REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 74, NUMBER 1
JANUARY 2003
Thermal lens temperature scanning for quantitative measurements
in transparent materials „invited…
J. H. Rohling, J. R. D. Pereira, A. N. Medina, A. C. Bento, and M. L. Baessoa)
Departamento de Fı́sica, Universidade Estadual de Maringá, Av. Colombo, 5790, 87020-900, Maringá,
PR, Brazil
J. A. Sampaio, S. M. Lima, and T. Catunda
Instituto de Fı́sica de São Carlos, Grupo de Espectroscopia de Sólidos, Universidade de São Paulo,
Av. Dr. Carlos Botelho, 1465, 13560-250, São Carlos, SP, Brazil
L. C. M. Miranda
Instituto Nacional de Pesquisas Espaciais, 12201-970, São José dos Campos, SP, Brazil
共Presented on 24 June 2002兲
In this work the ability of thermal lens spectrometry for different temperature studies in transparent
materials is discussed. The method was applied in polymers and optical glasses to measure the
thermo-optical properties as a function of the temperature. The focus of the discussion will be on the
temperature range where the glass transition occurs. The perspectives of future studies in this area
will be discussed. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1517157兴
I. INTRODUCTION
On the other hand, the ultimate goal in the study of the
thermal and optical properties as a function of the temperature is the determination of the role played by these parameters in the material phase transitions. It is well known that
thermal conductivity 共K兲, thermal diffusivity 共D兲, and the
coefficient of temperature of the refractive index (dn/dT)
present strong variations in their values when the temperature of the sample is close to the region where the phase
transition is expected to occur. It is recognized that the precise determination of phase transition temperature is still an
experimental challenge that needs to be accomplished. Conventional calorimetric methods demand the use of a reference sample and therefore a temperature lag between the
tested sample and the reference may take place. This does
not allow locating the phase transition temperature.
In this work an overview of the recent developments in
the application of the TL method to measure the quantitative
values of the thermal and optical properties of transparent
materials as a function of temperature will be presented. The
TL experimental data obtained in polyvinyl chloride 共PVC兲
in the temperature range from 22 to 70 °C, in polycarbonate
from 22 to 170 °C and in fluoride glasses from 22 to 300 °C
will be shown. These temperature ranges include the glass
transitions region of these materials. It is discussed how the
experimentally determined TL parameters can be used to locate the glass transitions in these materials. In addition, a
The accidental observation of the thermal lens effect in
the early 1960’s resulted in the development of the technique
called thermal lens spectrometry 共TLS兲.1–15 The TL effect is
created when a Gaussian laser beam passes through an absorbing medium and part of the absorbed energy is converted
into heat. As a consequence, the refractive index of the illuminated area is changed, producing a radial profile in the
sample equivalent to that of the beam intensity. This is the
so-called thermal lens effect. The propagation of a probe
laser beam through the TL will be affected and a phase shift
in its wavefront will be induced. The thermo-optical properties of the sample can be determined by measuring this induced phase shift.4 –30 Since the first observation, the TL experimental configuration and theoretical models have been
improved in order to provide both higher sensitivity and the
possibility of performing quantitative measurements of the
investigated samples.4,5,10,22 As a result, the TL method is
now employed as a sensitive analytical tool to determine the
thermo-optical properties of transparent materials. However,
despite the wide range of applications of this technique, so
far the measurements have been carried out mostly near
room temperature. Since the TL technique is an intrinsically
remote method the measurements on a sample placed inside
a harsh environment present, in principle, no additional difficulty. As illustrated in Fig. 1, the experimental arrangement
can be set in such a way that a space of about 0.5 m between
the two lenses can be used to introduce a thermal heating
device to control the sample temperature. Using this procedure we recently introduced the TL technique for quantitative measurements as a function of temperature. The experiments have been performed to study optical glasses,9,13,23
polymers,24 –27 and liquid crystals.28 –30
a兲
FIG. 1. Probe and excitation beams in the mode mismatched TL configuration.
Author to whom correspondence should be addressed; electronic mail:
[email protected]
0034-6748/2003/74(1)/291/6/$20.00
291
© 2003 American Institute of Physics
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292
Rohling et al.
Rev. Sci. Instrum., Vol. 74, No. 1, January 2003
␻ e is the excitation beam waist at the sample position. The
solution of the heat conduction equation depends on the employed boundary conditions. Shen et al.4,5 developed the infinitive aberrant model for the mode-mismatched configuration. Using the conditions ⌬T(r,0)⫽0,(r⬍⬁) and
⌬T(⬁,t)⫽0(t⬎0), the temporal evolution of the temperature profile ⌬T(r,t) induced by the TL in the sample is given
by4,5
⌬T 共 r,t 兲 ⫽
冕冋
␲ ␳␻
2 P eA e
t
1
2
e
0
1⫹ 共 2t ⬘ /t c 兲
c
冋
⫻exp
⫺2r 2 / ␻ 2e
1⫹ 共 2t ⬘ /t c 兲
册
册
dt ⬘ ,
共1兲
in which ␳ is the density, c the specific heat, A e the optical
absorption coefficient at the excitation beam wavelength, and
t c the characteristic TL time constant, defined as
FIG. 2. TL experimental setup.
t c⫽
comparison with conventional calorimetric methods is made.
Finally, it is proposed that the current thermal lens technique
can be adopted as a new tool, called differential thermal lens
temperature scanning, especially designed for the investigation of the phase transitions in transparent materials. The
perspectives of future studies in this area will be discussed.
II. EXPERIMENT
The employed TL experimental setup is shown in Fig. 2.
The two laser beam mode mismatched TL configuration was
used with the excitation beam at 514.5 nm and a probe beam
at 632.8 nm. The mode-mismatched configuration has been
shown to be the most sensitive experimental setup for the TL
measurements. This arrangement uses two laser beams with
different spot sizes at the sample position. TL measurements
can be performed for both time-resolved and steady state
mode. The time-resolved method permits the measurement
of the development of the thermal lens in a short period of
time, and the advantage of this procedure when compared
with steady state mode is that it allows to measure the
sample thermal diffusivity.4,6,10,17 The experiment is performed in the following way: first, the probe beam is aligned,
using mirror 5, in order to have its center passing through the
pinhole positioned in front of the detector; after using mirror
2 the excitation beam is used to induce the thermal lens in
the central part of the probe beam and a consequent change
in its intensity in the detector, photodiode 2. This signal
variation is recorded by the oscilloscope and the data thus
obtained are transferred to a microcomputer for subsequent
analysis.
The theoretical treatment of the thermal lens effect takes
into account the spherical aberration of the thermal lens and
also considers the whole optical path length change with
temperature.10 The first step in the development of the model
is to consider the heat source profile, Q(r), induced by the
laser beam. Q(r) is proportional to the Gaussian intensity
profile, which can be expressed as: I e (r)⫽(2 P e / ␲␻ 2e )
exp(⫺2r2/␻2e ), in which P e is the excitation beam power and
␻ 2e
4D
with D⫽
K
,
␳c
共2兲
where D is the thermal diffusivity and K is the thermal conductivity. This temperature rise, which carries a Gaussian
profile, induces a slight distortion in the probe beam wavefront that can be associated with the change in the optical
path length of the sample with respect to the axis of the
beam, as follows:10
冉 冊
⌽␭ p
ds
⫽l 0
2␲
dT
关 ⌬T 共 r,t 兲 ⫺⌬T 共 0,t 兲兴
共3兲
p
in which ⌽ is the phase shift induced when the probe beam
passes through the TL, ␭ p is the probe beam wavelength, l 0
is the sample thickness, and (ds/dT) p is the temperature
dependence of the optical path length of the sample. Finally,
using the Fresnel diffraction theory, the probe beam intensity
at the detector plane can be written as an analytical expression for absolute determination of the thermo-optical properties of the sample, as4,5
冋
␪
I 共 t 兲 ⫽I 共 0 兲 1⫺ tan⫺1
2
⫻
冉
2mV
关共 1⫹2m 兲 2 ⫹V 2 兴共 t c /2t 兲 ⫹1⫹2m⫹V 2
冊册
2
,
共4兲
in which
␪ ⫽⫺
冉 冊
P e A e l 0 ds
K␭ p dT
;
p
V⫽
Z1
;
Zc
m⫽
冉 冊
␻p
␻e
2
.
共5兲
In Eq. 共4兲 I(t) is the temporal dependence of the probe laser
beam intensity at the detector, I(0) is the initial value of
I(t), ␪ is the thermally induced phase shift of the probe
beam after its passing through the sample, ␻ p is the probe
beam spot size at the sample, Z c is the confocal distance of
the probe beam, Z 1 is the distance from the probe beam waist
to the sample. We should mention that the parameter ds/dT
describes the whole optical path length change induced by
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Rev. Sci. Instrum., Vol. 74, No. 1, January 2003
the excitation beam, which means that for solid material it
depends on several mechanisms such as the sample bulging
during the illumination and also on the stress-optical coefficient. In the case of liquid samples we have ds/dT
⫽dn/dT. Therefore, using least-square curve fitting of Eq.
共4兲 to the TL transient experimental data, ␪ and t c can be
obtained.
The poly共vinyl chloride兲 共PVC兲 film used was a 200 ␮m
thick with 12 mm diameter disk. The polycarbonate sample
consisted of a 1.4 mm thick, 12 mm diameter disk of
polysafe poly共carbonate兲 manufactured by Wilson Safety
Products Co. The fluoride glass compositions were fluoroindates 共PGIZCa, ISZn, InSBZnGdN兲 and fluorozirconates
共ZBLAN兲.23 The starting materials used for the preparation
of the glasses were fluorides 共BDH and Strem products兲 and
oxides In2 O3 , ZnO, Ga2 O3 共MetalEurop兲. The ammonium
bifluoride was used to transform oxides into fluorides. The
mixtures were heated in a platinum crucible for melting and
refining. Finally, the liquid was poured into a brass mold at
temperatures few degrees smaller than T g to prepare samples
4 mm thick. All these operations were performed in a glove
box with inert atmosphere whose moisture was kept below
10 ppm. These glasses have low optical absorption coefficients, ⬍10⫺3 cm⫺1 , and phonon energies of the order of
500 cm⫺1 . These characteristics make the TL signal of these
systems difficult to detect. To overcome this problem the
samples were doped with a small amount of CoF2 which
induces optical absorption bands in the samples, one of them
centered at about 540 nm. The CoF2 doping concentrations
were: 0.392 mol % for ISZn, 0.20 mol % for PGIZCa, 0.29
mol % for InSBZnGdN, and 0.155 mol % for the ZBLAN.
In order to validate and evaluate the sensitivity of the TL
method we carried out complementary measurements of differential scanning calorimetry 共DSC兲 for polymer samples
and differential thermal analysis 共DTA兲 for the optical
glasses.
III. RESULTS AND DISCUSSION
A. Thermal lens for glass transition analysis of
polymers
Figure 3 shows the resulting temperature dependence
obtained for the thermal diffusivity and the normalized thermal lens signal, ␪, for the PVC sample. To better understand
the temperature dependence of both D and ␪ / P, we have
also carried out DSC measurements. In Fig. 4共a兲 we present
a typical DSC curve for our PVC sample. This result indicates that the glass transition region extends from 50 to about
67 °C with a peak at 62.5 °C. In Fig. 3 we observe that both
D and ␪ show a marked change in their temperature dependence in this glass transition region. Between 50 to about
67 °C, D decreases by a factor of about 2 while ␪ / P experiences an increase of the order of 3.5.
The somewhat complex temperature dependence of ␪
may be better understood by looking at the temperature derivative of the solid line of curve b in Fig. 3, shown in Fig.
4共b兲. It shows us that, on increasing the temperature, (1/␪ )
⫻(d ␪ /dT) increases, reaching a peak around 55 °C, then
decreases very sharply to a minimum at roughly 61 °C, to
Photoacoustic and photothermal phenomena
293
FIG. 3. Temperature dependence of the thermal diffusivity and ␪ / P for
PVC.
finally reach a maximum at 64 °C as shown in the upper
curve. The comparison between the two plots in Fig. 4 is
such that the first peak in curve 共b兲, which occurs at 55.2 °C
corresponds to the beginning of the glass transition as determined by the DSC curve, whereas the minimum between the
two peaks of curve 共b兲 corresponds to the inflection point of
the DSC curve before reaching its minimum at 62.5 °C. We
adopted this minimum point as the glass transition temperature.
This result is indeed not as surprising as it may look at a
first glance. In a DSC experiment one has a reference material, the sample to be probed, and a predetermined heating
共or cooling兲 rate is imposed to the system for undergoing a
given temperature excursion. A servosystem makes the
sample to follow the temperature of the reference and the
heating power difference between the sample and the reference is recorded. That is, since dT/dt is fixed, one records
essentially dQ/dT.
Figure 5 shows the resulting temperature dependence
obtained for the thermal diffusivity and the TL signal parameter, ␪ / P, for the polycarbonate sample. The DSC data are
shown in curve 共c兲. As performed for the PVC sample, in
Fig. 6 we present the temperature derivative of ␪ / P. For
comparison, a plot of the temperature derivative of the DSC
data is also presented in Fig. 6, curve 共b兲. The existence of
two minima in the DSC temperature derivative data seems to
reflect the fact that the polycarbonate sample has two dominant phases. One phase corresponding to a glass transition
temperature of about 146 °C and another one with T g of
FIG. 4. Temperature derivative of ␪ / P and DSC for PVC.
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294
Rev. Sci. Instrum., Vol. 74, No. 1, January 2003
Rohling et al.
FIG. 5. Temperature dependence of the thermal diffusivity and ␪ / P for
polycarbonate.
about 148 °C. These two phases correspond to pure polycarbonate and to a PCA/ABS 共acrylonitrile–butadiene–styrene兲
blend. This ABS blend is used to improve the mechanical
shock resistance of the polycarbonate used as safety eyeglasses. In the temperature derivative of ␪ / P 关Fig. 6共a兲兴 in
the range between 140 and 160 °C the peaks exhibit a distinct correlation with those of the temperature derivative of
DSC 关Fig. 6共b兲兴. The maxima occur at 144.8 and 150 °C,
corresponding to the minima of the DSC temperature derivative. That is, the temperature dependence of ␪ / P apparently
has more information than that contained in the DSC curve.
The similarity between the behavior of (1/␪ )(d ␪ /dT)
and the DSC curve showed for both, PVC and polycarbonate, suggest that the TL technique can be used to perform an
equivalent differential scanning technique, namely, a differential thermal lens scanning, in which we set a given heating
rate and measure the rate of change of the TL signal. The
resulting signal of this technique would be d ␪ /dt
FIG. 6. Temperature derivative of ␪ / P and DSC for polycarbonate.
FIG. 7. Temperature dependence of the normalized TL phase shift for: 共a兲
InSBZnGdN, 共b兲 ISZN, 共c兲 PGIZCa, 共d兲 ZBLAN glasses.
⫽(d␪/dT)(dT/dt), that is proportional to d ␪ /dT, and can provide information regarding the onset of the glass transition.
B. Thermal lens for glass transition analysis of
optical glasses
Figure 7 shows the normalized TL phase shift 共␪兲 as a
function of temperature for the four different glasses studied
in this work, while Fig. 8 shows the temperature dependence
of their thermal diffusivities. The error for ␪ / P values is of
the order of 2%, while for D it is about 5%. From Eq. 共5兲 we
note that ␪ / P depends on several parameters, such as K, A e ,
l 0 , and ds/dT. Therefore, we adopted the room temperature
␪ / P as the reference datum for normalization. In this way the
data for different glasses can be compared in terms of the
temperature dependence of the TL signal amplitude. Similarly, to the observations for polymers, Figs. 7 and 8 show
variations in ␪ / P and D values when the temperature excursion passed through the regions where the glass transitions of
the samples were expected to occur.
Following the same procedure described above for polymers, to further explore the TL results the first derivative of
the resulting temperature dependence of the TL signal parameter, ␪ / P, was performed. The resulting curves are
shown in Figs. 9–12. For a better view, we show only the
temperature range between 220 and 300 °C, close to the temperature at which T g occurs. For comparison the DTA curve
of each glass is also shown in the respective (1/␪ )(d ␪ /dT)
figures. An additional point to be observed in the TL and
DTA data is that T g shifted when the different glasses are
compared. For example, T g changed from about 253 °C for
the ZBLAN glass to ⬃290 °C for the InSBZnGdN glass.
Again, similar to the previously described procedure employed for polymers, in Figs. 9–12 the temperature deriva-
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Rev. Sci. Instrum., Vol. 74, No. 1, January 2003
Photoacoustic and photothermal phenomena
295
FIG. 10. Temperature derivative of the thermal lens signal 共a兲 and the DTA
data 共b兲 for the ISZN glass.
FIG. 8. Temperature dependence of the thermal diffusivity for: 共a兲 InSBZnGdN, 共b兲 ISZN, 共c兲 PGIZCa, 共d兲 ZBLAN glasses.
tive of ␪ / P was used to attempt to locate the glass transition
temperature. These plots show quite well the glass transitions. We suggest therefore that the TL technique can also be
used to perform measurement equivalent to DTA. One of the
advantages of the TL method is that this technique does not
require a reference sample for the experiments. This prevents
the occurrence of the temperature lag between the sample
and the reference, especially in the region of T g , which occurs with the conventional calorimetric measurements. Here,
again, as shown for polymers samples, our results for optical
glasses show that by setting a given heating rate and measuring the rate of change of the TL signal, a differential
scanning can be obtained. The resulting signal of this scanning technique would be d ␪ /dt⫽(d ␪ /dT)(dT/dt), that is
proportional to d ␪ /dT, and can provide information regarding the onset of the glass transition. As is well known from
the literature, one of the procedures usually used to locate the
glass transition is to assume T g as the crossing point of the
FIG. 9. Temperature derivative of the thermal lens signal 共a兲 and the DTA
data 共b兲 for the InSBZnGdN glass.
two straight lines, as shown in the DTA data presented in the
figures above. This is an empirical way of identifying the
transition which has subjective errors. From the TL data we
observed that: (1/␪ )(d ␪ /dT) initially increases with increasing temperatures, goes through a maximum, then decreases
to a minimum, followed by a new maximum and a subsequent decrease. The minimum between the two maxima indicated in the curves 共a兲 of Figs. 9–12 could be assumed to
be the T g region, since it is the point at which ␪ / P in Fig. 7
has a smaller rate of variation as compared to the temperatures below and above this region. This could be a more
precise way to determine the glass transition as compared to
the usual procedure adopted in conventional calorimetric
measurements. The estimated error of our TL T g data is of
the order of 2 °C. A further reduction of the error may be
possible, since the temperature fluctuation induced by the
laser beam necessary to obtain the TL signal is of the order
of mK. This indicates therefore that accurate measurements
very close to the glass transition can be obtained by this
sensitive technique by reducing the temperature intervals between the data points.
IV. CONCLUSIONS AND PERSPECTIVES FOR
FUTURE WORK
In conclusion, in this article we discuss the use of thermal lens technique to determine the thermo-optical proper-
FIG. 11. Temperature derivative of the thermal lens signal 共a兲 and the DTA
data 共b兲 for the PGIZCa glass.
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296
Rohling et al.
Rev. Sci. Instrum., Vol. 74, No. 1, January 2003
6
FIG. 12. Temperature derivative of the thermal lens signal 共a兲 and the DTA
data 共b兲 for the ZBLAN glass.
ties of transparent materials as a function of the temperature.
It is also discussed how the experimentally determined TL
parameters can be used to locate the glass transition of these
materials. The results showed the ability of the thermal lens
technique to perform the measurements very close to the
phase transition. As compared to the conventional calorimetric methods we should emphasize that this technique does
not require the use of a reference sample. As a future work,
we propose that the technique could be adapted to a compact
device, called differential thermal lens scanning, especially
designed for the investigation of the phase transition in transparent materials.
ACKNOWLEDGMENTS
The authors are thankful to the Brazilian Agencies,
CNPq, CAPES, and Fundaç:o Araucária for the financial
support of this work.
1
J. P. Gordon, R. C. C. Leite, R. S. More, S. P. S. Porto, and J. R. Whinnery, Bull. Am. Phys. Soc. 9, 501 共1964兲.
2
J. P. Gordon, R. C. C. Leite, R. S. More, S. P. S. Porto, and J. R. Whinnery, Appl. Phys. Lett. 5, 141 共1964兲.
3
J. P. Gordon, R. C. C. Leite, R. S. More, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 共1965兲.
4
J. Shen, Ph.D. thesis, UMIST, Manchester, 1993.
5
J. Shen, R. D. Lowe, and R. D. Snook, Chem. Phys. 165, 385 共1992兲.
M. L. Baesso, J. Shen, and R. D. Snook, Chem. Phys. Lett. 197, 255
共1992兲.
7
M. L. Baesso, A. C. Bento, A. A. Andrade, T. Catunda, J. A. Sampaio, and
S. Gama, J. Non-Cryst. Solids 219, 165 共1997兲.
8
S. E. Bialkowsk, Photothermal Spectroscopy Methods for Chemical
Analysis 共Wiley, New York, 1996兲.
9
S. M. Lima, J. A. Sampaio, T. Catunda, A. C. Bento, L. C. M. Miranda,
and M. L. Baesso, J. Non-Cryst. Solids 273, 215 共2000兲.
10
M. L. Baesso, J. Shen, and R. D. Snook, J. Appl. Phys. 75, 3732 共1994兲.
11
S. Shen, M. L. Baesso, and R. D. Snook, J. Appl. Phys. 75, 3738 共1994兲.
12
M. L. Baesso, A. C. Bento, A. A. Andrade, T. Catunda, E. Pecoraro, L. A.
O. Nunes, J. A. Sampaio, and S. Gama, Phys. Rev. B 57, 10 545 共1998兲.
13
S. M. Lima, T. Catunda, R. Lebullenger, A. C. Hernandes, M. L. Baesso,
A. C. Bento, and L. C. M. Miranda, Phys. Rev. B 60, 15 173 共1999兲.
14
S. M. Lima, A. A. Andrade, R. Lebullenger, A. C. Hernandes, T. Catunda,
and M. L. Baesso, Appl. Phys. Lett. 78, 3220 共2001兲.
15
M. L. Baesso, A. S. Fontes, A. C. Bento, and L. C. M. Miranda, Anal. Sci.
17, S103 共2001兲.
16
M. L. Baesso, A. C. Bento, A. A. Andrade, T. Catunda, J. A. Sampaio, and
S. Gama, J. Non-Cryst. Solids 219, 165 共1997兲.
17
M. L. Baesso, A. C. Bento, A. R. Duarte, A. M. Neto, and L. C. M.
Miranda, J. Appl. Phys. 85, 8112 共1999兲.
18
S. M. Lima, J. A. Sampaio, T. Catunda, R. Lebullenger, A. C. Hernandes,
M. L. Baesso, A. C. Bento, and F. C. G. Gandra, J. Non-Cryst. Solids 256
& 257, 337 共1999兲.
19
J. A. Sampaio, T. Catunda, S. Gama, and M. L. Baesso, J. Non-Cryst.
Solids 284, 210 共2001兲.
20
J. A. Sampaio, T. Catunda, F. C. G. Gandra, S. Gama, A. C. Bento, L. C.
M. Miranda, and M. L. Baesso, J. Non-Cryst. Solids 247, 196 共1999兲.
21
E. Peliçon, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, D. F.
de Souza, S. L. Oliveira, J. A. Sampaio, S. M. Lima, L. A. O. Nunes, and
T. Catinda, J. Non-Cryst. Solids 304, 244 共2002兲.
22
S. J. Sheldon, L. V. Knight, and J. M. Thorne, Appl. Opt. 21, 1663 共1982兲.
23
J. A. Sampaio, S. M. Lima, T. Catunda, A. N. Medina, A. C. Bento, and
M. L. Baesso, J. Non-Cryst. Solids 304, 315 共2002兲.
24
J. H. Rohling, A. N. Medina, A. C. Bento, J. R. D. Pereira, M. L. Baesso,
and L. C. M. Miranda, J. Phys. D 34, 407 共2001兲.
25
J. H. Rohling, A. M. F. Caldeira, J. R. D. Pereira, A. N. Medina, A. C.
Bento, M. L. Baesso, and L. C. M. Miranda, J. Appl. Phys. 89, 2220
共2001兲.
26
J. H. Rohling, A. N. Medina, J. R. D. Pereira, A. F. Rubira, A. C. Bento,
L. C. M. Miranda, and M. L. Baesso, Anal. Sci. 17, S103 共2001兲.
27
J. H. Rohling, J. Mura, A. N. Medina, A. J. Palangana, A. C. Bento, J. R.
D. Pereira, M. L. Baesso, and L. C. M. Miranda, Braz. J. Phys. 32, 575
共2002兲.
28
J. R. D. Pereira, A. J. Palangana, A. M. Mansanares, E. C. da Silva, A. C.
Bento, and M. L. Baesso, Phys. Rev. E 61, 5410 共2000兲.
29
J. R. D. Pereira, A. J. Palangana, A. M. Mansanares, and M. L. Baesso,
Phys. Rev. E 64, 701 共2001兲.
30
J. R. D. Pereira, A. J. Palangana, A. M. Mansanares, E. C. da Silva, A. C.
Bento, and M. L. Baesso, Anal. Sci. 17, 175 共2001兲.
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