Classical model for energy transfer in microspherical droplets

Andrew C. Pineda; David Ronis

doi:10.1103/PhysRevE.52.5178

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Classical model for energy transfer in microspherical droplets

Andrew C. Pineda

David Ronis

Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138

Department of Chemistry, McGill University, 801 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6

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Phys. Rev. E 52, 5178 – Published 1 November, 1995

DOI: https://doi.org/10.1103/PhysRevE.52.5178

Abstract

A classical electrodynamic model for energy transfer between donor and acceptor molecules in which the molecules are modeled using Drude oscillators is presented for dye solutions in the form of micrometer-sized droplets. The model incorporates multiparticle scattering effects by means of a binary collision expansion. Enhanced energy transfer rates and nontrivial concentration effects appear due to the Mie resonances of the droplet. Theory is discussed in light of the experiments of L. M. Folan, S. Arnold, and S. D. Druger [Chem. Phys. Lett. 118, 322 (1985)].

References (18)

J. I. Gersten and A. Nitzan, Chem. Phys. Lett. \fB104\fP, 31 (1984).
R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, Phys. Rev. Lett. \fB44\fR, 475 (1980).
L. M. Folan, S. Arnold, and S. D. Druger, Chem. Phys. Lett. \fB118\fP, 322 (1985); S. Arnold and L. M. Folan, Opt. Lett. \fB14\fR, 388 (1989).
The exponent was obtained by least-squares fitting the data presented in Fig. 3 of Ref. [3]. The +- 0.08 estimate of the uncertainty can be justified in two ways: (1) drawing lines using the error bars appearing in the figure; and (2) fitting using the data points and using the standard formula for the error in a least-squares fit of the slope of a line [cf. E. L. Crow, F. A. Davis, and M. W. Maxfield, Statistics Manual (Dover Press, New York, 1960), p. 160]. In the latter event, the +- 0.08 figure represents the standard error in the slope.
(a) Th. Förster, Ann. Phys. \fB2\fP, 55 (1948); (b) S. D. Druger, S. Arnold, and L. M. Folan, J. Chem. Phys. \fB87\fP, 2649 (1987).
H. Eyring, S. H. Lin, and S. M. Lin, Basic Chemical Kinetics (John Wiley & Sons, New York, 1980), Chap. 7.
(a) S. W. Haan and R. Zwanzig, J. Chem. Phys. \fB68\fP, 1879 (1978); (b) C. R. Gochanour, H. C. Andersen, and M. D. Fayer, J. Chem. Phys. \fB70\fR, 4254 (1979); C. R. Gochanour and M. D. Fayer, J. Phys. Chem. \fB85\fR, 1989 (1981); R. F. Loring, H. C. Andersen, and M. D. Fayer, J. Chem. Phys. \fB76\fR, 2015 (1982); \fB80\fR, 5731 (1984).
Andrew C. Pineda, Ph.D thesis, Harvard University, 1993, Chap. 2.
A. J. F. Seigert and E. Teramoto, Phys. Rev. \fB110\fR, 1232 (1958).
(a) R. Zwanzig, Phys. Rev. \fB129\fP, 486 (1963); (b) J. T. Bartis and I. Oppenheim, Physica \fB74\fP, 1 (1974).
We model the radiative decay term as an exponential decay via the i ω ${\tilde{γ}}$ ${}_{i}$ ${}^{R}$ {p arrow} ${}_{i}$ term, thus avoiding the theoretical difficulties (e.g., runaway solutions) associated with using the Abraham-Lorentz equation as a model for the radiation damping (cf., e.g., Ref. [12(a)], Chap. 17).
(a) J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1975), Chap. 16; M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1975), Sec. 13.5; (b) A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, NJ, 1974), Chap. 5.
Note that if we did not ignore correlations, the diagrams could still be characterized by how they factorize or are connected once Ursell functions are introduced for the reduced configurational distribution functions. Now, however, some of the connectedness would be due to the equilibrium spatial distributions, which we neither know for these systems, nor are particularly interested in.
S. Weinberg, Phys. Rev. \fB131\fR, 440 (1963); S. Weinberg, Phys. Rev. \fB130\fR, 776 (1963); S. Weinberg, Phys. Rev. \fB133\fR, B232 (1964); C. J. Joachain and C. Quigg, Rev. Mod. Phys. \fB46\fR, 279 (1974).
A. Ishimaru, Wave Propagation and Scattering in Random Media. Volume 1. Single Scattering and Transport Theory (Academic, New York, 1978); A. Ishimaru, Wave Propagation and Scattering in Random Media. Volume 2. Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing (Academic, New York, 1978).
All calculations were done in double precision (64 bits) on 32-bit machines hence the machine precision is about 15 decimal digits. Hence, allowing for rounding errors, the Bessel functions that determine the Mie coefficients are probably good to at best 12 or 13 digits. The strongest resonances in these calculations have Q's (i.e., ω / Δ omega) in the $10^{10}$ - $10^{11}$ range. Hence, we expect some sensitivity to rounding errors to appear if the function being computed becomes very sensitive to the precise value of the resonant Mie coefficient.
This is easily shown using the definition of {Y arrow} ${}_{J, l, M}$ ( Ω ) in Edmonds and the properties of the Clebsch-Gordan coefficients.
CRC Handbook of Chemistry and Physics, 60th ed., edited by R. C. Weast (CRC Press, Boca Raton, 1974), p. C-318.

[c1] J. I. Gersten and A. Nitzan, Chem. Phys. Lett. \fB104\fP, 31 (1984).

[c2] R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, Phys. Rev. Lett. \fB44\fR, 475 (1980).

[c3] L. M. Folan, S. Arnold, and S. D. Druger, Chem. Phys. Lett. \fB118\fP, 322 (1985); S. Arnold and L. M. Folan, Opt. Lett. \fB14\fR, 388 (1989).

[c4] The exponent was obtained by least-squares fitting the data presented in Fig. 3 of Ref. [3]. The +- 0.08 estimate of the uncertainty can be justified in two ways: (1) drawing lines using the error bars appearing in the figure; and (2) fitting using the data points and using the standard formula for the error in a least-squares fit of the slope of a line [cf. E. L. Crow, F. A. Davis, and M. W. Maxfield, Statistics Manual (Dover Press, New York, 1960), p. 160]. In the latter event, the +- 0.08 figure represents the standard error in the slope.

[c5] (a) Th. Förster, Ann. Phys. \fB2\fP, 55 (1948); (b) S. D. Druger, S. Arnold, and L. M. Folan, J. Chem. Phys. \fB87\fP, 2649 (1987).

[c6] H. Eyring, S. H. Lin, and S. M. Lin, Basic Chemical Kinetics (John Wiley & Sons, New York, 1980), Chap. 7.

[c7] (a) S. W. Haan and R. Zwanzig, J. Chem. Phys. \fB68\fP, 1879 (1978); (b) C. R. Gochanour, H. C. Andersen, and M. D. Fayer, J. Chem. Phys. \fB70\fR, 4254 (1979); C. R. Gochanour and M. D. Fayer, J. Phys. Chem. \fB85\fR, 1989 (1981); R. F. Loring, H. C. Andersen, and M. D. Fayer, J. Chem. Phys. \fB76\fR, 2015 (1982); \fB80\fR, 5731 (1984).

[c8] Andrew C. Pineda, Ph.D thesis, Harvard University, 1993, Chap. 2.

[c9] A. J. F. Seigert and E. Teramoto, Phys. Rev. \fB110\fR, 1232 (1958).

[c10] (a) R. Zwanzig, Phys. Rev. \fB129\fP, 486 (1963); (b) J. T. Bartis and I. Oppenheim, Physica \fB74\fP, 1 (1974).

[c11] We model the radiative decay term as an exponential decay via the i ω ${\tilde{γ}}$ ${}_{i}$ ${}^{R}$ {p arrow} ${}_{i}$ term, thus avoiding the theoretical difficulties (e.g., runaway solutions) associated with using the Abraham-Lorentz equation as a model for the radiation damping (cf., e.g., Ref. [12(a)], Chap. 17).

[c12] (a) J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1975), Chap. 16; M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1975), Sec. 13.5; (b) A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, NJ, 1974), Chap. 5.

[c13] Note that if we did not ignore correlations, the diagrams could still be characterized by how they factorize or are connected once Ursell functions are introduced for the reduced configurational distribution functions. Now, however, some of the connectedness would be due to the equilibrium spatial distributions, which we neither know for these systems, nor are particularly interested in.

[c14] S. Weinberg, Phys. Rev. \fB131\fR, 440 (1963); S. Weinberg, Phys. Rev. \fB130\fR, 776 (1963); S. Weinberg, Phys. Rev. \fB133\fR, B232 (1964); C. J. Joachain and C. Quigg, Rev. Mod. Phys. \fB46\fR, 279 (1974).

[c15] A. Ishimaru, Wave Propagation and Scattering in Random Media. Volume 1. Single Scattering and Transport Theory (Academic, New York, 1978); A. Ishimaru, Wave Propagation and Scattering in Random Media. Volume 2. Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing (Academic, New York, 1978).

[c16] All calculations were done in double precision (64 bits) on 32-bit machines hence the machine precision is about 15 decimal digits. Hence, allowing for rounding errors, the Bessel functions that determine the Mie coefficients are probably good to at best 12 or 13 digits. The strongest resonances in these calculations have Q's (i.e., ω / Δ omega) in the $10^{10}$ - $10^{11}$ range. Hence, we expect some sensitivity to rounding errors to appear if the function being computed becomes very sensitive to the precise value of the resonant Mie coefficient.

[c17] This is easily shown using the definition of {Y arrow} ${}_{J, l, M}$ ( Ω ) in Edmonds and the properties of the Clebsch-Gordan coefficients.

[c18] CRC Handbook of Chemistry and Physics, 60th ed., edited by R. C. Weast (CRC Press, Boca Raton, 1974), p. C-318.

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Classical model for energy transfer in microspherical droplets

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