10948
J. Phys. Chem. 1995, 99, 10948-10951
Polarization Response of a Dielectric Continuum to a Motion of Charge Ranko Richert* and Hermann Wagner Max-Planck-Institut j 2 r Polymelforschung, Ackermannweg 10, 55128 Mainz, Gennany Received: February 6, 1995; In Final Form: March 7, 1995@
In cases where dielectric relaxation dominates the time scale of molecular processes which involve the motion of charge (e.g., solvation dynamics, electron transfer reactions, or chemical reactions), the relevant time scale is the longitudinal relaxation time r ~which , is generally faster than the dielectric relaxation time ZD. Numerical calculations of the polarizations PD(t) with dE(t)/dt = 0 and P E ( ~with ) dD(t)/dt = 0 for an electrical RC network equivalent to an arbitrary dielectric function E * ( @ ) are performed in order to generalize the relation between ZL and ZD which only for the Debye case reads ZL = ZDE&. The results for non-Debye systems as a function of relaxation time dispersion and relaxation strength are that (ZL) 0. The observed quantity is the temporal evolution of the current Z(t), which gives
using P,(t) = D(t) = A-' J Z(t) dt. Being interested in the functional forms only, we will focus on the normalized polarization functions P(r) where P(0) = 1 and P(-) = 0. Since one observes the decay Oft) of the dielectric displacement, the related time constants are denoted ZD and the Gaussian distribution of ln(tD) is characterized by its mean (ln(tD)) and width OD. In the constant charge (CC) case one would apply a certain displacement D = DO = QdA at time t = 0 to the capacitor while dD/dt = 0, i.e. Z = 0, for t > 0. Here, according to eq 2, P(t) is proportional to E f t ) so that one would monitor the gradual decay of the field Eft)to resolve &(t) which now relates to the quantities (ln(tE)) and DE. From simple arguments it follows that PE(t) f PD(t); Le., the different conditions result in distinct polarization relaxations. For a Debye-type dielectric, Frohlich'O has shown that both P&) and PD(t) are exponential functions of time but with t d t ~ = e&. In order to calculate PE(~) for a given but arbitrary pD(t), a capacitor filled with a lossy dielectric is represented by the equivalent electric circuit using ideal capacitors and resistors. For a certain value of the geometric capacitance COa capacitor C, mimics the value of e, in terms of e, = CJCo. Parallel to
C, a number of i = 1, ..., N Debye-type circuits are included, each one being a combination of Ci and Ri in series. A Gaussian density of ln(z) is now approximated by N values of ti equidistantly spaced on a ln(t) scale within the range -30 I ln(t) I+30. For each t i the value of Ci is given by Ci exp[(ln t i - (ln ~ ) ) ~ / 2and 0 ~ that ] of Ri by Ri = tJCi. The Ci's are normalized such that CCi = AC = C0(es - e,) = CoAe. For large enough N the circuit represents to a good approximation a continuous probability density for ln(tD), which in the limit N reads
-
--
For the CV case a relaxation time distribution according to eq 3 directly results in a polarization of the form'*
(4) For the electric circuit and a sufficiently large N eq 4 would emerge from calculating P a D = A-'S Z(t) dt for the condition of applying a step voltage UOfor t L 0 and assuring that the voltages of the Ci's are zero at t = 0, which is the trivial CV case. The quantities U(t) and Z(t) for all components of the network are obtained simply using Ohm's law UR = RZR and for capacitors CUc(t)= J Zc(t) dt. The net current is given by the sum over the N uncoupled RiCi paths, Z(t) = Di(t).For the CC case the procedure is similar, but with U(t)= UO- C,-'jC Zi(t) dt with integration limits 0 and t. Since now dU(t)/dt f 0, the decay behavior of one RiCi path is coupled to all N - 1 others. Using the circuit described above, we can now easily arrive at P E ( ~by) numical solution of the N 1 coupled differential equations. At time t = 0 a certain amount of charge QO is applied to the device, resulting in the initial voltage U(0) = UO = QdC,. Since the net current in this CC case vanishes the polarization leads to a decay of E(r) or U(t) toward the limiting AC. value U(-) = WOE& = UoCJCS, where Cs = C, Analytically, we have calculated that the function PE(t) must satisfy the integral eq 5 which can also be employed for a numerical solution.
+
+
The goal of the calculation is to find a representation of PE(?) in terms of its relaxation time distribution function G ( l n ZE) in the form
in order to compare G ( l n ZD) of eq 4 with G ( l n ZE) of eq 6: Figure 2 shows a set of typical relaxation profiles for D(t) and Eft), both being proportional to the polarization in the cases of CV and CC, respectively. The decays in Figure 2 are compared for a Debye dielectric (linear curves in the In P ( t ) versus t plot) with UD = 0, Le., G ( l n t o ) = d(ln t~ - (In ZD)), and for a non-Debye case with OD = 2, with both e, = 1, Ae = 4, and (In TD) = 0. A set of 24 relaxation functions for P&) and PE(t) have been calculated numerically for the parameters e, = l, Ae = 0.1, l , and 10, 0 5 UD 5 3.5, and (In t ~ =) ln(50), where the ranges of Ae and ffD are chosen to represent experimentally relevant systems. Note that e, = 1 is no restriction since the dielectric constants enter the calculation
Richert and Wagner
10950 J. Phys. Chem., Vol. 99,No. 27, 1995
ranges of Figure 3, A6 < 10 and UD < 3.5, the numeric results for (In t ~and ) OE can be approximated by the empirical relation
Discussion
0
0.5
1 0
TIME
Figure 2. Traces of In[P(r)] versus time, with P(t) normalized to P(0) = 1 and P(-) = 0. The calculationsrefer to a dielectric with a Gaussian density of In(tD) centered at t~ = 1 and having a Gaussian width UD. Dashed curves refer to the decay of the dielectric displacement D(t) = P&) for the case dE(f)/dr = 0, whereas solid curves refer to the decay of the electric field E(r) PE(t) for the case dD(r)/dt = 0. The linear and curved lines are calculated for the widths OD = 0 and OD = 2,
respectively. +2 A h
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+1
Ac= 1
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W
0 c
0,
0
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V
0
1
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3
4
-1
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1
2
The present analysis of the temporal evolution of polarization emphasizes the well-known fact that P ( f ) is not a specific property of a dielectric but is also dependent on the details of how the system is ~ e r t u r b e d . ~ In most cases the dielectric properties are measured in the frequency domain, where an harmonic oscillating voltage E(r) = EO exp(iwf) is applied to the capacitor for various angular frequencies @ . I 2 In this manner one obtains the complex dielectric function E * ( @ ) which relates to PD(t) via the appropriate integral tran~formation.'~The equivalent inverse transform of PE(f) into the frequency domain yields M*(w), the complex dielectric modulus, which by definition is linked to 6 via M*(u)c*(w)= 1. In the mechanical domain these relations are paralleled by the situations of strain relaxation y(t) at constant stress uoand stress relaxation a(f)at constant strain In perfect analogy the quantities shear modulus G and shear compliance J are related by G*(o)J*(w) = 1. A complete set of the formal relations of present interest have been compiled e1~ewhere.l~On the basis of the mathematical relations PD(f) PE(f) and M*(w) E*(w), it becomes clear that the different functions can in no way reflect different physical properties regarding the molecular processes which restore equilibrium conditions in the linear response regime, although different physical effects have been discussed occasionally. When a molecule is electronically excited, the charge distribution within the molecule is generally redistributed. In the dipole approximation this situation is accounted for by p~ f p ~where , p~ and ,UG are the excited (SIor TI) and ground (SO)state dipole moments, respectively. If this chromophore is surrounded by a more or less polar solvent, excitation will thus induce a dielectric relaxation of the solvent molecules (solvent dipoles). There are several aspects by which the dielectric response to excitation of a chromophore differs from the usual dielectric relaxation experiment employing a macroscopic capacitor: (i) the electric field generated by the chromophore is a dipole field, Le., not homogeneous, (ii) in a nonhomogeneous field translational motion may contribute to polarization, (iii) the static quantity in the molecular experiment is the charge distribution, not the field, (iv) structure breaking in the first solvation layer might complicate the situation, and (v) the molecular solvent structure will be more important in the local dipole field, where only several solvation layers are involved. Rigorous clarification of most of the details of these items involves the analysis of c*(k,w), where k is a wavevec-
3
OO Figure 3. Fit results for the mean relaxation time t~ in terms of (log(tds))from a decay analysis of the electric field as a function of the Gaussian width UD and parametric in A6 as indicated. The calculation is for 6- = 1 and (log(tD/s))= log(50) as marked by the dashed horizontal line. The inset plots the fit results for UE versus UD in the range 0-4 (linear) for both quantities.
only in terms of the ratio E ~ E , . Since by definition G ( l n ZD) is known, it remains to characterize GE(ln ZE), which was achieved by assuming a Gaussian &(ln ZE) and fitting the righthand side of eq 6 to the PE(f) data. Although GE(ln %) deviates from a true Gaussian profile, the fits regarding PE(f) are in most instances better than the expected experimental resolution. Strictly speaking, the Gaussian profile can be viewed as a second-order approximation with respect to the moments of GE(In ZE); Le., only the mean and variance of GE are analyzed. The results of such an analysis are indicated in Figure 3 in terms of the dependence of (In ZE) on A6 and UD, Le., on relaxation strength and relaxation time dispersion. The values of (In t ~ ) at UD = 0 comply with the well-known relation Z&D = do/^,. However, for polar non-Debye type dielectric materials ,Le., for values A6 >> 0 and OD >> 0, the calculations state that (In z~)/(lnZD) < E&. This implies that the mean relaxation time for PE(f) can be a factor of '10 faster compared to what is expected on the basis of the Debye approximation UD = 0. The inset of Figure 3 plots UE versus OD, indicative of the relation UE % OD within the ranges A€ 5 10 and UD 5 3.5. Regarding the polarization functions P,(t) and P E ( ~the ) results state that the decay patterns for the CV and CC cases are quite similar, whereas the average time scales differ by a factor in the range E&, to 0.03~&, for realistic dielectrics. Within the parameter
-
In the present investigation point iii is addressed only, which compares the conditions of a static field with that of a static charge distribution. The general results of the previous section thus mimic the reponse to a motion of charge in the case of a dielectric continuum without translational contributions. Qualitatively, the difference in the responses stems from the field at a certain distance from the charge assembly being diminished as the polarization proceeds. From the present numeric results it follows that the difference between the CV and CC cases has little impact on the polarization as a function of time. The commonly used relation Z&D = d e s , however, can severly underestimate the acceleration of P E ( ~relative ) to P d t ) for polar
J. Phys. Chem., Vol. 99, No. 27, 1995 10951
Response of Dielectric Continuum to Motion of Charge non-Debye systems. For practical purposes the calculations were restricted to a Gaussian density of In z in order to account for a non-Debye-type behavior, in which case the dielectric function € * ( O ) reads
of solvent structure become evident when comparing results from the dynamic MSA theory with the simple continuum model. Qualitatively, the screening of the electric field by polarizing the medium within a continuum picture must fail for at least the fust solvation layer, which will be subject to constant field like conditions. For this reason the MSA theory predicts a distribution of relaxation times ranging from ZD to In accord with experimental e v i d e n ~ e ,the ' ~ actual solvation time scale should thus be slower than predicted on the basis of continuum considerations. 2
It should be noted, however, that neither the numerical method nor the basic results are affected when going to other more asymmetric distribution functions. Several theories exist for relating the polarization relaxation to changes of a charge (ion solvation dynamics) or of a dipole moment (dipole solvation dynamics) to the dielectric properties c*(w) of the surrounding medium.I6 Within the mean spherical approximation (MSA) for ion solvation the relevant admittance X*MSA is given by the formula x*MSA(iw) = (2Ri)-'[1 - I/€*(iw)]/[l A*(iw)],where Ri is the ion radius and A*(iw) is a dynamic correction term which accounts for the microscopic structure of the ~ o l v e n t . ' ~ ~The ' * continuum picture is restored in the limit A*(iw) = 0, which yields
+
x*coNT(io) = l/€*(iO) = M * ( i o )
(9)
Translating the normalized admittance x*com(io) into the equivalent time domain function via the appropriate integral transformation yields the so-called Stokes shift correlation function C(f) defined as
~
.
~
Summary Straightforward numerical calculations for non-Debye dielectrics are employed in order to obtain numerical results on the decay of the electric field under constant charge conditions which allows comparing the normalized polarization relaxation for the constant field case ( P D ( ~as ) ) in dielectric spectroscopy with the constant charge case (P&) as in ion or dipole solvation dynamic experiments. An analytical time domain expression for relating PE(t) to an arbitrary P D ( f ) is given by eqs 4 and 5 . Within this continuum approach it is found that the decay profiles for the distinct polarization functions are similar, whereas their mean time scales differ by % J c s for Debye-like systems of weak polarity and by 50.03rJcs for polar systems subject to a significant relaxation time dispersion. By virtue of assuming a static charge distribution the normalized field decay PE(t) represents a continuum model prediction for the Stokes shift correlation function C(t) observed in solvation dynamics experiments in terms of the time-dependent shifts of mean emission energies.
References and Notes where v ( t ) denotes the time-dependent mean emission energy which is the relevant observable in solvation dynamics experiments. The equivalence of C(r) from eq 10 and PE(r) clarifies that the continuum model of relating solvation dynamics to dielectric relaxation precisely reflects the transition from the constant voltage to the constant charge case in a dielectric relaxation experiment. In the special case of a Debye-type continuum dielectric the simple relation T ~ T = D ZJZD = E& of eq 1 is sufficient to express the connection between dielectric and solvation response functions. The implicit identification of the time constant z~ with the longitudinal time ZL stems from the notion that the field of a static charge distribution is always longitudinal since it satisfies the condition curl E(r) = 0. In the more general case of a distribution of relaxation times, the solvation dynamic function is described by the dielectric modulus M*(w) as inferred above. Therefore, a continuum picture does not necessarily imply an exponential progress of solvation as expected in some instances. As stated above, the real situation encountered in solvation dynamics experiments is more complicated than regarding the constant charge conditions only. The main cause is believed to be the noncontinuum nature of the solvent or dielectric; i.e., microscopic aspects have to be included in model considerations. The quantitative effects
(1) Maroncelli, M. J. Mol. Liq. 1993, 57, 1. (2) Barbara, P. F.; Jarzeba, W. Adv. Photochem. 1990,15, 1. Barbara, P. F. Acc. Chem. Res. 1988, 21, 195. (3) Richert, R. In Disorder Effects on Relaxational Processes; Richert, R., Blumen, A,, Eds.; Springer: Berlin, 1994. (4) Marcus, Y. Ion Solvation; Wiley: Chichester, 1985. (5) Kosower, E. M.; Huppert, D. Annu. Rev. Phys. Chem. 1986, 37, 127; Chem. Phys. Lett. 1983, 96, 433. (6) Reichardt, C . Solvents and Solvent Effects in Organic Chemistry; VCH: Weinheim, 1988. (7) Streck, C . ; Richert, R. Ber. Bunsen-Ges. Phys. Chem. 1994, 98, 619. (8) Kivelson, D.; Friedman, H. J . Phys. Chem. 1989, 93, 7026. (9) Kubo, R.; Toda, M.; Hashitsume, N. Statistical Physics II; SolidState Sciences 31; Springer: Berlin, 1991. (10) Frohlich, H. Theory of Dielectrics; Clarendon: Oxford, 1958. (11) Loring, R. F.; Mukamel, S. J . Chem. Phys. 1987, 87, 1272. (12) McCmm, N. G.; Read, B. E.; Williams, G.Anelastic and Dielectric Effects in Polymeric Solids; Dover: New York, 1991. (13) Bbttcher. C. J. F.: Bordewiik. P. Theorv of Electric Polarization: Elsevier: Amsterdam, 1978; Vol. 2". (14) Gross. B. Kolloid Z . 1953. 131. 168: Kolloid 2. 1953, 134, 65. (15) Bagchi, B.; Chandra, A. Adv. Chem. Phys. 1991, 80, 1. (16) Zhou, H.-X.; Bagchi, B.; Papazyan, A.; Maroncelli, M. J . Chem. Phys. 1992, 97, 9311. (17) Rips, I.; Klafter, J.; Jortner, J. J . Chem. Phys. 1988, 88, 3246. (18) Rips, I.; Klafter, J.; Jortner, J. J. Chem. Phys. 1988, 89, 4288. (19) Richert, R.; Stickel, F.; Fee, R. S.; Maroncelli, M. Chem. Phys. Lett. 1994, 229, 302. .
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