3006
J . Phys. Chem. 1985,89, 3006-3010
Electron Localization in Low-Temperature Alcohol Glasses. A Theoretical Model Dorota Swiatla, Witold M. Bartczak,* and Jerzy Kroh Institute of Applied Radiation Chemistry, Technical University, Lbdi, WrBblewskiego 15, Poland (Received: November 2, 1984)
A statistical approach to electron localization in low-temperature glassy alcohols is presented. Apart from the capture by the ground state of the potential traps electron capture by the excited states of deep traps is also considered. The electron capture is treated as a radiationless transition between the conduction state and the trapped state of the electron assisted by the multiple emission of matrix phonons. The calculations of the e; spectra are performed for several alcohols and different temperatures. The effect of electron relaxation from the excited level of the deep trap on the optical spectra of the trapped electron is examined.
Introduction Pulse radiolysis at low temperatures enables one to obtain optical absorption spectra of trapped electrons in the very early stages of their existence. The electron localization in low-temperature alcohol glasses has recently been extensively studied.'" Briefly, the experimental picture is as follows. Electrons trapped in methanol at liquid helium temperature absorb in the visible region, e; in ethanol absorb in the infrared, and the higher alcohols show e; absorption increasing toward the infrared. The absorption spectra measured at higher temperatures as, e.g. 50 or 77 K, show a blue shift in comparison with the 6 K spectra. The higher the alcohol the larger the blue shift is observed. These experimental data are interpreted in terms of the small polaron model.' According to this model the infrared and visible spectra are ascribed to electrons in two different kinds of traps. The hydrogen-bonded O H chains and the alkane parts of the molecules play the role of two different trapping sites. However, certain phenomena as, e.g., the infrared tail of the e; spectrum in methan01,~~~ the significant effect of the freezing rate of glassy samples on the electron spectra,* or extremely low efficiency of electron trapping in polycrystalline methanol9 remain unclear in the framework of the small polaron model. Pulse radiolysis studies of a series of normal alcohols at temperatures down to 6 K suggest that there is a distribution of trap depths in each alcohol matrix and that the probability of trapping at a given site is a function of the t e m p e r a t ~ r e .Therefore ~ we propose a new model which includes the statistical distribution of electron Electron capture by the potential trap built up of a few matrix molecules is treated as a radiationless transition assisted by the multiple emission of matrix phonons. According to our model electron localization in deep potential traps may proceed via the excited level. After capture by the excited level the electron relaxes to the ground state. In this stage the energy excess is dissipated via generation of intramolecular modes of the trap-composing molecules. This paper presents the details of the calculations performed for alcohol glasses. The statistical distributions of the energy levels of empty and occupied traps as well (1) N. Klassen, H. A. Gillis, G. G. Teather, and L. Kevan, J. Chem. Phys., 62, 2474 (1975). (2) G. V. Buxton, J . Kroh, and G. A. Salmon, Chem. Phys. Lett., 68, 554 (1979). (3) L. M. Perkey and J. F. Smalley, J . Phys. Chem., 83, 2959 (1979). (4) G. V. Buxton and G. A. Salmon, Radial. Phys. Chem., 17,335 (1981). ( 5 ) G. V . Buxton, J. Kroh, and G. A. Salmon, J . Phys. Chem., 85, 2021 (1981). (6) N. V . Klassen and G. G. Teather, J . Phys. Chem., 87, 3894 (1983). (7) R. L. Bush and K. Funabashi, J . Chem. Soc., Faraday Trans. 2,73, 274 (1977). (8) M. Ogasawara, K. Shimizu, K. Yoshida, J. Kroh, and H. Yoshida, Chem. Phys. Lett., 64, 43 (1979). (9) H. Barzynski and D. Schulte-Frohlinde, Z . Naturforsch. A, 22, 2131 (1967). (10) W. M. Bartczak, D. Swiatla, and J. Kroh, Radiat. Phys. Chem., 21, 469 (1983). ( I I ) W. M. Bartczak, D. Swiatia, and J. Kroh, "Proceedings of 5th Tihany Symposium on Radiation Chemistry", AkadCmiai Kiadb, Budapest, 1982, p 13.
0022-3654/85/2089-3006$01.50/0
as the optical absorption spectra of the trapped electron are computed for methanol, ethanol, and propanol glass.
Model The dipolar glassy matrix is represented by a set of close-packed spheres with radius R (treated as a parameter). Each sphere contains a certain number N of molecular dipoles randomly distributed and randomly oriented inside the sphere. Generally, the number of dipoles differs from cell to cell but for simplicity the same value of N = N,, was assumed for each sphere. Consider the state of an electron in a given sphere. The influence of the surrounding cells is neglected. In our statistical approach the orientations Bi of the dipoles as well as their positions ri inside the sphere differ from cell to cell. Consequently, the energy of the electron varies with the cell. This energy is obtained by solving the following SchrGdinger equation
where the vectors Zi and are the dipole moment and the position of ith dipole. The potential term of this equation, V(r'),can be averaged over the sphere and expressed in the form independent of r'
Thus the solution of eq 1 can be easily obtained by solving the problem of the three-dimensional square-well potential.I2 The binding level appears when the potential depth exceeds the critical value Vco = 7r2fi2/8m,R2.The trap deeper than V,, = 7 r 2 f i 2 / 2m,R2, in addition to a ground state, has also an excited level. The spherical averaging of the potential V ( q allows us to replace the polar medium by a set of square-well potentials with a random distribution of depths. The polar matrix is thus characterized by a statistical distribution of unoccupied traps g( V). The mathematical form of the g( V) distribution is obtained from the respective distributions of ri and Bi. The details of these calculations for different alcohols are given in the next section. In the present approach electron localization is treated as the capture of a free carrier by a potential trap. This process is then described as a radiationless transition from the delocalized state to a localized state with simultaneous dissipation of the excess energy AE via multiphonon emission. The electron capture cross section is given by 27rw u, = ---IIA12G(Q) (3) fiUth
where W is the volume of the sample, uth is the thermal velocity of free carrier, and R = A E / f i is the electron excess energy in the conduction state in comparison with the level in the trap. The matrix element of the electronic transition A is obtained similarly (12) L. F. Schiff, "Quantum Mechanics", McGraw-Hill, New York, 1968.
0 1985 American Chemical Society
Electron Localization in Alcohol Glasses
The Journal of Physical Chemistry, Vol. 89, No. 14, 1985 3007
as in ref 13. The generalized line-shape function for linear electron-phonon coupling is expressed as follows14
1) exp(iot)
+ n ( o ) exp(-iwt)]A(w)
dwl dt (4)
where A(w) = p(w)A2(w)/2 is the effective phonon spectrum, Le., the product of the phonon density of states p(w) and the coupling function A 2 ( w ) / 2 . S = l ; A ( w ) coth (hw/k,T) dw is the Debye-Waller factor and n(w) = [exp(hw/kBZ') - l]-' is the phonon occupation number. At low temperatures the probability of electron capture rapidly decreases with an increase of the energy difference AE. Thus electron capture by traps deeper than Vc, proceeds mainly via the excited level, its energy being closer to the energy of the conduction electron. After capture by an excited level the electron relaxes to the ground state. The reemission is rather improbable at low temperatures. Electron relaxation requires again dissipation of the energy difference between the excited and the ground state of the trap. In contrast with the free electron, the localized charge interacts mainly with the nearest molecules. This allows us to assume that in the relaxation process the energy is dissipated by excitation of the intramolecular modes of the trap-forming molecules. The transition rate kNR between the two binding levels is given byI5
where w, is the frequency of the vibrational mode, S, is the electron-vibron coupling constant, & = [exp(ho,/k,T) - l]-I, Ips denotes the modified Bessel function of the order of p-0, and the coefficients g8 are expressed as follows: go = -[2&(ii, 1) - 1]S,
+
+ 1)(1 - 2S,) g-, = ii,(l + 2S,)
g, = (ii,
g2
(6)
= (% + 1)2Sv
g-2
= (&,)2Sv
The number p can be interpreted as the number of vibrational quanta needed to cover the energy difference between the excited and ground state; p = (Eexc- E,)/hw,. A more precise definition is given in ref 15. The statistical distribution of the occupied traps P( V) or P(Eb), where E b denotes the energy of the binding level of the trap with depth V, can be defined as the product of the distribution of unoccupied traps and the probability of electron capture
= d E b ) ac(Eb) (7) The distribution of occupied traps is in turn the starting point for the calculations of the absorption spectrum of e;. To calculate the q-spectrum we assumed that the most significant contribution to the spectrum results from the transitions from bound (ground or excited) levels to extended states. The differential cross section for the photodetachment of the electron into the solid angle d b is given by p(Eb)
where k is the wavevector of the ejected electron, Wis the volume of the sample, c is the light velocity, $, is the wave function of the trapped electron, and hv = Eb+ h2k2/2m, is the energy of incident light. The wave function of the ejected electron qk is (13) C. H. Henry and D. V. Lang, Phys. Rev. E, 15, 989 (1977). (14) Y. Weissman and J. Jortner, Phil. Mag. E , 37, 21 (1978). (15) E. Gutsche, Phys. Status Solidi B, 109, 583 (1982).
C, H, OH
-
+
-a 1
-1
-2
-3
' 1 -4
-5
potential , V , eV
Figure 1. The statistical distributions of the trap depth g(V) for unoccupied traps in glassy alcohols.
approximated by the plane wave. The polar axis is taken parallel to the direction of the electric vector of incident light. The total cross section gph can be obtained by the integration of (8) over the solid angle. The total absorption spectrum is obtained by summing the partial spectra (spectra for a given Eb)over all the occupied ground and excited states. The energy range of the occupied levels depends on the time of electron localization. This time can be interpreted as the capture time to for the traps with one bound level. For two-level traps the time of electron localization is better characterized by the time tl of capture by the excited level. The time needed for the subsequent relaxation to the ground state is denoted by T,+. The capture time can be expressed as the inverse of the capture rate constant t, = [u,,aC(E,)NT(Ei)]-l, i = 0, 1 (9) where uthis the thermal velocity of the carrier, NT(E,)is the density of the potential traps with energy level Ei, and ac(Ei)denotes the radiationless capture cross section, the index i = 0 or i = 1 corresponds to the ground and excited state, respectively. The relaxation time T ] + is obtained as the inverse of the rate constant kNRgiven by eq 5 .
Results and Discussion Statistical Distribution of Unoccupied Traps. In order to describe the details of the local molecular structures in alcohols we used, whenever possible, the available experimental data. The investigations of crystalline and glassy alcohols'6 suggest that the structure of glassy ethanol is similar to the crystal structure represented by the polymerlike chains of hydrogen-bonded molecules. The same is assumed also for propanol. Considering the crystal structure of ethanol1*we assumed the tetramers of the hydrogen-bonded alcohol molecules to be possible traps in ethanol and higher alcohols. The crystal structure provides the bond distances and bond angles and allows us to estimate the trap radii as 5.75 A for ethanol and 6.05 A for propanol. Interaction between the alkane parts is approximated by the Kihara potential VKh.21 This interaction hinders the free rotation around hydrogen bonds. The probabilities of spatial positions and angles of individual molecules can than be expressed by the Boltzmann distributions exp(-C Vu/kBTg) calculated for the glass transition temperature Tg.The sum in the exponent runs over all the alkane parts of the trap-building molecules. In contrast to ethanol, glassy methanol shows a structure quite different from the ordered crystalline form. Therefore we assumed for the methanol glass a random structure described by the molecular distribution function. The distribution function was calculated by the method given in ref 17 for a dense Lennard-Jones fluid. The trap radius R (sphere radius) is chosen as the first (16) J. F. Mammone, S . K. Sharma, and M. Nicol, J . Phys. Chem., 84, 3130 (1980). (17) S. Goldman, J . Phys. Chem., 83, 3033 (1979). (18) P. G. Jonson, Acta Crystallogr., Sect. E, 32, 232 (1976)
Swiatia et ai.
3008 The Journal of Physical Chemistry, Vol. 89, No. 14, 1985 05
F
2
.03
I
0
w
I
m
01
00
-1 0 ground-state
-2 0 exclted-state
energy E o , e V
E,,
energy
eV
Figure 2. The statisticaldistributions of the ground-stateenergy (left part) and the excited-state energy (right part) of unoccupied traps in glassy alcohols.
minimum of the distribution function: R = 4.18 A. The number of dipoles N = 7 was obtained as the product of the number density and the integral over the region of the first peak of this function. We assume also a rectangular distribution of the cos e,, where Oi is the angle between the ith dipole and the axis connecting it with the trap center. These assumptions allowed us to compute the statistical distribution of the spherically averaged potential of the trap (eq 2). Figure 1 shows these distributions. As can be seen,the frobability of finding a deep preexisting trap is highest in methanol and rapidly decreases for ethanol and propanol. Figure 2 presents the distributions of the ground-state energy (left) and the excited-state energy (right) of unoccupied traps with the trap depth distributed as shown in Figure 1. The distributions are calculated from th€ distribution g(V) and the relation between the trap depth Vand the electron energy levels Eoand E , for the rectangular potential trap. The trap energy distribution for the methanol glass are characterized by the relatively lower concentration of shallow binding levels and higher concentration of deep states in comparison with higher alcohols. Statistical Distribution of Occupied Traps. The statistical distribution of the occupied traps is determined by both the statistics of empty traps and the probability of electron capture. The capture probability depends on two parameters characterizing a given medium, the phonon density of states p(w) and the coupling constant L = SA(w) dw. The phonon density of states calculated by the approximate method described in ref 19 is expressed by D ( 0 ’ Il
r ,
=
””( - [ 4*cP
1
1
0
.a.
H. Eyring, D.Henderson, and W. Jost, Eds., Academic Press, New York, 1970, p 663.
2
3
4
5
w , iot35.’
excited-state
00
c
6
1
8
9
-05
E,
energy
-15
-10
-2.0
, eV
-2 5
-30
I
-
c
a
250-
”0
-
- cn
]
(19) W. M.Bartczak, M. Hilczer, and J. Kroh, Radial. Phys. Chem., 16, 237 (1980). (20) C. Duke and G. D. Mahan, Phys. Reu. A, 139, 1965 (19651. (21) T. Kihara in ‘Physical Chemistry. An Advanced Treatise”, Vol. V,
1
Figure 3. The phonon density of states in alcohol glasses as computed from eq 10.
.5 3 0
+ -43 ? r c ( K / ~exp[ ) ~ -$rC(K/w))
where C i s the number density of the medium, K = (12U0/M)’/2, Uois the hydrogen bond energy, and M is the mass of the molecule. Figure 3 presents the density of states p(w) for different alcohols. The phonon spectrum in methanol extends over a relatively broad range of frequencies. This frequency range becomes narrower for higher alcohols. On account of the existence df relatively high frequencies in the spectrum, methanol is characterized by the highest probability of electron capture by the deep traps. The coupling constant L describes the difference between the interaction of free and bound electrons with phonons. The use of statistical distributions of the trap depth requires the dculation of the coupling constant as a function of the binding energy. The polar potential*O is assumed for electron-phonon coupling. The results are shown in Figure 4. The lower part of this figure refers to the capture by the ground level of the trap and thus the cbupling constant is presented vs. the ground-state energy Eo.The upper part of the figure corresponds to electron capture by the excited
1
I
D
0
-
” IO
L I
l0L 00
I
4
I
,
I
I
I
1
J -0 5
-10
-1 5
ground-state
-2 5
-2 0
energy
Eo
-3.0
,eV
I4. The electron-phonop coupling constanbas a function of binding energy for the ground state (lower part) and the excited state (upper part) of the electron trap. F
level and the coupling constant L is shown as a function of the excited-state energy E , . For an energy level deeper than, say, 0.5 eV the coupling constant .& increases almost linearly with an increase of the binding energy while the coupling constant L1 is
Electron Localization in Alcohol Glasses
The Journal of Physical Chemistry, Vol. 89, No. 14, 1985 3009
H30H
0.0
-0.5
-1.0 ground-state
J
-1.5 -2.0 -2.5 e n e r g y Eo , e V
0.0
-0.5
-1.0
excited-state
-1.5 -2.0 energy 5,eV
Figure 5. The statistical distributionsof the ground-stateenergy -. (left part) and the excited-state energy (right part) of occupied electron traps in alcohol glasses at 4 K. practically independent of energy. The statistical distributions of the occupied traps for alcohol glasses at 4 K are shown in Figure 5 . The left and right parts of the figure show the statistics of the occupied ground and excited levels, respectively. The highest average phonon energy ( h w ) and the highest coupling constant L result in deeper trapping of electrons in methanol than in higher alcohols. Thus one can expect a blue shift of the spectrum in methanol as compared to the absorption of e; in ethanol and propanol. Calculations of Absorption Spectra. The optical absorption spectra of e,- were calculated a t 4, 40, and 77 K for methanol, ethanol, and propanol. The results are shown in Figure 6. The spectra are calculated under assumption that the electrons captured by the excited levels do not relax to the ground states. Thus these spectra correspond to the time shorter than the relaxation time T~~ (or shorter than about 10 ns in ethanol at 77 K, see Figure 9) and can be considered as the unrelaxed or “initial” spectra of the trapped electron. The intensity of the optical absorption spectrum of e; at very short time after localization can be expressed as
The different values of E M were obtained by variation of the coupling constant L and the phonon energy h w . The energy difference corresponding to the shortest capture time increases with EM The dissipated energy hE is a function of the trap depth V. Thus in both cases of electron capture, by the ground and by the excited level, the capture time depends on V. The example of this dependence for 77 K ethanol glass is shown in Figure 8. The curves labeled to and t l correspond to the capture by the ground and excited level, respectively. These two curves have a crossing point at V = 1.65 eV-electron capture by the excited level of traps deeper than 1.65 eV is faster than capture by the ground level. These curves allow us to obtain the energy range of the occupied ground and excited levels at different times. The total time of the electron localization on the ground level of the deep traps with two bound levels is the sum of the capture time t l and the relaxation time T ~ + . In order to calculate T ~ + , the low-energy vibration (827 cm-I) of the carbon skeleton was assumed as a vibrational mode. This mode was suggested by Bush and Funabashi in their small polaron model.’ The time f , T’+ calculated as a function of the trap depth Vis shown in Figure 9. The time-dependent energy range of the occupied ground levels obtained in the relaxation process may be deduced from this figure by finding the projections on the Vaxis of the cross-section points of the log (tl T ~ + ) curve and the horizontal line corresponding to a given t . By use of this method the curves presented in Figures 8 and 9 allow us to calculate the time-dependent optical spectra. The lower and upper limits, E)(t), E>(t), i = 0, 1, of energy of the occupied traps at a given time t can be estimated from the function tl T ~ + = f ( V ) of Figure 8 for i = 1 and from the of Figure 8 for i = 0. The integrals of eq 1 1 function to are then taken between the limits E&t), Eoh(t) and Ell(t), Elh(t), respectively. The time-dependent spectra of e; in ethanol at 77 K are shown in Figure 10. The 200-nsabsorption spectrum is produced by electrons localized in the shallow ground and excited levels. The gradual decay with time of the long wavelength part of the spectrum is associated with the appearance of an absorption maximum in the short wavelength region. The initial spectrum shows an absorption maximum at about 1200 nm. The maximum shifts to about 200 nm with time of the order of several hundred microseconds. The inset to Figure 10 shows the experimental spectra.’ As can be seen from this figure as well as from Figure 6 which shows the ‘initial” spectra at different temperatures, the proposed model gives the absorption spectra of trapped electron in glassy alcohols in qualitative or sometimes even in quantitative agreement with experiment. It should be underlined that Figures 6 and 10 show the spectral changes caused only by electron capture and relaxation from the excited level of deep trap without changing the trap structure or the electron transfer between traps. The present model does not use the concept of a trap “digging” or “seeking” mechanism. Our
+
+
where the integrals correspond to summation of the partial spectra over all the occupied ground and excited levels, respectively. The methanol spectrum at 4 K shows the absorption maximum at about 700 nm with the infrared tail extending to 2000 nm. At higher temperatures the maximum shifts toward shorter wavelengths and the infrared tail becomes lower. The spectra of higher alcohols appear in the infrared region. The absorption maxima shift from 1550 nm at 4 K to 1200 nm at 77 K for ethanol and from 2200 nm at 4 K to 1700 nm at 77 K for propanol. The obtained results show quite good agreement with the experimental spectra. The maxima of the experimental spectra are marked in the figure by vertical lines. To calculate the time-dependent spectra one should take into account both the kinetics of electron capture and relaxation from the excited levels. The time of electron capture as calculated from eq 9 depends on several parameters such as the energy difference AE,the medium reorganization energy EM= L h w , and the activation energy EA= (AE - E M ) ~ / ~ E M The . low value of the activation energy assures a high capture probability and thus the shortest capture time. The lowest value of EAappears when the dissipated energy AE is equal to the reorganization energy EM. The energy E M characterizes the response of the outer medium to the change in the charge distribution between the initial and final electron states. Figure 7 shows how the medium reorganization energy affects the dependence of the capture time on the dissipated energy AE.
+
Swiatka et al.
3010 The Journal of Physical Chemistry, Vol. 89, No. 14, 1985
-2
-4 y1
-6 I
+ I
m
2
-8
-1 0
-1 2 , eV
potentini V
Figure 8. The capture time by the ground level (to) and excited level (tl) as a function of the trap depth V for 77 K ethanol glass.
or
n ,nm
I
Figure 6. The spectra of the trapped electron calculated for glassy al-
cohols at 4, 40, and 77 K. These spectra have been calculated by assuming electron capture by the ground level of the shallow traps or by the excited level of the deep traps with no subsequent relaxation from the excited to ground state (the unrelaxed or “initial” spectra of trapped electron). The vertical bars show the positions of the maxima of the corresponding experimental spectra measured at approximately the same temperatures. e n e r g y d i f f e r e n c e AE
TI: 4 K nu = 0 0 2 e V
5
t
/
,eV E’HANOL
I
1
/ I
-
’!,
-1 7
-1 3
/
/
77K
1
-10
-2 1 potentiol V ,
-I
25
-2 9
eV
Figure 9. The total time for electron capture by an excited level and relaxation to the ground state tl + T+,, as a function of trap depth V . A 12 I
-13
00
01 energy
02 difference
03
a€
04
1
,
nm
700
900
500
ETHANOL
0.0I 05
,eV
Figure 7. The capture time as a function of the dissipated energy AE. (A): -, E M = 0.02 eV; E M = 0.1 eV; ---, EM = 0.2 eV. (B): -, E M = 0.025 eV; E , = 0.05 eV; ---, EM = 0.075 eV; ..., E M = 0.1 eV. -e-.,
-.-a,
estimations indicate that these mechanisms are too slow to explain the fast spectral changes at liquid helium temperatures. Trap reconstruction is obviously a very slow process at low temperatures. Tunnel transfer of electrons to deeper traps can also be relatively slow because of the low concentration of deep traps or, equivalently, very wide potential bamers. Thus the capture by the excited state in the trap and subsequent relaxation seem to be the fastest way of deep trapping delocalized electrons a t very low temperatures.
4
1
I
I
tl
TI 4 K L. 5
I
1500
1
77K
1
I
12
3
, cm-1.
20
20
36
10’
Figure 10. The theoretical absorption spectra of electrons trapped in ethanol glass at 77 K calculated at 200 ns, 70 ps, and 800 ps after the
radiation pulse. The inset shows the experimental spectra.l The model neglects the presence of positive ions in the glassy matrix. However, the concentration of polar traps is usually much higher than the concentration of positive ions and, consequently, the probability of direct capture of an electron by a positive ion is rather low. Electron reaction with positive ions is probably more significant at longer times leading to a net loss of electrons. This process should not, however, lead to very significant changes in the shape of the absorption spectra and the respective corrections are probably lower than the overall accuracy of the present model. Registry NO. CH3OH, 67-56-1;CIHjOH, 64-17-5;C3HTOH, 71-23-8.
