge of the methyl group (see Table 1). The minimum in water appears because of the existence of two stabilizing hydrogen bonds in the bridging solvent molecule, which is not possible in methanol.
4
-70 1
2
3
4
5
8
7
8
r/A Figure 1. Contributions to the mean force on the Na+-Na+, Na+C1-, and CI--CI- ion pairs in methanol, evaluated by using (2) and (3): (- -) Fd(r); (0)AF(r); (*) F(r); ( x ) AF(r) in the continuous model approximation (eq 9).
AF(r) results in an all distances repulsive force onto the C1-C1- pair. For the Na+-Cl- pair, the total force is repulsive at very short distances and attractive at intermediate distances. The total force on the Na+-Na+ pair has the same trends, but the attractive part has much lesser importance.
qlq2
S e d et al.
12650 J. Phys. Chem., Vol. 99,No. 33, 1995
-
30
320 h *lS 25
c
I
=c 3
2 L
-30
-40
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d
-so
-
I
0 0
0
3
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10
s-
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-60 1 -70
2
3
4
5
6
7
-5O 2
8
3
4
5
7
S9
8
r/A6
70
so
-
Na+- C I-
Na *-C I'
so
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$ 4 0
3
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30
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U
LL Q
20 10
0
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5
8
7
8
I
-I
-5
1
I
I
3
4
r/A5
I
I
I
6
7
8
Figure 3. Mean force potentials calculated using (4) for the Na+Na+, Na+-Cl-, and Cl--Cl- ion pairs in methanol (*) and in water
r/A Figure 2. Mean force on the Na+-Na+, Na+-Cl-, and Cl--Clpairs in methanol ( 0 ) and in water (0).12J3
I
2
(-
-).l2,13
ion
The Na+-Na+ mean force potential has a minimum at about 3.6 A, and a maximum at 4.6 A, and then it decreases monotonically to zero. Figure 5 displays the most probable orientation of the methanol molecules around the ion pair at 3.6 A that arises from the analysis of the solvent density profile.
The Na+-Na+ pair structure is stabilized at a distance corresponding to the minimum of W(r)by the methanol molecules surrounding it. They have their oxygen atoms oriented toward the Na+ ions. As in the water solution,'* the structure is stabilized by the ox gen of the solvent molecule in the central region. At r = 4.6 , the orientation of the methanol molecules
1
Na+-Cl-, Na+-Na+, and Cl--Cl-
in Methanol
J. Phys. Chem., Vol. 99, No. 33, 1995 12651
Hydrogen density Oxygen density
----
I
6
::e
4
8
-6
-6
-4
-2
0
2
4
6
I
I
I
1
1
I
1
Methyl density 2
6
-
W
8
8
i.
Figure 4. Contour plots of the methyl group and oxygen (-), and hydrogen (- -) density profiles around the Cl--Cl- ion pair at an interionic separation of 5.4 A.
around the ion pair is just like in the previous case, but there is a probability of them being between the ions. At such distances, the hydrogen-Na+ and the methyl-Na+ repulsion become important and they give instability to the resulting distribution. Finally, the corresponding W(r)in water has a second minimum, missing in methanol. The very general trends of W(r) for the Na+-Cl- pair in methanol are similar to those in water,I3 with the position of the relevant points shifted toward slightly lower ion pair distances. It presents a marked minimum at 2.6 A and a shallower one at 4.6 A. The first of them can be assigned to the contact ion pair (CIP) region and the second to the solvent separated ion pair (SSIP) region. We have verified such assessment by analyzing the solvent density profiles around the ion pair at several interionic separations. In the CIP, the pair of ions is surrounded by methanol molecules with their hydrogen atoms oriented toward C1- and their oxygen atoms oriented toward Na+, as shown in Figure 5 . In spite of the fact that there are no noticeable disagreements between the distributions of both types of solvent molecules around the ion pair, the second minimum of the potential corresponding to the SSIP position is less abrupt in methanol than in water. This can be related to the structure of the hydrogen bonds in both solvents. In methanol, the intervening solvent molecule in the SSIP distribution cannot belong to any hydrogen-bonded chain mainly
because of the bulky nature of the methyl group which prevents the oxygen from accepting a second hydrogen bond. In water, the hydrogen-bonding coordination is scarcely perturbed in the SSIP region, and the resulting configuration is much more stable. There is a maximum between the minima just mentioned, at about 3.4 A. The most likely distribution of methanol molecules around the ion pair at such interionic distance is plotted in Figure 5. It appears that there is a probability of finding methanol molecules between the ions. Information on the relevant points of mean force potentials calculated from simulation results (WSIM(T))and by using the RISM technique ( W ~ l s ~ ( r )is) "gathered in Table 3. Important disagreements appear in all the systems. For the Cl--Cl- pair, WsIM(r) decreases monotonically, whereas WRISM(~)has a shallow minimum followed by a very soft maximum. In the Na+-Na+ potential, the simulation and the RISM results agree in the number of maxima and minima, but not in their relevance. WSIM(T) has a very shallow minimum at about 3.6 A, and the one in WRISM(~), which is much more pronounced, appears at about 3.2 A. Accordingly, their potential barriers are of 1.22k~T and 8.8kBT, respectively. In the mean force potentials corresponding to the Na+-Cl- pair, the discrepancies between simulation and FUSM calculations are slightly less important than in the previous systems. The first minimum and the subsequent maximum are a bit more marked in WRISM(T), whereas
12652 J. Phys. Chem., Vol. 99, No. 33, 1995
SesC et al. 1 .a
1
Na*-CI-
0.8
r-5.4A
r-5.4A
0.6
@8@
-
-
0.4
r 3.8 A
r 3.6 A
r-26A
-0.2
J
I
1.o
-
NaC-Na*
r=3.6A --r=4.6A ------- r=6.OA
r 3.4 A
Figure 5. Schematic drawings of the most probable arrangements of solvent molecules around the ion pairs at several interionic separations. Methanol (on the left) and water (on the right) are the solvents.
0.4
-
TABLE 3: Relevant Points of the Mean Force Potentials Obtained in Our Simulations and in RISM Calculationdo
-0.2
ion pair Cl--ClNa+-Na+ Na+-Cl-
I
0.00
simulation RISM simulation RISM simulation RISM
4.4 3.6 3.2 2.6 2.8
3.25 9.38 -5.30 -5.70 -8.00
5.6 4.6 6.3 3.4 4.0
-1.5
1
0.50
I
0.75
1
1.00
1.15
t(PS>
4.3
10.6 3.5 -0.55
,
0.25
Figure 6. Normalized friction kemels for Na+-Cl- and Na+-Na+ in methanol at several interionic distances. 4.6 4.8
-5.0 -2.0
TABLE 4: Initial Values of the Friction Kernels at Several Interionic Distances
its second minimum is shallower than the one appearing in WSIM(~).In addition the position of the maximum in W R ~ S M ( ~ ) c1--c15.4 0.70 is shifted toward larger interionic distances in comparison with Na+-Na+ 3.6 2.20 the one in W.IM(I). According to previous studies of ion pairs 4.6 2.3 1 in water,'* these disagreements may be attributed to the 6.0 2.56 approximations made when using the RISM technique. HowNaf-C12.6 1.32 ever, it should be noted that the potential parameters employed 3.4 1.47 4.6 1.78 in the RISM calculations were slightly different from ours and that W(r) are sensitive to small changes in the interaction similar initial decay lasting less than 0.1 ps. Then follows a modelsaZ3 long-time decay that characterizes every system. Big differences between &(t) are correlated with important changes in the 4. Friction Kernels solvent distributionaround the ion pair. For instance, the solvent The results obtained for the three pairs of ions in methanol structure around the Na+-Cl- system at 3.4 8, has more are qualitatively very similar to the ones obtained for them in similarities with the one at 4.6 8, than with the one at 2.6 8, water.2' Firstly, the friction kemels depend on the ionic species and so have their corresponding friction kemels. Finally, the and on the interionic distance. Secondly, the results obtained &(t) at an interionic distance corresponding to the minimum in all our systems show that an increase in the interionic distance of W(r)for the Na+-Cl- pair has the slowest long-time decay. results in an increase on the initial value of the friction kemel In Figure 7 we have displayed &(t) for the Cl--Cl- ion (E(O)), as shown in Table 4. The normalized friction kemels pair in methanol at r = 5.4 8,, which corresponds to the (&(t) = &t)/E(O)) associated with the Na+-Cl- and with the minimum of W(r)for this ion pair in water. This function has Na+-Na+ pairs at several interionic distances are displayed in a slower initial decay and a faster long-time decay than the Figure 6. There are common features in the functions correfunctions in Figure 6. This result might be related to the fact sponding to both systems. All of them present a rapid and very that the residence time for methanol molecules around the C1-
Na+-Cl-, Na+-Na+, and Cl--Cl1.0
in Methanol
J. Phys. Chem., Vol. 99,No. 33, 1995 12653 I
5
0.8
0.6
0.4
TABLE 5: Na+-CI- Association-Dissociation Reactions in Methanol and in Water. Top Barrier Positions, Barrier Frequencies, Kramers and Grote-Hynes Transmission Coefficients, Nonadiabatic Frequencies, Reaction Time Scales, Solvent Time Scales, and Constant Friction Coeficients inMeOH i n H z 0 inMeOH i n H z 0 3.7 UNA' x 10-3(p~-2) -1.08 -1.7 r (4 3.4 0.28 0.10 WL, (PS-') 19.7 16.1 &-I (PS) 0.10 0.054 kkr 0.23 0.08 tc(ps) 80 199 kGH 0.49 0.22 t(ps-l)
0.2
0.0
-0.2
I
,
0.25
0.00
I
0.50
0.75
1
1.00
1.25
+(PSI 1.0
,
which has been obtained by fitting an inverted parabola to the maximum of the mean-force potential. In the Grote-Hynes theory, the reaction coordinate evolves following a generalized Langevin equation. A time dependent friction coefficient ( ( ( t ) ) represents the solvent contribution to the process. Then, we define ( as its integral value. The transmission coefficient can be obtained as
1
kGH
= 'cob
(11)
where the reactive frequency A, is the solution of the equation NaC-NaC
!*
AI = u;[il, +
0.4 j
0.2
i
t
:
/
XI
I
,
,,
O'Oi -0.2 0.00
I
::
;
5.
!
0.25
I
0.50
0.75
1.00
1.25
+(PSI
Figure 7. Normalized friction kemels for Cl--Cl- and Na+-Na+ in methanol (-) and ia water (- -). For the Cl--Cl- pair the interionic separation is 5.4 A; For the Na+-Na+ pair the interionic distance corresponds to the first minima of W(r) (3.6 and 3.8 A, respectively). ion is shorter than the one around the Na+ ion.24 In the same plot, ( N ( f ) for the Cl--cl- ion pair in water at the same interionic distance is displayed. No significant differences are observed. On the contrary, the long-time decay of ( ~ ( t for ) the Na+-Na+ pair in methanol is slower than in water (see bottom of Figure 7). This can be correlated with the result that the residence time of methanol molecules around the Na+ ion is much bigger than the one with water molecules.
5. Ion-Pair Interconversion in the Na+-Cl- System The mean force potential between Naf and C1- in methanol has a remarkable barrier at about 3.4 A. We have determined the transmission coefficient (k) of this barrier using Kramers and Grote-Hynes theories for chemical reactions in solution.25 A stochastic equation for the time evolution of the reaction coordinate is assumed in both theoretical frameworks. In the Kramer's theory this is the Langevin equation, which implies an instantaneous solvent response in the reactive process. According to Kramers, the transmission coefficient can be expressed as
where is the friction coefficient and a b is the barrier frequency,
(12)
The Grote-Hynes approach has given satisfactory results when compared with MD calculations for waterz2 and a model polar solvent.26 The coefficients involved in the previous theories for the Na+-Cl- system in methanol are summarized in Table 5 . For comparison purposes, the values obtained in a previous work taking water as the solvent2I have also been included. In that work, it was found that the friction kemels depend on the interionic distance. This is true in our systems too (see Figure 6). Then, we have considered the ( ( t ) function at the top of the potential barrier to evaluate 5 and 1,. As the barrier of the Na+-Cl- mean force potential in methanol is sharper than the one in water, the O b corresponding to the former is bigger. This is correlated with the fact that the reaction time value (Ar-') is shorter in methanol. Because of the negative value for the square of the nonadiabatic barrier frequency (UNA' = a b z we are in the so-called polarization caging regime. In this regime, the motion of the solvent molecules is very important during the passage across the barrier top. Accordingly, the integral value of (N(f), which gives the solvent time scale (T~),is shorter than the reaction time scale (Ar-[). In water we got the same qualitative results,21 but the characteristic time scales are slightly smaller in methanol than in water. Moreover, in the methanol solution k~~ is smaller than k G H which was obtained for Na+-Cl- in water, too, but both coefficients are remarkably bigger in methanol. In addition, the friction kemel, which represents the dynamical role of the solvent in the reaction, has smaller initial and integral values in methanol. This might be related to some differences in the behavior of both solvents during the reaction. In water, the solvent traps the reactants and moves with them during the reactive process. This is also true for the methanol solution, but in it the crossing of the barrier does not necessarily involve drastic changes in the linear structures of the hydrogen-bonded chains made up by methanol molecules. On the contrary, the more tied hydrogen-bonded star-shaped structure in water has to be necessarily broken. Finally, we have calculated the full rate constants for the dissociation (kf)and association (kb) processes. They are given
..-.----.--..---....,...----...-..-._.---.------.--
'
hw&) exp(-Art) dt1-l
12654 J. Phys. Chem., Vol. 99, No. 33, 1995
Sest et al.
by2225
time is smaller in the former, which is consistent with the intuitive idea that the number of recrossings decreases as the reactive time becomes shorter. Acknowledgment. Part of this work has been done thanks to the computational facilities provided by the Centre de SupercomputaciB de Catalunya (CESCA). Financial support of DGICYT through Project PB90-0613-C03 is also acknowledged.
where ,UI is the reduced mass of the ion pair, r* is the interionic distance at the transition state (Le. the top barrier), and r, is some value of r at which the diffusive motion is a good approximation. We have chosen r, = 8 A. By using the Grote-Hynes prediction for the transmission coefficient (kGH), the values for the full rate constants are kf = 0.025 ps-I and kb = 0.005 ps-’, and for the equilibrium constant kq= kf/kb = 5 , which indicates a greater stability of the SSIP complex compared to the CIP structure. The overall rate constant (kf k b ) corresponds to a relaxation time of 30 ps approximately.
+
6. Conclusions We have obtained the mean force potentials for Na+-Na+, Na+-Cl-, and Cl--Cl- in methanol. For the Na+-Cl- ion pair, the solvent allows for two stable interionic distances: a CIP and a SSIP. In the Na+-Na+ system, there is only one stable position, in which the ions are bridged by means of the oxygen of solvent molecules. For the Cl--Cl- ion pair, the potential is repulsive at all interionic distances. Our results show remarkable disagreement with the ones obtained by applying the RISM approximation. These discrepancies are bigger in the Na+-Na+ system. Our W(r) can be reasonably interpreted in terms of the solvent structure around every ion pair. In addition, there are some qualitative differences between our results and the ones obtained for the same ions in water, especially in the case of the Cl--Cl- ion pair. They are correlated with the arrangement of solvent molecules at small interionic distances. The friction kernels of pairs of ions in methanol have the same qualitative behavior as the ones obtained in water. For each pair of ions, they depend on the interionic distance. As it increases, the friction kernels increase their initial and their integral values. The ion pair interconversion process in the Na+-Cl- system has been analyzed. The crossing of the barrier takes place in the polarization caging regime. The transmission coefficient is bigger in methanol than in water, and the reaction
References and Notes (1) Robinson, H. L.; Symons, M. C. R. J . Chem. Soc., Faraday Trans. 11985, 81, 2131.
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