J. Phys. Chem. 1995,99, 17265-17268
17265
Analysis of the Energy Landscape for Charge Transport in Polar Glassy Materials Ranko Richert”” and Roger F. Loring* Max-Planck-Institut f i r Polymelforschung, Ackermannweg 10, 55128 Mainz, Gennany and the Department of Chemistry, Baker Laboratory, Come11 University, Ithaca, New York 14853 Received: July 11, 1995; In Final Form: September 14, 1 9 9 9
We apply the methods of equilibrium fluid theory to calculate the distribution of site energies experienced by a diffusing charge carrier in a polar molecular glass. The mean spherical approximation is used to express the distribution of ion-dipole interaction energies of a hard-sphere ion at infinite dilution in a hard-sphere dipolar solvent in terms of the medium’s dielectric constant. The resulting distribution is compared to that extracted from experimental charge transport data with the assumption that in the nonergodic regime for T < Tgthe medium is characterized by its equilibrium structure at Tg.
1. Introduction Understanding charge transport in molecular glasses is of practical importance because of its relevance to xerographic processes and is of fundamental importance because of the role played by disorder in determining the efficiency of transport. Electron or hole transport in disordered materials displays regimes of dispersive (Le. nonequilibrium) t r a n ~ p o r t ,Gaussian ~,~ transport which violates the Einstein relation between mobility ,u and diffusion coefficient D even for moderate field^,^^^ and other anomalies in the dependence of the mobility on temperature and electric Recent measurements of charge transport in disordered molecular solids using time-of-flight (TOF) techniques have been successfully interpreted with a model that emphasizes the role played by static d i ~ o r d e r .In ~,~ this picture, localized noninteracting charge carriers execute hopping motion on a lattice of sites, with the site energies e, taken to be Gaussian random variables. The rate of charge carrier transfer from site i to j is assumed to be proportional to exp[-(e, - e,)/keq for e, > e, and to be independent of e, - e, for e, < e,. The rate is also taken to fall off exponentially with the intersite distance multiplied by an inverse length (wave function overlap parameter) that is also a Gaussian random variable.’ While the mobility in this model may be calculated with approximate analytical approaches, notably the effective medium approximation8 of Movaghar et al., determination of the full field and temperature dependence of the mobility requires Monte Carlo simulation of this m0de1.~ Such simulations have been demonstrated to capture even subtle details of numerous experimental findings, such as deviations from Gaussian transport found in TOF signals, mobilities which depend on the electric field in a nomonotonic fashion, and the departure from an apparent Arrhenius temperature dependence of ,u.5-7 Within this simulation approach, the variance of the distribution of site energies is treated as an adjustable parameter whose value is determined through comparison of simulation results to measured mobilities. Dieckmann et aL9 have investigated the assumption of a Gaussian distribution of site energies by using Monte Carlo simulations of a randomly filled lattice of noninteracting dipoles to determine the distribution of the electrostatic potential at a site. The simulated distributions of site energies were well-represented by Gaussian forms over a Max-Planck-Institut fur Polymerforschung.
* Cornell University. +
@
Abstract published in Advance ACS Abstracts, November 1, 1995.
0022-365419512099-17265$09.00/0
wide range of dipole concentration^.^ Recently, Young has investigated this lattice model analytically and has analyzed the non-Gaussian form of the wings of the distribution and the effect of these wings on the conductivity.I0 In the present work, we apply methods of equilibrium fluid theory to compute the distribution of electrostatic interaction energies felt by an instantaneouslycreated ion in a dipolar fluid. This quantity and its analog for an instantaneously created dipole have been the subject of numerous theoretical and simulational studies,’ because of their relevance to solution phase optical absorption and luminescence experiment^.'^-^^ In addition, several recent works have addressed the general problem of calculating the distribution of local fields in a fluid at equilibrium for interactions of arbitrary type and have considered the circumstances under which the distribution is approximately Gaussian in form.24,25 We show that within any linearized solvation theory this distribution is Gaussian, and we relate its variance to the static dielectric constant E of the medium within the mean spherical approximation (MSA).26 Application of this result for a fluid in thermodynamic equilibrium to a glass requires invoking the assumption that a glass at T < Tg is characterized by its structure at T = Tg.22327Below, we outline the calculation of the distribution of site energies and compare the results to experimental findings. 2. Method and Results
The distribution of electrostatic energies experienced by a newly created ion in a polar medium may be determined with the strategy applied by Loring to the calculation of inhomogeneously broadened optical line shapes in solution.I6 The potential energy of the solution containing an electrically neutral solute molecule at infinite dilution in a polar solvent in configuration R has the form:
+
V(R) = Vvv(R) V,,;”(R)
(1)
The potential energy of the neat solvent is denoted Vvv(R),and the contribution to the potential energy from interactions between the neutral solute and solvent molecules is represented by VuJo)(R).When the solute is ionized, the fluid’s potential energy-gainsan additional term, VUv(’)(R), which represents the electrostatic part of the solute-solvent interaction. We wish to determine P(w), the probability density of the dimensionless 0 1995 American Chemical Society
Richert and Loring
17266 J. Phys. Chem., Vol. 99, No. 47, I995 solute-solvent electrostatic interaction energy, given by
P(w)= (d(w
-B
+ p Vu,,(')(R)))
V,,P)(R), (2)
where p ( k ~ T ) - l .The angular brackets in eq 2 represent an average over configurations R with the distribution function associated with the potential V(R) in eq 1. By employing the configurational distribution function associated with the neutral solute, we assume that on time scales relevant to the hopping of a charge carrier the dipoles in the medium are frozen and do not adjust their orientations to solvate the newly created charge. Application of the integral representation of the Dirac delta function to eq 2 yields:
J-L
P(w)= (2n)-'
(34
dy
The Fourier transform of P(w), ab),is expressed in terms of the dimensionless free energy difference, 4A). 4A)is defined in eq 3c to be the free energy change, multiplied by -p, associated with transforming the fluid's potential energy function from V,, Vuvfo)to V,, Vuvfo) AV,,vfl), According to eqs 3, P(w) may be calculated by determining the solvation Helmholtz free energy associated with a solute whose interaction with the solvent is complex-valued.I6 In principle, the integral equation methods of equilibrium fluid theory may be analytically continued into the complex plane to determine this complexvalued free energy." This task of analytical continuation into the complex plane is trivial for the class of approaches known as linearized solvation theories,28such as the MSA,26in which solvation free energies are calculated to lowest order in perturbation theory in the solvation potential Vuv(l). The resulting free energy 4 A ) is quadratic in A:
+
+
term -fl VUv(l)would be added to the exponents in both integrands in eq 3c. For a linearized solvation theory, the result would be a Gaussian distribution function with the same variance as that in eq 5 but with a nonzero mean ( w ) = 241) = aw2. We specialize the distribution in eq 5 to a model in which the solute is represented by a hard sphere of radius ri, and the solvent molecules are represented by hard spheres of radius r, and with point dipoles of moment ps. For this case, the potential in eq 1 represents the hard sphere contribution to the solute-solvent interaction and Vu,(') represents the ion-dipole contribution. For this model, the MSA integral equation has been solved for an ion of charge q by Chan, Mitchell, and Ninham to yield26
+
a(A) = A2,(1)
where cgs units are employed in the right-hand side of eq 6a. In eq 6a, the solvation free energy 41) is expressed in terms of a dimensionless function a(€), which depends on the ratio of the solute and solvent molecular radii and on the solvent dielectric constant E. The dependence of a on thermodynamic state is carried by the state dependence of E. The explicit dependence of E on molecular dipole moment, particle density, and temperature may be determined within the MSA but is known to be only qualitatively accurate for this model.30 We will use the MSA to relate the distribution of interaction energies to E and will then regard 6 as a property to be determined in the laboratory. This approach has been applied successfully to optical line shapes in polar systems by Richert and Wagener.** For application to charge carrier transport, it is convenient to express the standard deviation of the distribution P(w) in eq 5 in practical units with the definition u k ~ T u ~ :
(4)
Substitution of eq 4 into eqs 3a and 3b yields a Gaussian form for P(w):
1 P(w) = [4na(l)i 112 exp[ -
&]
The interaction energy w has a distribution characterized by a mean value of zero and a variance ow2= 2 41). This analysis reveals that application of any linearized solvation theory to eqs 3 yields a Gaussian probability density for w. In principle, application of a nonperturbative approach such as the reference hypemetted chain (RHNC) equation will result in a non-Gaussian dist~ibution.~~ For the case of the solvation of a dipole in a dipolar solvent, Shemetulskis, Ladanyi, and Loring have shown that the RHNC distribution of solvation energies is well-represented by a Gaussian form, which in turn agrees well with simulation data for strongly interacting systems.I7 As noted following eq 2, the distribution in eq 5 is calculated assuming that the solvent does not have time to relax about the newly created charge. The steps leading to eq 5 may be repeated under the assumption that the polar solvent is fully equilibrated with respect to the ion, In this case, an additional
The dielectric constant E has been written as a function of T to emphasize the full temperature dependence of u.
3. Discussion It has been shown by analytical theory and by computer simulation that the mobility of a charge carrier moving among sites with a Gaussian distribution of site energies in the limit of a weak electric field has the form'
In this expression p, is independent of temperature, c is a number whose value is approximately */3, and u is the standard deviation of site energies. Mobility measurements on a variety of glassy materials have been shown to be consistent with this f o r m ~ l a . ~The - ~ form of this result emphasizes the importance of the magnitude of a2in determining the efficiency of charge transport in disordered solids. Before analyzing our predictions
Charge Transport in Polar Glassy Materials
J. Phys. Chem., Vol. 99, No. 47, I995 17267 4
1 .0 I \
I
\
h
2
0.5
8
‘ 0
0
20
1
60
40
a0
1
E
20
40
60
80
€
Figure 1. Dimensionless free energy of solvation, a(€), for an ion in a polar fluid plotted versus static dielectric constant e for various values of the ratio of solvent radius r, to ion radius r,. a(€) is defined in eq 6a within the MSA. For r, = 0 (upper curve) the continuum limit is recovered.
for u in eqs 7, we note some of the simplifications that led to this result. Systems that have been studied experimentally fall into two classes: neat amorphous films composed of the charge carrying species and polymer matrices doped with relatively high concentrations of polar charge canying molecule^.^^-^^ We have treated a nonpolar dopant at infinite dilution in a polar matrix and have thus neglected dipolar interactions among dopant molecules and dipolar interactions between dopant and solvent. The MSA may be solved for a two-component system of dipolar spheres in which both components, matrix and dopant, are present at significant concentration^.^^ In addition, the model may be refined to include the dipole moment of the ionized dopant. Here we analyze the simplest physically relevant model, with the knowledge that such refinements may be included. Because of the self-averaging effects of long-ranged ion-dipole interactions, we expect such additional contributions to introduce only minor changes. In this analysis we do not consider polaron effects, which have been treated in the context of charge carrier m o b i l i t i e ~ .The ~ ~ effect of a substantial polaron binding energy would be to introduce an extra energy barrier to charge transfer, in addition to the effects of the random site energies treated here. According to eqs 7, o2= a(€), whose dependence on solvent radius and dielectric constant is shown by the curves in Figure 1. Each curve is labeled by the ratio of solvent radius r, to ion radius q. As rs/ri 0 at fixed r,, the absolute value of the solvation free energy increases. In the limit rJri = 0, eq 6a reproduces the conventional expression for the Born energy within the dielectric continuum appr~ximation.~~ According to Figure 1, a(€)is relatively insensitive to small variations of rs/q about the value unity. Also, within realistic limits for the static dielectric constant, say 3 5 E 5 30, a(€)and hence u2 vary only by a factor of -1.7. This calculation allows us to examine the relation between the distribution of site energies experienced by a mobile charge carrier and the polarity of the medium as measured by an optical technique. A useful measure of polarity is the line shift induced by the medium in the optical absorption spectrum of a probe molecule whose dipole moment changes substantially upon e x c i t a t i ~ n . Within ~~ any linear solvation theory, this shift is proportional to a function W ( E that ) is analogous to a(€) defined in eqs 6 for ionic solvation. The dimensionless free energy change associated with introducing a dipole moment p d on a nonpolar solute in a polar solvent is defined to bel6
-
(9) in analogy to eq 6a, with rd denoting the solute radius. In order
Figure 2. The dimensionless free energy of solvation for an ion in a polar fluid, a(€) in eq 6a, is compared to its analog for solvation of a dipole, ~ ( 6 in) eq 9, within the MSA. In each case, solute and solvent radii are taken to be equal, r, = r,, and r, = rd. a and are plotted using the left-hand and right-hand scales, respectively. Their ratio, a/&, is shown by the dashed curve, which is plotted using the left-hand scale.
to illustrate the different dependences of a and @ on E , we show in Figure 2 the functions a(€), @ ( E ) , and a(c)/@(E). The lower solid curve in Figure 2 shows the function % ( E ) for rd = r,, calculated within the MSA from eq 9 of ref 16 with the renaming of quantities a@) a and a,(€) w(E). Comparison of @ ( E ) for rd = r, to a(€)for ri = r, in Figure 2 shows that the dipolar solvation free energy depends more sensitively on E over the physically relevant range of E values than does its analog for an ion. In addition, the dependence of on rJrd shown in Figure 1 of ref 16 indicates that is a more sensitive function of solvent radius than is a. These trends reflect the long range of ion-dipole interactions relative to dipole-dipole interactions. The comparison between ionic and dipolar solvation shown as a / by~ the dashed curve in Figure 2 suggests that care must be taken in applying optical polarity data in interpreting charge transport dynamics and efficiency. The absolute temperature enters explicitly in eq 7a for T L Tg but also indirectly through c(T), which decreases approximately with increasing temperature above Tg as E Below the glass transition at Tg, characterized by structural relaxation times on the order of 100 s, T and E ( T ) are replaced by Tg and E(T,)in eq 7b. The justification for T Tg in eq 7b is the structural freezing at T,, which prohibits equilibration for T < TgS2’ Consequently, even charge carriers with residence times on a particular site of up to 100s will experience a nonequilibrated environment for T < T,. At elevated temperatures, T >> T,, a situation is reached where the solvent is capable of equilibrating on the time scale that a charge resides on a particular site. Assuming average charge carrier waiting times on the order of 10 n ~ , this ~ * situation is usually attained only near the melting temperature Tm. Although we have shown that the Gaussian width u is identical for the cases of frozen and fully relaxed solvent, an additional energy barrier of height (W)relued = u2/kgTis to be expected whenever solvent relaxation is effective within the average hopping time scale, in perfect analogy to the solvent induced Stokes shift between absorption and emission ~ p e c t r a . ~ ~ , ~ ~ We next consider the comparison between the predictions of eqs 6 and 7 and charge transport data for polar molecular glasses. Measurements of mobilities p(T) and of the Gaussian width u obtained from TOF signals j ( t ) indicate correlations between these quantities and the polarity of the material, which is customarily expressed in terms of the dipole moment p d of the charge transporting molecules. For instance, Sugiuchi et aL3’ found log@) at a certain reference temperature and field to decrease approximately linearly with increasing pd of the active molecules, present at 50 wt % in a polycarbonate resin.
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17268 J. Phys. Chem., Vol. 99,No. 47, 1995
Together with the expression for the mobility in eq 8, this result implies that u2 increases linearly with increasing molecular dipole over the range studied. Dieckmann et aL9 Young et al.,32and Borsenberger et al.33have determined a values from TOF data for a variety of charge transport materials differing in dipole moment and concentration of the active molecules as well as in the polarity of the polymer matrix. In these systems, u is shown to increase with increasing dipole moment and concentration of the active molecules. According to our present results, which account for the interaction of an ion with an environment composed of interacting dipoles, the dipole moment of the dopant is not the only relevant measure of medium polarity. Our expression in eq 7 relates the Gaussian width u to the more macroscopic property E of the charge transport material, which is also easier to measure than is p d . The present lack of r(T,) data for the samples studied prevents us from making a complete quantitative comparision of the predictions of eq 7 to experimental results. On a qualitative level, eq 7 describes the increase of u with increasing medium polarity. The following approximate calculation demonstrates semiquantitative agreement between the predictions of eq 7 and measured results. Charge transport molecules with 0.9 D Ip d I4.3 D at various concentrations between 30% and 100% in polymer matrices yield values for u in the range 80- 140 meV.32 According to eq 7 with ri = r, = 10 A and Tg= 350 K for these systems, u is calculated to span the range 130 meV Ia I 170 meV for E in the range 3 5 E I30. Because the polarities of the studied transport materials vary significantly, from pd = 0.9 D at 30% in polystyrene to p d = 4.3 D at 100% (binderless), the corresponding results 80 meV Iu I140 meV confirm our prediction of E affecting u only within a factor of -1.3. Thus the predictions of eq 7 agree with measured values of a both in absolute magnitude and with regard to the weak dependence of u on E . We conclude that this simplified calculation has captured the essential features of the distribution of site energies experienced by a diffusing charge carrier in a polar molecular glass. The results emphasize the impact of disorder inherent in polar liquids or glassy materials on the diffusivity of a migrating charge, since thermal activation within the distribution of site energies govems the transport efficiency.
Acknowledgment. We thank Professor H. Bassler for stimulating discussions. R.F.L. thanks Professor G. Wegner for making possible his stay at the Max-Planck-Institut fiir Polymerforschung, Mainz. He acknowledges support from the Deutscher Akademischer Austauschdienst. The financial support from the Deutsche Forschungsgemeinschaft and Fonds der Chemischen Industrie is gratefully acknowledged. References and Notes (1) Muller-Horsche, E.; Haarer, D.; Scher, H. Phys. Rev. B 1987, 34, 1273.
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