Quantitative Photothermal Spectroscopy:
Interpretation of Photodynamic
Processes and Parameters Through
Nonlinear Signal Measurement

A synopsis of the research paper presented at the 9th International Conference on Photoacoustic and Photothermal Phenomena, Nanjing, P. R. China, 1996

Agnès Chartier and Stephen Bialkowski
Department of Chemistry and Biochemistry
Utah State University
Logan, UT 84322-0300

email: agnes@cc.usu.edu, Stephen.Bialkowski@usu.edu
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Also see the talks to be given at the 10th ICPPP in Rome, 1998

Abstract

The laser sources used in photothermal spectroscopy of homogeneous samples often have irradiances in excess of those required for singlet and triplet state optical bleaching of organic and biologically important molecules. These dynamic, nonlinear effects affect photothermal signal magnitudes in different ways, depending on the method used to detect the temperature change. In any case, signal magnitudes obtained at high irradiance or energy do not reflect the "small signal" absorbance, and apparatus calibration must take into account the change in effective absorption coefficient as a function of excitation irradiance and/or energy. This paper will present experimental methods for measuring nonlinear effects, methods and considerations for calibration for analytical measurement, and progress we have made towards interpretation of the nonlinear data in terms of the photophysics and excited state relaxation dynamics of the condensed phase species under study.

Nonlinear absorption can occur using either pulsed and continuous wave laser excitation sources. For molecules with known physical properties and dynamics, the nonlinear absorption may be modeled by means of simple kinetic processes which take into account the rates of the slower processes and the transition rates produced by sample irradiation. It is relatively straightforward to determine the rate of heat production, or energy density in the case of pulsed excitation, by solving the governing rate equations, either exactly or by numerical integration. Modeling the effects that rate limited processes have on the measured photothermal response is slightly more complicated because the energy density profile deviates from that of the excitation source when nonlinear, irradiance dependent absorption occurs. Since signals produced in photothermal and photoacoustic spectrometry depend on both the spatial and the temporal energy density, details concerning the excitation laser beam propagation parameters and time-dependent irradiance profile must be known or determined by calibration with linear absorbing species in order to relate the physics of photon absorption to experimental measurements. Simple relationships between excitation irradiance and expected signals produced in photothermal and photoacoustic spectrometric measurements will be shown. These relationships may be used to facilitate interpretation of nonlinear data.

Analysis of nonlinear data produced using continuous laser excitation are easier to interpret than those using pulsed excitation. This is because in the long-time limit, the inverse photothermal lens is directly proportional to the absorption coefficient. The thermal lens signal does not depend on the specific beam profile used to excite the sample or distortions due to nonlinear absorption effects in this case. Subsequently, excitation laser power dependent photothermal lens signal magnitudes can be analyzed directly in terms of the effective absorption coefficient.

It is found that the effective absorption coefficient changes as a function of excitation power for a large number of organic dye substances. This is interpreted as being due to a bleaching of the singlet electronic state system in favor of population of the triplet states. Modeling the effective excitation laser power dependent absorption coefficient data allows determination of the ground singlet and lower triplet state absorption cross sections and the rate constant for triplet to singlet relaxation.

Advantages to using Photothermal Lens Spectrometry

• Signals are finite even in the event of optical bleaching
• Photothermal signals are relatively linear and independent of excitation and probe laser beam focus geometries
• Excitation energy or power dependent signals may be easily measured and modeled yielding information regarding ground and excited state absorption cross sections and relaxation rate constants
• Data is complimentary to relaxation kinetic measurements but yield more information
• Sensitivity allows the use of thin optical cells

Experiments described herein address the nonlinear absorption photophysics of typical organic dye substances; erythrosin, eosin, and pseudoisocyanin

Molecular Model

Kinetic model for irradiance-dependent photothermal signal for organic dyes. Molecules are optically excited from the ground, S₀, state to the excited singlet, S₁. Vibrationally excited molecules relax rapidly. Excited singlet molecules may return to the ground state by fluorescence or by radiationless internal conversion (IC). Alternatively, they may undergo inter-system crossing (ISC) to the triplet state, T₁. Triplet-to-singlet relaxation is slow. Triplet absorption produces molecules in the excited triplet, T₂, which rapidly relax to T₁.

Kinetic Model (Theory)

The kinetic equations for this model are

N₀(t), N₁(t), N₃(t), N₄(t) (m^-3) are the time-dependent number densities in S₀, S₁, T₁, and T₂, respectively, W_1,3 (s^-1) = Es_1,3/hv, the rate-constants for optical excitation from S₀ and T₁;s_1,3 (m²) are the absorption cross sections for S₀ and T₁; and the k (s^-1) are first order relaxation rate constants.

Approximate Number Density Solution

• Steady-state S₁ assumption; W ₁N₀(t)=(k₁₀+k₁₃)N₁(t)
• The fast k₁₃ assumption results in

where f_T=k₁₃/(k₁₀+k₁₃) is the triplet state quantum yield and N_tot is the total number density.

Absorbed Energy Density Solution

• The energy density is the time-integral over the irradiance absorbance product
• Energy densities are obtained separately for singlet and triplet absorbance

• For continuous excitation, e.g., E(r,t)=E(r) for 0£t£t_p, else E(r,t)=0

where U_S and U_T (J m^­3) are energy densities due to S₀ and T₁ absorption.

Thermal Yields

• Incomplete heat recovery is modeled using the unitless heat yield terms, Y_H^(S) and Y_H^(T)
• Y_H^(S) is the yield for singlet relaxation and Y_H^(T) is that for triplet relaxation associated with ISC.
• Y_H^(S) fraction energy above the T₀₀ transition energy and Y_H^(T)=1-DE_S-T/hv

Photothermal Lens Strength

• U_total=U_S+U_T
• dT=U_total/rC_P
• The focal length, f (m), is found using the relationships

• After substituting the Gaussian irradiance profile,

• where F₀ (W) is the average power of the square-pulse laser, w (m) the electric field beam waist radius, and r (m) is radius.

Singlet State Bleaching

• Assuming fast T₂®T₁ relaxation, e.g., k₄₃»W₃
• The inverse photothermal lens strength (TLS signal) is

• In the limit of long excitation duration, k₃₀t _p»1, the continuous laser-excited inverse focal length is

• This results in the convenient expression for the excitation irradiance-dependent inverse photothermal lens strength

where

and E_S (W m^-2) is the saturation irradiance given by

Prediction Examples

• The organic dye, erythrosin, is used for prediction calculation.
  This dye has the following photo-physical parameters;
  s₁~4x10^-20 m² @ 526 nm (e~10⁵ M cm^-1)
  f₃~1.0
  s₃/s₁~0.25
  and t_T-S~10 - 50 ms
• For the continuous laser excitation model, singlet-state bleaching effects occur for F₁~1. Since F₁=Es₁f₃/hvk₃₀, the irradiance for S₀ bleaching for erythrosin is E_S~1 MW m^-2. This seems like a lot. But consider a continuous laser focused to w=8 mm. Since E=2F/pw², the laser power needed to obtain bleaching is F_S~0.1 mW.
• In the pulsed-laser excitation case, singlet bleaching occurs for H_S~s₁^-1 or 4x10^-19 photon m^-2. Subsequently, H_S~16 J m^-2 @ 526 nm. For a pulsed source with t_p«k₃₀^-1 and focused to w=8 mm, the energy required for bleaching is Q_S~10^-3mJ.

Chopped-Laser Lens Experiment

Features

1. Time-dependent "chopped" data collected with analog-to-digital converter and processed with matched filter
2. Spatial filter at reference detector results in accurate estimates of excitation energy at the sample
3. Samples flow continuously through a short (1 mm) optical cell

Experimental Procedure Summary

Experimental Data

1. Obtain laser power dependent PTS signal for cobalt solution
2. Determine linear regression coefficients for cobalt data
3. Obtain laser energy or power dependent analyte signal
4. Calibrate analyte signal with cobalt regression coefficients

Model

1. Dynamic absorption model based on simple 4-level organic with coupled singlet and triplet states
2. Determine energy density from kinetic model
3. Find excitation energy or irradiance dependent photothermal elements
4. Deduce effective energy or irradiance dependent absorption coefficients

Parameter Determination

1. Use nonlinear regression to determine model parameters
2. Compare to known sample parameters

Procedures

1. Kinetic and signal equations solved with symbolic language processor (Macsyma)
2. Data collection with matched filter written in FORTRAN/MASM
3. Data regression programs in FORTRAN and C
4. Results plotted using spreadsheet (Quattro)