J . Phys. Chem. 1989, 93, 6965-6969
6965
Excess Electron Migration in Liquid Water Jurgen Schnitkert and Peter J. Rossky* Department of Chemistry, University of Texas at Austin, Austin, Texas 78712 (Received: February 28, 1989)
The transport behavior of an excess electron in liquid water at room temperature has been studied via quantum molecular dynamics simulation. In correspondence with experiment, the excess electron diffuses substantially faster than is observed for comparable halide anions in a series of corresponding classical simulations. However, nonlocal transitions involving adiabatic quantum mechanical tunneling and nonadiabatic hopping are absent and thus need not be invoked to explain the unusually high electronic mobility. Rather, the diffusion rate can be attributed to the instantaneous, rather than inertial, response of the electronic solute to the solvent configuration, as is distinctly manifest in the short-time dynamics.
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I. Introduction The solution chemistry of excess electrons in simple and molecular liquids has recently received increasing theoretical attention due to the advent of new quantum simulation techniques such as path integral simulations14 and quantum molecular dynamicss9 (QMD). The application of these new techniques to this class of system is particularly appealing since, for many purposes, the quantum statistical description can be reasonably well limited to a single quantum particle in a classical, fluctuating, bath.'O For the case of polar fluids, these studies have given overwhelming support to the so-called cavity model of electron solvation.2-8 In this picture, the roughly spherical excess electronic wave function is solvated in an ion-like manner by the solvent molecules. It is now of obvious interest to explore the range and limitations of this analogy between excess electronic and ionic solvation. The purely structural aspects of this question have been examined el~ewhere.~,'In the present study, we focus on some of the dynamical ramifications. Our particular test case is the hydrated electron (e,) for which we calculate transport properties in thermodynamic equilibrium at room temperature. This is done within a framework assuming strictly adiabatic dynamics of the excess electronic ground state. Nonadiabatic transitions involving excited states, as well as very long range adiabatic hopping transitions (>- 10 %.)via quantum mechanical tunneling, are precluded from the outset." We then ask whether the experimentally observed accelerated diffusion of eaq- can be reproduced even with such restrictions. We mention that in another study' we have applied an approach that is similar in spirit to the problem of nonequilibrium electron solvation dynamics after injection of an excess electron into pure water. Either the successes or the shortcomings can indicate important aspects of the physics of the problem that had not been previously anticipated. The study reported here is a similar effort to explore the diffusion behavior of e,- by a well-defined computer experiment that combines a microscopic Hamiltonian with explicit thermal averaging and a quantum dynamical description. We emphasize at the outset that we are not the first to apply the basic methods described above to the problem of electronic diffusion in liquids. The excess electron diffusion in both a molten salt5s6and in liquid ammonia*J2has been considered. In both cases, values of the diffusion coefficient in rough accord with experiment were obtained. However, also in both cases, the role of tunneling and, more generally, the basic mechanism for enhanced transport were not fully addressed. In the present study, we go significantly beyond these earlier studies by carrying out comparable simulations of classical anions and explicitly comparing the time-dependent displacement distributions characterizing the quantum and classical solute cases. This allows a clear interpretation of the observations in terms of a diffusive mechanism. In section 11, we outline the scheme of our computer simulations. The results are presented and discussed in section 111, with emphasis on the similarities and differences between excess electronic 'Present address: AT&T Bell Laboratories, Murray Hill, NJ 07974.
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and simple ionic mobility in solution. In section IV, we draw the conclusions. 11. Methods
The main subject of our analysis is a 38-ps equilibrium QMD simulation of eaq-, using a reasonably detailed electron-water pseudopotential for the electronsolvent interaction and calculating the real time quantum evolution of this system. For comparison, we also perform several new purely classical molecular dynamics simulations of simple, point charge like, spherical anions where the nonelectrostatic interaction with the solvent is of the Lennard-Jones type. We first briefly describe the eaq- simulation (the same as that which underlies a study of optical hole burning reported elsewhereI3). The simulated system consists of one (quantum mechanical) excess electron and 500 classical SPC water m0lec~le.s~~ in a cubic simulation cell of side length 24.66 %. and under periodic boundary conditions. The interaction of the electron with the solvent molecules is described in a pseudopotential f r a m e ~ o r k ' ~ with repulsive terms associated with the molecular atoms. These repulsion terms reflect the electron-molecule wave function orthogonality constraints that would arise in a full quantum mechanical description. The electrostatic interaction between excess electron and water molecules is evaluated with the partial charges (1) Chandler, D.; Wolynes, P. G. J . Chem. Phys. 1981, 74, 4078. (2) (a) Parrinello, M.; Rahman, A. J. Chem. Phys. 1984, 80, 860. (b) Sprik, M.; Impey, R. W.; Klein, M. L. J. Chem. Phys. 1985,83, 5802. (c) Wallqvist, A.; Thirumalai, D.; Berne, B. J. J. Chem. Phys. 1987, 86, 6404. (3) Schnitker. J.: Rosskv. P. J. J . Chem. Phvs. 1987. 86. 3471. (4) Review article: Ber& B. J.; Thirumalai,b. Annu'. Rev. Phys. Chem. 1986, 37, 401. (5) Selloni, A.; Carnevali, P.; Car, R.; Parrinello, M. Phys. Rev. Lett. 1987, 59, 823. (6) Selloni, A.; Car, R.; Parrinello, M.; Carnevali, P. J. Phys. Chem. 1987, 91, 4947. (7) Rossky, P. J.; Schnitker, J. J . Phys. G e m . 1988, 92, 4277. (8) Sprik, M.; Klein, M. L. J . Chem. Phys. 1988,89, 1592. Marchi, M.; Sprik, M.; Klein, M. L. Discuss. Faraday Soc., in press. (9) Kosloff, R. J. Phys. Chem. 1988, 92, 2087. (10) Chandler, D. J . Phys. Chem. 1984.88, 3400. (1 1) Long-range tunneling is not possible because of the finite size of the grid that is used for the representation of the wave function in physical space.; see section 11. (12).In the study of an electron in ammonia in ref 8, an error in the conversion of time units from internal program units to those used in the published data led to a dilation of all times in that text by a factor of 2 r and a contraction of all frequencies by a factor of 1/2r. This led to inappropriate conclusions. First, the corrected electronic energy fluctuation power spectrum indeed shows components in the solvent librational region in accord with the present results (see section 111). Second. the corrected diffusion constant is about 50% of the experimental estimate. Therefore, conclusions regarding the possible role of tunneling in that process are less clear than is indicated in that text (Klein, M. L., private communication). (13) Motakabbir, K. A,; Schnitker, J.; Rossky, P. J. J. Chem. Phys., in press. (14) Berendsen, H. J. C.; Postma, J. P. M.; Van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces; Pullman, B.,Ed.; Reidel: Dordrecht, 1981. (15) Schnitker, J.; Rossky, P. J. J . Chem. Phys. 1987, 86, 3462.
0 1989 American Chemical Society
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The Journal of Physical Chemistry, Vol. 93, No. 19, 1989
of the SPC model. Molecular polarization is also taken in account with a semiempirical expression taken from the technology of electron-molecule scattering calculations. The excess electronic wave function is represented on a grid of 163points that spans a cube of side length 13.46 A. The wave function is represented in the dual spatial-momentum Fourier space for efficacy in the method of eigenstate determination employed The time evolution of this system is calculated treating the excess electron and solvent in the Born-Oppenheimer approximation and following the solvent dynamics on the electronic ground state potential energy surface. Since the time dependence of the excess electronic wave function arises solely from the time dependence of the classical solvent coordinates, the quantum dy- , namical propagation can be done relatively easily as f01lows.~ Given a solvent configuration, the electronic ground state is evaluated by solving the respective time-independent one-electron eigenvalue problem.17J8 A new solvent configuration is then obtained by solving the classical equations of motion, accounting for the forces exerted by the excess electronic distribution via the Hellmann-Feynman t h e ~ r e m . ~By . ~ use of the old eigenvector from the previous configuration as a trial wave function, the new electronic ground state for the new set of solvent coordinates is determined, and the process is repeated. Starting out with a room-temperature thermal velocity distribution for the solvent molecules, the solvent equations of motion are integrated via standard method^'^?^^ with a time step of 2 fs. We also note that we allow the wave function grid to follow the diffusive motion of the electron through the simulation box, by updating the numerical grid so that its center is always close to the electronic center of the mass. After sufficient aging, a trajectory of 19 000 configurations, corresponding to an elapsed time of 38 ps, was computed and every tenth configuration was stored for analysis. Such a simulation takes about 17 h on a Cray X-MP.The bulk of this time is spent for the evaluation of the electron-water pair energies and forces ( -2/3 of the total time) and for the fast Fourier transforms used in solving for the electronic wave function^'^*'^ (- 1/3 of the total time). The relative time needed for the simulation of the water alone is negligible. Finally, several classical molecular dynamics simulations of hydrated conventional anions were performed. As far as applicable, all simulation conditions were chosen to be exactly the same as in the QMD case. In particular, this includes the use of an 8-A cutoff for the evaluation of all pair interactions, to avoid any potential inbalance due to an inequivalent treatment of the long-ranged contributions to so-called dielectric f r i ~ t i o n . ~ l -The ~~ nonelectrostatic part of the ion-water interactions was of the Lennard-Jones 6-12 type, with parameters either chosen ad hoc or as given by Heinzinger for the series of halide ions.24 In the latter case, no attempt was made to account for differences in the charge distributions of the water models since the original Lennard-Jones parameters24 were evaluated without regard to the electrostatic interactions. The length of these classical simulations, after initial equilibration, was always at least 12 ps and in the case of Br- as long as 96 ps. 111. Results and Discussion
In this section, the equilibrium dynamics of eaq- is evaluated and the comparison with classical simulations of hydrated hal(16) Feit, M. D.; Fleck, Jr., J. A.; Steiger, A. J. Comput. Phys. 1982,47, 412. (17) Schnitker, J.; Motakabbir, K.; Rossky, P. J.; Friesner, R. A. Phys. Rev. Lett. 1988, 60, 456. (18) Motakabbir, K. A.; Rossky, P. J. Chem. Phys. 1989, 129, 253. (19) Verlet, L. Phys. Rev. 1967, 159, 98. (20) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C . J. Comput. Phys. 1977, 23, 327. (21) Zwanzig, R. J. Chem. Phys. 1963, 38, 1603. (22) Hubbard, J.; Onsager, L. J. Chem. Phys. 1977,67,4850. (23) Wolynes, P. G. Annu. Rev. Phys. Chem. 1980,31,345, and references
therein. (24) Palinkas, G.; Riede, 32A, 1137.
W.0.;Heinzinger, K. 2.Nuturforsch. 1977,
3 PS
7 PS
-
8 PS
9 PS
5A
Figure 1. Examples from the time evolution of solvated excess electronic density (the water molecules are not drawn). The linearly spaced countour lines (increment of 0.003 37 e/A3) refer to the square of the ground-state wave function, after integration along the slight line. The outermost countour corresponds to about 3% of the projected density a t the center.
2.4
2.2 r -
I
A 2.0
1.8
0
I
I
I
10
20
30
U
t / ps Figure 2. Fluctuation of electronic radius of gyration during course of equilibrium QMD simulation of e,P.
ide-like anions is made. After an initial qualitative visualization, we consider the equilibrium fluctuations of some geometrical and energetical properties briefly and then proceed to the net transport of the species through the medium. A. Microscopic View. The density distribution of the excess electron in liquid water is visualized in Figure 1 for a sequence of examples that are equally spaced over a total time span of 8 ps. The contour lines are computed from the electron density after projection into the sight plane; this renders an intuitively appropriate picture of the time-dependent wave function. In each frame, the outermost contour plotted corresponds to about 3% of the maximum. In this sequence of distributions,the frame has always been automatically recentered around the electronic center of mass at each time point. The total net motion of the center of mass during the time span shown is about 4 A. The figure shows that the excess electronic distribution is roughly spherical at any given time. As was already noted in earlier nondynamical computer studies of eW-,’-l7 the average shape of the electronic distribution is to a very good approximation Gaussian.25
The Journal of Physical Chemistry, Vol. 93, No. 19. 1989 6967
Excess Electron Migration in Liquid Water
3
L
1
- 8
I
E eV
-1
t
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Figure 3. Fluctuation of electronic energies in Q M D simulation.
As one might anticipate, the electron continuously deforms its shape as the system evolves in time, with the wave function adapting to the changing microstructure of the surrounding solvent. However, the picture demonstrates that these deformations, albeit clearly discernible, are not very pronounced and strongly distorted wave functions are unlikely to be encountered. We have indeed no evidence that our simulation contains any configurations that are not qualitatively represented by the pictures shown, Nevertheless, these seemingly minor fluctuations are of crucial importance for some aspects of the physics of eaq-, as has been emphasized earlier.7*15q*7 B. Fluctuations. The magnitude of the size fluctuations of eaq- is illustrated in Figure 2 by displaying the time dependence of the radius of gyration, calculated from the squared ground-state wave functions. The fluctuation amplitude is only about 0 . 1 A, in agreement with the visual impression given by Figure 1. There is a wide range of time scales which are typically manifest also in the energetic fluctuations, discussed next, as well as in simulated transient spectroscopy.i3 The fluctuation of the potential EPt, kinetic EEn,and total binding energy Ebind (potential + kinetic energy) of eaq-is shown in Figure 3. The fluctuation amplitude is very large compared to thermal energies (on the order of 2 0 k T and 1 5 k T for E , and ,!ni$!, respectively). That the local energy fluctuations in pure liquid water are very large has recently been stressed by Ohmine et alaz6 The quantum kinetic energy, generally reflecting the cavity size, varies less than the potential energy, the latter reflecting more sensitively the changing solvent orientation. The shortest time scale evident in Figure 3 is clearly related to the librational motion of the solvent (see Figure 1 in ref 7).12 At the other extreme, there is also a long-time fluctuation period of the order of 10 ps that has been recognized as characteristic for the collective dynamics of liquid water in quite different contexts2' An issue relevant for the assessment of the potential accuracy of ground-state adiabatic dynamics is the fluctuation of the ground-state excitation energies. Analysis of every tenth of all configurations in this respect reveals that the smallest ground-state excitation energy ever encountered is 1.5 eV (see also Fig. 2 in ref 13). Accordingly, the Boltzmann factors connected with the excited states are very small and the probability for spontaneous radiationless transitions out of the ground state is always van(25) For ideal Gaussian distributions, the two average radii defined in ref 17 should differ by about 9%, as they indeed do for the simulated eaq- in ref 17 and in the present study. (A concluding remark in ref 17 concerning deviations from Gaussian statistics is not correct.) The radii given in ref 2c would imply quite pronounced deviations from Gaussian shape. (26) Tanaka, H.; Ohmine, I. J . Chem. Phys. 1987,87,6128. Ohmine, I.; Tanaka, H.; Wolynes, P. G. J . Chem. Phys. 1988, 89, 5852. (27) See ref 26 and earlier references therein.
- 4 A Z
2 A2
0 0
2
1
3
t / ps
Figure 4. Mean-square displacement of centers of mass as a function of time for excess electron and water, from Q M D simulation, and corresponding results for bromide ion from classical molecular dynamics simulation.
ishingly small. Hence, the strictly adiabatic simulation scheme used here should be an excellent approximation. This view of the energy level spacing is reinforced by the concordance with results found from path integral simulation of the same s y ~ t e m . ~ In .'~ that case, the complete thermal average is evaluated, corresponding to the room-temperature quantum partition function, and no evidence for an even temporary narrowing of the energy gap between ground state and lowest excited states were encountered." Interestingly, simulations of solvated electrons in molten KC1 yield a somewhat different result with a measurable probability for the Occurrence of nearly degenerate ground and first excited It remains to be explored whether this indicates a basic difference in the behavior of molten salts as compared to polar molecular liquids or whether it is a purely statistical difference in the simulations. C. Diffusion. Since the electron is very well localized at all times, it is sufficient to analyze the dynamics semiclassically by using the time dependence of the expectation value of the position (the center-of-mass motion) for this purpose. The mean-square displacement of the electronic center of mass is displayed in Figure 4,together with the mean-square displacement for the diffusion of all water molecules in the simulation cell (solid lines). The diffusion constant of e,, following from this is about 3.3 X cmz/s. This is about as fast as the solvent self-diffusion constant for the water in the same simulation, which is about 3.5 X cmz/s. The experimental estimatez8for the diffusion coefficient of e,; (from the conductivity via the Nernst-Einstein equation) is 4.9 X cm2/s which is a factor of about 2.5 larger than the diffusion coefficients of the halide ions C1-, Br-, and I- and about a factor of 3.5 larger than the corresponding number for F.29 We have investigated if the same is also true in the computer experiment. As discussed in a previous path integral study,7 the size of our simulated eaq-solute is comparable to that of a halide ion somewhere between C1- and Br-. Accordingly, we also show in Figure 4 the mean-square displacement for a typical model for Br- (dashed line). For this classical ion, a diffusion coefficient of about 0.5 X cm2/s is obtained. The computed mobility of eaq- is thus bigger by a factor of -7, and the qualitative difference between the two species is evident. The experimental observation concerning the relatively high mobility of excess electrons in liquid water has given rise to ~~
~~~~~
(28) Hart, E. J.; Anbar, M. The Hydrated Electron; Wiley: New York, 1970. (29) Robinson, R. A,; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1959.
6968 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 At 1
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c-
0.0
0.5
1.0
1.5
2.0
2.5
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Figure 5. Relative probability for center-of-mass displacements for indicated particle during a time interval of 20 fs (upper frame) and 200 fs (lower frame).
considerable speculation concerning nontrivial transport mechanisms involving genuinely quantum mechanical transitions from solvent site to solvent site.z8-3*32 Even proton-like (Grotthuss) transport mechanisms have been In the present study, a comparable speedup of the e, diffusion is seen without invoking such mechanisms. Due to the almost vanishing mass of the solute itself, a Brownian motion per se of an excess electron in liquid water cannot be expected. The comparable behavior of the mean-square displacements of the electron and of the solvent in Figure 4 for times longer than about 0.2 ps indicates, though, that the motion of the e,, complex (with its hydration shell) is effectively Brownian on such time scales. Only for very short times is the displacement behavior of the excess electron distinct, as seen in the inset of Figure 4. On this time scale, the mean-square displacement of e,, is not quadratic, as would be expected for the ballistic motion of a particle with a given thermal velocity. This distinct behavior in the normal inertial time regime is illustrated more clearly in Figure 5 , where we show the probability distribution for displacements of the electronic center of mass during a time span of 20 fs (upper frame), in comparison to the corresponding distribution for the water, and for the Br- in the classical ion simulation. For the electron, the distribution is not only shifted to larger distances but is also asymmetric. For longer times (lower frame, Figure 5), there is no such distinction between the distributions, demonstrating that there are not even occasional “hops” of the electron over relatively large distances. One is led to conclude from this analysis that the enhanced diffusion rate for the electron arises from the essentially instantaneous response of the electronic distribution to the changes in solvent configuration. In particular, the electron is capable of following the high-frequency solvent polarization fluctuations associated primarily with the rapid solvent librational motions, predominantly occurring on a time scale of 200 fs and less. For a localized species that was not confined by the repulsive forces associated with the surrounding solvation shell, the po(30) McHale, J.; Simons, J. J . Chem. Phys. 1978, 68, 1695. (3 1) Schmidt, W. F. In Electron-Solvent and Anion-Solvent Interactions; Kevan, L., Webster, B. C., Eds.; Elsevier: Amsterdam, 1976. (32) Schindewolf, U.Angew. Chem., Inr. Ed. Engl. 1978, 17, 887. (33) Hameka, H. F.; Robinson, G.W.; Marsden, C. J. J . Phys. Chem. 1987, 91, 3 150.
Schnitker and Rossky larization fluctuations alone can be argued to lead to a diffusion rate comparable to that observed here.34 However, in the present case, the localized state is so confined. In fact, in the absence of solvent center-of-mass motion, one would expect an electronic diffusion constant that would be negligible compared to that seen here. What appears clear from the observations made from the current simulation is that the consequences of the coupling between solvent dynamics and that of the solute are rather different in the inertial and adiabatic cases. In the electronic case, the solvent responds to a substantially shifted electrostatic source on short time scales, while in the inertial case, the ionic source provides a restoring force that leads the solvent to tend to return more closely to its initial configuration. An “amoeba type” of motion35has occasionally been suggested before to explain the diffusion behavior of solvated electron^.^^,^^ However, this characterization alone, without the recognition that the solvent dynamics in this case is different from that for a normal anion, would not provide an explanation for the enhanced diffusion rate. The absolute diffusion constants computed in the present study deserve comment. They do not compare well with experiment, in general. This applies most notably to the diffusion constant of the classical Br- ion which experimentallyz9is 2.08 X lV5cm2/s, Le., a factor of about 4 larger than simulated here. It is remarkable that the diffusion constants of ions in aqueous solution almost always appear to be. severely underestimated, with the ST2 water mode137,38as well as with the TIP4P mode139940or the MCY modeL4’ (An exception-for unclear reasons-is the study of Impey et al.4z) In the present study, with SPC water, the same result was observed not only for Br- but also for the whole series of halide ions from F to I-, using the ion-water Lennard-Jones parameters of ref 24. In an effort to corroborate this pattern, we have also compared the classical diffusion of four different anions that were all assigned the mass of C1-, but whose Lennard-Jones parameters were systematically varied ( u = 3.0 or 4.5 A; c = 0.1 or 1.0 kJ/mol). The computed diffusion coefficients fall in the range of (0.35-0.7) X cmz/s, with nontrivial trends if a correlation with the variation of the potential parameters is attempted. Nevertheless, all of these values fall well below the experimental observations. In general, the ability to predict dynamical properties of aqueous systems quantitatively remains relatively elusive. The self-diffusion coefficients of most water models are well off the experimental value.43 The recently proposed flexible SPC model4 appears to improve on these matters,45but it remains a significant issue to systematically understand the model dependence of dynamical
proper tie^.^^ IV. Conclusions We have studied the diffusive behavior of an illustrative example of a solvated electron. The results show that the excess electron follows the collective solvent dynamics instantaneously, leading to quasi-Brownian motion on time scales longer than about 0.2 ps. The analysis indicates that the quantum mechanical transport mechanisms proposed in the past28,3+33need not be invoked for (34) Cohen, M. L. Can. J. Chem. 1977, 55, 1906. (35) Kestner, N. R. In Electrons in Fluids; Jortner, J., Kestner, N. R., Eds.; Springer: Berlin, 1973. (36) Thompson, J. C. Electrons in Liquid Ammonia; Clarendon: Oxford, 1976. (37) Szasz, G . I.; Ride, W. 0.; Heinzinger, K. Z . Naturforsch. 1979,34A, 1083. (38) Migliore, M.; Fornili, S. L.; Spohr, E.; Heinzinger, K. Z . Nuturforsch. 1982, 42A, 227. (39) Berkowitz, M.; Wan, W. J. Chem. Phys. 1987, 85, 376. (40) Reddy, M. R.; Berkowitz, M. J . Chem. Phys. 1988,88, 7104. (41) Wilson, M. A.; Pohorille, A,; Pratt, L. R. J . Chem. Phys. 1985, 83, 5382. (42) Impey, R. W.; Madden, P. A,; McDonald, I. R. J . Phys. Chem. 1983, 87, 5071. (43) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. P.; Impey, R. W.; Klein, M. L.J . Chem. Phys. 1983, 79, 926. (44) Toukan, K.; Rahman, A. Phys. Rev. B 1985, 31, 2643. (45) Anderson, J.; Ullo, J. J.; Yip, S. J . Chem. Phys. 1987, 87, 1726. (46) Teleman, 0.;Jonsson, B.; Engstrom, S. Mol. Phys. 1987, 60, 193.
6969
J . Phys. Chem. 1989, 93, 6969-6975 an adequate portrait of the diffusion process. Correspondingly, it seems clear the the use of a classical diffusive description of the electronic motion with a phenomenological diffusion constant is faithful down to quite small distances and time scales. A new challenge for analytical theory is thus presented. The prevalent theory of ionic dynamics in solution derives from the hydrodynamic approach due to Zwanzigzl that evaluates the additional dielectric friction term arising in a polar solvent. In its sophisticated extended formulation by Hubbard and cow o r k e r ~ ,this ~ ~ theory * ~ ~ is relatively successful compared to experimental data. Analytical treatments along these lines for the classical diffusion of a particle that responds instantaneously to polarization fluctuations, but lacks any classical inertial behavior, are clearly of great interest in light of the picture presented here for the electronic motion. It is an interesting question whether
the deformability of the electronic distribution from a spherical shape plays any important role. Computer simulation investigations of such alternative pseudoclassical models are accessible in any case, and it appears likely that our understanding of the transport properties of electrons in complicated dense polar fluids will make further strides in the near future. Acknowledgment. Partial support of this work by a grant from the Robert A. Welch Foundation is gratefully acknowledged, as is computational support from the University of Texas System Center for High Performance Computing. P.J.R. is the recipient of an NSF Presidential Young Investigator Award and a Camille and Henry Dreyfus Foundation Teacher-Scholar Award. Registry No. Water, 7732-18-5.
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Implicatlons of the Triezenberg-Zwanzig Surface Tension Formula for Models of Interface Structure John D. Weeks* and Wim van Saarloos AT& T Bell Laboratories, Murray Hill, New Jersey 07974 (Received: February 28, 1989; In Final Form: May 9, 1989)
While the Triezenberg-Zwanzig (TZ) formula and a modified version introduced by Wertheim both relate the surface tension of a liquid-vapor interface to interfacial correlation functions, they appear to support very different models of interface structure. We resolve this paradox by showing that although the Wertheim and the TZ formulas are equivalent in the thermodynamic limit, their underlying physics as well as their range of validity is different. The former is valid only on very long length scales (requiring system sizes much larger than the capillary length), while the latter continues to be relatively accurate down to scales of the order of the bulk correlation length. Thus the TZ formula is consistent both with the capillary wave model, which gives correlation functions affected by the long-wavelength interface fluctuations that occur in a large system, and with the classical van der Waals picture, which holds for small system sizes. We also discuss some more general issues involved in the use of the density functional formalism for a two-phase system.
I. Introduction The derivation of the Triezenberg-Zwanzig (TZ) formula for the surface tension’ represented a major conceptual advance in the modern theory of interfacial phenomena. This was one of the first, and certainly one of the most elegant and instructive, applications of density functional theory to a two-phase system, and today it remains the starting point for almost all theoretical work in interfaces.2 In this paper we reexamine some of the results and implications of the density functional methods pioneered by T Z and their connection to the qualitative pictures of interface structure suggested by the capillary wave modeP5 and the van der Waals That some subtle issues arise can be seen from an examination of the original TZ formula (generalized to d-dimensions5) for the surface tension u of a planar liquid-vapor interface whose normal vector is along the z axis: Po =
--
JdZl
Jdz2
Jdr2 ~122C~Zl,Z27~12) P’O(Z1) P’O(Z2)
2(d - 1) (1.1)
Triezenberg, D. G.; Zwanzig, R. Phys. Rev. Left. 1972, 28, 1183. (2) For a recent review, see: Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarify;Clarendon: Oxford, 1982. (3) Buff, F. P.; Lovett, R. A,; Stillinger, F. H. Phys. Rev. Letr. 1965, 15, (1)
621.
(4) Weeks, J. D. J . Chem. Phys. 1977, 67, 3106. (5) Bedeaux, D.; Weeks, J. D. J . Chem. Phys. 1985, 82, 972.
(6)Widom, B. In Phase Transitions and Critical Phenomena; Domb, C . ,
Green, M. S., Eds.; Academic: New York, 1972; Vol. 2.
(7) van der Waals, J. D. Z . Phys. Chem. 1894, 13, 657; for an English translation, see: Rowlinson, J. S . J . Stat. Phys. 1979, 20, 197.
0022-3654/89/2093-6969$01.50/0
Here @ is the inverse temperature (kBT)-l,C the (generalized) direct correlation function, r a (d - 1) dimensional vector in the interface plane, rI2= Irl - rzl, and dpo(z)/dz 3 ~ ’ ~ (the z ) derivative of the density profile po(z) (see section I11 for precise definitions). Implicit in (1.1) is the existence of a weak external field +o(z) that produces macroscopic phase separation. According to capillary wave theory, in dimensions d I3, ~ ’ ~ ( z ) has a sensitive dependence on the strength of that external (gravitational) field and vanishes as the field strength g tends to If this prediction is correct, then implicitly C must also have a strong field dependence so that the surface tension u has O+. a finite limit, becoming essentially independent of g as g This is physically required and is predicted by the exact Kirkwood-Buff formula8 for u. For d = 2, Bedeaux and Weeks5 explicitly showed that the capillary wave model satisfied (1.1) as an identity, with a C that indeed had a strong field dependence. Along with the strong field dependence, the capillary wave model also predicts a strong dependence on system size for C and ~ ’ ~ ( z ) in (1.1). One of the main goals of this paper will be to examine the approximate validity of (1.1) and (1.2)below for finite systems as well as in the thermodynamic limit. This predicted strong field and size dependence of the correlation functions in (1.1) is quite different from what is suggested by the classical van der Waals picture. Here one envisions an “intrinsic” profile of finite width and correlation functions resembling those of the bulk phases, which are essentially independent of system size or the strength of a weak external field.6 Perhaps this point can be seen even more clearly from the formally equivalent expression for u involving the (generalized)
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(8) Kirkwood, J.
G.; Buff, F. P. J . Chem. Phys. 1949, 17, 338.
0 1989 American Chemical Society
