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OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
Simple way to analyze Brillouin spectra from turbid liquids Mikolaj Pochylski* and Jacek Gapinski ´ 1
Faculty of Physics, Adam Mickiewicz University, ul. Umultowska 85, Poznan 62-614, Poland *Corresponding author: [email protected] Received December 12, 2014; revised February 27, 2015; accepted February 28, 2015; posted March 3, 2015 (Doc. ID 230484); published March 27, 2015
The shape of the Brillouin light-scattering spectrum recorded from turbid liquids is distinctly distorted compared to that from a transparent sample. The reason for this is the multiple scattering of light within the medium. The usual expression for the Brillouin spectrum does not apply to the multiple scattering situation. In this Letter, we consider a Brillouin spectrum from opaque samples composed of a distribution of spectra resulting from elementary scattering events, each occurring in single scattering vector conditions. We introduce a one-parameter test function to define the probability distribution of scattering events occurring at a given value of the scattering vector. The proposed procedure was tested on model liquids that consisted of suspensions of sub-micrometer spherical particles of different size and concentration, dispersed in different carrier liquids and studied as a function of temperature. Our analysis made it possible to account for the effect of multiple scattering and to recover the values of mechanical parameters describing the pure solvents. © 2015 Optical Society of America OCIS codes: (000.2170) Equipment and techniques; (300.6250) Spectroscopy, condensed matter; (290.4210) Multiple scattering; (290.5820) Scattering measurements; (290.5830) Scattering, Brillouin; (290.7050) Turbid media. http://dx.doi.org/10.1364/OL.40.001456
Even small amounts of one kind of material dispersed in an optically unmatched transparent liquid can make the whole system turbid. This is a common situation in multiphase media where inhomogeneity size reaches the range of the light wavelengths. If the system components differ in the values of the speed of light, the multiple scattering processes take place making the photon diffuse inside the material rather than propagating in straight directions. As a consequence, neither the incident nor the scattered photon directions are well defined, and the same refers to the scattering vector. Such a situation generates serious problems in inelastic scattering techniques like Brillouin scattering for which the exact value of scattering vector is usually fixed by experimental geometry. In the multiple scattering situation, the external scattering geometry does not ensure the unique value of the scattering vector anymore. Consequently, the shape of the Brillouin spectrum is distinctly distorted [1–3], and the usual description derived for a single scattering event is not a proper one. Also the equations for scattering from opaque solid surfaces [4–6], spatial filtering procedures for avoiding spurious geometrical scattering-vector contributions [7], or other experimental approaches to mitigate multiply scattered light [8,9], do not apply to the opaque liquid systems. In fact a proper mathematical description for such a case has not been found, yet. As a consequence, utilization of Brillouin scattering for mechanical characterization of the materials that are inherently heterogeneous (micro-phase separated systems [1,2], liquid suspensions [3], oceanic waters [10], or biological tissues [11–13]) is difficult and gives at best uncertain results. In this Letter, we make a first attempt to describe the influence of multiple scattering processes on Brillouin scattering phenomenon. The Brillouin scattering occurs when light is inelastically scattered on the spontaneously (thermally) induced sound waves (acoustic phonons). These density fluctuations propagate inside the material in every direction, and their wavelengths range from macroscopic down 0146-9592/15/071456-04$15.00/0
to atomic scales. In a single scattering event, the scattering vector, q, of the analyzed fluctuation is uniquely chosen by external scattering geometry, given by the angle, θ, between the incident and the scattered radiation. The magnitude of the scattering vector is given by qθ 4πn∕λ sinθ∕2, where n is the refractive index of the medium, and λ is the incident wavelength. If the relaxation frequency of any process active in the sample is far from the frequency window of Brillouin method (a few GHz), the density fluctuation of a given scattering vector, behaves like a damped harmonic oscillator (DHO). The appropriate spectrum takes the following form [14–16]: Iθ; ω I 0
2ω2q Γ2q Γq : ω ω2q Γ2q 2 2ωΓq 2 2
(1)
Equation (1) reproduces the Brillouin doublet with ωq and Γq corresponding approximately to the frequency position and to the half width at half maximum of Brillouin peaks, respectively. These parameters are related to speed of sound, v, and dynamic longitudinal viscosity, ηL , of the medium through ωq vq and Γq 1∕2ηL q2 . Thus, the position and the width of Brillouin lines both depend on the value of the scattering vector, q (which changes with the scattering angle θ), characterizing the elementary scattering event. In transparent samples, selection of the scattering vector is performed exclusively by the geometry (scattering angle) of the experimental setup. The situation changes in heterogeneous systems where multiple elastic scattering additionally occurs. Let’s consider a simple model situation of noninteracting solid particles dispersed in a liquid of unmatched refractive index. Both components are nonabsorbing for the incident wavelength. The volume fraction of particles is low enough to assume that the behavior of density fluctuations (phonon transport) of the pure solvent is intact, and the inelastic scattering occurs mostly in the liquid © 2015 Optical Society of America
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
phase. In this case, there is a probability that the incident photon will be elastically scattered at the liquid/particle boundary before the inelastic scattering occurs. In each elastic scattering event, the incident photon changes its direction, initially well defined by the external (laboratory) scattering geometry. During this process, the photon diffuses inside the sample until the inelastic (Brillouin) event occurs. After that, the scattered photon diffuses again until it leaves the sample and eventually reaches the detector. As a result, the external scattering geometry does not ensure the unique value of the internal scattering vector. Instead, every possible scattering angle may be realized, and the measured spectrum contains the contributions from all the scattering geometries (from the forward- to the back-scattering). The single scattering approximation is obviously not valid in the analysis of such data. To the best of our knowledge, there is still no analytical formula for a Brillouin spectrum obtained in turbid liquids. However, if the system meets the assumptions described above, the requested profile should be composed of a distribution of elementary spectra, each describing the single scattering event occurring in a liquid at every possible angle. The task is to find the proper distribution function describing the probability that the photon that reached the detector was internally scattered at an angle θ. Such an angular distribution function in principle could be found by considering diffusion of photons (similar to what is done in diffusive wave spectroscopy [17]) together with the particle form factor from Mie theory. For full description, information about the experimental geometry should also be included. Even if the proper distribution function could be derived, it would surely be expressed by many parameters. The experimental Brillouin spectrum covers a relatively narrow frequency range, and the recorded spectra are rather uniformly distorted (see Fig. 1). The description of such a profile with a multi-parameter function is simply unpractical. The purpose of this report is to present an easy method for the analysis of real experimental data. Any angular distribution function describing the internal scattering process recorded in the externally back-scattering geometry needs to fulfill the following conditions: (1) the function must be defined in the range of angles 0 < θ < π (corresponding to the range from forward- to backscattering); (2) for transparent samples, the function is a delta peak located at θ π (pure back-scattering); (3) for opaque samples, the delta peak spreads, and the function takes finite values for angles θ < π. As a test function, we will use an expression of the Gaussian form. Although this is an arbitrary choice, it is justified, the analytic solution of the multiple scattering in the small angle approximation [18]. The function below meets all three conditions listed above: θ−π 2 . Pθ; c c 1 − c exp − c
(2)
Here, c is called the opacity parameter. It can take values from the range 0 < c ≤ 1, where the limit of 0 corresponds to the single back-scattering scattering event (transparent samples). The value of 1 describes the
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Fig. 1. Anti-Stokes line of the Brillouin spectrum recorded for a turbid sample (open circles). Only half of all data points are shown for clarity. The small full circles represent the spectrum recorded for pure (transparent) carrier liquid (bulk water). The long-dashed line is the best result of fitting to the opaque sample data the model of damped harmonic oscillator, DHO (Eq. 1). The solid line is the best fitting result assuming the finite distribution of scattering vectors (Eq. 3). The shape of the distribution function is shown in inset. The short-dashed line is the DHO spectrum calculated using the acoustic parameters of the solvent obtained from fitting with Eq. (3). The agreement of this line with the experimental spectrum obtained for pure water is excellent, which demonstrates the usability of our model in the analysis of Brillouin spectra from turbid systems.
situation where scattering at any angle from 0 to π is equally probable. It should be stressed that the opacity parameter was introduced a priori (as Eq. 2) just to account in an approximate way for the spread of angular distribution. It is not directly related to any physical parameters of the system in a quantitative way. In fact, from the formal point of view, the denominator in the exponent of Eq. (2) needs a constant to convert dimensionless c into the distribution width value expressed in the units of angle (radian in our case). Our tests have shown that the value of this constant close to one gave good results in our procedure, and small variations did not change this conclusion. Since our procedure is purely numerical and we aimed at reducing the number of additional fit parameters, Eq. (2) was used in this simplified form. The distribution function Pθ; c expresses the probability that individual scattering event, occurring internally at angle θ (at a given value of q), will be detected in external back-scattering geometry. The integrated intensity of the spectrum describing elementary scattering process in hydrodynamic regime is the same for every scattering angle (does not depend on q) [19]. Hence, the experimental spectrum is simply the distribution of individual scattering events with contributions given by Pθ; c: Iω
Rπ 0
Pθ; cIθ; ω ⊗ Aωdθ Rπ B. 0 Pθ; cdθ
(3)
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In Eq. (3), Aω is the instrumental resolution function of the spectrometer, ⊗ stands for the convolution operation, and B is the spectrum background. The profile generated by Eq. (3) can be directly compared with the experimental spectrum. This should allow to extract the values of acoustical (mechanical) parameters of the carrier liquid. All the distortion produced by multiple scattering processes is included in the value of a single additional parameter, c. To test the usefulness of Eq. (3) in the analysis of Brillouin spectra recorded in moderately turbid samples, we performed scattering experiments in solutions of particles of different size and material, dispersed in different carrier liquids, and studied as a function of concentration and temperature. The two examined systems were: (1) latex spheres (210 nm) in water (purchased from Invitrogen) and (2) silica spheres (84 nm) in ethanol-based solvent (synthesized and kindly delivered by Dr. Johan Buitenhuis from Forschungszentrum Juelich). In Brillouin scattering experiment, the beam from a DPSS Excelsior laser (Spectra Physics, λ0 532 nm), working at a mean power of about 80 mW, was used as the probe. The scattered light has been collected in the back-scattering geometry and analyzed by a Sandercock-type (3 3) pass Tandem Fabry–Perot interferometer. The acquisition time of a single spectrum was about 15 min. The strong elastic component has been cut out from the center of the spectrum using the standard mechanical shutter synchronized with the scanning ramp of the spectrometer. A typical anti-Stokes side spectrum recorded for a turbid system (latex particles in water) is presented in Fig. 1. For comparison, the spectrum corresponding to the pure (transparent) carrier liquid (bulk water) is also included. The Brillouin peak for both kinds of sample is clearly visible, however the one from the opaque system is obviously distorted. The intensity of the low-frequency part of the spectrum is distinctly increased. As can be seen in Fig. 1, the formula derived for transparent systems (for harmonic oscillator; Eq. 1) cannot describe correctly the spectra measured in turbid systems. DHO description gives a peak whose frequency position (and so, the speed of sound) is lower than expected. At the same time, the width of the peak (the sound attenuation) is much too high. The situation improves substantially when our Eq. (3) is applied. Now the shape of the experimental profile is much better reproduced. At the same time, the acoustic parameters obtained closely resemble those of bulk water. All of this is at the cost of only one additional fitting parameter. In Fig. 2 [panels (a) and (b)], we present the values of the acoustic parameters characterizing dispersions of latex particles at different concentrations (and so different opacity) at a single temperature (25°C). We found that the DHO function was losing validity as the sample became more and more opaque. The shift of the DHO peak toward the center of spectrum (to compensate the elevated low-frequency part of the data) generates underestimated sound velocity values. Similar geometrical effect broadens the peak of the model function, which results in apparently stronger attenuation. These problems apparently disappear when Eq. (3) is used as the fitting model. The systematic concentration dependence
Fig. 2. Acoustical parameters of the carrier liquid (water) as obtained from fitting Eq. (3) to Brillouin spectra recorded for latex/water liquid dispersions. Panels (a) and (b) present volume fraction dependencies of the speed of sound (scaled by refractive index, n) and the longitudinal viscosity (scaled by n2 ), respectively. Panels (c) and (d) show the same parameters as a function of temperature for ϕ 0.005. The inset in panel (a) shows the concentration dependence of the opacity parameter. The inset in panel (b) compares the χ 2 values (describing the quality of the fit) when Eq. (1) or Eq. (3) were fitted to experimental data.
is actually lost. The same procedure has been applied to the analysis of data obtained for a sample with fixed concentration (ϕ 0.5%) at a series of temperatures [panels (c) and (d)]. Again, the DHO analysis provided the speed of sound values being systematically underestimated compared to those measured in bulk water. The apparent sound attenuation was instead strongly overestimated. Bulk water parameters were much better reproduced when Eq. (3) was used to fit the spectra. Using our simple approach, the average error in evaluation of bulk water acoustic parameters was reduced to about 1% for speed of sound and to about 18% for longitudinal viscosity. Taking into account that for transparent samples, usually the uncertainties for these parameters are 1% and 5%, respectively, the above result is very encouraging. To further test our method, we recorded Brillouin spectra of dispersions of silica particles suspended in ethanol. Also in this case, the recorded spectra were distorted in a manner similar to that shown in Fig. 1. We prepared dispersions with two different concentrations of particles and performed measurements in a broad temperature range. In order to have a reference for further analysis and comparisons, we also took measurements of the pure (transparent) solvent. As follows from Fig. 3, the viscosity of the liquid exhibits a maximum at about T −130°C, while the plot of the sound velocity deviates from the linear behavior. These indications suggest that some relaxation process is taking place in the sample at the time scale of picoseconds. Such a process has already been recognized in pure ethanol using Brillouin spectroscopy and was related to some intramolecular degrees of freedom [20].
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
Fig. 3. Acoustic parameters of the carrier liquid as obtained from fitting Eq. (3) to Brillouin spectra recorded for silica/ethanol liquid dispersions. Panel (a) shows the temperature dependence of sound velocity presented. The dashed line shows the behavior of bulk solvent. The symbols represent values obtained for opaque liquids. The open symbols were obtained from fitting Eq. (1) (DHO). Closed symbols show the results of the analysis with Eq. (3). The circles are for ϕ 0.3%, and squares correspond to ϕ 1%. The inset shows the difference between sound velocity derived from turbid samples and that expected for bulk solvent. Panel (b) shows the temperature dependence of longitudinal viscosity.
Similar to the latex/water case, also for silica/ethanol, the DHO description is far from the expected one. When the analysis is based on Eq. (3), the speed of sound values obtained for pure solvent are well recovered, and the viscosity values are much closer to the ones describing pure ethanol. Clear overestimation of the viscosity is observed only for the more concentrated solution. For the more diluted sample (ϕ 0.3%), the reproduction of the solvent properties is very good, at least for temperatures higher than about −70°C (inset of Fig. 3). In this range, the deviation for the speed of sound value is about 1% and for viscosity only about 7%. As follows from Fig. 3 (see the inset), for temperatures lower than −70°C, Eq. (3) seems to lose its descriptive performance. The reason for this is the interference of the relaxation process. As was mentioned before, Eq. (3) assumes a distribution of harmonic oscillator spectra that correctly describes the Brillouin profile only for nonrelaxing fluids. When a relaxation process enters the explored frequency scale, an additional spectral component (so-called Mountain line) appears [14,19]. It influences the central part of the spectrum, distorting the low-frequency wings of the Brillouin lines. Our description does not take into account changes in the spectral shape generated by relaxation processes. Any distortion in the Brillouin line shape is assigned to the distribution of scattering vectors. Therefore, Eq. (3) holds only when all the relaxation
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processes in the liquid are far from the spectral window of the Brillouin measurement. We are aware that the proposed description can be subject to criticism, particularly with regard to the functional form of the scattering vector distribution function and the physical meaning of parameter c. The proposed expression is based on obvious qualitative assumptions and many comparisons made among different distribution functions of similar shape. Due to lack of any better justified models and taking advantage of surprisingly good results obtained with the help of our approach, we suggest to use it for initial analysis of the Brillouin spectra obtained in turbid samples in the back-scattering geometry. The method proposed here could be easily adjusted, e.g., to fitting the results of Brillouin microscopy in different kinds of soft matter samples. References 1. M. Pochylski and J. Gapinski, ´ J. Phys. Chem. B 114, 2644 (2010). 2. M. Philipp, U. Müller, R. Aleksandrova, R. Sanctuary, P. Müller-Buschbaum, and J. K. Krüger, Soft Matter 8, 11387 (2012). 3. S. Sirotkin, A. Mermet, M. Bergoin, V. Ward, and J. L. Van Etten, Phys. Rev. E 90, 022718 (2014). 4. J. R. Sandercock, Top. Appl. Phys. 51, 173 (1982). 5. A. Dervisch and R. Loudon, J. Phys. C 9, L669 (1976). 6. N. L. Rowell and G. I. Stegeman, Phys. Rev. B 18, 2598 (1978). 7. A. Battistoni, F. Bencivenga, D. Fioretto, and C. Masciovecchio, Opt. Lett. 39, 5858 (2014). 8. E. Berrocal, E. Kristensson, M. Richter, M. Linne, and M. Alden, Opt. Express 16, 17870 (2008). 9. M. A. Linne, M. Paciaroni, E. Berrocal, and D. Sedarsky, Proc. Combust. Inst. 32, 2147 (2009). 10. B. D. Joelson and G. W. Kattawar, Appl. Opt. 35, 2693 (1996). 11. G. Scarcelli, R. Pineda, and S. H. Yun, Invest. Ophthalmol. Vis. Sci. 53, 185 (2012). 12. S. Reiß, G. Burau, O. Stachs, R. Guthoff, and H. Stolz, Biomed Opt. Express 2, 2144 (2011). 13. G. Scarcelli and S. H. Yun, Opt. Express 19, 10913 (2011). 14. J. P. Boon and S. Yip, Molecular Hydrodynamics (McGrawHill, 1980). 15. F. Aliotta, J. Gapinski, ´ M. Pochylski, R. C. Ponterio, F. Saija, and C. Vasi, Phys. Rev. E 84, 051202 (2011). 16. C. J. Montrose, V. A. Solovyev, and T. A. Litovitz, J. Acoust. Soc. Am. 43, 117 (1968). 17. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988). 18. H. A. Bethe, Phys. Rev. 89, 1256 (1953). 19. B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley Interscience, 1976). 20. J.-H. Ko and S. Kojima, J. Non-Cryst. Solids 307, 154 (2002).
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
Simple way to analyze Brillouin spectra from turbid liquids Mikolaj Pochylski* and Jacek Gapinski ´ 1
Faculty of Physics, Adam Mickiewicz University, ul. Umultowska 85, Poznan 62-614, Poland *Corresponding author: [email protected] Received December 12, 2014; revised February 27, 2015; accepted February 28, 2015; posted March 3, 2015 (Doc. ID 230484); published March 27, 2015
The shape of the Brillouin light-scattering spectrum recorded from turbid liquids is distinctly distorted compared to that from a transparent sample. The reason for this is the multiple scattering of light within the medium. The usual expression for the Brillouin spectrum does not apply to the multiple scattering situation. In this Letter, we consider a Brillouin spectrum from opaque samples composed of a distribution of spectra resulting from elementary scattering events, each occurring in single scattering vector conditions. We introduce a one-parameter test function to define the probability distribution of scattering events occurring at a given value of the scattering vector. The proposed procedure was tested on model liquids that consisted of suspensions of sub-micrometer spherical particles of different size and concentration, dispersed in different carrier liquids and studied as a function of temperature. Our analysis made it possible to account for the effect of multiple scattering and to recover the values of mechanical parameters describing the pure solvents. © 2015 Optical Society of America OCIS codes: (000.2170) Equipment and techniques; (300.6250) Spectroscopy, condensed matter; (290.4210) Multiple scattering; (290.5820) Scattering measurements; (290.5830) Scattering, Brillouin; (290.7050) Turbid media. http://dx.doi.org/10.1364/OL.40.001456
Even small amounts of one kind of material dispersed in an optically unmatched transparent liquid can make the whole system turbid. This is a common situation in multiphase media where inhomogeneity size reaches the range of the light wavelengths. If the system components differ in the values of the speed of light, the multiple scattering processes take place making the photon diffuse inside the material rather than propagating in straight directions. As a consequence, neither the incident nor the scattered photon directions are well defined, and the same refers to the scattering vector. Such a situation generates serious problems in inelastic scattering techniques like Brillouin scattering for which the exact value of scattering vector is usually fixed by experimental geometry. In the multiple scattering situation, the external scattering geometry does not ensure the unique value of the scattering vector anymore. Consequently, the shape of the Brillouin spectrum is distinctly distorted [1–3], and the usual description derived for a single scattering event is not a proper one. Also the equations for scattering from opaque solid surfaces [4–6], spatial filtering procedures for avoiding spurious geometrical scattering-vector contributions [7], or other experimental approaches to mitigate multiply scattered light [8,9], do not apply to the opaque liquid systems. In fact a proper mathematical description for such a case has not been found, yet. As a consequence, utilization of Brillouin scattering for mechanical characterization of the materials that are inherently heterogeneous (micro-phase separated systems [1,2], liquid suspensions [3], oceanic waters [10], or biological tissues [11–13]) is difficult and gives at best uncertain results. In this Letter, we make a first attempt to describe the influence of multiple scattering processes on Brillouin scattering phenomenon. The Brillouin scattering occurs when light is inelastically scattered on the spontaneously (thermally) induced sound waves (acoustic phonons). These density fluctuations propagate inside the material in every direction, and their wavelengths range from macroscopic down 0146-9592/15/071456-04$15.00/0
to atomic scales. In a single scattering event, the scattering vector, q, of the analyzed fluctuation is uniquely chosen by external scattering geometry, given by the angle, θ, between the incident and the scattered radiation. The magnitude of the scattering vector is given by qθ 4πn∕λ sinθ∕2, where n is the refractive index of the medium, and λ is the incident wavelength. If the relaxation frequency of any process active in the sample is far from the frequency window of Brillouin method (a few GHz), the density fluctuation of a given scattering vector, behaves like a damped harmonic oscillator (DHO). The appropriate spectrum takes the following form [14–16]: Iθ; ω I 0
2ω2q Γ2q Γq : ω ω2q Γ2q 2 2ωΓq 2 2
(1)
Equation (1) reproduces the Brillouin doublet with ωq and Γq corresponding approximately to the frequency position and to the half width at half maximum of Brillouin peaks, respectively. These parameters are related to speed of sound, v, and dynamic longitudinal viscosity, ηL , of the medium through ωq vq and Γq 1∕2ηL q2 . Thus, the position and the width of Brillouin lines both depend on the value of the scattering vector, q (which changes with the scattering angle θ), characterizing the elementary scattering event. In transparent samples, selection of the scattering vector is performed exclusively by the geometry (scattering angle) of the experimental setup. The situation changes in heterogeneous systems where multiple elastic scattering additionally occurs. Let’s consider a simple model situation of noninteracting solid particles dispersed in a liquid of unmatched refractive index. Both components are nonabsorbing for the incident wavelength. The volume fraction of particles is low enough to assume that the behavior of density fluctuations (phonon transport) of the pure solvent is intact, and the inelastic scattering occurs mostly in the liquid © 2015 Optical Society of America
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
phase. In this case, there is a probability that the incident photon will be elastically scattered at the liquid/particle boundary before the inelastic scattering occurs. In each elastic scattering event, the incident photon changes its direction, initially well defined by the external (laboratory) scattering geometry. During this process, the photon diffuses inside the sample until the inelastic (Brillouin) event occurs. After that, the scattered photon diffuses again until it leaves the sample and eventually reaches the detector. As a result, the external scattering geometry does not ensure the unique value of the internal scattering vector. Instead, every possible scattering angle may be realized, and the measured spectrum contains the contributions from all the scattering geometries (from the forward- to the back-scattering). The single scattering approximation is obviously not valid in the analysis of such data. To the best of our knowledge, there is still no analytical formula for a Brillouin spectrum obtained in turbid liquids. However, if the system meets the assumptions described above, the requested profile should be composed of a distribution of elementary spectra, each describing the single scattering event occurring in a liquid at every possible angle. The task is to find the proper distribution function describing the probability that the photon that reached the detector was internally scattered at an angle θ. Such an angular distribution function in principle could be found by considering diffusion of photons (similar to what is done in diffusive wave spectroscopy [17]) together with the particle form factor from Mie theory. For full description, information about the experimental geometry should also be included. Even if the proper distribution function could be derived, it would surely be expressed by many parameters. The experimental Brillouin spectrum covers a relatively narrow frequency range, and the recorded spectra are rather uniformly distorted (see Fig. 1). The description of such a profile with a multi-parameter function is simply unpractical. The purpose of this report is to present an easy method for the analysis of real experimental data. Any angular distribution function describing the internal scattering process recorded in the externally back-scattering geometry needs to fulfill the following conditions: (1) the function must be defined in the range of angles 0 < θ < π (corresponding to the range from forward- to backscattering); (2) for transparent samples, the function is a delta peak located at θ π (pure back-scattering); (3) for opaque samples, the delta peak spreads, and the function takes finite values for angles θ < π. As a test function, we will use an expression of the Gaussian form. Although this is an arbitrary choice, it is justified, the analytic solution of the multiple scattering in the small angle approximation [18]. The function below meets all three conditions listed above: θ−π 2 . Pθ; c c 1 − c exp − c
(2)
Here, c is called the opacity parameter. It can take values from the range 0 < c ≤ 1, where the limit of 0 corresponds to the single back-scattering scattering event (transparent samples). The value of 1 describes the
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Fig. 1. Anti-Stokes line of the Brillouin spectrum recorded for a turbid sample (open circles). Only half of all data points are shown for clarity. The small full circles represent the spectrum recorded for pure (transparent) carrier liquid (bulk water). The long-dashed line is the best result of fitting to the opaque sample data the model of damped harmonic oscillator, DHO (Eq. 1). The solid line is the best fitting result assuming the finite distribution of scattering vectors (Eq. 3). The shape of the distribution function is shown in inset. The short-dashed line is the DHO spectrum calculated using the acoustic parameters of the solvent obtained from fitting with Eq. (3). The agreement of this line with the experimental spectrum obtained for pure water is excellent, which demonstrates the usability of our model in the analysis of Brillouin spectra from turbid systems.
situation where scattering at any angle from 0 to π is equally probable. It should be stressed that the opacity parameter was introduced a priori (as Eq. 2) just to account in an approximate way for the spread of angular distribution. It is not directly related to any physical parameters of the system in a quantitative way. In fact, from the formal point of view, the denominator in the exponent of Eq. (2) needs a constant to convert dimensionless c into the distribution width value expressed in the units of angle (radian in our case). Our tests have shown that the value of this constant close to one gave good results in our procedure, and small variations did not change this conclusion. Since our procedure is purely numerical and we aimed at reducing the number of additional fit parameters, Eq. (2) was used in this simplified form. The distribution function Pθ; c expresses the probability that individual scattering event, occurring internally at angle θ (at a given value of q), will be detected in external back-scattering geometry. The integrated intensity of the spectrum describing elementary scattering process in hydrodynamic regime is the same for every scattering angle (does not depend on q) [19]. Hence, the experimental spectrum is simply the distribution of individual scattering events with contributions given by Pθ; c: Iω
Rπ 0
Pθ; cIθ; ω ⊗ Aωdθ Rπ B. 0 Pθ; cdθ
(3)
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OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
In Eq. (3), Aω is the instrumental resolution function of the spectrometer, ⊗ stands for the convolution operation, and B is the spectrum background. The profile generated by Eq. (3) can be directly compared with the experimental spectrum. This should allow to extract the values of acoustical (mechanical) parameters of the carrier liquid. All the distortion produced by multiple scattering processes is included in the value of a single additional parameter, c. To test the usefulness of Eq. (3) in the analysis of Brillouin spectra recorded in moderately turbid samples, we performed scattering experiments in solutions of particles of different size and material, dispersed in different carrier liquids, and studied as a function of concentration and temperature. The two examined systems were: (1) latex spheres (210 nm) in water (purchased from Invitrogen) and (2) silica spheres (84 nm) in ethanol-based solvent (synthesized and kindly delivered by Dr. Johan Buitenhuis from Forschungszentrum Juelich). In Brillouin scattering experiment, the beam from a DPSS Excelsior laser (Spectra Physics, λ0 532 nm), working at a mean power of about 80 mW, was used as the probe. The scattered light has been collected in the back-scattering geometry and analyzed by a Sandercock-type (3 3) pass Tandem Fabry–Perot interferometer. The acquisition time of a single spectrum was about 15 min. The strong elastic component has been cut out from the center of the spectrum using the standard mechanical shutter synchronized with the scanning ramp of the spectrometer. A typical anti-Stokes side spectrum recorded for a turbid system (latex particles in water) is presented in Fig. 1. For comparison, the spectrum corresponding to the pure (transparent) carrier liquid (bulk water) is also included. The Brillouin peak for both kinds of sample is clearly visible, however the one from the opaque system is obviously distorted. The intensity of the low-frequency part of the spectrum is distinctly increased. As can be seen in Fig. 1, the formula derived for transparent systems (for harmonic oscillator; Eq. 1) cannot describe correctly the spectra measured in turbid systems. DHO description gives a peak whose frequency position (and so, the speed of sound) is lower than expected. At the same time, the width of the peak (the sound attenuation) is much too high. The situation improves substantially when our Eq. (3) is applied. Now the shape of the experimental profile is much better reproduced. At the same time, the acoustic parameters obtained closely resemble those of bulk water. All of this is at the cost of only one additional fitting parameter. In Fig. 2 [panels (a) and (b)], we present the values of the acoustic parameters characterizing dispersions of latex particles at different concentrations (and so different opacity) at a single temperature (25°C). We found that the DHO function was losing validity as the sample became more and more opaque. The shift of the DHO peak toward the center of spectrum (to compensate the elevated low-frequency part of the data) generates underestimated sound velocity values. Similar geometrical effect broadens the peak of the model function, which results in apparently stronger attenuation. These problems apparently disappear when Eq. (3) is used as the fitting model. The systematic concentration dependence
Fig. 2. Acoustical parameters of the carrier liquid (water) as obtained from fitting Eq. (3) to Brillouin spectra recorded for latex/water liquid dispersions. Panels (a) and (b) present volume fraction dependencies of the speed of sound (scaled by refractive index, n) and the longitudinal viscosity (scaled by n2 ), respectively. Panels (c) and (d) show the same parameters as a function of temperature for ϕ 0.005. The inset in panel (a) shows the concentration dependence of the opacity parameter. The inset in panel (b) compares the χ 2 values (describing the quality of the fit) when Eq. (1) or Eq. (3) were fitted to experimental data.
is actually lost. The same procedure has been applied to the analysis of data obtained for a sample with fixed concentration (ϕ 0.5%) at a series of temperatures [panels (c) and (d)]. Again, the DHO analysis provided the speed of sound values being systematically underestimated compared to those measured in bulk water. The apparent sound attenuation was instead strongly overestimated. Bulk water parameters were much better reproduced when Eq. (3) was used to fit the spectra. Using our simple approach, the average error in evaluation of bulk water acoustic parameters was reduced to about 1% for speed of sound and to about 18% for longitudinal viscosity. Taking into account that for transparent samples, usually the uncertainties for these parameters are 1% and 5%, respectively, the above result is very encouraging. To further test our method, we recorded Brillouin spectra of dispersions of silica particles suspended in ethanol. Also in this case, the recorded spectra were distorted in a manner similar to that shown in Fig. 1. We prepared dispersions with two different concentrations of particles and performed measurements in a broad temperature range. In order to have a reference for further analysis and comparisons, we also took measurements of the pure (transparent) solvent. As follows from Fig. 3, the viscosity of the liquid exhibits a maximum at about T −130°C, while the plot of the sound velocity deviates from the linear behavior. These indications suggest that some relaxation process is taking place in the sample at the time scale of picoseconds. Such a process has already been recognized in pure ethanol using Brillouin spectroscopy and was related to some intramolecular degrees of freedom [20].
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
Fig. 3. Acoustic parameters of the carrier liquid as obtained from fitting Eq. (3) to Brillouin spectra recorded for silica/ethanol liquid dispersions. Panel (a) shows the temperature dependence of sound velocity presented. The dashed line shows the behavior of bulk solvent. The symbols represent values obtained for opaque liquids. The open symbols were obtained from fitting Eq. (1) (DHO). Closed symbols show the results of the analysis with Eq. (3). The circles are for ϕ 0.3%, and squares correspond to ϕ 1%. The inset shows the difference between sound velocity derived from turbid samples and that expected for bulk solvent. Panel (b) shows the temperature dependence of longitudinal viscosity.
Similar to the latex/water case, also for silica/ethanol, the DHO description is far from the expected one. When the analysis is based on Eq. (3), the speed of sound values obtained for pure solvent are well recovered, and the viscosity values are much closer to the ones describing pure ethanol. Clear overestimation of the viscosity is observed only for the more concentrated solution. For the more diluted sample (ϕ 0.3%), the reproduction of the solvent properties is very good, at least for temperatures higher than about −70°C (inset of Fig. 3). In this range, the deviation for the speed of sound value is about 1% and for viscosity only about 7%. As follows from Fig. 3 (see the inset), for temperatures lower than −70°C, Eq. (3) seems to lose its descriptive performance. The reason for this is the interference of the relaxation process. As was mentioned before, Eq. (3) assumes a distribution of harmonic oscillator spectra that correctly describes the Brillouin profile only for nonrelaxing fluids. When a relaxation process enters the explored frequency scale, an additional spectral component (so-called Mountain line) appears [14,19]. It influences the central part of the spectrum, distorting the low-frequency wings of the Brillouin lines. Our description does not take into account changes in the spectral shape generated by relaxation processes. Any distortion in the Brillouin line shape is assigned to the distribution of scattering vectors. Therefore, Eq. (3) holds only when all the relaxation
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processes in the liquid are far from the spectral window of the Brillouin measurement. We are aware that the proposed description can be subject to criticism, particularly with regard to the functional form of the scattering vector distribution function and the physical meaning of parameter c. The proposed expression is based on obvious qualitative assumptions and many comparisons made among different distribution functions of similar shape. Due to lack of any better justified models and taking advantage of surprisingly good results obtained with the help of our approach, we suggest to use it for initial analysis of the Brillouin spectra obtained in turbid samples in the back-scattering geometry. The method proposed here could be easily adjusted, e.g., to fitting the results of Brillouin microscopy in different kinds of soft matter samples. References 1. M. Pochylski and J. Gapinski, ´ J. Phys. Chem. B 114, 2644 (2010). 2. M. Philipp, U. Müller, R. Aleksandrova, R. Sanctuary, P. Müller-Buschbaum, and J. K. Krüger, Soft Matter 8, 11387 (2012). 3. S. Sirotkin, A. Mermet, M. Bergoin, V. Ward, and J. L. Van Etten, Phys. Rev. E 90, 022718 (2014). 4. J. R. Sandercock, Top. Appl. Phys. 51, 173 (1982). 5. A. Dervisch and R. Loudon, J. Phys. C 9, L669 (1976). 6. N. L. Rowell and G. I. Stegeman, Phys. Rev. B 18, 2598 (1978). 7. A. Battistoni, F. Bencivenga, D. Fioretto, and C. Masciovecchio, Opt. Lett. 39, 5858 (2014). 8. E. Berrocal, E. Kristensson, M. Richter, M. Linne, and M. Alden, Opt. Express 16, 17870 (2008). 9. M. A. Linne, M. Paciaroni, E. Berrocal, and D. Sedarsky, Proc. Combust. Inst. 32, 2147 (2009). 10. B. D. Joelson and G. W. Kattawar, Appl. Opt. 35, 2693 (1996). 11. G. Scarcelli, R. Pineda, and S. H. Yun, Invest. Ophthalmol. Vis. Sci. 53, 185 (2012). 12. S. Reiß, G. Burau, O. Stachs, R. Guthoff, and H. Stolz, Biomed Opt. Express 2, 2144 (2011). 13. G. Scarcelli and S. H. Yun, Opt. Express 19, 10913 (2011). 14. J. P. Boon and S. Yip, Molecular Hydrodynamics (McGrawHill, 1980). 15. F. Aliotta, J. Gapinski, ´ M. Pochylski, R. C. Ponterio, F. Saija, and C. Vasi, Phys. Rev. E 84, 051202 (2011). 16. C. J. Montrose, V. A. Solovyev, and T. A. Litovitz, J. Acoust. Soc. Am. 43, 117 (1968). 17. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, Phys. Rev. Lett. 60, 1134 (1988). 18. H. A. Bethe, Phys. Rev. 89, 1256 (1953). 19. B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley Interscience, 1976). 20. J.-H. Ko and S. Kojima, J. Non-Cryst. Solids 307, 154 (2002).
