Dynamics of semiconductor-to-dye electron transfer for anthracene dyes bound to different sized TiO2 particles

Article abstract of DOI:10.1021/jp990302r

The photosensitization of different sized TiO₂ nanoparticles by anthracenecarboxylic acid dye molecules has been studied using transient absorption spectroscopy. These experiments primarily yield information about the semiconductor-to-dye electron transfer reaction. Our measurements show that for TiO₂ particles in the 4 to 40 nm size range, the rate of this reaction does not depend on the particles' size. The results were analyzed using a model where the electrons are assumed to be evenly distributed over the surface of the particle. To reproduce our experimental results with this analysis, the scaling parameter that describes the distance dependence of the semiconductor-to-dye electron transfer reaction must be > 180 nm. We do not consider this to be physically reasonable. An alternative explanation is that the injected electrons are in localized trap sites that are spatially close to the dye radical cation, i.e., they are not randomly distributed over the particle surface. The observation of single exponential decays in the transient absorption data implies that these trap sites have a narrow energy distribution.

Full text of DOI:10.1021/jp990302r

9104
J. Phys. Chem. B 1999, 103, 9104-9111
Dynamics of Semiconductor-to-Dye Electron Transfer for Anthracene Dyes Bound to
Different Sized TiO2 Particles
†
Ignacio Martini, Jos e´ H. Hodak, and Gregory V. Hartland*
Department of Chemistry and Biochemistry, UniVersity of Notre Dame, 251 Nieuwland Science Hall,
Notre Dame, Indiana 46556-5670
ReceiVed: January 25, 1999; In Final Form: April 2, 1999
The photosensitization of different sized TiO
been studied using transient absorption spectroscopy. These experiments primarily yield information about
the semiconductor-to-dye electron transfer reaction. Our measurements show that for TiO particles in the 4
2
nanoparticles by anthracenecarboxylic acid dye molecules has
2
to 40 nm size range, the rate of this reaction does not depend on the particles’ size. The results were analyzed
using a model where the electrons are assumed to be evenly distributed over the surface of the particle. To
reproduce our experimental results with this analysis, the scaling parameter that describes the distance
dependence of the semiconductor-to-dye electron transfer reaction must be > 180 nm. We do not consider
this to be physically reasonable. An alternative explanation is that the injected electrons are in localized trap
sites that are spatially close to the dye radical cation, i.e., they are not randomly distributed over the particle
surface. The observation of single exponential decays in the transient absorption data implies that these trap
sites have a narrow energy distribution.
1. Introduction
Kelley and co-workers examined MoS2 nanoparticles of 3.0 and
4
.5 nm diameter and found that the electron-hole recombination
Dye sensitization of wide band gap semiconductor electrodes
1
9
dynamics are slower for the larger particles. These workers
also presented a kinetic model that successfully explained their
results. In this model, the electrons and holes are assumed to
be randomly distributed around the edge of the disklike nano-
clusters (MoS2 is a two-dimensional layered semiconductor) and
diffusion between trap sites is assumed to be slow compared to
electron-hole recombination. This gives a distribution of reac-
tion distances between the electron and the hole and, therefore,
has gained significant attention in recent years, largely because
of the demonstration of dye-sensitized solar energy cells with
conversion efficiencies as high as 10%.1,2 An important factor
in the development of these solar cells was the introduction of
nanoporous electrodes made from semiconductor particles,
which greatly increased the density of bound dye molecules
per unit area of electrode. In addition to the large surface-to-
volume ratio, semiconductor nanoparticles may present quantum
confinement effects, i.e., have electronic and optical properties
that strongly depend on the particle size.3 The efficiency of
dye-sensitized solar cells depends critically on the rates of the
forward (dye-to-semiconductor) and back (semiconductor-to-
dye) electron transfer reactions, thus, it is significant to ask how
the rates of these reactions depend on the size of the nanopar-
ticles. In this paper we are primarily concerned with the back
electron transfer reaction in the sensitization process. This
reaction has been extensively studied for TiO2 particles, and
19
multiexponential decays. Recently, Kelly and co-workers have
used this model to analyze the dynamics of semiconductor-to-
dye electron transfer for a cyanine dye bound to MoS2 and WS2
,4
20
nanoclusters. The distance scaling parameters obtained for both
the interfacial electron transfer reaction and the electron-hole
recombination process were found to be equal to the radius of
1
9,20
the bulk excitation in MoS2/WS2.
In this paper we present a study of the electron transfer
dynamics for the 1-, 2-, and 9-isomers of anthracenecarboxylic
acid bound to two different sized anatase TiO2 particles. We
have found that there is no size dependence for the rate of the
back electron transfer reaction for particles in the 4 to 40 nm
diameter size range. Moreover, all of the experimental data can
be fit to single exponential decays, indicating that the back
reaction takes place from trap sites with a narrow distribution
it’s rate varies from ca. 10 ps to µs depending on the specific
dye/semiconductor system.5
-17
In many cases the dynamics
show complex multiexponential decays, which could be the
result of either a spatial or an energetic distribution of the
trapped electrons in the semiconductor particles.
To the best of our knowledge no definitive study on the size
dependence of dye sensitization has been published, however,
some related work on the charge recombination after optical
excitation in small semiconductor nanoclusters has been per-
formed. For example, Serpone and co-workers studied the
relaxation dynamics of TiO2 in aqueous solution for three
different nanoparticle sizes and demonstrated that the smaller
the particle, the greater the fraction of electron/hole pairs that
1
6
of energies. An extension of Kelley’s model to three-dimen-
sional, spherical particles is applied to our results. It is important
to note that the time scales for back electron transfer in the
anthracenecarboxylic acid/TiO2 systems (ca. 20-30 ps) are
much faster than the average back electron transfer time for
5
-9,15,17
systems that show distributed kinetics.
1
8
2. Experimental Section
have recombined by the first 20 ps after band gap excitation.
The experimental apparatus used in these studies has been
described in detail elsewhere. It consists of a regeneratively
*
Corresponding author email, hartland.1@nd.edu; fax, (219) 631-6652.
Present address: Department of Chemistry and Biochemistry, University
1
1
†
of California, Los Angeles, CA 90095-1569.
amplified mode-locked Ti-sapphire laser system (Clark-MXR
1
0.1021/jp990302r CCC: $18.00 © 1999 American Chemical Society
Published on Web 05/13/1999

Dynamics of Semiconductor-Dye Electron Transfer
J. Phys. Chem. B, Vol. 103, No. 43, 1999 9105
2
CPA-1000; 100-120 fs fwhm sech deconvolution; 0.6 mJ/
pulse; λ ) 780 nm) pumped by a CW argon ion laser (Coherent
I-90/6). The output of the regenerative amplifier is split by a
7
0/30 beam splitter (CVI). The 70% portion is doubled in a 1
mm type-I BBO crystal to supply the pump laser pulses (390
nm). The 30% portion is used as the probe beam, after being
strongly filtered by neutral density filters. The cross-correlation
between the 390 nm pump and the 780 nm probe pulses was
measured by sum frequency generation in a 0.3 mm KDP
2
crystal, producing a typical fwhm of 150-200 fs (sech
deconvolution). The polarization of the laser pulses was
controlled by passing the pump and probe beams through λ/2
achromatic waveplates (Special Optics) and Glan-Laser calcite
polarizers (Karl-Lambrecht). The probe beam was split into
signal and reference beams which were detected by matched
photodiodes (Thorlabs) for shot-to-shot normalization. The
normalization, and data collection and analysis routines have
been described elsewhere.11 Numerical solutions to the equations
presented in section 4 were obtained using Mathematica (ver.
2
.2.2; Wolfram Research).
The TiO2 nanoparticles were prepared by controlled hydroly-
sis in acidified water of a 20 mL solution of 1 mL titanium(IV)
1
8,21-23
isopropoxide (ACROS, 98%) dissolved in 2-propanol.
The hydrolysis was carried out under N2 at different tempera-
tures. For the smaller particles, lower temperatures were required
and the hydrolysis was performed at 1 °C.1
8,21-23
For the larger
particles, the water was kept at room temperature (20 ( 1 °C)
during the addition of the titanium(IV) isopropoxide solution
(
∼10 to 15 min). Ten minutes after the end of the addition the
temperature was lowered to less than 5 °C to avoid precipitation
Figure 1. (a) Absorption spectra of 1AC in ethanol (s) and 1AC bound
of the TiO2 particles.21 Note that the titanium(IV) isopropoxide/
to small (‚‚‚) and large (- - -) TiO
of 1AC in ethanol (s) and 1AC bound to ZrO
Excitation spectra of 1AC bound to TiO nanoparticles (‚‚‚) and to
ZrO nanoparticles (- - -) are also shown (see text for details).
2
nanoparticles. (b) Absorption spectra
2
-propanol solution was always at room temperature. After the
2
nanoparticles (-‚-‚-).
reaction mixture was stirred for 3 h, the water was removed by
rotatory evaporation yielding a white powder that was dried
overnight and redissolved in absolute ethanol. Both preparations
yielded TiO2 particles with an anatase crystal structure, as judged
from X-ray powder diffraction measurements and the absorption
onset in the UV/vis spectra. ZrO2 nanoparticles were prepared
in a similar way using zirconium(IV) isopropoxide (Aldrich,
2
2
absorption spectrum for the molecules bound to the larger (ca.
0 nm diameter) TiO2 particles. Both spectra are red shifted
with respect to the free dye in solution, as shown in Figure 1
4
1
6
(
a) for 1AC. The TiO2 absorption at λ < 350 nm shows a
16
redder onset for the big particles; this is not due to the particle
size but to the fact that a larger concentration of TiO2 was used
for the solutions of the larger particles. This was done to ensure
that the majority of dye molecules are bound in both solutions
99.9%). Dynamic light scattering measurements performed on
both a Protein Solutions DynaPro-MSDC spectrometer and a
Coulter N4 plus spectrometer showed that the smaller particles
have a hydrodynamic radius of ca. 1.4 to 2 nm and the larger
particles have a hydrodynamic radius of 20 to 30 nm. The larger
TiO2 particles were also examined by transmission electron
microscopy (TEM, 100,000× magnification). To help locate the
TiO2 particles on the TEM grid, a small amount of a Au colloid
solution was added to the TiO2 particle solution before deposi-
tion. The Au particles (ca. 15 nm diameter) have a strong
contrast and can be clearly seen by TEM. These measurements
show that the larger TiO2 particles are cylindrical, with an
average short axis length of 20 nm, an average long axis length
of 40 nm, and a fair amount of polydispersity. These particle
sizes are consistent with the sizes reported in the literature for
(the larger particles have less surface sites per unit volume of
semiconductor and, therefore, a higher concentration is needed
to bind a given amount of dye). Freshly prepared dye/TiO2
solutions show a significant quenching of their fluorescence
emission compared to the free dye in solution, and to the dye
1
6
molecules bound to ZrO2.
Although the fluorescence quenching is considerable (>
90%), it is not complete. To determine whether the remaining
fluorescence is due to free molecules in solution or to bound
molecules, we recorded excitation spectra of 1AC bound to the
three different kinds of semiconductors (small and large TiO2
particles and ZrO2 particles). The results of these measurements,
shown in Figure 1(b), suggest that for the two TiO2 solutions
there is only one fluorescing species s the free dye molecules.
This assignment is indicated by the perfect overlap of the rising
portion of the fluorescence excitation spectra of the 1AC-TiO2
solutions with the absorption spectrum of the free dye in
solution. On the other hand, for the ZrO2 solution there is a
substantial red-shift in the excitation spectrum that perfectly
matches the red-shift of the absorption spectrum of the bound
dye molecules, indicating that in this case the fluorescence
emission has an important contribution from the bound mol-
1
8,23
the above recipe.
AC and 2AC were purchased from TCI (Tokyo-Kasei) and
AC was purchased from Aldrich. All three isomers were used
1
9
without further purification. Steady-state absorption spectra were
recorded with a Perkin-Elmer Lambda 6 UV/vis spectropho-
tometer, and fluorescence emission and excitation spectra were
obtained with a SLM Instruments MC-320 spectrometer.
3. Results
The absorption spectrum of the dye molecules bound to the
small (ca. 4 nm diameter) TiO2 nanoparticles is identical to the

9106 J. Phys. Chem. B, Vol. 103, No. 43, 1999
Martini et al.
Figure 2. Solid line: transient absorption data for 1AC (top) and 2AC
bottom) bound to small and large TiO nanoparticles. Dashed line:
Figure 3. Transient absorption data for 1AC (top) and 2AC (bottom)
(
2
bound to freshly prepared large TiO
2 2
nanoparticles (s) and large TiO
fits to the data using a single-exponential decay plus an offset, see text
for details.
nanoparticles that had been aged for ca. 1 week (‚‚‚).
offset, which accounts for 13 ( 3% of the signal for 1AC and
5 ( 3% for 2AC, is assigned to the free dye in solution, as
shown by the excitation spectra. Experimental conditions were
such that > 90% of the dye molecules are bound.
ecules, see Figure 1(b). We have previously shown that the
absorption spectra of the dye molecules bound to anatase TiO2
particles are identical to the spectra obtained when the molecules
are attached to the ZrO2 particles, see ref 16. Thus, these results
show that when the dye molecules are bound to TiO2 they do
not fluoresce. Similar results were obtained for the other two
anthracenecarboxylic acid isomers.
Transient absorption measurements for all three anthracen-
ecarboxylic acid isomers show that there is no difference in the
dynamics for freshly prepared samples of the different sized
nanoparticles. Figure 2 shows the magic angle transient absorp-
tion experiments for 1AC and 2AC performed with 390 nm
pump and 780 nm probe pulses. This probe wavelength monitors
After aging the samples for periods of time ranging from a
couple of days to a couple of weeks, the transient absorption
results for the dye molecules bound to the large TiO2 particles
present a considerable increase in the background signal. This
is shown in Figure 3 for 1AC and 2AC. These changes are
accompanied by an increase in the fluorescence quantum yield
of the dye-semiconductor solution. There is a 1:1 correspondence
between the increase in the fluorescence and the increase in
the background transient absorption signal. In addition, there
is no change in the excitation spectra of the samples, i.e., the
fluorescence from the dye-TiO2 samples is still due to free dye
molecules in solution. These results indicate that the increase
in the offset is due to an increase in the concentration of free
dye. This aging effect is not observed for the smaller TiO2
particles.
the electronically excited dye molecules, the dye radical cation,
and the trapped conduction band electrons.1
1-13
By using a
combination of transient absorption, time-resolved anisotropy
and bleach recovery measurements we have determined that the
decay in the transient absorption signal is due to the back
electron transfer reaction which destroys the radical cation and
We have previously shown that water molecules effectively
compete with the anthracenecarboxylic acid dye molecules for
adsorption sites at the surface of the semiconductor nanopar-
the trapped electrons.1
1,13,16
Figure 2 shows that the data for
the large and small particles almost perfectly overlap for 1AC
and 2AC; similar results were obtained for 9AC. Thus, the time
scale for the back electron transfer reaction is the same for the
1
3
ticles. For example, adding g 0.1% H2O by volume to an
ethanolic solution of small TiO2 particles with adsorbed 9AC
results in a noticeable increase in the background transient
absorption signal. The large TiO2 particles have approximately
10 times less surface area than the 4 nm diameter particles for
a given TiO2 concentration, hence, they are more susceptible
to contamination by water. Thus, we propose that the increase
in the background signal shown in Figure 3 is due to adsorption
of water from the atmosphere onto the nanoparticle surface. This
process depends on how tightly the samples are sealed, how
they are stored, the relative humidity in the laboratory, etc.,
which also explains why different samples take different periods
of time to develop the larger background.
4
and 40 nm particles.
Also shown in Figure 2 are the best fits to the experimental
data obtained from the convolution of a sum of exponentials
with a Gaussian instrumental response function. For the freshly
prepared large particle samples, as well as for the small TiO2
particle samples, the data recorded over a 100 ps delay range
fits well to a single-exponential decay plus an offset. The time
constants obtained for the small (4 nm) particles are 19 ( 1 ps
for 1AC, 14 ( 1 ps for 2AC, and 33 ( 2 ps for 9AC. For the
large (40 nm) particles the time constants are 19 ( 1 ps, 15 (
1
ps and 36 ( 4 ps for 1AC, 2AC, and 9AC, respectively. The

Dynamics of Semiconductor-Dye Electron Transfer
J. Phys. Chem. B, Vol. 103, No. 43, 1999 9107
small TiO2 particles in the 0-200 ps time range. The latter
observation is the most important point for this paper. In general,
we expect larger distances (on average) between the dye radical
cation and the trapped electrons for the larger TiO2 particles
and, therefore, longer back electron transfer times.1
9,20
Thus,
the observation that the transient absorption decays are identical
for the small and large TiO2 particles is somewhat surprising.
Before analyzing the experimental results, it is important to
address the issue of quantum-size effects in TiO2 particles. There
is some controversy about whether TiO2 shows size quantization
at the typical sizes used in solar cells (2-50 nm diameter).
Predictions of the threshold of quantum size effects depend
3
critically on me, the effective mass of the electron. Early studies
reported me values for TiO2 ranging from 0.01 to 30 m0, where
2
4-26
m0 is the electron rest mass.
where quantum size effects are expected for TiO2 vary from
.6 to 2000 nm. More recent studies suggest that TiO2
nanoparticles do not present quantum size effects for particles
Thus, estimates of the region
0
2
7
as small as 2 nm diameter. In addition, recent electrical
resistivity measurements of anatase TiO2 films indicate that the
effective mass of the electrons is approximately 1 m0, which
implies a Bohr radius for the exciton of ca. 1.6 nm for the
Figure 4. Transient absorption data for 2AC bound to large (ca. 40
nm) TiO nanoparticles (solid line) recorded over a 0-200 ps time
2
range. Fits to the data using a single-exponential decay plus an offset
are also presented (dashed line), as well as the difference between the
fit and the experimental data (residuals).
2
8
conduction band electrons. Thus, we do not believe that
quantum size effects are important for TiO2 particles in the 4
nm to 40 nm size range. For TiO2, the surface charge density
and, therefore, the conduction band energy also depend on the
To fit the data for the aged, large particle samples, a function
that contains two exponential decays plus an offset must be used.
This is necessary because of the greater amount of free dye
present in these samples. The transient absorption signal from
the free dye molecules shows a growth that distorts the decay
due to the back electron transfer reaction for the bound dye
molecules. Because of this complication, only the results from
the freshly prepared solutions will be analyzed in the following
discussion, as these results are more reliable (and meaningful)
than the results from the aged samples.
Transient absorption data for 2AC bound to the large TiO2
particles collected over a 200 ps time delay range is presented
in Figure 4. In this experiment the data can be satisfactorily
fitted using a single-exponential decay of 15 ( 1 ps plus an
offset. In several other scans recorded over this longer delay
range, multiexponential decay functions gave slightly better fits
to the data compared to single exponential decays. However,
the deviations from a single exponential were always small, and
the weighted average of the time constants from the multiex-
ponential fits always gave a decay time that was only 10% to
2
9-32
pH of the solution.
In our experiments, the small and large
particles were synthesized under identical pH conditions so that
the surface charge density should be the same for the two
samples. Thus, in the following analysis we will assume that
the conduction band energy and, therefore, the energetics for
the back electron transfer reaction are the same for the different
sized TiO2 particles.
In the following discussion we will assume that the larger
particles are spherical with a mean diameter of 40 nm, even
though the TEM results show that they have an aspect ratio of
∼
2. This approximation is not expected to change the conclu-
sions of this paper. The starting point in our analysis is to extend
1
9,20
the model of Kelly et al.
to spherical particles. This model
describes radiative recombination between photoexcited elec-
trons and holes in MoS2 nanoclusters; however, it can be easily
refined to account for the reaction between trapped electrons
2
0
and an adsorbed dye molecule. In this model the electrons
are assumed to be randomly distributed over the surface of the
particle. The distribution is described by the function g˜ (x,t),
where x is the distance between the trapped electron and the
radical cation. The back electron transfer reaction occurs with
a distance dependent rate constant given by
1
5% longer than that obtained from the single-exponential fits.
Thus, we believe that a single-exponential decay function is
sufficient to account for our experimental data in the 0-100 ps
time range, which accounts for > 90% of the back electron
transfer reaction, i.e., more complex multiexponential or
stretched exponential fitting functions are not justified.
k(x) ) k exp[-2x/a]
(1)
0
where k0 gives the magnitude of the back electron transfer rate
constant, and a is a distance scaling factor. For the MoS2
particles examined in ref 19, the value of a was found to be
4
. Discussion
The photoexcited dye molecules rapidly transfer an electron
to the TiO2 nanoparticles. The electrons are subsequently trapped
into sites at the surface of the particle that can have a distribution
of energies as well as a distribution of distances from the dye
radical cation. The initial trapping process is believed to be very
similar to the Bohr radius of the bulk exciton in MoS . Assuming
2
that there is no diffusion between trap sites on the time scale of
the back electron transfer reaction, the distribution function
simply decays in time as
15,17,22
fast (ca. 100 fs),
and we note here that very little
information is available about the character of the trap sites.
Our transient absorption experiments monitor the reaction
between the trapped electrons and the dye radical cation. The
results from these measurements show that the majority of the
back electron transfer occurs with a single-exponential decay,
and that there is no difference in the dynamics for the large and
d g˜ (x,t)
dt
)
-k(x) g˜ (x,t)
(2)
In this model the close electrons (small values of x, large values
of k(x)) react first and the far away electrons react at latter times,
i.e., the distribution function changes its shape during the course

9108 J. Phys. Chem. B, Vol. 103, No. 43, 1999
Martini et al.
of the reaction, which causes multiexponential decay kinetics.
The probability of finding an electron in the particle at time t
2
0
is given by
2
R
N(t) )
∫
g˜ (x,t) dx
(3a)
(3b)
0
2
R
g˜ (x) e- dx
k(x)t
)
∫
0
0
where g˜ 0(x) ≡ g˜ (x, t ) 0), and the distribution function is
2
R
normalized so that ∫ g˜ 0(x) dx ) 1. In these equations R is the
0
radius of the particles, so that the maximum distance between
the trapped electrons and the dye radical cation is 2R. The total
number of the trapped electrons is equal to the number of dye
radical cations, thus, N(t) is proportional to the absorbance of
the sample, which is the observable in our experiments
(disregarding the contribution from the free dye molecules in
solution, which produce an offset in the transient absorption
data).
To solve eq 3, an expression is needed for the distribution
function g˜ 0(x). To derive this function we assume that the dye
radical cation is at a fixed point on the surface of a sphere and
the trapped electrons are randomly distributed over this surface.
In this case it can be shown that g˜ 0(x) is given by
2
g˜ 0(x) ) x/2R
(4)
where 0 e x e 2R (the derivation of this function is given in
Appendix A). Substitution of eq 4 into eq 3b yields
Figure 5. (a) Time-dependent populations for small (4 nm) and large
1
2Rxe-k(x)t dx
N(t) )
∫
(5)
(40 nm) particles calculated using eq 5 with a ) 1.6 nm and k ) 2.5
0
2
0
-1
2R
ps . (b) Ratio of rate constants for the large and small particles
calculated using eq 7 versus a, the distance scaling parameter. See text
for details.
which can be easily integrated using Mathematica (for example).
Equation 5 contains two adjustable parameters: k0 and a. In
MoS2 the value of a was found to be the same as the Bohr
radius of the bulk exciton, i.e., a ) ꢀa0/(me/m0) where ꢀ is the
static dielectric constant of the semiconductor and a0 ) 0.053
nm. Reasonable values of ꢀ and me/m0 for anatase TiO2 are ꢀ
that used to derive eq 5). In this case it can be shown that the
probability of finding a trapped electron in the particle at a time
t is given by the relatively simple expression
-
k(R)t
31 and me/m0 ) 1,28 which yields a ) 1.6 nm. Note that this
N(t) ) N0e
(6)
)
value of the distance scaling parameter is strictly appropriate
for an electron in the conduction band of TiO2.
where the particle size dependent rate constant k(R) is given by
Calculated decays for the small and large TiO2 particles using
this value of a are shown in Figure 5 (a). In this calculation, k0
was adjusted to give a decay that is consistent with a 19 ps
back electron transfer time for the small particles (this value is
appropriate for 1AC). Equation 5 clearly predicts very different
dynamics for the 4 and 40 nm diameter particles. In addition,
the calculated decay for the 4 nm particles is clearly multiex-
ponential, which is not observed in our experiments. Increasing
the value of a reduces the difference in the decay times for the
large and small particles. For values of a g 180 nm the decay
constants for the different sized particles are essentially equal.
2
R
k(R) )
∫
g˜ 0(x)k(x)dx
(7)
0
In this equation g˜ 0(x) is given by eq 4, i.e., the electrons are
still assumed to be randomly distributed over the surface of the
particle. Equations 6 and 7 are derived in Appendix B. In this
model we would always expect to see single exponential decays
(which is essentially what is observed experimentally), but the
rate constant should be different for different sized particles.
For example, choosing a ) 1.6 nm and k0 ) 0.69 ps gives a
back electron transfer time of 19 ps for the 4 nm particles and
1800 ps for the 40 nm particles. As was found for eq 5, using
larger values of a reduces the difference in the electron transfer
times. A plot of the ratio of the rate constants calculated using
eq 7 for the small and large particles k(4 nm)/k(40 nm) is shown
in Figure 5(b) as a function of the distance scaling parameter
a. Note that this ratio is independent of the value of k0. This
plot shows that the rate constants are essentially equivalent for
values of a g 200 nm. However, once again such a large value
for the distance scaling parameter for an electron in TiO2 is
unrealistic. Thus, allowing for fast diffusion between trap sites
does not lead to a physically reasonable model. The two cases
considered (no diffusion between trap sites and diffusion much
-
1
-
1
For example, using a ) 200 nm and k0 ) 0.06 ps gives a
decay time of ca. 19 ps for both the 4 and 40 nm particles.
(Larger values of a and correspondingly smaller values of k0
can give the same decay time.) Note that when a . 2R the
calculated decays are essentially single exponential.
The large value of a obtained from the above analysis is
unphysical: if the Bohr radius of an electron in TiO2 was g
180 nm, then TiO2 should show enormous quantum size effects.
This result means that the assumptions used to derive eq 5 must
be incorrect. An alternative approach to this problem is to
assume that diffusion between trap sites is much faster than
the back electron transfer reaction (this is the opposite limit to

Dynamics of Semiconductor-Dye Electron Transfer
J. Phys. Chem. B, Vol. 103, No. 43, 1999 9109
faster than back electron transfer) represent two opposite limits
for the mobility of the trapped electrons. Thus, we conclude
that all models where the electrons are assumed to be initially
evenly distributed over the surface of the particle cannot explain
our experimental results with physically realistic parameters.
An alternative approach is to assume that the electrons are
randomly distributed within the volume of the particle rather
than distributed over the particle surface (i.e., the electrons are
in the conduction band and not surface trap sites). This situation
is more difficult to quantitatively analyze. However, we would
still expect to see a difference in the back electron transfer times
for the different sized particles for this model. We note that the
average time for a species that is randomly distributed inside a
2
2
sphere to diffuse to the surface of the sphere is τ ) R /π D,
where D is the diffusion constant.33 This equation describes the
time scale for a reaction between a species inside a sphere and
a second species that is located at the surface, assuming that
Figure 6. Comparison of experimental transient absorption data for a
9
AC-TiO
2
solution (s) and the signal calculated using eqs 8-10
(‚‚‚). The 9AC-TiO
2
sample was in ethanol with ca. 1% added water.
the reaction occurs as soon as the diffusing species reaches the
The water produces nonsingle exponential decay kinetics for these
samples, see text for details and ref (13).
surface.3
3,34
In our experiments the electrons must diffuse to
the dye radical cation, which is fixed at a specific position at
the surface of the nanoparticle, i.e., the reaction does not occur
as soon as the electron reaches the surface. Therefore, the
reaction time will be longer than predicted by the above
equation. We expect that the time scale for the reaction should
increase by a factor that is related to the total surface area of
the particle divided by the area of an adsorbed dye molecule.
of the conduction band levels in electron transfer for the
anthracenecarboxylic acid-anatase TiO2 systems is unclear at
the present time.
It should be noted that, in general, a distribution of energies
2
is expected for the electron trap sites, which should also lead
to nonsingle exponential decay kinetics. The effect of a
distribution of trap site energies can be modeled by writing the
2
This factor is proportional to R , which means that the difference
3
7
transient absorption signal as
in reaction times for different sized particles should scale as
4
R . (Note that this treatment neglects Coulombic interactions
∞
S(t) )
∫
P(E)e- dE
k(E)t
(8)
4
between the electron and the dye.) Thus, we expect a 10
-
∞
difference in the time scales for diffusion of the electron to the
dye radical cation for the two particles, if the electrons are
randomly distributed inside the particles. This estimate of the
difference in reaction times is not consistent with our experi-
mental data.
where
2
e-
(E-λ) /4λkT
(9)
k(E) ) k
0
In these equations P(E) is the energy distribution function for
the sites, λ is the reorganization energy, E is the energy
difference between the trap site and the redox potential of the
dye, and we have assumed that the classical Marcus theory for
The final possibility considered is that the electrons injected
into the semiconductor particles are localized into sites that are
spatially close to the dye radical cation. Furthermore, the back
electron transfer reaction occurs before the electrons can escape
from these sites. In this case the size of the particles would not
be expected to have an effect on the rate of the semiconductor-
to-dye electron transfer reaction. The dye molecules used in
this study attach to the TiO2 particles through the carboxylate
group, which binds to titanium atoms at the surface of the
3
6-40
electron transfer is sufficient.
To simulate the experiments
P(E) is assumed to be given by a Gaussian distribution function:
2
2
-
(E-∆E) /σ
P(E) ) e
(10)
where ∆E is the average energy difference between the trap
2
particles. The surface electron trap sites are also located on
2
sites and the redox potential of the dye, and σ gives the width
the titanium atoms, and typical Ti-Ti distances in TiO2 are ca.
of the energy distribution. The choice of a Gaussian function is
somewhat arbitrary; however, the exact function used in the
present analysis is not expected to change the conclusions given
below. Equation 8 is simply a sum of exponential decays for
the different trap sites that is correctly weighted for the trap
site energies. Note, we have assumed that the coupling element
for electron transfer is identical for the different trap sites.
Equations 8-10 have been used previously to simulate
transient absorption data for 9AC-anatase TiO2 in ethanolic
solutions with small (<2 vol %) amounts of added water.37
3
5
3
Å. Thus, if the electrons are localized on nearest neighbor,
or next nearest neighbor Ti’s, the back electron transfer reaction
would be expected to be rapid and independent of the size of
the particles. Of the three models considered for the electron
sites, random distribution over the surface or within the volume
of the semiconductor, or trapping into localized sites adjacent
to the dye radical cation, the last is the only one that is consistent
with our experimental data.
An interesting consequence of this conclusion is that if the
dye molecules only interact with localized sites at the semi-
conductor surface, and not the delocalized conduction band
levels, then the density of accepting states for electron transfer
should be small. The ultrafast time scales for forward (dye-to-
semiconductor) electron transfer observed in these systems are
usually attributed to the high density of states in the conduction
band of the semiconductor.36 The anthracenecarboxylic acid dye
molecules that were examined in this paper show forward
Adding water to the ethanol/TiO solutions changes the energy
2
of the electron trap sites at the surface of the particles and
13
produces nonexponential decays. For example, Figure 6 shows
data collected for 9AC in a TiO /ethanol solution with 1% added
2
water. Also shown is a fit to the data using eqs 8-10 and the
following parameters: ∆E ) 1.6 eV, λ ) 0.7 eV, and σ )
0.043 eV (the value of ∆E corresponds to the difference in
energy between the flatband potential of the semiconductor and
the redox potential of the dye). An offset has also been added
16
electron transfer times that are < 200 fs. Thus, the exact role

9110 J. Phys. Chem. B, Vol. 103, No. 43, 1999
Martini et al.
to account for the free dye molecules in solution. The calculated
and experimental transient absorption signals are clearly in
excellent agreement. A multiexponential fit to the experimental
data shown in Figure 6 yielded time constants of 33 ps (35%),
SCHEME 1
8.7 ps (39%), and an offset (26%). These results are presented
here to demonstrate that eqs 8-10 provide a good description
of the multiexponential decays that can be observed in semi-
conductor-to-molecule electron transfer reactions. Note that for
a given ∆E the deviation of the calculated decay from a single
exponential increases as σ increases. For the experimental data
shown in Figures 2 and 4, the values of σ used in eqn 10 must
be less than 0.02 eV to obtain a reasonable fit to the data. This
means that the fwhm of the trap site energy distribution must
be less than 0.03 eV, which is on the order of kT. Thus, the
back electron transfer reaction occurs from a fairly narrow
energy distribution of trap sites for the anthracenecarboxylic
acid-anatase TiO2 systems.
proportional to the area of the ring around the sphere defined
by x and x + dx.
The area of this ring can be calculated by considering the
2
area element in spherical coordinates, dA ) R cosθ dθ dæ,
and then integrating over the azimuth angle æ, to give
2
g(x)dx ∝ 2πR cosθ dθ
(A1)
5
. Summary and Conclusions
where x is the distance between P and A. From simple geometry
and it can be shown that x ) R[2(1 - sinθ)] . Equation (A1)
can be rearranged to give
The electron transfer dynamics for the 1-, 2-, and 9-isomers
1/2
of anthracenecarboxylic acid bound to different sized anatase
TiO2 particles has been examined by ultrafast transient absorp-
tion experiments. The results from these measurements show
that (i) greater than 90% of the back (semiconductor-to-dye)
electron transfer reaction is complete within 100 ps, and that in
this time range the observed decays in the transient absorption
data can be fitted using a single exponential decay plus an offset.
2
2
πR cosθ
g(x) ∝
(A2)
|dx/dθ|
Performing the differentiation dx/dθ yields g(x) ∝ 2πx, which
in turn gives the normalized distribution function
(
The offset accounts for the unbound dye molecules.) (ii) The
2
size of the TiO2 particles does not affect the electron transfer
dynamics for particles in the 4 to 40 nm size region. These
experimental results can only be explained by assuming that
the electrons are trapped in localized states that are adjacent to
the binding site for the dye radical cation, i.e., their spatial
distribution is not random. The trapped electrons undergo back
electron transfer before they can diffuse away from the dye
radical cation. These results also show that the shape of the
nanoparticles should have very little effect on the semiconductor-
to-dye electron transfer times for this system. The observation
of essentially single-exponential decays for the back electron
transfer reaction in our data also implies that the trap sites for
the electrons at the surface of the particle have a fairly narrow
distribution in energy.
g˜ (x) ) x/2R
(A3)
2R
where the normalization condition is ∫ g˜ (x)dx ) 1.
0
Appendix B: Derivation of Equations 6 and 7
To derive the rate law for the case where diffusion between
trap sites is much faster than the back electron transfer time,
consider the distribution at time t1 ) t0 + ∆t. From the rate
law d g˜ (x, t)/dt ) -k(x) g˜ (x, t) we have
g˜ (x, t ) ) g˜ (x, t ) - k(x) g˜ (x, t )∆t
(B1)
1
0
0
Integration of g˜ (x, t1) over x gives N(t1), the probability of
finding an electron on the particle at time t1
2
R
2R
Acknowledgment. This work was supported by the U.S.
Department of Energy, Office of Energy Research (Grant # DE-
FG02-98ER14898). The authors thank Prof. Nancy Levinger
of Colorado State University, Mr. David Dolak of Protein
Solutions, and Mr. Mark Halas of Coulter for performing the
dynamic light scattering experiments. We also thank Prof. Paul
McGinn of the University of Notre Dame for recording the
X-ray powder diffraction data.
N(t ) )
g˜ (x, t ) dx ) N(t ) - ∆t
k(x) g˜ (x, t ) dx (B2)
0
0
1
∫
1
0
∫
0
and by comparison at time t2 ) t1 + ∆t
2
R
N(t ) ) N(t ) - ∆t
∫
k(x) g˜ (x, t ) dx
(B3)
2
1
1
0
If we assume that g˜ (x, t1) ) N(t1) × g˜ (x, t0), i.e., the distribution
at time t1 has the same form as that at t0 but is weighted by the
population N(t1), then we can write
Appendix A: Derivation of Equation 4
2
R
N(t ) ) N(t )[1 - ∆t
∫
k(x) g˜ (x, t ) dx]
(B4)
The distribution function for the distance between a species
fixed at a point on the surface of a sphere and a second species
randomly distributed over the surface can be derived as follows.
The fixed species is considered to be located at the point P in
Scheme 1. The second species (the trapped electron in our case)
is located at a random position A, at the surface of the sphere
of radius R. The angle θ goes from -π/2 to π/2 and is defined
as the polar angle, and æ is the azimuth angle which goes
from 0 to 2π.41 The probability of finding the species A at a
distance between x and x + dx from the point P, g(x)dx, is
2
1
0
0
and in general
2
R
N(t
) ) N(tn-1)[1 - ∆t
∫
k(x) g˜ (x, t ) dx]
0
(B5)
n
0
The condition used to derive eqs B4 and B5 is equivalent to
assuming that the electrons rapidly diffuse between trap sites,
so that they are always randomly distributed over the surface
of the particle. Equation B5 can be rewritten as

Dynamics of Semiconductor-Dye Electron Transfer
N(t)
J. Phys. Chem. B, Vol. 103, No. 43, 1999 9111
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