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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 6, Iss. 3 — Mar. 18, 2011
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Soret effect and photochemical reaction in liquids with laser-induced local heating

L. C. Malacarne, N. G. C. Astrath, A. N. Medina, L. S. Herculano, M. L. Baesso, P. R. B. Pedreira, J. Shen, Q. Wen, K. H. Michaelian, and C. Fairbridge  »View Author Affiliations


Optics Express, Vol. 19, Issue 5, pp. 4047-4058 (2011)
http://dx.doi.org/10.1364/OE.19.004047


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Abstract

We report a theoretical model and experimental results for laser-induced local heating in liquids, and propose a method to detect and quantify the contributions of photochemical and Soret effects in several different situations. The time-dependent thermal and mass diffusion equations in the presence and absence of laser excitation are solved. The two effects can produce similar transients for the laser-on refractive index gradient, but very different laser-off behavior. The Soret effect, also called thermal diffusion, and photochemical reaction contributions in photochemically reacting aqueous Cr(VI)-diphenylcarbazide, Eosin Y, and Eosin Y-doped micellar solutions, are decoupled in this work. The extensive use of lasers in various optical techniques suggests that the results may have significance extending from physical-chemical to biological applications.

© 2011 Optical Society of America

1. Introduction

Thermodiffusion or the Soret effect, which arises from a thermally induced concentration gradient in a fluid mixture, has been known for more than a century [1

1. C. Soret, “Concentrations differentes d’une dissolution dont deux parties sont a’ des temperatures differentes,” Arch. Sci. Phys. Nat. 2, 48–61 (1879).

]. The Soret effect is observed in many molecular systems, including gases and simple liquid mixtures. Moreover, thermodiffusion of complex fluids such as polymer solutions [2

2. B. J. deGans, R. Kita, S. Wiegand, and J. Luettmer-Strathmann, “Unusual thermal diffusion in polymer solutions,” Phys. Rev. Lett. 91, 245501 (2003). [CrossRef]

, 3

3. A. Würger, “Molecular-weight dependent thermal diffusion in dilute polymer solutions,” Phys. Rev. Lett. 102, 078302 (2009). [CrossRef] [PubMed]

], colloidal dispersions [4

4. R. Piazza and A. Guarino, “Soret effect in interacting micellar solutions,” Phys. Rev. Lett. 88, 208302 (2002). [CrossRef] [PubMed]

6

6. S. N. Rasuli and R. Golestanian, “Soret motion of a charged spherical colloid,” Phys. Rev. Lett. 101, 108301 (2008). [CrossRef] [PubMed]

] and magnetic fluids [7

7. J. Lenglet, A. Bourdon, J. C. Bacri, and G. Demouchy, “Thermodiffusion in magnetic colloids evidenced and studied by forced Rayleigh scattering experiments,” Phys. Rev. E 65, 031408 (2002). [CrossRef]

, 8

8. B. Hoffmann, W. Köhler, and M. Krekhova, “On the mechanism of transient bleaching of the optical absorption of ferrofluids and dyed liquids,” J. Chem. Phys. 118, 3237–3242 (2003). [CrossRef]

] has been investigated experimentally in recent years. After several decades of research the phenomenon has been framed in terms of nonequilibrium thermodynamics concepts [9

9. S. R. De Groot and P. Mazur, Nonequilibrium Thermodynamics (North Holland, 1962)

]; however, a satisfactory microscopic description incorporating a quantitative model that accounts for the complexity of the interparticle interactions in the mixture has not yet been achieved [10

10. F. Huang, P. Chakraborty, C. C. Lundstrom, C. Holmden, J. J. G. Glessner, S. W. Kieffer, and C. E. Lesher, “Isotope fractionation in silicate melts by thermal diffusion,” Nature (London) 464, 396–400 (2010). [CrossRef]

22

22. D. Vigolo, S. Buzzaccaro, and R. Piazza, “Thermophoresis and thermoelectricity in surfactant solutions,” Langmuir 26, 7792–7801 (2010). [CrossRef] [PubMed]

].

Identification of individual contributions is essential for a proper understanding of the dominant physical mechanisms in an experiment. For this reason, it is highly desirable to have a unified and quantitative (and ideally simple) method to distinguish among the effects that occur during laser excitation. Previously, we developed a theoretical model to accommodate PCR in thermal lens measurement, and the model was experimentally proved [27

27. P. R. B. Pedreira, L. R. Hirsch, J. R. D. Pereira, A. N. Medina, A. C. Bento, M. L. Baesso, M. C. Rollemberg, M. Franko, and J. Shen, “Real-time quantitative investigation of photochemical reaction using thermal lens measurements: theory and experiment,” J. Appl. Phys. 100, 044906 (2006). [CrossRef]

29

29. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, P. A. Santoro, and M. L. Baesso, “Arrhenius behavior of hydrocarbon fuel photochemical reaction rates by thermal lens spectroscopy,” Appl. Phys. Lett. 95, 191902 (2009). [CrossRef]

]. In this communication, we report a theoretical model and experimental measurements that consider TL, PCR and Soret effects, and propose a method to detect and identify the contribution of each one during a laser-induced local heating experiment. Photochemical reactions and the TL effect in aqueous Cr(VI)-diphenylcarbazide and Eosin Y solutions as well as the PCR and Soret effects that accompany the TL in dye-doped micellar solutions, are investigated.

2. Thermal Lens theory in the presence of PCR and Soret effects

Thermal lens spectrometry is a well-established highly sensitive photothermal technique, which can be used for the thermal, optical and chemical analysis of solids, liquids, and gases [28

28. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, A. N. Medina, and M. L. Baesso, “Thermal-lens study of photochemical reaction kinetics,” Opt. Lett. 34, 3460–3462 (2009). [CrossRef] [PubMed]

32

32. N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20K,” Phys. Rev. B 71, 214202 (2005). [CrossRef]

]. TL experiments are based on measurement of the temperature rise in a sample that results from non-radiative de-excitation occurring after the absorption of optical energy. The generated transverse temperature gradient induces a refractive index gradient which behaves like an optical lens. The refractive index gradient produces an optical path length variation and then a phase shift on a probe beam, causing a change in the intensity profile of the probe beam that propagates through the TL. Quantitative information on the physical properties of the sample can be obtained by measuring the variation of the probe beam intensity.

The effect of a photo-induced chemical reaction has recently been introduced into the TL theoretical model for liquids [27

27. P. R. B. Pedreira, L. R. Hirsch, J. R. D. Pereira, A. N. Medina, A. C. Bento, M. L. Baesso, M. C. Rollemberg, M. Franko, and J. Shen, “Real-time quantitative investigation of photochemical reaction using thermal lens measurements: theory and experiment,” J. Appl. Phys. 100, 044906 (2006). [CrossRef]

29

29. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, P. A. Santoro, and M. L. Baesso, “Arrhenius behavior of hydrocarbon fuel photochemical reaction rates by thermal lens spectroscopy,” Appl. Phys. Lett. 95, 191902 (2009). [CrossRef]

]. During laser excitation of a photo-reacting substance, the concentration of the absorbing species in the excited volume is reduced, generating a radial gradient from outside to inside this volume. Mass diffusion may then compensate for part of the consumed species. After prolonged excitation the amplitude of the TL signal reaches a steady state, reflecting an equilibrium in the concentration of the absorbing species. The time-dependent optical absorption coefficient is Ae(t) = C(t)ɛ, where ɛ is the molar optical absorption coefficient of the sample and C(t) [29

29. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, P. A. Santoro, and M. L. Baesso, “Arrhenius behavior of hydrocarbon fuel photochemical reaction rates by thermal lens spectroscopy,” Appl. Phys. Lett. 95, 191902 (2009). [CrossRef]

] denotes the time dependence of the species concentration. This model has been used to investigate the temperature dependence of thermo-optical and photochemical reaction properties of hydrocarbon fuels, yielding quantitative information about these liquids [27

27. P. R. B. Pedreira, L. R. Hirsch, J. R. D. Pereira, A. N. Medina, A. C. Bento, M. L. Baesso, M. C. Rollemberg, M. Franko, and J. Shen, “Real-time quantitative investigation of photochemical reaction using thermal lens measurements: theory and experiment,” J. Appl. Phys. 100, 044906 (2006). [CrossRef]

29

29. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, P. A. Santoro, and M. L. Baesso, “Arrhenius behavior of hydrocarbon fuel photochemical reaction rates by thermal lens spectroscopy,” Appl. Phys. Lett. 95, 191902 (2009). [CrossRef]

].

2.1. Temperature and concentration gradients

We first consider the case in which a TEM00 laser excites the sample at 0 < tξ (laser-on). No excitation exists when t > ξ (laser-off). It will be shown later that the laser-off TL transient is required to distinguish between Soret and PCR effects in the TL transient signal. For an isotropic, weakly absorbing material, the temperature rise distribution in a sample is described by the heat conduction differential equation
T(r,z,t)tD2T(r,z,t)=Q(r,t),
(1)
where D = k/(ρc) is the thermal diffusivity, k is thermal conductivity, ρ is mass density and c is the specific heat of the sample. The source term Q(r,t), described for on-off excitation under the assumption that the sample exhibits a photochemical reaction [28

28. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, A. N. Medina, and M. L. Baesso, “Thermal-lens study of photochemical reaction kinetics,” Opt. Lett. 34, 3460–3462 (2009). [CrossRef] [PubMed]

], can be written as
Q(r,t)=Q0e2r2/ω2C(t)ɛ[1H(tξ)],
(2)
in which H, the Heaviside Theta function, accounts for the laser-on/off excitation. C(t) = (C0Ceq) exp(−KTt) + Ceq is the concentration of absorbing species, C0 denotes the initial concentration, and Ceq is the equilibrium concentration in the illuminated area. The total reaction rate constant KT is given by KT = kr + kd, where kr and kd represent the reaction and species diffusion rates, respectively. Q0 = 2Peφ/(ρcπω2), where ω and Pe are the radius and power of the excitation laser, respectively. φ is the fraction of the absorbed energy available for conversion to heat. The heat produced by absorbed excitation beam is treated as a line heat source, and the sample as an infinite medium with respect to the excitation beam radius ω. The heat conduction equation is solved using integral transform methods (Laplace, Fourier cosine, and Hankel transform methods) and Eq. (2), yielding
T(r,t)=Q0ɛ[Ceqt0te2r2/ω21+2τ/tc1+2τ/tcdτ+(C0Ceq)eKTtt0teKTτe2r2/ω21+2τ/tc1+2τ/tcdτ],
(3)
with t0 = 0 for tξ and t0 = tξ for t > ξ. Eq. (3) describes the temperature rise in a photo-reacting sample during laser-on and laser-off periods. The characteristic heat diffusion time constant is tc = ω2/(4D).

In addition to the PCR occuring during laser excitation, the thermal gradient may also induce a concentration gradient - the Soret effect. The concentration gradient for a binary mixture in the absence of convection is given by the the mass flow diffusion equation [5

5. R. Rusconi, L. Isa, and R. Piazza, “Thermal-lensing measurement of particle thermophoresis in aqueous dispersions,” J. Opt. Soc. Am. B 21, 605–616 (2004). [CrossRef]

]:
c(r,t)tDm2c(r,t)=STc¯(1c¯)Dm2T(r,t)
(4)
where Dm is the mass diffusion coefficient, is the initial average concentration and ST = DT/Dm is the Soret coefficient. DT denotes the coefficient of thermal diffusion. Writing the characteristic mass diffusion time as tm = ω2/(4Dm) ≫ tc, for ttc [5

5. R. Rusconi, L. Isa, and R. Piazza, “Thermal-lensing measurement of particle thermophoresis in aqueous dispersions,” J. Opt. Soc. Am. B 21, 605–616 (2004). [CrossRef]

], one can replace ∇2T(r,t) in Eq. (4) with the stationary solution of the heat conduction equation, Eq. (1). Within this approximation, the mass diffusion equation is formally the same as the heat conduction equation. We assume that the sample is sufficiently thick that the axial null thermal flux approximation can be applied as shown in Ref. [28

28. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, A. N. Medina, and M. L. Baesso, “Thermal-lens study of photochemical reaction kinetics,” Opt. Lett. 34, 3460–3462 (2009). [CrossRef] [PubMed]

], and that viscous surface effects can be neglected. In addition, the laser-excited area is much smaller than the sample dimensions. The solution of the mass flow diffusion equation, Eq. (4), using Eqs. (1) and (2), is
c(r,t)=STc¯(1c¯)DmDQ0ɛ[Ceqt0te2r2/ω21+2τ/tm1+2τ/tmdτ+(C0Ceq)eKTtt0teKTτe2r2/ω21+2τ/tm1+2τ/tmdτ].
(5)

2.2. Probe beam phase shifts

When a TEM00 Gaussian probe laser beam propagates through the illuminated volume of the liquid sample, its wave front is slightly distorted, and the distortion can be expressed as an additional phase shift, which can be calculated by considering the problem from the point of view of optical path length variation regarding the axis as ϕ(r,t)=(2π/λp)0L[n(r,z,t)n(0,z,t)]dz. Here n is the refractive index, L is the sample thickness, and λp the probe beam wavelength. The total additional phase shift on the probe beam when both the PCR and Soret effects occur is the superposition of the phase shifts, ϕPCR and ϕSoret, caused by the temperature gradient [28

28. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, A. N. Medina, and M. L. Baesso, “Thermal-lens study of photochemical reaction kinetics,” Opt. Lett. 34, 3460–3462 (2009). [CrossRef] [PubMed]

] and the mass gradient (Soret effect) [5

5. R. Rusconi, L. Isa, and R. Piazza, “Thermal-lensing measurement of particle thermophoresis in aqueous dispersions,” J. Opt. Soc. Am. B 21, 605–616 (2004). [CrossRef]

], respectively. Both effects give rise to z-independent refractive index gradients that act as optical elements in the sample. The phase shifts are given by
ϕPCR(r,t)=2πλpLdndT[T(r,t)T(0,t)],
(6)
and
ϕSoret(r,t)=2πλpLdndc[c(r,t)c(0,t)].
(7)
The temperature and concentration coefficients of the refractive index at the probe beam wavelength λp are dn/dT and dn/dc, respectively. In other words, the phase shift describes the distortions of the probe beam caused by the temperature and concentration changes in the medium. From Eq. (6) using Eq. (3), ϕPCR can be calculated as
ϕPCR(g,t)=θthtc[crt0t1e2mg1+2τ/tc1+2τ/tcdτ+(1cr)eKTt0teKTτ(1e2mg1+2τ/tc)1+2τ/tcdτ].
(8)
From Eq. (7) using Eq. (5) the phase shift produced by the Soret effect can be written
ϕSoret(g,t)=θmtm[crt0t1e2mg1+2τ/tm1+2τ/tmdτ+(1cr)eKTt0teKTτ(1e2mg1+2τ/tm)1+2τ/tmdτ].
(9)
In Eqs. (8) and (9), the variables g=r2/ω1P2, m = (ω1P/ω)2, and cr = Ceq/C0 have been introduced. In addition, θth is defined as
θth=PeC0ɛLkλpdndT,
(10)
and θm is
θm=STc¯(1c¯)PeC0ɛLkλpdndc.
(11)
ω1P is the radius of the probe laser beam in the sample. When a PCR and the Soret effect both occur, the total phase shift is
ϕTL(g,t)=ϕPCR(g,t)+ϕSoret(g,t).
(12)

2.3. Thermal lens intensity at the detector plane

The propagation of the emergent probe beam from the sample with the additional phase shift to the detector plane can be treated as a diffraction phenomenon, and thus calculated using Fresnel diffraction theory [30

30. J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992). [CrossRef]

]. The complex amplitude at the centre of the probe beam spot at the detector plane is given by [30

30. J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992). [CrossRef]

]
U(Z1+Z2,t)=C0e(1+iV)giϕTL(g,t)dg,
(13)
where Z1 and Z2 are the distances from the probe beam waist to the sample and the sample to the detector plane, respectively. In Eq. (13), V = Z1/ZC, ZC is the confocal distance of the probe beam, and C is a constant [27

27. P. R. B. Pedreira, L. R. Hirsch, J. R. D. Pereira, A. N. Medina, A. C. Bento, M. L. Baesso, M. C. Rollemberg, M. Franko, and J. Shen, “Real-time quantitative investigation of photochemical reaction using thermal lens measurements: theory and experiment,” J. Appl. Phys. 100, 044906 (2006). [CrossRef]

]. The probe beam intensity I(t) at the detector plane can be calculated as
I(t)=|U(Z1+Z2,t)|2.
(14)
As special cases, pure Soret or PCR effects are obtained in this model by taking KT = 0 or θm = 0, respectively, in Eq. (12). I(t) with Eq. (12) is used to fit the experimental data, where m, V, ω, Pe, L and λp are experimental setup parameters previously determined and θth, θm, tc, tm, cr and KT are obtained from the fits.

3. Experimental results and discussion

Figure 1 shows a schematic illustration of the experimental apparatuses used in this work. In setup A, a solid-state 532.0-nm laser and a 543.5-nm He-Ne laser provided the excitation and probe beams, respectively. Setup B used a 514.5-nm Argon ion excitation laser and a 632.8-nm He-Ne probe laser. A shutter controlled the exposure of the samples to the excitation beams, and the signal from photodiode P1 triggered the digital oscilloscope that recorded the TL signal. Converging lens L1 focused the excitation beams, with the sample placed at its focal plane. The probe beams were focused by lens L2, and the sample was positioned near its confocal plane. A small angle γ < 1.5° existed between the excitation and probe beams. After passing through the TL the probe beam propagated to photodiode P2, positioned in the far field (Z2 ≈ 5m). A pinhole was placed in front of photodiode P2, such that only the central part of the probe beam was detected and recorded by the oscilloscope. The probe-beam power absorbed by the sample was assumed to be negligible as compared to the power of the excitation beam in both cases. Sample temperatures were maintained at 28°C. For setup B, ω = 55μm was measured while m = 23.6 and V = 3.05 were calculated using the relations presented above. Setup A was used for the Cr(VI) solution with ω = 72μm, m = 45.2 and V = 8.3. These geometrical parameters, which were constant during the measurements, were determined as described in Ref. [30

30. J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992). [CrossRef]

].

Fig. 1 Schematic diagrams of the time-resolved TL experimental apparatuses. Mi, Li, and Pi denote mirrors, lenses, and photodiodes, respectively.

The present model was used to evaluate the thermo-optical, PCR and mass transport properties of four aqueous solutions: 40ppb Cr(VI)-diphenylcarbazide; 2.33ppm Eosin Y (EY) [27

27. P. R. B. Pedreira, L. R. Hirsch, J. R. D. Pereira, A. N. Medina, A. C. Bento, M. L. Baesso, M. C. Rollemberg, M. Franko, and J. Shen, “Real-time quantitative investigation of photochemical reaction using thermal lens measurements: theory and experiment,” J. Appl. Phys. 100, 044906 (2006). [CrossRef]

, 33

33. S. Hazebroucq, F. Labat, D. Lincot, and C. Adamo, “Theoretical insights on the electronic properties of eosin Y, an organic dye for photovoltaic applications,” J. Phys. Chem. A 112, 7264–7270 (2008). [CrossRef] [PubMed]

]; 6.5wt% Brij 35 [CH3(CH2)11(OCH2CH2)23OH, polyoxyethylene 23 lauryl ether] in cobalt nitrate (absorbance 0.04) [34

34. N. Arnaud and J. Georges, “On the analytical use of the Soret-enhanced thermal lens signal in aqueous solutions,” Anal. Chim. Acta 445, 239–244 (2001). [CrossRef]

]; 5.6wt% Brij 35 in 6.66ppm Eosin Y. The latter two solutions are denoted below as Brij and EY-Brij, respectively. The Cr(VI) and Eosin Y solutions were selected because their photo-reaction characteristics are well known [27

27. P. R. B. Pedreira, L. R. Hirsch, J. R. D. Pereira, A. N. Medina, A. C. Bento, M. L. Baesso, M. C. Rollemberg, M. Franko, and J. Shen, “Real-time quantitative investigation of photochemical reaction using thermal lens measurements: theory and experiment,” J. Appl. Phys. 100, 044906 (2006). [CrossRef]

, 33

33. S. Hazebroucq, F. Labat, D. Lincot, and C. Adamo, “Theoretical insights on the electronic properties of eosin Y, an organic dye for photovoltaic applications,” J. Phys. Chem. A 112, 7264–7270 (2008). [CrossRef] [PubMed]

]; the Soret effect does not occur in these solutions. By contrast, Brij solutions are known to exhibit the Soret effect [34

34. N. Arnaud and J. Georges, “On the analytical use of the Soret-enhanced thermal lens signal in aqueous solutions,” Anal. Chim. Acta 445, 239–244 (2001). [CrossRef]

]. Micellar solutions of Brij 35 were prepared by weighing the required amounts of Brij 35 in distilled water. Cobalt nitrate was chosen because it does not fluoresce or exhibit a photochemical reaction, and is not expected to affect local or bulk solvent properties significantly. These solutions were placed in L = 0.5cm quartz cuvettes. Photo-reactive dyes such as Cr(VI) and Eosin as well as micellar Brij solutions are commonly used to improve spectroscopic sensitivity [33

33. S. Hazebroucq, F. Labat, D. Lincot, and C. Adamo, “Theoretical insights on the electronic properties of eosin Y, an organic dye for photovoltaic applications,” J. Phys. Chem. A 112, 7264–7270 (2008). [CrossRef] [PubMed]

]; consequently the study of these substances is thought to be of considerable importance.

Micellar solutions consist of molecular assemblies that are in dynamic equilibrium with surfactant monomers and the solvent. In water, each aggregate is composed of a liquid-like apolar core and of more-polar head groups oriented outward toward the bulk solvent [26

26. N. Arnaud and J. Georges, “Thermal lens spectrometry in aqueous solutions of Brij 35: investigation of micelle effects on the time-resolved and steady-state signals,” Spectrochim. Acta A 57, 1085–1092 (2001). [CrossRef]

]. Micellar solutions are microscopically heterogeneous and can be considered as pseudo-biphasic systems where the two components have different physico-chemical properties. The nonionic Brij 35 used in this work has a critical micelle concentration (cmc) of 0.04 – 0.10g/dm3. The geometry of the micelles is ambiguous, although their structures have recently been modelled using spherical and/or ellipsoidal cores. The radius of the micelles is about 40Å. As mentioned above, the Cr(VI) and EY samples analyzed here are expected to display only a PCR while the Brij solution exhibit only the Soret effect. Accordingly, when EY and Brij solutions are mixed, both PCR and Soret effects should be observable.

Two examples of normalized TL signals for the Cr(VI) and Brij solutions are presented in Fig. 2. A transient for the 40ppb Cr(VI) sample is shown in Fig. 2(a). For the laser on-transient, one can observe that the probe beam centre intensity sensed by the detector decreases due to the existence of the pure TL effect (without PCR modification) for a duration less than 200ms. Subsequently, the PCR causes a reduction in the concentration of the absorbing species; the pure TL effect diminishes and the probe beam intensity increases. Mass diffusion causes the PCR induced gradient in the concentration of the absorbing species in the excited region to eventually approach an equilibrium value, and the probe beam intensity is affected correspondingly. On the other hand, qualitatively different behavior is observed for the micellar Brij sample [Fig. 2(b)]; the longer time scale in this graph should also be noted. This result is discussed below.

Fig. 2 Normalized experimental TL signals I(t)/I(0) (open circles) for (a) 40ppb Cr(VI), and (b) 6.5wt% Brij 35 in aqueous cobalt nitrate (absorbance 0.04) solutions. Dotted (green) and dashed (red) lines denotes on-transient least-square curve fits using Eq. (14) and Eq. (12) with θm = 0 and KT = 0, respectively. Off-transient curves were generated using the parameters obtained in the fits.

Turning to the laser-off transient for Cr(VI) [Fig. 2(a)], one finds that use of Eq. (14) and Eq. (12) with KT = 0 (Soret) and the parameters obtained from the on-transient does not yield a satisfactory fit of the experimental data. On the other hand, the parameters derived from the PCR model, Eq. (12) with θm = 0, produce very good agreement between the calculated curve (off-transient dotted line) and the experimental data. This shows that the off-transient signal must be taken into account when processing the experimental data if the effect taking place in the sample is to be correctly identified. The fitted parameters obtained using Eq. (12) with θm = 0 (estimated uncertainty ±5%) are as follow: θth/PeL = 18.1W−1cm−1; cr = 0.59; and KT = 2s−1. The thermal diffusivity is D = 1.45 × 10−3cm2s−1.

The off-transient becomes also important for the micellar solution. Fig. 2(b) shows a typical TL on/off-transient for a micellar Brij solution. The initial rapid change in probe beam intensity, for t < 200ms, corresponds to a the thermally induced refractive index gradient with a time constant of approximately 5.3ms. The Soret effect then occurs under the influence of the temperature gradient, causing species to migrate and establish a concentration gradient. The on/off-transient was fitted to Eq. (14) using Eq. (12) with KT = 0 (Soret), yielding the results listed in Table 1. The agreement between theory and experimental results is very good and the obtained parameters agree well with the expected values for this sample [34

34. N. Arnaud and J. Georges, “On the analytical use of the Soret-enhanced thermal lens signal in aqueous solutions,” Anal. Chim. Acta 445, 239–244 (2001). [CrossRef]

].

Table 1. Results of Experimental Measurements for Aqueous Solutions

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Fig. 3 Normalized experimental TL signals I(t)/I(0) (open circles) for (a) aqueous 2.33ppm Eosin Y (EY), and (b) 5.6wt% Brij 35 in aqueous 6.66ppm Eosin Y solutions. Solid (blue) line, least-squares curve fit to Eq. (14) using both the PCR and Soret models with Eq. (12); dotted (green) and dashed (red) lines, on-transient fits using Eq. (12) with θm = 0 (only PCR) and KT = 0 (only Soret), respectively. The off-transient curves were generated with the same equations and the parameters obtained from the on-transient curves.

When Eosin Y is mixed with Brij 35 solution to form the EY-Brij sample, one might expect to observe both PCR and Soret effects in the TL transient. Fig. 3(b) presents the on/off-transient for this sample. Again, the on-transient could erroneously be attributed to only the PCR or Soret effect; however, the off-transient displays an intriguing trend as compared with that in Fig. 3(a). The on-transient was separately fitted to Eq. (14) using Eq. (12) with θm = 0 (PCR) and KT = 0 (Soret) and the resulting parameters were used to generate off-transient curves for the PCR (dotted line) and Soret (dashed line) contributions. As might be expected, the PCR contribution alone does not fit the experimental data well; this suggests that the off-transient may incorporate both PCR and Soret components. In fact, the use of Eq. (14) and Eq. (12) makes it possible to fit all of the data (both on- and off-transients). The result (continuous line) agrees very well with experiment, again indicating that - although the on-transient appears to be due to either the PCR or Soret effects - the off-transient must be taken into consideration to arrive at a correct interpretation of the results.

In fact, most complex fluids used to study mass transport properties (particularly dye-doped mixtures [33

33. S. Hazebroucq, F. Labat, D. Lincot, and C. Adamo, “Theoretical insights on the electronic properties of eosin Y, an organic dye for photovoltaic applications,” J. Phys. Chem. A 112, 7264–7270 (2008). [CrossRef] [PubMed]

]) exhibit the PCR effect to some extent. This situation can lead to an incorrect interpretation when the experimental data are analyzed. The theoretical model presented here clearly shows that it is possible to distinguish between the PCR and Soret phenomena that accompany laser excitation of fluid mixtures.

Application of the above-mentioned procedure to the EY-Brij solution for longer times is depicted in Fig. 4. This figure shows that the PCR and Soret effects are concurrent and that both contributions eventually reach a steady-state for times greater than 3s (on-transient). The solid line in Fig. 4 displays the least-squares on/off curve fit obtained with Eq. (14) using Eq. (12). Several transients were recorded at different excitation powers; the resulting parameters are plotted in Fig. 5. Reasonably, the thermal diffusivities (D) for the samples are very close to the accepted value of 1.45 × 10−3cm2s−1 for pure water. The mass diffusion coefficients (Dm) of the samples are listed in Table 1. These values were calculated using the relations D = ω2/4tc and Dm = ω2/4tm, respectively. Dotted and dashed lines in Fig. 4 represent the on/off calculated contributions of the PCR and Soret effects using the same equations with θm = 0 and kT = 0, respectively, and parameters obtained for the on- and off-transient curves.

Fig. 4 Normalized experimental TL signal I(t)/I(0) (open circles) for the 5.6wt% Brij 35 in aqueous 6.66ppm Eosin Y solution. Solid line (blue), least-squares curve fit using both the PCR and Soret models (Eq. (14) using Eq. (12)); dotted (green) and dashed (red) lines, on/off-contributions of the PCR and Soret effects using the same equations with θm = 0 and kT = 0, respectively.
Fig. 5 Excitation power dependence of (a) θth and (b) θm for the samples.

Additional information on reaction kinetics and transport can also be obtained from the cr, θth, and θm data. Figs. 5(a) and 5(b) show the excitation power dependencies of θth and θm. Analysis of solutions using the adopted procedure described here provides information correlated to their thermal, optical and chemical, and transport properties which is fundamental for the characterization of fluids [29

29. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, P. A. Santoro, and M. L. Baesso, “Arrhenius behavior of hydrocarbon fuel photochemical reaction rates by thermal lens spectroscopy,” Appl. Phys. Lett. 95, 191902 (2009). [CrossRef]

]. Therefore, systematic study is necessary for an accurate characterization of binary mixtures. In particular, the concentration dependencies of the refractive index and the influence of sample thickness and other geometric parameters on the TL should be taken into consideration.

4. Conclusions

In this work, we have distinguished between the contributions to the laser-induced refractive index gradients in liquid samples due to Soret effect and photochemical reactions. A theoretical model that incorporates laser-induced thermal, Soret and PCR effects was introduced. Thermal and mass diffusion equations in the presence and absence of laser excitation were solved. The PCR and Soret effects can produce similar behavior for laser-on transients, but very different behavior for laser-off transients. This situation, in fact, can lead to erroneous interpretation of experimental results.

The theoretical model presented here decouples the PCR and Soret contributions in liquid mixtures subject to laser excitation. Aqueous Cr(VI)-diphenylcarbazide, Eosin Y and micellar Brij solutions were investigated, and quantitative results for their thermal, optical and transport properties were obtained. Finally, it was demonstrated that TL spectrometry and the theoretical model presented here furnish information regarding the primary mechanisms during laser excitation of these complex samples that is not obtainable by traditional means. In view of the extensive use of lasers in a variety of optical techniques and configurations, our results may prove to have significance in both physical-chemical experiments and biological applications.

Acknowledgments

The authors are thankful to the Brazilian Agencies CAPES, CNPq, and Fundação Araucária for financial support of this work.

References and links

1.

C. Soret, “Concentrations differentes d’une dissolution dont deux parties sont a’ des temperatures differentes,” Arch. Sci. Phys. Nat. 2, 48–61 (1879).

2.

B. J. deGans, R. Kita, S. Wiegand, and J. Luettmer-Strathmann, “Unusual thermal diffusion in polymer solutions,” Phys. Rev. Lett. 91, 245501 (2003). [CrossRef]

3.

A. Würger, “Molecular-weight dependent thermal diffusion in dilute polymer solutions,” Phys. Rev. Lett. 102, 078302 (2009). [CrossRef] [PubMed]

4.

R. Piazza and A. Guarino, “Soret effect in interacting micellar solutions,” Phys. Rev. Lett. 88, 208302 (2002). [CrossRef] [PubMed]

5.

R. Rusconi, L. Isa, and R. Piazza, “Thermal-lensing measurement of particle thermophoresis in aqueous dispersions,” J. Opt. Soc. Am. B 21, 605–616 (2004). [CrossRef]

6.

S. N. Rasuli and R. Golestanian, “Soret motion of a charged spherical colloid,” Phys. Rev. Lett. 101, 108301 (2008). [CrossRef] [PubMed]

7.

J. Lenglet, A. Bourdon, J. C. Bacri, and G. Demouchy, “Thermodiffusion in magnetic colloids evidenced and studied by forced Rayleigh scattering experiments,” Phys. Rev. E 65, 031408 (2002). [CrossRef]

8.

B. Hoffmann, W. Köhler, and M. Krekhova, “On the mechanism of transient bleaching of the optical absorption of ferrofluids and dyed liquids,” J. Chem. Phys. 118, 3237–3242 (2003). [CrossRef]

9.

S. R. De Groot and P. Mazur, Nonequilibrium Thermodynamics (North Holland, 1962)

10.

F. Huang, P. Chakraborty, C. C. Lundstrom, C. Holmden, J. J. G. Glessner, S. W. Kieffer, and C. E. Lesher, “Isotope fractionation in silicate melts by thermal diffusion,” Nature (London) 464, 396–400 (2010). [CrossRef]

11.

M. Giglio and A. Vendramini, “Thermal-diffusion measurements near a consolute critical-point,” Phys. Rev. Lett. 34, 561–564 (1975). [CrossRef]

12.

C. Debuschewitz and W. Köhler, “Molecular origin of thermal diffusion in benzene plus cyclohexane mixtures,” Phys. Rev. Lett. 87, 055901 (2001). [CrossRef] [PubMed]

13.

P. A. Artola and B. Rousseau, “Microscopic interpretation of a pure chemical contribution to the Soret effect,” Phys. Rev. Lett. 98, 125901 (2007). [CrossRef] [PubMed]

14.

D. Jung and M. Lücke, “Localized waves without the existence of extended waves: oscillatory convection of binary mixtures with strong Soret effect,” Phys. Rev. Lett. 89, 054502 (2002). [CrossRef] [PubMed]

15.

K. I. Morozov, “Soret effect in molecular mixtures,” Phys. Rev. E 79, 031204 (2009). [CrossRef]

16.

A. Parola and R. Piazza, “Particle thermophoresis in liquids,” Eur. Phys. J. E 15, 255–263 (2004). [CrossRef] [PubMed]

17.

N. Ghofraniha, C. Conti, G. Ruocco, and F. Zamponi, “Time-dependent nonlinear optical susceptibility of an out-of-equilibrium soft material,” Phys. Rev. Lett. 102, 038303 (2009). [CrossRef] [PubMed]

18.

S. Fayolle, T. Bickel, S. LeBoiteux, and A. Würger, “Thermodiffusion of charged micelles,” Phys. Rev. Lett. 95, 208301 (2005). [CrossRef] [PubMed]

19.

S. Duhr and D. Braun, “Thermophoretic depletion follows Boltzmann distribution,” Phys. Rev. Lett. 96, 168301 (2006). [CrossRef] [PubMed]

20.

S. A. Putnam, D. G. Cahil, and G. C. L. Wong, “Temperature dependence of thermodiffusion in aqueous suspensions of charged nanoparticles,” Langmuir 23, 9221–9228 (2007). [CrossRef] [PubMed]

21.

R. Piazza, “Thermophoresis: moving particles with thermal gradients,” Soft Matter 4, 1740–1744 (2008). [CrossRef]

22.

D. Vigolo, S. Buzzaccaro, and R. Piazza, “Thermophoresis and thermoelectricity in surfactant solutions,” Langmuir 26, 7792–7801 (2010). [CrossRef] [PubMed]

23.

D. Braun and A. Libchaber, “Trapping of DNA by thermophoretic depletion and convection,” Phys. Rev. Lett. 89, 188103 (2002). [CrossRef] [PubMed]

24.

S. Duhr and D. Braun, “Optothermal molecule trapping by opposing fluid flow with thermophoretic drift,” Phys. Rev. Lett. 97, 038103 (2006). [CrossRef] [PubMed]

25.

M. Ichikawa, H. Ichikawa, K. Yoshikawa, and Y. Kimura, “Extension of a DNA molecule by local heating with a laser,” Phys. Rev. Lett. 99, 148104 (2007). [CrossRef] [PubMed]

26.

N. Arnaud and J. Georges, “Thermal lens spectrometry in aqueous solutions of Brij 35: investigation of micelle effects on the time-resolved and steady-state signals,” Spectrochim. Acta A 57, 1085–1092 (2001). [CrossRef]

27.

P. R. B. Pedreira, L. R. Hirsch, J. R. D. Pereira, A. N. Medina, A. C. Bento, M. L. Baesso, M. C. Rollemberg, M. Franko, and J. Shen, “Real-time quantitative investigation of photochemical reaction using thermal lens measurements: theory and experiment,” J. Appl. Phys. 100, 044906 (2006). [CrossRef]

28.

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, A. N. Medina, and M. L. Baesso, “Thermal-lens study of photochemical reaction kinetics,” Opt. Lett. 34, 3460–3462 (2009). [CrossRef] [PubMed]

29.

N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, P. A. Santoro, and M. L. Baesso, “Arrhenius behavior of hydrocarbon fuel photochemical reaction rates by thermal lens spectroscopy,” Appl. Phys. Lett. 95, 191902 (2009). [CrossRef]

30.

J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992). [CrossRef]

31.

M. L. Baesso, J. Shen, and R. D. Snook, “Mode-mismatched thermal lens determination of temperature-coefficient of optical-path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732–3737 (1994). [CrossRef]

32.

N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20K,” Phys. Rev. B 71, 214202 (2005). [CrossRef]

33.

S. Hazebroucq, F. Labat, D. Lincot, and C. Adamo, “Theoretical insights on the electronic properties of eosin Y, an organic dye for photovoltaic applications,” J. Phys. Chem. A 112, 7264–7270 (2008). [CrossRef] [PubMed]

34.

N. Arnaud and J. Georges, “On the analytical use of the Soret-enhanced thermal lens signal in aqueous solutions,” Anal. Chim. Acta 445, 239–244 (2001). [CrossRef]

OCIS Codes
(350.5340) Other areas of optics : Photothermal effects
(350.6830) Other areas of optics : Thermal lensing

ToC Category:
Materials

History
Original Manuscript: January 13, 2011
Revised Manuscript: February 2, 2011
Manuscript Accepted: February 3, 2011
Published: February 15, 2011

Virtual Issues
Vol. 6, Iss. 3 Virtual Journal for Biomedical Optics

Citation
L. C. Malacarne, N. G. C. Astrath, A. N. Medina, L. S. Herculano, M. L. Baesso, P. R. B. Pedreira, J. Shen, Q. Wen, K. H. Michaelian, and C. Fairbridge, "Soret effect and photochemical reaction in liquids with laser-induced local heating," Opt. Express 19, 4047-4058 (2011)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-5-4047


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References

  1. C. Soret, “Concentrations differentes d’une dissolution dont deux parties sont a’ des temperatures differentes,” Arch. Sci. Phys. Nat. 2, 48–61 (1879).
  2. B. J. deGans, R. Kita, S. Wiegand, and J. Luettmer-Strathmann, “Unusual thermal diffusion in polymer solutions,” Phys. Rev. Lett. 91, 245501 (2003). [CrossRef]
  3. A. Würger, “Molecular-weight dependent thermal diffusion in dilute polymer solutions,” Phys. Rev. Lett. 102, 078302 (2009). [CrossRef] [PubMed]
  4. R. Piazza and A. Guarino, “Soret effect in interacting micellar solutions,” Phys. Rev. Lett. 88, 208302 (2002). [CrossRef] [PubMed]
  5. R. Rusconi, L. Isa, and R. Piazza, “Thermal-lensing measurement of particle thermophoresis in aqueous dispersions,” J. Opt. Soc. Am. B 21, 605–616 (2004). [CrossRef]
  6. S. N. Rasuli and R. Golestanian, “Soret motion of a charged spherical colloid,” Phys. Rev. Lett. 101, 108301 (2008). [CrossRef] [PubMed]
  7. J. Lenglet, A. Bourdon, J. C. Bacri, and G. Demouchy, “Thermodiffusion in magnetic colloids evidenced and studied by forced Rayleigh scattering experiments,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65, 031408 (2002). [CrossRef]
  8. B. Hoffmann, W. Köhler, and M. Krekhova, “On the mechanism of transient bleaching of the optical absorption of ferrofluids and dyed liquids,” J. Chem. Phys. 118, 3237–3242 (2003). [CrossRef]
  9. S. R. De Groot and P. Mazur, Nonequilibrium Thermodynamics (North Holland, 1962)
  10. F. Huang, P. Chakraborty, C. C. Lundstrom, C. Holmden, J. J. G. Glessner, S. W. Kieffer, and C. E. Lesher, “Isotope fractionation in silicate melts by thermal diffusion,” Nature 464, 396–400 (2010). [CrossRef]
  11. M. Giglio and A. Vendramini, “Thermal-diffusion measurements near a consolute critical-point,” Phys. Rev. Lett. 34, 561–564 (1975). [CrossRef]
  12. C. Debuschewitz and W. K¨ohler, “Molecular origin of thermal diffusion in benzene plus cyclohexane mixtures,” Phys. Rev. Lett. 87, 055901 (2001). [CrossRef] [PubMed]
  13. P. A. Artola and B. Rousseau, “Microscopic interpretation of a pure chemical contribution to the Soret effect,” Phys. Rev. Lett. 98, 125901 (2007). [CrossRef] [PubMed]
  14. D. Jung and M. Lücke, “Localized waves without the existence of extended waves: oscillatory convection of binary mixtures with strong Soret effect,” Phys. Rev. Lett. 89, 054502 (2002). [CrossRef] [PubMed]
  15. K. I. Morozov, “Soret effect in molecular mixtures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79, 031204 (2009). [CrossRef]
  16. A. Parola and R. Piazza, “Particle thermophoresis in liquids,” Eur. Phys. J. E 15, 255–263 (2004). [CrossRef] [PubMed]
  17. N. Ghofraniha, C. Conti, G. Ruocco, and F. Zamponi, “Time-dependent nonlinear optical susceptibility of an out-of-equilibrium soft material,” Phys. Rev. Lett. 102, 038303 (2009). [CrossRef] [PubMed]
  18. S. Fayolle, T. Bickel, S. LeBoiteux, and A. Würger, “Thermodiffusion of charged micelles,” Phys. Rev. Lett. 95, 208301 (2005). [CrossRef] [PubMed]
  19. S. Duhr, and D. Braun, “Thermophoretic depletion follows Boltzmann distribution,” Phys. Rev. Lett. 96, 168301 (2006). [CrossRef] [PubMed]
  20. S. A. Putnam, D. G. Cahil, and G. C. L. Wong, “Temperature dependence of thermodiffusion in aqueous suspensions of charged nanoparticles,” Langmuir 23, 9221–9228 (2007). [CrossRef] [PubMed]
  21. R. Piazza, “Thermophoresis: moving particles with thermal gradients,” Soft Matter 4, 1740–1744 (2008). [CrossRef]
  22. D. Vigolo, S. Buzzaccaro, and R. Piazza, “Thermophoresis and thermoelectricity in surfactant solutions,” Langmuir 26, 7792–7801 (2010). [CrossRef] [PubMed]
  23. D. Braun and A. Libchaber, “Trapping of DNA by thermophoretic depletion and convection,” Phys. Rev. Lett. 89, 188103 (2002). [CrossRef] [PubMed]
  24. S. Duhr and D. Braun, “Optothermal molecule trapping by opposing fluid flow with thermophoretic drift,” Phys. Rev. Lett. 97, 038103 (2006). [CrossRef] [PubMed]
  25. M. Ichikawa, H. Ichikawa, K. Yoshikawa, and Y. Kimura, “Extension of a DNA molecule by local heating with a laser,” Phys. Rev. Lett. 99, 148104 (2007). [CrossRef] [PubMed]
  26. N. Arnaud and J. Georges, “Thermal lens spectrometry in aqueous solutions of Brij 35: investigation of micelle effects on the time-resolved and steady-state signals,” Spectrochim. Acta [A] 57, 1085–1092 (2001). [CrossRef]
  27. P. R. B. Pedreira, L. R. Hirsch, J. R. D. Pereira, A. N. Medina, A. C. Bento, M. L. Baesso, M. C. Rollemberg, M. Franko, and J. Shen, “Real-time quantitative investigation of photochemical reaction using thermal lens measurements: theory and experiment,” J. Appl. Phys. 100, 044906 (2006). [CrossRef]
  28. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, A. N. Medina, and M. L. Baesso, “Thermal-lens study of photochemical reaction kinetics,” Opt. Lett. 34, 3460–3462 (2009). [CrossRef] [PubMed]
  29. N. G. C. Astrath, F. B. G. Astrath, J. Shen, J. Zhou, K. H. Michaelian, C. Fairbridge, L. C. Malacarne, P. R. B. Pedreira, P. A. Santoro, and M. L. Baesso, “Arrhenius behavior of hydrocarbon fuel photochemical reaction rates by thermal lens spectroscopy,” Appl. Phys. Lett. 95, 191902 (2009). [CrossRef]
  30. J. Shen, R. D. Lowe, and R. D. Snook, “A model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992). [CrossRef]
  31. M. L. Baesso, J. Shen, and R. D. Snook, “Mode-mismatched thermal lens determination of temperaturecoefficient of optical-path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732–3737 (1994). [CrossRef]
  32. N. G. C. Astrath, J. H. Rohling, A. N. Medina, A. C. Bento, M. L. Baesso, C. Jacinto, T. Catunda, S. M. Lima, F. G. Gandra, M. J. V. Bell, and V. Anjos, “Time-resolved thermal lens measurements of the thermo-optical properties of glasses at low temperature down to 20K,” Phys. Rev. B 71, 214202 (2005). [CrossRef]
  33. S. Hazebroucq, F. Labat, D. Lincot, and C. Adamo, “Theoretical insights on the electronic properties of eosin Y, an organic dye for photovoltaic applications,” J. Phys. Chem. A 112, 7264–7270 (2008). [CrossRef] [PubMed]
  34. N. Arnaud and J. Georges, “On the analytical use of the Soret-enhanced thermal lens signal in aqueous solutions,” Anal. Chim. Acta 445, 239–244 (2001). [CrossRef]

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