1238
J. Phys. Chem. 1996, 100, 1238-1245
Solvation Dynamics in Slow, Viscous Liquids: Application to Amides† Ranjit Biswas and Biman Bagchi* Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012, India ReceiVed: May 3, 1995; In Final Form: July 11, 1995X
As the viscosity of a liquid increases rapidly in the supercooled regime, the nature of molecular relaxation can exhibit dynamics rather different from the fast dynamics observed in the normal regime. In this article, we present theoretical studies of solvation dynamics and orientational relaxation in slow liquids. As the local short-range correlations are important in the slow liquids, we have extended our previous theory to take into account the short-range pair correlations between the polar solute and the dipolar solvent molecules. Application of the generalized theory to the study of solvation dynamics of amide systems gives nice agreement with the experimental results of Maroncelli and co-workers (J. Phys. Chem. 1990, 94, 4929). The theory also provides valuable insight into the orientational relaxation processes in the viscous liquids.
I. Introduction When the viscosity of a liquid becomes large near its glass transition temperature, several interesting anomalies are observed in its transport properties.1-4 In this slow regime, local, shortrange correlations (both spatial and orientational) can play an important role in determining the nature of the response of the liquid to a given perturbation. The latter is often provided by instantaneously changing the nature of a suitable probe placed inside the liquid. The response of the liquid may now derive significant contributions from a collection of a small number (say 10-100) of molecules which are spatially close to the solute probe. However, understanding the nature of this response is often difficult. Fortunately, the contribution of the translational modes of the liquid can be unambiguously studied at the molecular length scale by using the inelastic neutron scattering experiments. On the other hand, there does not seem to exist any such efficient experimental technique at this point which probes the collective orientational dynamics at a local level, as inelastic neutron scattering does for the translational dynamics. This has severely limited our understanding of molecular relaxation and hence that of chemical dynamics in slow liquids. Thus, we may need to study chemical dynamics in slow liquids with the twin goal of understanding both the natural dynamics of these liquids and their effects on chemical relaxation processes. Recently, Angell5 proposed an elegant classification of a large number of seemingly unrelated results on relaxation in glassy liquids. He showed that most of the liquids fall between the two extreme limits of liquid behavior, termed as fragile and strong, respectively. Liquids made of simple rigid molecules, like benzene and neopentane, are in the fragile limit, while network liquids, like silica, correspond to the strong limit. The fragile liquids show anomalous behavior, such as adherence to the Vogel-Fulcher temperature dependence of relaxation time5 and nonexponential dynamics. Strong liquids, on the other hand, show Arrhenius temperature dependence and nearly exponential kinetics. Most of the chemically important solvents fall in the fragile limit. Thus, the study of chemical relaxation in these liquids may provide a valuable tool to understand and further characterize the relaxation in the slow, viscous liquids. Note
that from a theoretical point of view, the study of slow liquids has remained a difficult and also challenging problem for several decades, and the progress itself has been slow. Of the various chemical relaxation processes studied in supercooled liquids, orientational relaxation, diffusion-limited chemical reactions, solvation dynamics, and ionic mobility are known to exhibit interesting dynamics.5 In this article, we present theoretical studies of orientational relaxation and solvation dynamics. Recent advances in the study of solvation dynamics in the normal regime6-15 make one optimistic that these techniques can also be used to obtain valuable information regarding the dynamics in supercooled liquids. We have chosen a specific system, namely, the amides, for which experimental results are available. Since the nature of relaxation in slow liquids is quite different from that in fast solvents, the theoretical formulation which is successful in the ultrafast limit11,15-24 may not be adequate in the opposite limit of slow liquids. Fortunately, we find that the molecular hydrodynamic theory developed in the recent years can be suitably extended to treat the slow regime as well. The extended theory properly includes the short-range correlations between the polar solute and the dipolar solvent molecules. Professor Friedman and co-workers have already made important contributions in this aspect.8,19-21 The extended theory is remarkably successful in explaining the solvation dynamics of coumarin in amides. This agreement is over many decades of time evolution and is particularly pleasing because it does not involve any adjustable parameter. Furthermore, our study of orientational relaxation reveals the following interesting fact. We find that even though the relaxation is highly non-Markovian and the decay of orientational relaxation markedly nonexponential, the average relaxation times for the first- and second-rank correlation functions still follow closely the Debye l(l + 1) law. The reason for this has been discussed. The organization of the rest of this article is as follows. The next section contains the theoretical formulation. Section III contains the numerical results and the comparison with experiments. Section IV concludes with a brief discussion of the results. II. Theoretical Formulation
* Also at the Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, India. † Dedicated to Prof. H. L. Friedman. X Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-1238$12.00/0
Let us first review our earlier work in brief. This will help in understanding the extended theory which is presented later in this section. © 1996 American Chemical Society
Solvation Dynamics in Slow, Viscous Liquids
J. Phys. Chem., Vol. 100, No. 4, 1996 1239
A. Brief Review of the Earlier Work. The progress of solvation of a newly created ion in a dipolar liquid is usually described by a time correlation function known as solvationenergy time correlation function (STCF). This is defined as
S(t) )
Esol(t) - Esol(∞) Esol(0) - Esol(∞)
(1)
where Esol(t) is the time-dependent solvation energy due to the interaction of the nascent ion with the surrounding dipolar solvent molecules. In our previous work, the electrical part of the sovlation energy was assumed to be given by the following expression
Esol(t) ) -
1 ∫dr E0(r)‚P(r,t) 2
(2)
Here, E0(r) is the bare electric field of the charged solute at a position r and P(r,t) is the space- and time-dependent polarization of the solvent molecules due to the presence of the ionic field. This polarization can be obtained from a molecular hydrodynamic theory (MHT) by solving the coupled conservation equations of number and momenta densities. The above approach led to the following expression for Esol(t)6,11,18
Esol(t) ) -
[ ][
sin krc 2Q2 ∞ dk ∫ 0 πσ krc
2
1-
] [
]
1 1 L -1 L(k) z + Σ(k,z) (3)
where rc is the distance of closest approach between the solute ion and a dipolar solvent molecule and Q is the charge of the ion. L(k) is the wave vector (k) dependent dielectric function characterizing the space dependence of the static correlation between the solvent molecules. L -1 stands for the Laplace inversion, and z is the Laplace frequency conjugate to time. Σ(k,z) is the wave vector (k) and frequency (z) dependent generalized rate of solvent polarization, which was shown to be given by11,22
Σ(k,z) )
2kBTfL(k) I[z + ΓR(k,z)]
+
kBTk2fL(k) mσ2[z + ΓT(k,z)]
(4)
where I is the average moment of inertia of a solvent molecule of diameter σ and mass m. kBT is the Boltzmann constant times the absolute temperature. ΓR(k,z) and ΓT(k,z) are the rotational and the translational dissipative kernels, respectively. The relevant dynamical responses of the solvent are solely governed by these two kernels, and they play the most crucial role in describing accurately the dynamics of the medium. Note that eq 4 retains its form when the moment of inertia I is a tensorial quantity. Then the denominator of the first term is replaced by I[z1 + ΓR(k,z)], where 1 is the unit tensor and ΓR is the dissipative tensor. The static structural correlations of the pure solvent are expressed by
fL(k) ) 1 -
( )
F0 c(110;k) 4π
(5)
where F0 is the average number density of the neat liquid. c(llm;k) denotes the llmth component of the direct correlation function in the intermolecular frame with k parallel to the z axis. fL(k) is also related to the longitudinal part of the wavevector-dependent dielectric function by the following relation11
1-
1 3Y ) L(k) fL(k)
(6)
Here, 3Y is the polarity parameter defined as 3Y ) (4π/3)βµ2F0 for the solvent molecules with permanent dipole moment µ. β ) (kBT)-1. Equation 6 provides an accurate estimate of fL(k) in the small wave-vector (that is, k f 0) limit. It is evident from the above discussion that the most important parameters for the calculation of time-dependent solvation energy (eq 3) are the static orientational correlations and the dissipative kernels. Fortunately, recent advances in the equilibrium theory of dipolar liquids have made accurate estimation of the static correlations possible.19-21 On the other hand, the theoretical calculations of ΓR(k,z) and ΓT(k,z) are highly nontrivial. An approximate scheme for the calculation of these kernels for underdamped liquids has been developed and succesfully applied in many of our earlier studies.22-28 The details in this regard have been discussed in great length elsewhere22-24 and need not be repeated here. We would like to mention that the theory thus formulated has been remarkably successful in explaining the solvation dynamics in ultrafast liquids like acetonitrile23 and water24-26 and in Stockmayer liquid22 and also in Brownian dipolar lattice.27 In the above theory, the static correlation, needed to evaluate fL(k), is actually the equilibrium orientational correlation between two dipolar solvent molecules. The ion-dipole interaction is assumed to be given by the interaction between the bare electric field of the ion with the solvent dipoles (see eq 2). Thus, the structural distortion of the solvent due to the presence of the charge on the ion is neglected. This description can only be valid in the t f 0 limit, that is, immediately after the charge is switched on a neutral solute molecule. This is because in that limit only, the solvent distortion is negligibly small. As the solvent orientational and spatial structures change due to the presence of the ion, there develops an orientational pair correlation between the solute and the solvent dipoles. This will lead to a breakdown of the above description in the limit t f ∞. So a better description is needed to quantify this deviation, and this can only be done by taking into account the proper ion-dipole correlation. In fact, a proper theory should include the time evolution of the ion-dipole orientational correlation function, which, however, requires a nonlinear theory. In the long time, one can obviously again employ a linear theory which takes into account a proper ion-dipole pair correlation. This should certainly be a better approach than the earlier “bare electric-field” description, as we shall demonstrate later. Another drawback of the earlier theory was that of the self-motion of the ion was not taken into account consistently. It is known that the translational motion of the ion can significantly enhance its own rate of solvation.28 This effect is quite large for the lighter ions, and a self-consistent description has now been developed. However, in the present case, we find the effect of self-motion is negligible. For subsequent discussions, we shall refer to the earlier theory as the linear theory and denote the solvation time correlation function, given by eq 3, as SL(t). B. Extended Theory. It should be clear from the above discussion that we need a theory that properly includes the effects of ion-dipole pair correlation on the solvation energy of the ion. The situation is somewhat complicated because we are considering a strongly interacting many-body system. The present theoretical formulation is based on the well-known density functional theory (DFT).11 Let us consider a system where the solute ion is translationally mobile, while the surrounding dipolar molecules are free to rotate and translate. All these motions can contribute to the process of solvation of the ion. One can then use the DFT to obtain a general freeenergy functional (of density) from statistical mechanics. The
1240 J. Phys. Chem., Vol. 100, No. 4, 1996
Biswas and Bagchi
free-energy functional leads to the following expression for the time-dependent solvation energy
Esol(r,t) ) -kBTnion(r,t)∫dr′ dΩ′ cid(r,r′,Ω′) δF(r′,Ω′,t) (7) where nion(r,t) is the position-dependent number density of the solute ion and δF(r,Ω,t) is the fluctuation in the position (r), orientation (Ω), and time (t) dependent number density of the dipolar solvent. cid(r,r′,Ω′) is the ion-dipole direct correlation function (DCF).28,29 This gives the required structure of the solvent around the ion. Equation 7 leads to an expression of solvation energy that includes the effects of self-motion of the ion. Thus, this approach provides a description superior to the earlier treatment of selfmotion by Roy and Bagchi.30 However, we found that the effect of self-motion is negligible for slow liquids considered in this work, and we need not discuss the full formulation here. Thus, we shall consider the solvation of a fixed solute. As a result, the time-dependent solvation energy is represented by
Esol(t) ) -kBT∫dr′ dΩ′ cid(r′,Ω′) δF(r′,Ω′,t)
(8)
r′ being the position vector from the ion. Next we express eq 8 as an integral over the wave vector k. Spherical harmonic expansions of the cid and δF and subsequent integration over Ω leads to the following expression lm (k) alm(k,t) Esol(t) ) -kBT∫dk cid
(9)
where clm id is the lmth component of the ion-dipole direct correlation function when expanded in the intermolecular frame and alm(k,t) is the lmth component of the solvent polarization fluctuation.11 For solvation of an ion, where only the longitudinal (that is, the l ) 1, m ) 0) component is involved, the above equation takes the following form 10 (k) a10(k,t) Esol(t) ) -kBT∫dk cid
(10)
Now, we need the solvation energy-energy time correlation function (EETCF) which will form the basis of our solvation study in the amide systems. It will be denoted by CEE(t) and is given as follows 3
CEE(t) )
3(kBT) 2
4πµ
∫0∞dk k2|cid10(k)|2 〈a10(-k)a10(k,t)〉
(11)
where 〈a10(-k)a10(k,t)〉 is obtained by using the following expression
〈a10(-k)a10(k,t)〉 )
[
] [
]
1 N 1 1L -1 4π3Y L(k) z + Σ(k,z)
(12)
As before, Σ(k,z) stands for the wave vector and frequencydependent generalized rate and L -1 denotes the Laplace inversion with respect to the frequency z. The insertion of eq 12 into eq 11 gives the following form
CEE(t) )
3(kBT)3 2
4πµ
∫0∞dk k2|cid10(k)|2
[
1-
]
1 1 L -1 L(k) z + Σ(k,z) (13)
Consequently, the normalized solvation energy time correlation function (SETCF) takes up the following form
∫0∞dk k2|cid10(k)|2[1 - 1/L(k)]L -1[z + Σ(k,z)]-1 SE(t) ) ∫0∞dk k2|cid10(k)|2[1 - 1/L(k)]
(14)
This SE(t) provides the information which is equivalent to the time-dependent fluorescence stokes shift (TDFSS) of a probe molecule. This expression reduces correctly to the preViously used expression of solVation energy (eq 3) if the asymptotic form of cid is used. Note that eq 13 is a fully microscopic expression which does not involve any cavity or other continuum model concepts. Detailed numerical predictions of eq 14 and comparisons with relevant experiments will be presented in the next section. C. Orientational Relaxation. The orientational relaxation in dipolar liquids exhibits rich dynamics, much of which is still ill-understood.31,32 For example, we do not yet fully understand the reason for the difference between the single-particle and the collective dynamics in dense dipolar liquids.32-34 Another example is the orientational relaxation in supercooled liquids, which shows a very interesting dynamical phenomenon known as the R - β bifurcation. These problems have been addressed and discussed at length.11,35-37 Although some aspects have been clarified,35-37 much remains unclear. This is because a detailed theoretical calculation of the orientational correlation functions has not yet been possible due to the complexity of the systems involved. Another problem has been the lack of experimental techniques to measure the first-rank (that is, l ) 1) single-particle orientational correlation function, C1(t), directly in the time domain. In the absence of this information, the relaxation time of C1(t) has often been equated to the time constant of the dielectric relaxation. But these two may be quite different. This is because the collective memory function contains the contribution only from the k ) 0 mode, whereas the single-particle friction is more susceptible to the short-range orientational correlations. In principle, we need the full wavevector and frequency-dependent rotational dissipative kernel, ΓR(k,z), which, unfortunately, is not available yet. In order to understand orientational relaxation in the slow liquids, we have carried out the following calculation in order to gain insight into the single-particle orientational relaxation. The orientational correlation functions, Clm(t), are defined as11
Clm(t) ) 〈Y* lm[Ω(0)]Ylm[Ω(t)]〉
(15)
where Ylm(Ω) is the spherical harmonic of rank l and projection m. Y*lm(Ω) is the complex conjugate to Ylm(Ω), and 〈 〉 denotes the average over the initial orientations. For isotropic systems, the m dependence can be ignored, and we refer to the orientational correlation functions as Cl(t). On rather general grounds, this correlation function can be represented in terms of the memory function as follows
[
Cl(t) ) L -1 z +
l(l + 1)kBT
]
I[z + ΓR(z)]
-1
(16)
where the orientational memory function, ΓR(z) is related to ΓR(k,z), defined previously. In our work, this has been obtained from experiment by using an inversion procedure described below. III. Numerical Results and Discussion We first discuss calculations of the memory functions, ΓR(k,z) and ΓT(k,z). As mentioned earlier, reliable microscopic calculations of these quantities are prohibitively difficult. It was
Solvation Dynamics in Slow, Viscous Liquids
J. Phys. Chem., Vol. 100, No. 4, 1996 1241
TABLE 1: Solvent Parameters Needed for the Theoretical Calculation solvent
molar vol, A3
µ, D
T, K
0
∞
τD, ns
η, cP
NMF NMA NMP NMP NMP
60.2 73.0 93.0 93.0 93.0
3.82 3.71 3.59 3.59 3.59
234 302 273 253 244
304 182 216 270 300
10 10 6 6 6
1.50 0.39 4.02 11.1 18.6
6.7 4.0 10.8 20.7 28.7
observed in our previous work24-27 that certain meaningful approximations of these quantities can lead to remarkably good agreements with known experimental results. If the k dependence of the rotational memory function is neglected, then ΓR(k,z) can be replaced by ΓR(k)0,z) (≡ΓR(z)). The latter can now be obtained directly from frequency-dependent dielectric function (z) by using the molecular hydrodynamic theory.22 This leads to the following expression for ΓR(z)22,26
kBT I[z + ΓR(z)]
)
z 0[(z) - ∞] 2fL(0) [0 - (z)]
(17)
where 0 is the static dielectric constant of the medium and ∞ is the limiting dielectric constant at infrared frequency. fL(0) is the long wavelength limit of fL(k). The translational kernel ΓT(k,z) can be obtained directly from the dynamic structure factor of the liquid using the following expression35
k BT 2
)
S(k)[S(k) - zS(k,z)]
mσ ΓT(k,z)
k2S(k,z)
(18)
where the dynamic structure factor is assumed to be given as S(k,t) ) S(k) exp[(-DTk2)/S(k)], with DT as the translational diffusion coefficient of a solvent molecule calculated from the experimentally determined viscosity by using the StokesEinstein relation with the stick boundary condition. In order to calculate ΓR(z), we need the frequency-dependent dielectric function (z). For slow liquids, the latter has been assumed to be given by the Cole-Davidson formula30,31 as follows
(z) ) ∞ +
0 - ∞ [1 + zτD]β
(19)
where τD is the Debye relaxation time and β is the DavidsonCole fitting parameter. Fortunately, for amides, all these parameters are available experimentally.38-40 For the three amide systems studied, namely, N-methylacetamide (NMA), N-methylformamide (NMF), and N-methylpropionamide (NMP), β = 0.91.39 The radii of the different solvent molecules have been calculated from their respective van der Waal’s molar volumes. The other parameters needed for the calculations are given in the Table 1. In order to understand the dynamical response of the liquid, we have plotted in Figure 1 the collective rotational memory function ΓR(z) as a function of the frequency z for NMP. It shows a biphasic dependence which is typical of slow liquids where non-Debye relaxation assumes a greater significance. Note the large value in the z f 0 limit and the relatively smaller friction at higher frequencies. With the above dynamical scheme at hand, let us now look at the other characteristics of the three amide systems, namely, NMA, NMF, and NMP. The most interesting feature of these three amides is that they possess very high static dielectric constants as well as very large Debye relaxation times. Actually, these high values of static dielectric constants are indicative of
Figure 1. Frequency-dependent orientational memory function, ΓR(z), plotted as a function of the Laplace frequency, z, for NMP at 253 K. We have used eq 17 to calculate ΓR(z). The frequency-dependent dielectric function, (z), needed in eq 17, has been obtained using eq 19. The other parameters needed to characterize the solvent are summarized in Table 1. Note the biphasic character of ΓR(z) which is typical of slow liquids like NMP at lower temperatures.
an extensive intermolecular hydrogen-bonded network that provides a suitable geometry to align the molecular dipoles producing a large dipole moment. These large values of 0 and τD makes these solvents considerably slower in responding to the external perturbation. The static correlations in these liquids at the intermediate (that is, kσ = 2π) and large (that is, kσ > 2π) wave vectors also play an important role in slowing down the rate of solvation at long time. For ultrafast liquids like water and acetonitrile, these correlations had been taken from the XRISM calculation of Raineri et al.41 For these liquids, it was found that the solvation was dominated by the k = 0 modes. But for amide systems, the solvent structure at small wavelengths (that is, kσ g 2π) may also be important for the proper description of the slower dynamics inherent to these solvents. In the absence of any microscopic calculation, we have followed the following systematic scheme to find fL(k) and, hence, 1 1/L(k). At the intermediate wave vectors, we have taken the orientational correlation functions from the mean spherical approximation (MSA) model.28 In the k f 0 limit, we have used eq 6 to obtain the correct low-wave-vector behavior of these static correlations. In the k f ∞ limit, we have used a Gaussian function which begins at the second peak height of 1 - 1/L(k) to describe the behavior at large k. This is because at large k, fL(k) should be a Gaussian function of k.20 This Gaussian function has replaced the MSA beyond the second peak of 1 - 1/L(k) and thereby eliminates the wrong large k behavior of the MSA. This should be fairly accurate in estimating the electrostatic correlations of these solvents. Moreover, we also found from our earlier studies that a semiquantitatively accurate (with the correct limiting properties) L(k) seems capable of producing accurate S(t). This is again borne out by the calculated results (see, for example, Figures 3-5). We found that the results are insensitive to the details of the Gaussian function. In Figure 2, we have plotted the function 1 - 1/L(k) against k to show the static correlations used in our calculations. Here, the wave vector k is scaled by the solvent diameter σ. We next describe the calculation of the ion-dipole direct correlation function (DCF), cid(k). This is taken directly from the solution of Chan et al.29 of the mean spherical model of electrolyte solution. We have, of course, used the zero concentration limit. Note that the MSA for ion-dipole correlation is known to be fairly accurate.42 Both in the evaluation of cid(k) and in dynamical mean spherical approximation (DMSA), one requires the knowledge of the solute-solvent size
1242 J. Phys. Chem., Vol. 100, No. 4, 1996
Figure 2. Wave-vector dependence of 1 - 1/L(k) in NMP. Here, the longitudinal dielectric function, L(k), has been obtained from MSA with proper corrections at both k f 0 and k f ∞ limits. The wave vector k is scaled by σ, where σ is the diamater of the solvent molecule. The other parameters remain the same as in Figure 1.
Biswas and Bagchi
Figure 4. Comparison between the prediction of the present extended molecular theory (solid line) and the experimental results (filled circles). Here, the normalized energy-energy time correlation function, SE(t), is plotted against time (t) for the solvation of the excited coumarin in NMA at 302 K. The dashed line represents the predction of DMSA of the same liquid at the same temperature. For the theoretical calculation, the solute-solvent size ratio is taken as 1.6. The other parameters needed to characterize the solvent are given in the Table 1. Note that the ordinate is in the logarithmic scale.
Figure 3. Comparison between the prediction of the present extended molecular theory (solid line) and the experimental results (filled circles). Here, the normalized energy-energy time correlation function, SE(t), is plotted against time (t) for the solvation of the excited coumarin in NMF at 234 K. The prediction of the same by the DMSA model (dashed line) has also been compared. The theoretical calculations have been carried out with the solute-solvent size ratio 1.73. For the calculation of SE(t), we have made use of eq 14. The other parameters needed for computation are given in Table 1. Note that the ordinate is in the logarithmic scale.
Figure 5. Normalized energy-energy correlation function, SE(t), predicted by the present extended molecular theory for NMP at 253 K (solid line) compared with that of experiment (filled circles). The solute is the excited coumarin molecule. The prediction of the DMSA is also shown (dashed line). In this case, the solute-solvent size ratio is 1.5. For other parameters, see Table 1.
ratio. This is equal to 1.73, 1.6, and 1.5 for the solvation of coumarin in its excited state in NMF, NMA, and NMP, respectively. Figures 3, 4, and 5 contain the results of our calculations for solvation dynamics of excited coumarin in NMR, NMA, and NMP, respectively. In order to study the solvation dynamics in these systems, we have used eq 14 where the Laplace inversion has been done numerically by using the Stehfest algorithm.43 In all three figures, we show the experimental results40 and also the prediction of the dynamic mean spherical approximation (DMSA) model.44,45 From the three figures, it is clear that the agreement between the present extended molecular theory and the experimental results is quite good. The agreement for NMF is particularly good, for reasons not very clear yet. It is obvious from these three figures that our theory predicts somewhat faster solvation at short times than what has been observed in experiments. This may arise from the fact that the experiments missed an initial part in the relaxation of the total solvation energy.40 This early response may come from an ultrafast component of the solvent response, as revealed by the recent studies of Chang et al.46 We shall come back to this point later. It is also clear from these three figures that the DMSA fails completely to describe the solvation dynamics. The reason for
this is not very clear but appears to arise from two factors. First, the mean spherical approximation gives the wrong correlations both at the long and the short wavelengths. The error in the short wavelength is that it predicts large correlations to persist even when k f ∞.47,50,51 This will certainly slow down the decay. Second, the way the dynamics is introduced in DMSA is rather adhoc, as discussed in ref 22. In order to understand the effects of the ion-dipole correlation function, we have compared CEE(t) with the earlier linear theory, where the solvation energy, Esol(t), has been calculated by using eq 3. This is shown in Figure 6. The linear theory seems to break down in the longer times. This is expected, as this theory ignored the solvent structure around the ion which is incorporated in this extended molecular theory via cid. We have also studied the temperature dependence of the solvation dynamics in NMP. In this regard, the required values of the necessary parameters are available in Table 1.38-40 As the temperature is reduced, the rate of solvation is expected to be progressively slower as all the relaxation times of the medium becomes larger. In fact, there can even be an interesting competition between 0 and τDsthe former tends to make the solvation faster while the latter does the reverse. In Figure 7, we have compared the solvation dynamics at the three different
Solvation Dynamics in Slow, Viscous Liquids
Figure 6. Comparison between the predictions of the earlier linear theory (using eq 3) (dashed line) and the present extended molecular theory (solid line) for the solvation time correlation function against time, t, for the solvation of excited coumarin in NMP at 253 K. Note the deviation of the linear theory from the present extended theory in the long time. The solute-solvent size ratio is 1.5. Other parameters used in the theoretical calculations are given in Table 1.
J. Phys. Chem., Vol. 100, No. 4, 1996 1243
Figure 8. Decay of the solvation time correlation functions, SE(t), calculated by using the fits to the Davidson-Cole (dashed line) and the biexponential (solid line) forms of dielectric function, (ω), compared with the experimental results (filled circles). The liquid is dimethylformamide (DMF) at 298 K. The solute-to-solvent size ratio is 1.70. For the Davidson-Cole form, β ) 0.91 and τD ) 10.8 ps. The other parameters needed for this calculation are given in ref 38.
Figure 7. Temperature dependence of the rate of solvation energy relaxation in NMP. Here, logarithmic values of the normalized energyenergy correlation function, SE(t), at temperatures T ) 244 K (dashed dot line), T ) 253 K (dashed line), and T ) 273 K (solid line) have been plotted as a function of time t. Note that the dynamics is considerably slower at 244 K. The parameters used to characterize the solvent are given in Table 1.
Figure 9. Relaxation of the single-particle orientational correlation function, C1(t) (dashed line) in NMP at T ) 253 K plotted against time t and compared with the relaxation of the solvation energy-energy correlation function, SE(t) (solid line), in the same solvent. C1(t) has been obtained using eq 16, while SE(t) was calculated from eq 14. For other solvent parameters, see Table 1.
temperatures, 244, 253, and 273 K. The relaxation slows down considerably at T ) 244 K. In view of the good agreement obtained between the theory and experiment, one wonders about the sensitivity of the theoretical calculations on the experimental parameters used. In particular, one must have an estimate of the effects of small deviations in the dielectric function on the calculated solvation time correlation function (STCF). It is important to note here that there is considerable uncertainties in the values of Davidson-Cole parameters employed here. Also note that the Davidson-Cole form of the dielectric function tends to overemphasize the slower decay in the long time. All these points suggest that we should use an alternative form of (z) such as a biexponential (known as the Budo formula48) to check the robustness of the present theoretical scheme. Unfortunately, however, the necessary parameters to be used in such a form are not available for any of the amide systems at the temperatures studied here. We could find only one amide, namley, the dimethylformamide (DMF) for which both the fits of the Davidson-Cole and the biexponential forms are available.39 This is, however, at room temperature (298 K) only. We have, therefore, carried out a calculation for this amide system using both of these fits in order to study the robustness of the present scheme. In Figure 8, we have compared the predicted STCFs thus obtained with the available experimental result.46 It is clear from this figure that though the Davidson-Cole form is
successful in predicting the decay of STCF at shorter times, the biexponential form of (z) provides a better description in the long time. Thus, we suggest that in dielectric relaxation experiments, fits to the Budo formula should also be performed. This will perhaps allow a better description of the solvation dynamics in these otherwise slow liquids. The above analyses seem to suggest that the present theoretical scheme can be regarded at least semiquantitatively successful for a long range of time. It also suggests that a more reliable form of dielectric relaxation function for these liquids is needed to describe the dynamics more accurately. In Figure 9, the first rank (that is, l ) 1) single-particle orientational relaxation rate, C1(t), has been compared with the normalized SETCF, SE(t). In the short time, SE(t) decays faster than C1(t), while the rates of the decays are comparable in the long time. This is because in the short times, SE(t) derives contribution from the long-wavelength (that is, k = 0) modes which relaxes much faster than the single-particle orientation. In the long time, on the other hand, the intermediate wave vectors (that is, the molecular length scale processes) control the dynamics, and consequently, the rate of the decay becomes comparable to C1(t). In order to understand the orientational relaxation in the amide system, we have also calculated C1(t) and C2(t) by using the memory function ΓR(z) in eq 16. The results of these calculation are shown in Figure 10. Note that both of the functions are
1244 J. Phys. Chem., Vol. 100, No. 4, 1996
Biswas and Bagchi
Figure 10. Comparison between the rates of relaxation of the orientational correlation functions, Cl(t) (where l denotes the rank of the correlation function), for l ) 1 (solid line) and l ) 2 (dashed line). The solvent is NMP at 273 K. We have used eq 16 to calculate both correlations. Note the strong nonexponentiality of C2(t). The other parameters needed for calculations are given in Table 1.
TABLE 2: Comparison between the Calculation Average Relaxation Times of the Orientational Correlation Functions, Cl(t) solvent
T, K
τD, ps
〈τ1〉, ps
〈τ2〉, ps
〈τ1〉/〈τ2〉
NMF NMA NMP
234 302 273
1500 390 4020
756.24 215.98 1927.6
251.903 71.95 635.241
3.00 3.00 3.03
nonexponential. The nonexponentiality in C2(t) is more pronounced than that in C1(t). This nonexponentiality indicates the marked non-Markovian effects and can be understood from Figure 1, where we have shown the frequency dependence of the memory function, ΓR(z). Since, the decay of C1(t) is slower, it probes mostly the low-frequency value. The C2(t), on the other hand, may probe a substantial range of the frequency (z), along which the ΓR(z) changes considerably. Thus, the C1(t) is expected to be strongly nonexponential. What is most surprising, however, is that the average relaxation time, 〈τl〉 (where, 〈τl〉 ) ∫∞0 dt Cl(t)) closely follows the l(l + 1) law prescribed by the rotational diffusion model. This is shown in Table 2. This, we think, is rather interesting and should be studied in detail. Now we shall turn our attention to the missing component of the inertial response of the solvation dynamics. It was mentioned earlier that the initial experimental study might have missed up to 40% of the total response due to the limited temporal resolution used in those experiments.40 Subsequent experiments by Chang and Castner46 revealed that both in formamide (FA) and dimethylformamide (DMF), there is a dominant ultrafast component which may carry even up to 6070% of the total strength. Note that while formamide is a strongly hydrogen-bonded system, no such bonding should be present in dimethylformamide. However, even this system shows ultrafast solvation. In previous theoretical studies for water, acetonitrile, and methanol, the origin of the ultrafast component and the biphasic decay is now fairly wellunderstood.17,22-28 In the amide systems, the reason is somewhat less clear. We have shown in this work that the extended molecular hydrodynamic theory (EMHT) can explain the long time decay satisfactorily. Therefore, the question that immediately needs to be answered is the importance of the ultrafast component in these systems. We now turn our attention to this problem. In order for the ultrafast component to make a significant contribution, the following criteria should be satisfied. First, and the most important, is that the difference between ∞ and n2 (n is the refractive index of the medium) must be large. For
Figure 11. Normalized energy-energy correlation function, SE(t), of coumarin in NMF at T ) 234 K plotted as a function of time t. Note the presence of the ultrafast component which was missed by Chapman et al. in their experimental studies. An important contribution to the ultrafast polar response of the solvent comes from the high-frequency librational modes of the solvent, assumed to be centered at 110 cm-1. For further details, see the text.
the amide systems, ∞ = 10 and n2 = 2.1 so that the difference between them is quite significant. Second, most of the fast relaxation responsible for the decrease of the dielectric constant from ∞ to n2 should come from the high-frequency librational modes of the system. The experiments carried out by Chang et al.46 reveal that these amide systems may contain more than one librational mode with frequencies between 50 and 100 cm-1. We have, therefore, carried out the following theoretical investigation which is motivated purely by the desire to explore the possibility of ultrafast component in otherwise slow amides. We have taken only one librational mode49 at 110 cm-1, and we have assumed that this is responsible for the decrease of the dielectric constant from ∞ to n2. The rest of the parameter values remain the same. In addition, we have assumed that this librational mode is overdamped. The result is shown in Figure 11, which suggests that the ultrafast component is indeed responsible for about 70% of the solvation dynamics. This estimate seems to be in surprisingly good agreement with that of Chang and Castner (see Figure 9 of ref 46), though their approach was completely different from ours. We should also caution the reader that our estimate is rather crude because of the approximations involved. IV. Conclusion Let us first summarize the main results of this paper. We have considered solvation dynamics and orientational relaxation in the slow amide liquids at low temperatures. In order to describe the local, short-range correlations that are probed by the polar solute in a slow liquid, we have extended the molecular hydrodynamic theory to include the specific solute-solvent spatial and orientational correlations. The extended theory has been remarkably successful in describing the solvation dynamics over many decades of temporal evolution. We find that the memory function for these slow liquids has an interesting biphasic structure which reflects the marked nonexponentiality of the orientational correlation functions. Another interesting finding is that although the Debye model of rotational diffusion is totally inadequate to describe the decay of the orientational correlation functions, the average relaxation time still approximately obeys the l(l + 1) Debye relation for rotational diffusion. We have also explored the possible magnitude of the ultrafast component in the solvation dynamics in these liquids. A simple calculation indicates that the ultrafast component may contribute up to 70% of the total energy, and this decay may be over within 500 fs. This estimate seems to be in agreement with the estimate of Chang and Castner46 for
Solvation Dynamics in Slow, Viscous Liquids DMF. Although the time scale and the magnitude of the ultrafast component is comparable with that in acetonitrile, the slow, long-time component is entirely different. For amides, the long-time component is slower by many orders of magnitudes than the fast component. This makes the solvation here truely biphasic. In the present work, we have used the ion-dipole direct correlation function (DCF) to describe the solvent structure around the dipolar solute. This approximation is superior to our earlier studies11,22-26 where an electric field with a spherical cut-off (to include the size of the ion) was employed. However, a realistic molecule often have extended charge distribution.15 In such a case, the solute-solvent interaction is more complex. The formalism employed here can be recast in terms of solutesolvent direct correlation functions. However, the calculation of such correlation functions may be very difficult. Some progress toward this goal has been made by Raineri, Friedman, and co-workers.19-21 However, the satisfactory agreement between the theoretical investigations and the existing experimental results obtained here and elsewhere50,51 seems to suggest the existence of a significant ionic character of the probe molecule in its excited state. But the story may be quite complicated in actual practice, and certainly, a thorough and proper investigation is needed to understand this aspect in detail. Another limitation of the present work is that the specific solute-solvent interaction effects like hydrogen bonding (Hbonding) have not been included. This implies that the longtime part of the solvation dynamics may depend on the nature of the specific solute-solvent interaction. Again, the satisfactory agreement obtained in all four liquid systems studied here indicates that even if these specific effects are important, their dynamics may occur at the same time scale as generated by the present theory. The present investigation seems to indicate that the study of solvation dynamics can be a powerful tool to explore the relaxation spectrum of fragile molecular liquids. More importantly, the solvation dynamics can directly reveal the collective orientational motion of a small number of molecules. This information is not easily available by other techniques. One possibility is to use both quenched52 and instantaneous53,54 normal mode analyses, which have been found useful to reveal many interesting dynamical properties of complex liquids. Acknowledgment. It is a great pleasure to dedicate this work to Prof. H. L. Friedman, who has taught us much about electrontransfer reactions and ion-solvent dynamics. We join his innumerable colleagues in wishing him a long, happy, and productive life. We thank the anonymous referee for his constructive criticisms and suggestions. We thank Ms. Srabani Roy and Mr. S. Ravichandran for help and discussions and Dr. E. W. Castner, Jr., for sending us his papers, especially ref 46. This work was supported in part by a grant from the Council of Scientific and Industrial Research, India. We also thank the Supercomputer Education and Research Center of the Indian Institute of Science, Bangalore, for computational facilities. References and Notes (1) Angell, C. A. Chem. ReV. 1990, 90, 523. Mohanty, U. AdV. Chem. Phys. 1995, 89, 89. (2) Gotze, W. In Liquids freezing in glass transitions; Hansen, J. P., et al. Eds.; Elsevier: Amsterdam, 1991. (3) Madan, B.; Keyes, T.; Seeley, G. J. Chem. Phys. 1990, 92, 7565; 1991, 95, 3847. (4) See the papers in: Dynamics of disordered materials; Richter, D., et al., Eds.; Springer-Verlag: Berlin, Heidelberg, 1989. (5) Angell, C. A.; et al. AIP Proc. 1992, No. 256, 3. (6) Bagchi, B. Annu. ReV. Phys. Chem. 1989, 40, 115.
J. Phys. Chem., Vol. 100, No. 4, 1996 1245 (7) Fleming, G. R.; Wolynes, P. G. Phys. Today 1990, 43, 36. (8) Raineri, F. O.; Perng, B. C.; Friedman, H. L. Chem. Phys. 1994, 183, 187. (9) Barbara, P. F.; Jarzeba, W. AdV. Photochem. 1990, 15, 1. (10) Maroncelli, M.; McInnis, J.; Fleming, G. R. Science 1990, 243, 1674. (11) Bagchi, B.; Chandra, A. AdV. Chem. Phys. 1991, 80, 1 and references therein. (12) Maroncelli, M. J. Mol. Liq. 1993, 57, 1. (13) Gauduel, Y. J. Mol. Liq. 1995, 63, 1. (14) Rossky, P. J.; Simon, J. D. Nature 1994, 370, 263. (15) See the papers in: Ultrafast Reaction Dynamics and SolVent effects; Gauduel Y., Rossky, P. J., Eds.; AIP: New York, 1994. (16) See the articles in Ultrafast Dynamics of Chemical Systems; Kluwer Academic: 1994; Vol. 1. (17) Ladanyi, B. M.; Stratt, R. M. J. Phys. Chem. 1995, 99, 2502. (18) Chandra, A.; Bagchi, B. J. Chem. Phys. 1989, 91, 1829; 1989, 93, 6996. (19) Resat, H.; Raineri, F. O.; Friedman, H. L. J. Chem. Phys. 1993, 98, 7277; 1992, 97, 2618. (20) Raineri, F. O.; Resat, H.; Perng, B. C.; Hirata, F.; Friedman, H. L. Chem. Phys. 1994, 100, 1477. Raineri, F. O.; Zhou, Y.; Friedman, H. L. Chem. Phys. 1991, 152, 201. (21) Raineri, F. O.; Resat, H.; Friedman, H. L. J. Chem. Phys. 1993, 98, 3068. (22) Roy, S.; Bagchi, B. J. Chem. Phys. 1993, 98, 1310. (23) Roy, S.; Komath, S.; Bagchi, B. J. Chem. Phys. 1993, 98, 1. (24) Roy, S.; Bagchi, B. J. Chem. Phys. 1993, 99, 9938. (25) Nandi, N.; Roy, S.; Bagchi, B. J. Chem. Phys. 1995, 102, 1390. (26) Roy, S.; Bagchi, B. In Ultrafast Chemical Reactions; Gauduel, Y., Rossky, P. J., Eds.; AIP: New York, 1994; p 296. (27) Komath, S. S.; Bagchi, B. J. Chem. Phys. 1993, 98, 8987. (28) Grey, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Oxford University: Oxford, 1984; Vol. I. (29) Chan, D. Y. C.; Mitchell, D. J.; Ninham, B. W. J. Chem. Phys. 1979, 70, 2946. Blum, L. J. Chem. Phys. 1974, 61, 2129. (30) Roy, S.; Bagchi, B. Chem. Phys. 1994, 183, 207. (31) Debye, P. Polar Molecules; Dover: New York, 1929. (32) Frohlich, H. Theory of Dielectrics; Clarendon: Oxford, 1958. (33) Berne, B. J.; Pecora, R. Dynamic Light Scattering; John Wiley and Sons: New York, 1976. (34) Madden, P.; Kivelson, D. AdV. Chem. Phys. 1984, 56, 467. (35) Ravichandran, S.; Roy, S.; Bagchi, B. J. Phys. Chem. 1995, 99, 2489. (36) Zohari, G. P.; Goldstein, M. J. J. Chem. Phys. 1970, 53, 2372. (37) Brawer, S. Relaxation in Viscous Liquids and Glasses; The American Ceramic Society, Inc.: Columbus, OH, 1985. (38) Barthel, J.; Buchner, R. Pure Appl. Chem. 1991, 63, 1473. (39) Barthel, J.; Bachhuber, K.; Buchner, R.; Gill, J. B.; Kleebauer, M. Chem. Phys. Lett. 1990, 167, 62. (40) Chapman, C. F.; Fee, R. S.; Maroncelli, M. J. Phys. Chem. 1990, 94, 4929. (41) Raineri, F. O.; Resat, H.; Friedman, H. L. J. Chem. Phys. 1992, 97, 3058. (42) Wertheim, M. S. J. Chem. Phys. 1971, 55, 4291. (43) Stehfest, H. Commun. ACM 1970, 13, 624. (44) Wolynes, P. G. J. Chem. Phys. 1987, 86, 5133. (45) Rips, I.; Klafter, J.; Jortner, J. J. Chem. Phys. 1988, 88, 3246; 1989, 89, 4288. (46) Chang, Y. J.; Castner, E. W., Jr. J. Phys. Chem. 1994, 98, 9712. (47) Chandra, A.; Bagchi, B. J. Chem. Phys. 1989, 91, 3056. (48) Bottcher, C. J. F.; Bordewijk, P. Theory of electric polarization; Elsvier: Amsterdam, 1978; Vol. 2. (49) Chang, Y. J.; Castner, E. W., Jr. J. Chem. Phys. 1993, 99, 113. (50) Roy, S.; Bagchi, B. J. Chem. Phys. 1994, 101, 4150. (51) Roy, S.; Bagchi, B. J. Phys. Chem. 1994, 98, 9207. (52) Ohmine, I. J. Phys. Chem., in press. (53) Stratt, R. M.; Cho, M. J. Chem. Phys. 1994, 100, 6700. (54) Cho, M.; Flemming, G. R.; Saito, S.; Ohmine, I.; Stratt, R. M. J. Chem. Phys. 1994, 100, 6672.
JP951241+
