## 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]. 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, 3], colloidal dispersions [4–6] and magnetic fluids [7, 8] has been investigated experimentally in recent years. After several decades of research the phenomenon has been framed in terms of nonequilibrium thermodynamics concepts [9]; 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 [10–22].

Thermophoresis, a closely related process involving the drift of dispersed particles due to a thermal gradient, gives rise to interesting effects such as contact-free molecular segregation with potential applications in biotechnology [20,21]. Thermophoresis has also been used for optical manipulation of molecules, with special relevance for DNA trapping [23–25]. In these applications, laser-induced local heating allows all-optical microscale access to small molecules. However, the local heating also directly produces a radial refractive index change (known as the thermal lens (TL) effect) in the sample [5, 26–29]. Moreover, photo-induced chemical reaction (PCR) may take place in the illuminated volume [27–29]. As a result of the PCR, the number of the absorbing species in the liquid decreases in the illuminated volume, causing a variation of optical absorption coefficient of the liquid at the wavelength of the excitation laser. It modifies the local heating and then the thermal lens effect [27–29]. These refractive index changes produce lens-like optical components. Each optical component introduces an optical path length variation and then an additional phase shift that can be probed by a laser beam that passes through the illuminated volume in the liquid, resulting in its intensity profile change. Therefore, all these experimental effects can be laser-induced and may contribute to the transient signal in complex fluid mixtures.

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 [27–29]. 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.

In this work, we show that Soret effect and PCR can give similar trends in the transient signals. In fact, this situation may lead to erroneous interpretation of experimental results if only one effect is assumed to occur. To overcome this difficulty, we solve the time-resolved temperature and concentration equations when laser excitation is present (at time 0 < *t* ≤ *ξ*) and absent (*t* > *ξ*). The phase shifts produced by the local heating and concentration gradients are also calculated. Fresnel diffraction theory is employed to attain the intensity change of the probe beam. We show that the complete on-off transients can be used to discriminate among the phenomena and determine the physical parameters related to each one.

## 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 [28–32]. 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 [27–29]. 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 *A _{e}*(

*t*) =

*C*(

*t*)

*ɛ*, where

*ɛ*is the molar optical absorption coefficient of the sample and

*C*(

*t*) [29] 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 [27–29].

#### 2.1. Temperature and concentration gradients

We first consider the case in which a TEM_{00} laser excites the sample at 0 < *t* ≤ *ξ* (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

*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], can be written as

*C*(

*t*) = (

*C*

_{0}−

*C*) exp(−

_{eq}*K*) +

_{T}t*C*is the concentration of absorbing species,

_{eq}*C*

_{0}denotes the initial concentration, and

*C*is the equilibrium concentration in the illuminated area. The total reaction rate constant

_{eq}*K*is given by

_{T}*K*=

_{T}*k*+

_{r}*k*, where

_{d}*k*and

_{r}*k*represent the reaction and species diffusion rates, respectively.

_{d}*Q*

_{0}= 2

*P*/(

_{e}φ*ρcπω*

^{2}), where

*ω*and

*P*are the radius and power of the excitation laser, respectively.

_{e}*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*

_{0}= 0 for

*t*≤

*ξ*and

*t*

_{0}=

*t*−

*ξ*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

*t*=

_{c}*ω*

^{2}/(4

*D*).

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]:

*D*is the mass diffusion coefficient,

_{m}*c̄*is the initial average concentration and

*S*=

_{T}*D*/

_{T}*D*is the Soret coefficient.

_{m}*D*denotes the coefficient of thermal diffusion. Writing the characteristic mass diffusion time as

_{T}*t*=

_{m}*ω*

^{2}/(4

*D*) ≫

_{m}*t*, for

_{c}*t*≫

*t*[5], one can replace ∇

_{c}^{2}

*T*(

*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], 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

#### 2.2. Probe beam phase shifts

When a TEM_{00} 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
$\varphi \left(r,t\right)=\left(2\pi /{\lambda}_{p}\right){\int}_{0}^{L}\left[n\left(r,z,t\right)-n\left(0,z,t\right)\right]dz$. Here *n* is the refractive index, *L* is the sample thickness, and *λ _{p}* 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,

*ϕ*and

_{PCR}*ϕ*, caused by the temperature gradient [28] and the mass gradient (Soret effect) [5], respectively. Both effects give rise to

_{Soret}*z*-independent refractive index gradients that act as optical elements in the sample. The phase shifts are given by

*λ*are

_{p}*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),

*ϕ*can be calculated as

_{PCR}*m*= (

*ω*

_{1}

*/*

_{P}*ω*)

^{2}, and

*c*=

_{r}*C*/

_{eq}*C*

_{0}have been introduced. In addition,

*θ*is defined as and

_{th}*θ*is

_{m}*ω*

_{1P}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

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

*Z*

_{1}and

*Z*

_{2}are the distances from the probe beam waist to the sample and the sample to the detector plane, respectively. In Eq. (13),

*V*=

*Z*

_{1}/

*Z*,

_{C}*Z*is the confocal distance of the probe beam, and

_{C}*C*is a constant [27]. The probe beam intensity

*I*(

*t*) at the detector plane can be calculated as As special cases, pure Soret or PCR effects are obtained in this model by taking

*K*= 0 or

_{T}*θ*= 0, respectively, in Eq. (12).

_{m}*I*(

*t*) with Eq. (12) is used to fit the experimental data, where

*m*,

*V*,

*ω*,

*P*,

_{e}*L*and

*λ*are experimental setup parameters previously determined and

_{p}*θ*,

_{th}*θ*,

_{m}*t*,

_{c}*t*,

_{m}*c*and

_{r}*K*are obtained from the fits.

_{T}## 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 *P*_{1} triggered the digital oscilloscope that recorded the TL signal. Converging lens *L*_{1} focused the excitation beams, with the sample placed at its focal plane. The probe beams were focused by lens *L*_{2}, 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 *P*_{2}, positioned in the far field (*Z*_{2} ≈ 5m). A pinhole was placed in front of photodiode *P*_{2}, 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* = 55*μm* 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* = 72*μm*, *m* = 45.2 and *V* = 8.3. These geometrical parameters, which were constant during the measurements, were determined as described in Ref. [30].

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, 33]; 6.5wt% Brij 35 [CH_{3}(CH_{2})_{11}(OCH_{2}CH_{2})_{23}OH, polyoxyethylene 23 lauryl ether] in cobalt nitrate (absorbance 0.04) [34]; 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, 33]; the Soret effect does not occur in these solutions. By contrast, Brij solutions are known to exhibit the Soret effect [34]. 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]; 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]. 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 – 0.10*g*/*dm*^{3}. 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 40*Å*. 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.

Although Cr(VI) solutions are known to demonstrate only a PCR [28], the laser-on transient for these solutions can also be fitted to Eq. (14) using Eq. (12) with *K _{T}* = 0, which actually pertains to the Soret effect. It should be noted that this use of Eq. (12) with

*K*= 0 is effected only to show that the PCR and Soret contributions to the TL on-signal can have similar trends. The dotted line in the Fig. 2(a) on-transient thus represents either the Soret (Eq. (12) with

_{T}*K*= 0) or PCR (Eq. (12) with

_{T}*θ*= 0) fit. In other words, both mass diffusion and PCR effects would give similar trends in the TL on-transient signal. Further experimental evidence is therefore required to ascertain whether the Soret or PCR effect is responsible for the transient TL signal.

_{m}Turning to the laser-off transient for Cr(VI) [Fig. 2(a)], one finds that use of Eq. (14) and Eq. (12) with *K _{T}* = 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

*θ*= 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

_{m}*θ*= 0 (estimated uncertainty ±5%) are as follow:

_{m}*θ*/

_{th}*P*= 18.1

_{e}L*W*

^{−1}

*cm*

^{−1};

*c*= 0.59; and

_{r}*K*= 2

_{T}*s*

^{−1}. The thermal diffusivity is

*D*= 1.45 × 10

^{−3}

*cm*

^{2}

*s*

^{−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* < 200*ms*, 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 *K _{T}* = 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].

Consistent with the Cr(VI) solution, the EY on/off-transient in Fig. 3(a) demonstrates the well known PCR [33]. One again observes that the laser-on transient can be fitted to Eq. (14) using Eq. (12) with *K _{T}* = 0; the dotted line in the Fig. 3(a) on-transient represents either the Soret or the PCR fit. As noted for Cr(VI), both PCR and mass diffusion yield similar trends in the TL on-transient signal. However, use of the parameters obtained with the Soret equation for the laser-off transient produces a curve that does not fit the experimental data. By contrast, the parameters obtained from the PCR model, Eq. (14) using Eq. (12) with

*θ*= 0, provide very good agreement between the calculated curve and the experimental off-transient data (dotted line).

_{m}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 *θ _{m}* = 0 (PCR) and

*K*= 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.

_{T}In fact, most complex fluids used to study mass transport properties (particularly dye-doped mixtures [33]) 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 3*s* (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^{−3}*cm*^{2}*s*^{−1} for pure water. The mass diffusion coefficients (*D _{m}*) of the samples are listed in Table 1. These values were calculated using the relations

*D*=

*ω*

^{2}/4

*t*and

_{c}*D*=

_{m}*ω*

^{2}/4

*t*, 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

_{m}*θ*= 0 and

_{m}*k*= 0, respectively, and parameters obtained for the on- and off-transient curves.

_{T}Additional information on reaction kinetics and transport can also be obtained from the *c _{r}*,

*θ*, and

_{th}*θ*data. Figs. 5(a) and 5(b) show the excitation power dependencies of

_{m}*θ*and

_{th}*θ*. 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]. 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.

_{m}## 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.

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