Role of Photophysics Processes in Thermal Lens Spectroscopy of Fluids

Article pubs.acs.org/JPCA

Role of Photophysics Processes in Thermal Lens Spectroscopy of Fluids: A Theoretical Study L. C. Malacarne, E. L. Savi, M. L. Baesso, E. K. Lenzi, and N. G. C. Astrath* Departamento de Física, Universidade Estadual de Maringá, Maringá, Paraná 87020-900, Brazil ABSTRACT: Photophysics processes are ubiquitous in nature and difficult to be quantitatively characterized by conventional spectroscopy. Alternatively, pump−probe methods have been widely applied to study these complex processes. In this context, the thermal lens technique is a precise spectroscopic tool for material characterization and presents a wide range of applications in chemical analysis. Here, we present an all numerical approach to analyze the dynamics of photophysics processes and to identify the role of individual contributions of photoreaction and mass diffusion in the thermal lens experiments. The results are essential for a proper understanding of the dominant physical mechanisms in laser-induced photodegradation, which allow precise data analysis of the effects in photosensitive fluids.

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INTRODUCTION Photoinduced chemical reaction is a common process in a variety of materials in nature, and the study of the photophysical processes induced by absorption of light is of fundamental importance in different areas that range from chemical analysis to microscopy.1−9 In microscopy, for instance, a major problem accompanying the use of dyes is photoinduced degradation, appearing as fading of the emitted fluorescent light.4,5 However, photodegradation can be used as a method of imaging, as in photobleaching imprinting microscopy.6,7 In addition, the photobleaching control is necessary to keep the quality of food, medicines, and other commercial products. A better understanding of the photoinduced processes allows appropriated control of experiments where these effects are present. The use of lasers to study photophysics processes presents some advantages when compared to conventional spectroscopy, since its specific wavelength and energy density are higher than other light sources. The thermal lens (TL) method is a precise spectroscopic tool for material characterization,10−12,14−24,26−29 especially because of its high sensitivity that overcomes the limits of traditional transmittance spectroscopy. This method has been used in a wide range of applications, such as trace analysis12,14−18 and flow injection,19 characterization of chemical reactions,20−22,28,29 and determination of fluorescence quantum efficiency.23−25 Despite this range of applications, a large number of approximations are usually employed in the analytical treatment of the effects occurring during laser excitation and relaxation processes. Particularly important for fluids, when localized excitation induces photoreaction, a mass diffusion process needs to be considered in addition to the thermal effects. The identification of the individual contributions of each effect is essential for a proper understanding of the dominant physical mechanisms in the TL experiment. We stress here that the thermal and mass diffusion mechanisms can be identified using this method since their characteristics evolution times are usually distinct. For this © 2014 American Chemical Society

reason, it is highly desirable to have a unified and quantitative method to distinguish among the effects that occur during laser excitation. In this work, we introduce an all numerical approach to describe the thermal and mass diffusion dynamics and use this model to quantify the contribution of the different effects to the TL signal. Our results are of fundamental importance to the improvement of data analysis of photoinduced chemical reaction in fluids.

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THEORY In the thermal lens spectroscopy, the intensity in the center of the probe beam at the far field detector plane is given by the following:13 I (t ) = |

∫0

∞

exp[(1 + iV )g − i Φ(g , t )]dg |2

(1)

in which V is a geometric parameter from the experimental setup, g = (r/w1p)2, and w1p is the probe beam radius in the sample.14 In processes where the excitation laser induces photomodification in the solution, the phase shift induced by the change in the refractive index could have contribution from thermal and concentration gradients as follows: Φ(r , t ) =

2π λp

∫0

L

⎤ ⎡ dn dn C R (r , t )⎥ d z ⎢ T (r , t ) + dC R ⎦ ⎣ dT

(2)

in which T(r,t) and CR(r,t) are the thermal and concentration gradients, respectively, induced by absorption of light. dn/dT and dn/dCR are the temperature and concentration coefficients of the refractive index at the probe beam wavelength, λp, respectively. To describe the evolution of the TL signal, the changes in the temperature and concentration of excited molecules by the Received: May 28, 2014 Revised: July 10, 2014 Published: July 14, 2014 5983

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approximation for the spatial concentration implies in some difficulties to separate the effects of photoreaction rate and mass diffusion in the TL signal and, in addition, it becomes invalid when the contribution from dn/dCR is significant. We report here a straightforward study of the effects by solving numerically eqs 3 and 5. The role of photoreaction, mass diffusion, and concentration lens in the total TL signal is explored.

light-matter interaction must be considered. As the TL presents high sensitivity, we consider the study of fluid presenting low optical absorption. For low optical absorption fluids, the attenuation of light along the axial direction can be neglected, and the thermal process is described by the thermal diffusion equation: 2 2 2Peφ ∂ T (r , t ) − Dth ∇r2 T (r , t ) = β(r , t )e−2r / w0ef (t ) 2 ∂t πw0eρc p

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NUMERICAL ALGORITHM The numerical solutions for the partial differential equations, eqs 3 and 5, are calculated employing the “NDSolve” input in the Mathematica software, with the internal method “MethodOf Lines”, under the boundary conditions T(r0,t) = 0, ∂T(r0,t)/ ∂r = 0, CR(r0,t) = 1, and ∂Cr(r0,t)/∂r = 0, in which r0 ≫ w1p. This numerical method of lines is a technique for solving partial differential equations (PDE) by discretizing in all but one dimension, and then integrating the semidiscrete problem as a system of ordinary differential equation (ODE). It is necessary that the PDE problem be well-posed as an initial value (Cauchy) problem in at least one dimension. The intensity of TL signal, eq 1, is obtained by numerical integration. The Mathematica software works with the numerical solution for T(r,t) and CR(r,t) inside of eq 1 as an ordinary function. In the simulations, we consider characteristic parameters of soybean oil. Experimental evidence show the high potentiality of the thermal lens method in the characterization of vegetable oils, mainly due to its photosensibility30 and the fact that this fluid presents thermal and concentration contributions for the total phase shift. In addition, we also analyze aqueous solution of Eosin, where only thermal effects are significative. To understand the effect of each physical parameter in the TL signal, we introduce fluctuations in the parameters given in Table 1.

(3)

Assuming first or pseudo-first order reaction, in which the photoreaction process is as follows: C R (0) + hν → C R (t ) + C P(t )

(4)

with the constraint that the sum of the concentration of reactant, CR(t), and the concentration of the product, CP(t), are constant in time, C0 = CR(t) + CP(t) and CR(0) = C0, the time dependence of the reactant concentration is governed by mass diffusion equation, 2 2 2P σ ∂ C R (r , t ) − Dm∇r2 C R (r , t ) = − 2e e−2r / w0e ∂t πw0ehν

C R (r , t )f (t )

(5)

Here f(t) = [1 − H(t − ξ)], H(t − ξ) is the Heaviside Theta function, which accounts for the laser-on/off excitation. Dth = k/(ρcp) is the thermal diffusivity, k is the thermal conductivity, ρ is the mass density, cp is the specific heat of the sample, σ is the photoreaction cross-section, which characterizes the ability of a species to absorb light of a particular wavelength and undergo to photoreaction, h is Planck’s constant, ν is the optical frequency, Dm is the mass diffusion coefficient, w0e and Pe are the radius and the optical power of the excitation laser beam, respectively, and φ is the heat yield. The total absorption coefficient in the illuminated volume, β(r,t) = β0[(1 − ε)CR(r,t) + ε], can be written in terms of the equilibrium ratio between the molar absorptivities of the products (εP) and reactants (εR), ε = εP/εR, in which β0 = β(r,t = 0) is the optical absorption coefficient of the reactants. The diffusion term in eq 5 accounts for the molecular reposition by Brownian motion from the nonirradiated to the irradiated volume. Here, we assume that no additional effects, such as Soret effect and convection, act on the induced concentration gradient. The Soret effect is a concentration gradient induced by the thermal gradient, and does not depend on the excitation wavelength, and also could contribute to the total thermal lens signal.27 However, the absence of this effect could easily be checked by performing the TL experiments in different excitation wavelengths.27 The sample is assumed to be sufficiently thick, thus axial null thermal flux approximation can be applied, and viscous surface effects can be neglected. In addition, the area excited by the laser is much smaller than the sample dimensions. Although it is highly desirable to have a closed-form solution to eqs 3 and 5, analytical expressions are only possible assuming spatial average of each term of the concentration equation as CR(t) = ⟨CR(r,t)⟩r. This results in a simple form for the effective rate equation, dCR(t)/dt = −KTCR(t). Here, the total reaction rate KT represents the averaged rates of photobleaching and movement by the molecular diffusion. This approximation has been employed in the investigation of laser-induced chemical reaction in ionic aqueous solutions of Fe(II)-TPTZ28 and photobleaching of Eosin Y in aqueous solution.29 This averaged

Table 1. Standard Physical Properties Used in the Simulations.29,30a parameter σ β0 Dm Dth dn/dT dn/dCr ρ cp φ λe w0e

units 2

m m−1 m2 s−1 m2 s−1 K−1 L kg−1 kg m−3 Jkg−1K−1 nm μm

soybean oil −24

3 × 10

16 0.5 1.2 −4.3 −0.8 920 2100 1 476 80

× × × ×

10−9 10−7 10−4 10−5

eosin 5 × 10−26 0.73 1 × 10−9 1.4 × 10−7 −4 −1 × 10 0 998 4182 0.73 514 68

The TL setup parameters are λp = 632.8 nm, V = 6.5, and w1p = 366 μm.

a

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RESULTS AND DISCUSSION In TL transients in the presence of photoreaction, the role of each physical parameter is not easily identified. Similar trends could be observed with different set of parameters, especially if we consider only the transient with the excitation laser-on. During laser excitation of a photoreacting 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 5984

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consumed species. Figure 1 shows that complete laser-on/off transients signal must be taken into account when processing

Figure 2. TL signal for soybean oil with different values of dn/dCr. Additional parameters are presented in Table 1.

affects both the thermal and the concentration lenses, as expected from the diffusion eqs 3 and 5.

Figure 1. Characteristic TL signals presenting similar on-transient for different set of parameters. The laser-off transient discriminates completely the presence of the concentration lens effect, indicated by the crossing of the TL intensity above the initial values. Additional parameters are presented in Table 1.

the experimental data in order to discriminate among the phenomena taking place and to correctly identify the physical parameters related to each effect. The initial rapid decreasing in probe beam intensity, for t < 200 ms, corresponds to the thermal contribution to the TL signal. The dynamic of this short time transient gives information related to the thermal diffusion process, Dth, the amplitude is related to the thermal coefficient of refractive index, dn/dT, and the fraction of absorbed energy converted in heat, φ. Thereafter, changes in the optical absorption and the effect of concentration lens contribute to the signal that increases with time during the laser-on transient (from 0 to 4s). In the laser-off transient (from 4 to 12s), no photodegradation is occurring, and the evolution of the TL signal is determined by the temperature relaxation and mass diffusion. The contribution of the thermal and concentration lenses are easily identified. When the concentration lens is negligible, the relaxation process is governed by the thermal relaxation, and the TL signal returns to its initial intensity, I(0). The effect of the concentration lens on the TL transient can be observed on the particular behavior of the relaxation process. After the fast thermal relaxation, the contribution of the concentration lens still affects the TL signal, and the TL intensity goes over its value at t = 0, following the slower relaxation governed by the mass diffusion coefficient Dm. As for most liquids, the thermal characteristic time is smaller than the mass diffusion characteristic time; these two processes are easily discriminated in the TL signal. Figure 2 presents the laser-on/off transients for the set of parameters given in Table 1 for soybean oil, with artificial fluctuation only in the concentration coefficient of the refractive index, dn/dCR. Note that the laser-off transients produce curves that overcome the initial values, following the relaxation process when the concentration lens effect is not negligible. The amplitude of the laser-off transient over the initial values increases almost linearly with dn/dCR. The effect of the reaction cross section, σ, in the TL transient is illustrated in Figure 3. It shows that the photoreaction rate

Figure 3. TL signal for soybean oil with fluctuation in the values of σ. Additional parameters are presented in Table 1. Inset plot gives only the thermal contribution for the TL transients, including the optical absorption variation by the photoreaction.

The effect of the mass diffusion coefficient, Dm, is explored in the simulation presented in Figure 4. Clearly, the relaxation behavior of the laser-off transient is determined by the mass diffusion characteristic time. However, the reposition by Brownian motion from the nonirradiated to the irradiated volume also affects the laser-on transient, although it is more relevant for the contribution from concentration lens. Finally, the effect of ratio between the molar absorptivities of the products and reactants are explored in Figure 5. As expected, it affects only the laser-on transient when the photoreaction induced by excitation beam is occurring. The regression analysis of experimental data with the proposed model allows the access to a large number of physical properties. However, the usefulness of any model depends in part on the accuracy and reliability of its output. Yet, because all fluctuation in the experimental data, the output values are subject to imprecision, especially in the case of multiparameter regression analysis. As shown in the simulations presented, there are a large number of parameters involved in the process of photoreaction in fluids, in addition to other physical parameters. We explored the role of the concentration 5985

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between the molar absorptivities of the product and reactant, ε. In addition to the density, specific heat, and optical absorption coefficient of the reactant, β0, the thermal parameters as the temperature coefficient of the refractive index, dn/dT and the thermal diffusivity, Dth, were kept fixed during the fit. dn/dT can be easily determined by interferometric measurements31 and, if the thermal characteristic time is smaller than the reaction rate and mass diffusion characteristic time, the thermal diffusivity and the fraction of absorbed energy converted in heat, φ, can be determined performing regression analysis of TL measurement with short time transients. The other fixed parameters can be determined by measurements with conventional techniques. Reducing the free parameters to a minimum decreases uncertainty and increases stability of multiparameters fitting procedure. In addition, as showed in the simulation, to consider the laser-on/off transients is of fundamental importance to obtain correct values from the output regression analysis. In order to check the stability of the regression analysis, we generated a TL transient adding a small random fluctuation to the intensity and performed the fitting with the parameters related to photoreaction. The regression analysis becomes more unstable if both parameters σ and dn/dCr are kept as free parameters. This is easily understood from the simulations presented in Figures 2 and 3, as the two parameters induce similar trends on the laser-on/off transients. The parameters Dm and ε are stable since they are well-defined by the laser-on/ off transients. The mass diffusion coefficient characterizes the relaxation behavior of the transient. The ratio between the molar absorptivities only affects the laser-on transient. This suggests the need of additional technique to determine either σ or dn/dCR. In the absence of concentration lens, i.e., in case of negligible dn/dCR contribution to the TL signal, the concentration gradient will only contribute to the TL signal by changing the radial dependence of β(r,t). The laser-off transient is governed only by thermal relaxation and does not give information related to the photoreaction process, and can be neglected in the fit. However, the laser-on transient has contributions by the influence of σ, ε, and Dm. Figure 6 shows transients for parameters of low concentration of Eosin in water, including artificial fluctuations in the parameters σ, ε, and Dm. Indeed, introducing small random values in the intensity and

Figure 4. TL signal for soybean oil with fluctuation in the values of Dm. Additional parameters are presented in 1. Inset plot gives only the thermal contribution for the TL transients, including the optical absorption variation by the photoreaction.

Figure 5. TL signal for soybean oil with fluctuation in the values of ε. Additional parameters are presented in Table 1. Inset plot gives only the thermal contribution for the TL transients, including the optical absorption variation by the photoreaction.

coefficient of the refractive index, dn/dCR, the reaction cross section, σ, the mass diffusion coefficient, Dm, and the ratio

Figure 6. TL signal for Eosin in water including fluctuation in (a) Dm, (b) σ, and (c) ε. Additional parameters are presented in Table 1. 5986

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performing the regression analysis to check the stability, we note that the fitted values become more stable if only two parameters are considered free. This suggests the need of additional measurements as well. The parameter ε could be determined, for instance, by measuring the optical absorption coefficient of a sample before and after long time exposure to the excitation laser.

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CONCLUSIONS We presented a complete numerical study on the role of the physical parameters involved in photoreaction process to the total thermal lens signal. The influence and stability of each parameter in the regressions analysis of a data set are explored. The analyses presented are of fundamental importance to the improvement of experimental investigation of fluid samples presenting photoreaction and mass diffusion. The results and the proposed numerical approach increase the potential application of the thermal lens technique for the analysis of laser-induced effects in photoreactive solutions.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support for this work from the Brazilian agencies CAPES, CNPq, and Fundaçaõ Araucária.

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REFERENCES

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