Rayleigh-Debye-Gans as a model for continuous monitoring of biological particles: Part I, assessment of theoretical limits and approximations

Alicia C. Garcia-Lopez; Arthur David Snider; Luis H. Garcia-Rubio

doi:10.1364/OE.14.008849

1. Introduction

Real time continuous monitoring of micron and sub-micron bio-particulates for biomedical and industrial applications requires both suitable instrumentation and sophisticated models that relate the measurements to the desired properties of biological particles. A rapid tool for submicron particle characterization is light spectroscopy [1, 2], which yields information on their joint property distribution: chemical composition, size, shape, and orientation. These properties are inferred from the spectroscopy measurements through mathematical models derived from the theory of electromagnetic radiation (Maxwell’s equations), in particular from the theory of light scattering. To date rigorous computational techniques have been developed that enable the estimation of properties relevant to biological systems such as particle size and particle shape [3]. Unfortunately, these methods (T-matrix and Purcell-Pennypacker methods) are computationally intensive and do not yet lend themselves to the evaluation of particle ensembles for real-time continuous monitoring applications. Therefore there is a need for approximations that can provide adequate particle characterization results in relatively short computational times. Mie theory and the Rayleigh-Debye-Gans (RDG) approximation are special solutions and limiting cases of the electromagnetic theory that can be effectively used in the context of real-time applications using spectrophotometric measurements. In particular, RDG enables the estimation of the particle shape, an important feature when characterizing biological systems.

Rayleigh-Debye-Gans is an approximate theory for particles of any shape and size having a relative refractive index near unity and has found extensive use in direct light scattering applications [2] and other techniques such as flow cytometry [4]. Mie theory is the exact solution to the boundary value problem for light scattering by a sphere and is used as a reference [5, 6]. There are inherent limitations to using Rayleigh-Debye-Gans theory as a characterization tool. The relative complex refractive index m must be close to one and the size of the particle must be much smaller than λ/|m-1, where λ is the wavelength. There exists a trade off for the limits of RDG theory; first, if m is close to one and no absorption is present then the size of the particle can be the same order of magnitude as the wavelength. Conversely, if absorption is present and m is greater than one, the particle size must be smaller than the wavelength. It is noteworthy that RDG has been used for the characterization of blood [7] and bacteria-like particles [2, 8] which clearly have sizes and optical properties beyond the expected range of application of the theory.

The importance of the RDG approximation for biological particles lies on the flexibility of this approximation for generating the scattering structures and shapes of complex particles through the formulation of the form factors [4, 9, 13]. An important motivation for the study of RDG in the spectral window of 200–1100 nm has to do with the fact that most of the key components of biological particles (proteins, DNA, Chlorophyll, etc.) absorb and scatter light in UV portion of the spectrum. Therefore, if biological particles are to be characterized in terms of their absorption and scattering properties both, the spectral regions where absorption and scattering are significant have to be considered. A complication brought about by the use of a broad wavelength spectral region (200–1100 nm) is that particles may not adhere to the constraints required by RDG across the complete spectral range and it is therefore desirable, if not completely necessary, to increase the range of application of the RDG approximations.

In this paper, RDG and Mie theory are explored in the context of multiwavelength spectroscopy over a wavelength range where strict adherence to the conditions implicit in the approximations could not be met, but where important compositional information can be acquired [1]. As a result from this study, the differences between RDG and Mie theory are generalized and their range of application more accurately defined as a function of both wavelength and magnitude of the optical properties. Reported corrections to RDG proposed to obtain a better agreement with Mie theory have also been evaluated. These corrections include the use of effective optical properties [9] and the inclusion of hypochromic effects [10–12]. The results from this study indicate that the range of application of RDG is considerably more limited than suggested by the single wavelength evaluations reported in the literature [2, 6, 13], and that neither the correction of the optical properties for hypochromicity, nor the use of effective optical properties result in better agreement between RDG and Mie theories. It is evident that to have a theoretically based spectroscopy interpretation model suitable for real-time characterization of complex bio-particles the formulation of RDG must be revisited.

2. Methods

2.1 Simulations

The programs for Mie theory, Rayleigh-Debye-Gans theory, and hypochromicity were developed in Matlab v6.5.1. Computations for these programs were conducted using a Dell Inspiron 4100 with 1GHz Pentium III processor and 512 MB RAM. The experimentally determined optical properties (refractive indices) utilized were provided by Dr. Garcia-Rubio and the SAPD laboratory through the College of Marine Science at the University of South Florida¹⁴.

The ranges of particle volumes were chosen between 12700 nm³ and 87000e6 nm³. The spherical diameter equivalents to the volume range are between 25 nm–5500 nm. Table 1 gives the wavelength, concentration and density values used to define the suspensions for the analyses conducted in this manuscript. The simulation spectra are graphed by optical density (a standard unit for intensity ratio per unit pathlength * concentration) and wavelength.

Table 1. Simulation Parameters for Turbidity using Mie and Rayleigh-Debye-Gans Theory.

View Table

2.2 Materials

The computer codes developed for the analysis of Rayleigh-Debye-Gans and Mie particles were tested against published values of the scattering functions [5, 15]. In testing and exploring the algorithms for Rayleigh-Debye-Gans the refractive indices selected were those of soft bodies and hemoglobin, where soft bodies are defined here as particles whose relative refractive index is close to one with no absorption component. The values of the index of refraction n+iκ for biological particles commonly used are soft bodies (1.04≤n≤1.45) and hemoglobin (1.48≤n≤1.6, 0.01≤κ≤0.15) [7]. Polystyrene (1.5≤n≤2.2, 0.01≤κ≤0.82), silver bromide (2.6≤n≤3.5, 0.001≤κ≤1.6) and silver chloride (2≤n≤2.7, 0.001<κ≤0.85) are materials found in industrial applications whose properties are used as standards for the calibration and evaluation of optical instruments [7]. Water (1.3≤n≤1.4) was used as the suspending medium.

The optical properties are graphed as function of wavelength in the Appendix, numerical values are available upon request from Prof. Garcia-Rubio.

3. Theory

3.1 Turbidity

The turbidity equation is an energy balance equation applicable to small acceptance angle transmission measurements. Turbidity has been traditionally defined as an attenuation coefficient due to scattering (only) for the transmission of the incident beam. Herein, turbidity is described as the total attenuation observed due to scattering and absorption. The expression for the turbidity of a monodisperse system is

τ (λ) = N_{p} l Q_{ext} = N_{p} l (Q_{sca} + Q_{abs})

where N _p is the number of particles, l is the pathlength of the sample, Q _ext is the extinction efficiency, Q _sca is the scattering efficiency, and Q _abs is the absorption efficiency. The efficiencies are determined using Rayleigh-Debye-Gans theory, from which the turbidity equation can be expressed explicitly by the following equation [16].

τ (λ) = N_{p} ℓ (\frac{9 k^{4} V^{2}}{16 π} {∣ \frac{m^{2} - 1}{m^{2} + 2} ∣}^{2} \int_{0}^{π} P (θ) (1 + \cos^{2} θ) \sin θ d θ + 3 k V Im (\frac{m^{2} - 1}{m^{2} + 2}))

where P(θ) is the form factor (a function of wavelength, λ) for spheres as defined by Kerker6

P (θ) = \frac{9 π J_{3 ⁄ 2}^{2} (u)}{2 u^{3}}, u = h a, a = radius

Note that the turbidity is a function of m, the relative complex refractive index (a function of wavelength, λ) and θ the angle of observation.

3.2 Complex Refractive Index.

The connection between light scattering and absorption phenomena and particle’s joint property distribution (size, shape, orientation, chemical composition, and internal structure) is made through the optical properties that are characteristic of the materials contained in the particle. The complex refractive index is given by

N = n + i κ

where n and κ are non negative values, n is the refractive index (real), and κ is the absorption coefficient (imaginary). The scattering of light is due to differences in refractive indices between the medium and the particle. The refractive index of the particle (N ₁) relative to the suspending medium (N ₀) is

m = \frac{N_{1}}{N_{0}} = \frac{n_{1} + i κ_{1}}{n_{0} + i κ_{0}}

The real and imaginary parts of the complex refractive index expressed as function of frequency are not independent; these are related through the integral Kramers-Kronig transforms [5].

n (ω) - 1 = \frac{1}{π} P \int_{0}^{\infty} \frac{Ω k (Ω)}{Ω^{2} - ω^{2}} d Ω

κ (ω) = \frac{- 2 ω}{π} P \int_{0}^{\infty} \frac{n (Ω,)}{Ω^{2} - ω^{2}} d Ω

Here ω is the angular frequency and P is the principal value of the integral. In principle, if either n(ω) or κ(ω) is known or can be measured, the other can be calculated directly through Eq. (6) and Eq. (7). Measurements over the complete range of frequency (0 to ∞) are required when applying this transform.

3.3 Optical Property Corrections

Differences in the scattering behavior predicted by Mie and by RDG have been previously investigated within the context of the effect of shape and monochromatic angular scattering measurements [6, 9, 17], and it has been suggested that it is possible to compensate RDG through the use of effective refractive indices; in other words, the refractive indices become adjustable parameters. Latimer et al [9] have reported formulas to calculate the effective refractive indices for a variety of shapes. The differences in behavior observed between the spectra calculated with Mie and with RDG theory suggests that a similar approach may be used to correct RDG to obtain closer approximations to Mie theory. The corrections made to the optical properties can be approached empirically and/or theoretically. The theoretical models developed by Veshkin [10–12] to account for hypochromic effects offer a distinct possibility since these models compensate the absorption coefficient in the direction of decreasing intensity, as it would be required by RDG to approximate Mie. Both the empirical and the theoretical approaches are explored in this paper.

3.4 Hypochromicity as a Correction for RDG

The observable light scattering phenomena depends on the configuration of the instrumentation and on the optical properties of the material investigated. The optical properties (real and imaginary parts of the refractive index) are wavelength-dependent and intrinsic properties of matter. It is known that the optical properties depend on the state of aggregation [5]. However, under certain conditions (i.e. infinite dilution) the optical properties are additive and independent of concentration. The presence of absorbing groups (chromophores) in high concentration within particles gives rise to a concentration dependence of the observed optical phenomena. This phenomenon generally results in a decrease of the observed imaginary component of the refractive index–relative to its value in solution (hypochromicity) [18]. Note that any change in value of the imaginary component of the refractive index (–) will be reflected in the absorption component of the turbidity spectra.

A consequence of Mie theory is that the absorption component of the turbidity spectrum decreases relative to the scattering efficiency as the particle size increases [5]. Similarly, in the context of hypochromicity, as the concentration of chromophores increases, there is a decrease in the observed imaginary component (absorption) of the complex refractive index. Therefore, it seems plausible that the mathematical structure of successful models developed to account for hypochromic effects may be able to account for the decrease in the absorption component present in Mie theory, but absent in the RDG approximation. The attenuation of light due to absorption through a particle is shown schematically in Fig. 1 for Mie theory and RDG approximation.

The most recent models for hypochromicity are those developed by Veshkin [10–12] and take into consideration the molecular structure, the molecular orientation, and the number of chromophoric groups per unit volume of particle. Vekshin’s model describes screening of chromophores when stacked along the molecular chain axis. In this paper the chromophores were considered to be stacked within the particle, by doing so the model developed by Veshkin has been extended to include the effect of the wavelength and implemented for multiwavelength spectroscopy. The concentration of chromophoric groups per particle size per wavelength can be related as depicted in Fig. 2 [see Eq. (12)].

Fig. 1. The attenuation of light due to absorption through a particle shown schematically for Mie theory and RDG approximation.

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Fig. 2. Schematic of chromophoric groups per particle size per wavelength within a particle.

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3.5 Wavelength Dependent Hypochromicity

Experimentally the hypochromism value h at a given wavelength is defined by:

h = 100 % \frac{ε - \hat{ε}}{ε}

where ε is the extinction coefficient for the situation of single chromophore absorption in units of 1/M cm and ε̃ is the average extinction coefficient per chromophore. From the screening model [10, 11]

\hat{E} = \frac{s}{2.3 q k} (1 - {(1 - 2.3 \frac{E q}{s})}^{k})

This equation predicts the hypochromic extinction coefficient in a solution of stack chromophores (cluster) if the values E, s, q, and k are known. E is in units of molecular extinction coefficients (Å/molecule) and is a function of wavelength, Ẽ is the average extinction coefficient, s is the effective geometric area of a chromophore (Å ²), q is the orientation factor, and k (not to be confused with the wave number) is the quantity of chromophores.

Transforming Vekshin’s model from units 1/M cm to Å results in rewriting the equation above as

{\hat{ε}}_{m} = \frac{s}{2.3 q k} N_{A} [1 - {(1 - 2.3 \frac{E q}{s 6.022 E 4})}^{k}]

where N _A is Avogadro’s number. Equation (10) can now be used to correct the imaginary component of the refractive index by first calculating the extinction coefficient using the equation for absorptivity, $ε = \frac{4 π κ}{λ}$ . The extinction coefficient is then transformed to molar units

ε_{m} = \frac{ε M_{w}}{V}

where M _w is the molecular weight of the chromophore and V is the unit volume transformation of 1000 cm³/L. The number of chromophores k calculated from the volume fraction or concentration of the sample

k (λ) = \frac{v_{f} λ}{d}

where v _f is the volume fraction, λ is the wavelength, d is the diameter of the sample. The probability of absorption of a photon by a molecule P can be presented as

P = 2.3 \frac{q \dot{E}}{s}

\dot{E} = \frac{{ε_{m} V (1 E 8)}^{2}}{N_{A}}

where Ė is the calculated value from Eq. (9). Vekshin’s screening equation can therefore be written in the following form

{\hat{ε}}_{m} = \frac{N_{A} s}{V 1 E 8^{2} k \dot{E} q} [1 - {(1 - p)}^{k}]

where the extinction coefficient for one chromophore is calculated by

\hat{ε} = \frac{{\hat{ε}}_{m} V}{M_{w}}

From this, the hypochromicity can be calculated using Eq. (8). The corrected imaginary part of the refractive index κ _c can be computed using the following equation

κ_{c} = \frac{\hat{ε} λ}{4 π}

Notice that the hypochromicity models given by Eqs. (15)–(17) can be used directly within their theoretical context, or empirically as a calibration model to correct the absorption component.

4. Results

4.1 Exploration of theoretical limits.

Three approaches were taken to explore, through simulation, the constraints of Rayleigh-Debye-Gans theory for spheres. First, the sizes of the spherical particles were kept small compared to the wavelengths, but the wavelength-dependent relative refractive index was allowed to significantly exceed the value of one. Refractive index ratios greater than one are typical of actual materials. Second, the relative refractive index was kept close to one while the absorption was held at zero, and particle sizes comparable to the wavelengths were considered. Third, the contribution of absorption in the relative refractive index, kept close to 1, was investigated for particle sizes comparable to the wavelengths. The following subsections describe in more detail the parameters used and the conclusions and observations drawn.

4.2 Particle diameter ≪ wavelength.

The first of the sensitivity studies conducted tested the limits of Rayleigh-Debye-Gans for relative refractive indices greater than 1 and the absorption greater than zero, while keeping small sized spherical particles, compared to the wavelengths (200 nm–900 nm). The multiwavelength transmission spectra were calculated for Mie and Rayleigh-Debye-Gans using spheres of silver bromide (1.1≤n/n _o≤2.4, 0.0001≤κ≤0.85) and spheres of silver chloride (1.1≤n/n _o≤2.4, 0.0001≤κ≤ 0.6). The spherical diameter sizes chosen were 25 nm and 50 nm. Particle concentration, particle density, and wavelength range were kept constant

Figures 3–4 show that Rayleigh-Debye-Gans gives an adequate approximation to Mie for particle sizes much smaller than the wavelength; notice that across the UV-Vis spectral region and shape and the amplitude of the spectral features are quite similar. Figures 5–6 reveal that for slightly larger particles Rayleigh-Debye-Gans no longer closely follows Mie Theory. Notice that in the spectral region where absorption is small (300 nm–900 nm) both theories coincide even though n/n _o>1. However, where strong absorption is present, the theories rapidly diverge, clearly suggesting that absorption plays an important role in the disparity between the theories, and that RDG is not a good approximation whenever absorption is present.

Fig. 3. Comparison of the calculated transmission of Mie and Rayleigh-Debye-Gans for 25 nm AgBr spheres suspended in water.
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Fig. 4. Comparison of the calculated transmission of Mie and Rayleigh-Debye-Gans for 25 nm AgCl spheres suspended in water.
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Fig. 5. Comparison of the calculated transmission of Mie and Rayleigh-Debye-Gans for 50 nm AgBr spheres suspended in water. Spectral differences in RDG are visible below 400 nm wavelength compared to those is Fig. 1.
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Fig. 6. Comparison of the calculated transmission of Mie and Rayleigh-Debye-Gans for a 50 nm AgCl spheres suspended in water. The spectral differences in RDG are visible below 400 nm wavelength compared to those in Fig. 4.
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4.3 Particle diameter≈wavelength, no absorption.

The restriction of Rayleigh-Debye-Gans theory with respect to size was tested through the calculation of transmission spectra at wavelength range between 200 nm and 900 nm for non-absorbing spherical particles with relative refractive index close to one. The refractive indices chosen were soft bodies (n=1.04) and hemoglobin (1.01≤n/n _o≤1.2); only the real part of the refractive index was used for hemoglobin. Particle diameters used were 500 nm, 1 µm, and 5.5 µm. Note that the refractive index spectrum of hemoglobin changes as function of wavelength and that it is not possible to meet the conditions for the application of RDG across the spectrum.

Figures 7 and 8 show that Rayleigh-Debye-Gans theory produces similar spectral patterns as Mie theory for relatively small particle sizes. However, RDG begins to diverge at the lower wavelengths (200–300 nm) for the particle size of 1 µm. Figure 9 shows that for a particle size of 5.5 µm Rayleigh-Debye-Gans theory simply does not approximate Mie theory. The combination of zero absorption and relative refractive index ratio close to 1 does not increase the particle size ranges for which RDG is said to be applicable; this is in disagreement with the results reported in Kerker [6].

Fig. 7. Comparison of Mie and Rayleigh-Debye-Gans calculated spectra of 500 nm Soft Body spheres suspended in water. Notice that the spectrumcalculated with Rayleigh-Debye-Gans theory does not approximate Mie theory.
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Fig. 8. Calculated transmission of Mie and Rayleigh-Debye-Gans for 1 µm Soft Body spheres suspended in water. Shape of the spectra by both theories remains similar.
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Fig. 9. Transmission spectra calculated using Mie and Rayleigh-Debye-Gans for 5.5 µm Soft Body spheres suspended in water. Rayleigh-Debye-Gans spectra does not coincide with Mie theory at such a large particle size.
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The multiwavelength transmission calculations conducted with only the real part of the refractive index of hemoglobin show that for 500 nm diameter particles (Fig. 10), the theories follow one another closely in spectral shape but there are quantifiable differences in amplitude. If turbidity is used for analysis, the spectral differences between the two theories would result in considerable variation in the estimate of particle size and concentration. With increasing of the particle diameter to 1 µm (Fig. 11), the spectral shape for Mie theory relative to Rayleigh-Debye-Gans flattens considerably at the shorter wavelengths. Figure 12 shows a semi-logarithmic turbidity plot of 5.5 µm particles to show the differences in shape and amplitude for the two theories. The effect of a relative refractive index greater than one with no absorption results in a limited particle size range for the application of RDG theory, in contrast to particles with a refractive index close to one with no absorption. It is noteworthy that RDG is being used for the characterization of microorganisms and red blood cells at wavelengths where the discrepancies are considerable.

Fig. 10. Calculated spectra of Mie and Rayleigh-Debye-Gans for 500 nm Hemoglobin spheres with no absorption (κ=0) suspended in water. Note: the shape of the spectra are similar.
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Fig. 11. Spectral differences between Mie and Rayleigh-Debye-Gans theory for 1 µm Hemoglobin spheres with no absorption (κ=0) suspended in water. At shorter wavelengths the spectra calculated by Mie theory flattens considerably.
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Fig. 12. Semi log plot spectra for 5.5 µm Hemoglobin spheres with no absorption (κ=0) calculated using Mie and Rayleigh-Debye-Gans theory. The limits of RDG have been met for particle size.
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4.4 Particle diameter ~ wavelength, absorption κ>0.

The limits of validity of Rayleigh-Debye-Gans theory with a relative refractive index close to 1 and an absorption coefficient greater than zero were tested through the calculation of the transmission spectra or spherical particles whose sizes are comparable to those of the wavelengths. The refractive indices of whole hemoglobin corresponding to wavelengths between 200 nm and 900 nm (1.01≤n/n _o≤1.2, 0.001≤κ≤0.1), were considered. The particle diameter sizes used were 100 nm, 500 nm, and 1 µm. Figure 13 show that Rayleigh-Debye-Gans and Mie closely follow one another for a 100 nm sphere. As the particle size was increased to 500 nm and 1 µm the calculated turbidity from Rayleigh- Debye-Gans slowly deviates from Mie (see Fig. 14 and Fig. 15). As the size increases, the features of the spectra calculated with Mie theory flatten yielding completely different spectra. These differences are due primarily to the fact that absorption efficiency Q _abs which is directly proportional to the volume in the case of RDG theory.

Fig. 13. Close approximation of the calculated transmission of Rayleigh-Debye-Gans to Mie theory for 100 nm Hemoglobin spheres suspended in water.
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Fig. 14. Comparison of calculated transmission for Rayleigh-Debye-Gans to that of Mie theory shows the divergence of theories for 500 nm Hemoglobin spheres suspended in water. Note the difference of RDG compared to Fig. 10 where no absorption is present.
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Fig. 15. The calculated spectra for 1 µm Hemoglobin spheres suspended in water shows that Mie theory appears to flatten when compared to Rayleigh-Debye-Gans.
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4.5 Compensation of RDG through the modification of the Optical Properties

The practical range of application of Rayleigh-Debye-Gans has been considered limited to small deviations from the relative refractive index (n(λ)/n ₀(λ)~1) and small values of the absorption coefficient (κ(λ)~0) [6]. The derivation of Rayleigh-Debye-Gans theory assumes each dipole absorbs and scatters independently and only considers the interference of the scattered wave. As a result, the angular scattering intensity is shape and orientation dependent, whereas the absorption efficiency is independent of the particle shape; in other words, the total absorption is only dependent on the particle volume (the total number of chromophoric groups in the particle) [5, 6]. When comparing the total absorption calculated from RDG and Mie theories for κ(λ)>0, a large discrepancy can be observed; the absorption efficiencies calculated with Mie theory always being smaller (hypochromic) than the values calculated with RDG. This holds particularly true for large absorption coefficients (i.e., Hemoglobin, DNA). The apparent hypochromicity observed for Mie theory suggests that the theoretical models developed to account for hypochromic or “screening” effects may be able to compensate RDG and bring it into a better agreement with Mie theory.

To explore the effect of hypochromicity corrections, the volume fraction of chromophoric groups is treated as an adjustable parameter in Veshkin’s model. Two cases are considered: v _f=0 which corresponds to 100% hypochromicity and translates to the corrected κ _c(λ) being equal to zero; and v _f=1 which corresponds to using the value of κ _c(λ) equal to κ(λ). Spherical hemoglobin particles with a diameter of 1 µm were used as test cases where Veshkin’s correction was applied only to κ(λ). The volume fraction values used in this study were 0.15, 0.20, 0.33, and 0.50. The molecular parameters for hemoglobin are: characteristic length=68 Å, with a cross sectional area of 20 Å, and a molecular weight of 16100 [19]. The orientation value q was set to one, meaning the molecules are randomly oriented [11].

The results of the hypochromicity corrections implemented in RDG theory are shown in Fig. 16 and Fig. 17. Figure 16 shows the spectra calculated with RDG and Mie without any corrections (NC) for hypochromicity, together with the spectra calculated with RDG and several levels of hypochromicity (i.e., volume fractions). Notice that although intermediate levels of hypochromicity result in improved RDG-calculated spectra, Veshkin’s model is not very effective in reducing the differences between Mie and RDG theories. This point is demonstrated more dramatically when 100% hypochromicity is considered. Figure 17 shows the extreme cases of 0% and 100% hypochromicity applied to both theories.

Fig. 16. Intermediate levels hypochromicity using Veshkin correction for Rayleigh-Debye-Gans compared to no hypochromicity correction in Mie and Rayleigh-Debye-Gans for 1 µm Hemoglobin spheres suspended in water.
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Fig. 17. Hypochromicity of 0% and 100% calculated using Mie and Rayleigh-Debye-Gans theory for a 1 µm Hemoglobin spheres suspended in water. Note that at 100% hypochromicity Rayleigh-Debye-Gans does not achieve reduced difference to Mie theory.
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The use of Veshkin’s model for the correction of the absorption coefficient brings about the problem of the inconsistency in terms of the Kramers-Kronig transforms since, after the correction, Eq. (6) and Eq. (7) will no longer hold. To demonstrate this inconsistency, an effective value of n _eff (λ) was calculated through the Kramer-Kronig transform after κ(λ) was corrected, using Veshkin’s model. All the conditions were kept the same for calculating the transmission as previously in this section. Figure 18 shows the results of calculating the transmission for Rayleigh-Debye-Gans with an effective n _eff and a corrected κ _c using Veshkin’s model, compared to uncorrected values of κ and n for RDG and Mie theory.

Fig. 18. Calculated transmission of Mie and Rayleigh-Debye-Gans for 1 µm Hemoglobin spheres with Veshkin Correction to k _c and an n _eff calculated through Kramers-Kronig Transform.
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Using the effective n calculated from corrected κ _c, one would expect the transmission by RDG to more closely match the transmission calculated by Mie; however, the contrary is observed. At different volume fractions of the chromophore relating to κ _c and n _eff values, there are distinct differences in the shape of the spectra. The differences in the spectra are rooted in determining the values for n _eff from the κ _c using the Kramers-Kronig transform.

An alternate approach for bringing together the spectra calculated from Mie and RDG theories is the mathematical adjustment of the refractive indices at each wavelength [17]. This approach explored in the next section.

4.6 Correction to RDG through the estimation of Effective Refractive Indices

The concept behind calculating effective refractive indices is that, given the extinction efficiency Q _ext (λ) calculated from Mie theory there may be a set of effective n(λ) and κ(λ) that can compensate for the differences between the extinction efficiencies calculated from RDG and Mie theories. This approach has been successfully applied by Latimer et al [17] to compensate for the effect of particle shape in monochromatic angular scattering measurements. The absorption and scattering efficiencies for RDG can be expressed as:

Q_{abs} = 4 k a Im (\frac{m^{2} - 1}{m^{2} + 2}) \approx 4 k a Im (\frac{2}{3} (m - 1)) = \frac{8}{3} k a κ

and

Q_{sca} = {(k a)}^{4} {∣ \frac{m^{2} - 1}{m^{2} + 2} ∣}^{2} \int_{0}^{π} f^{2} (θ) (1 - \cos^{2} θ) \sin θ d θ

Introducing $Λ = \int_{0}^{π} f^{2} (θ) (1 - \cos^{2} θ) \sin θ d θ$ , and realizing that Λ is independent of n and κ, for m≈1 Eq. (19) can be expressed as:

Q_{sca} \approx {(k a)}^{4} Λ {∣ \frac{2}{3} (n - 1) + i \frac{2}{3} κ ∣}^{2} = \frac{4}{9} {(k a)}^{4} Λ ({(n - 1)}^{2} + κ^{2})

A simplified form of the extinction efficiency at any wavelength can be obtained from the sum of Eq. (18) and Eq. (20)

Q_{ext} = \frac{8}{3} (k a) κ + \frac{4}{9} {(k a)}^{4} Λ ({(n - 1)}^{2} + κ^{2})

Notice that Eq. (21) is a quadratic in terms of n(λ) and κ(λ). Also notice that there is only one degree of freedom at each wavelength, and therefore only one parameter, either n _eff (λ) or κ _eff (λ), can be estimated from the observed Q _ext. Replacing the extinction efficiency in Eq. (21) with the extinction efficiency evaluated from Mie theory (Q _Mie), explicit expressions for both n _eff (λ) and κ _eff(λ) can be obtained.

If Q _ext(λ)=Q _Mie(λ) and κ(λ) is estimated, then n _eff (λ) can be obtained directly from:

n_{eff} = \sqrt{\frac{Q_{ko}}{q_{3}} + 1}

Alternatively, if Q _ext(λ)=Q _Mie(λ) and n(λ) is estimated then κ _eff(λ) can be obtained directly from:

κ_{eff} = \frac{- q_{2} \pm \sqrt{q_{2}^{2} + 4 q_{1} Q_{no}}}{2 q_{1}}

The terms Q _ko and Q _no are given by

Q_{ko} = Q_{Mie} (n, κ) - q_{1} κ^{2} - q_{2} κ

Q_{no} = Q_{Mie} (n, κ) - q_{3} {(n - 1)}^{2}

where $q_{1} = \frac{4}{9} {(k a)}^{2} Λ, q_{2} = \frac{8}{3} (k a), and q_{3} = \frac{4}{9} {(k a)}^{4} Λ$

Previous reports on the range of validity of RDG⁶ and the simulation results using Eq. (18) and Eq. (19) 125

[16] have shown that the difference Q _ext(RDG)-Q _ext(Mie) is always positive for the particle size range of interest; on this basis it can be readily shown that

(n_{eff} - 1) = {(\frac{Q_{Mie} (n, κ) - Q_{RDG} (n, κ)}{{2 q_{3} (n - 1)}^{2}} + 1)}^{\frac{1}{2}} (n - 1)

Therefore, there exists a value for 0≤n _eff≤n that will equate the Mie and RDG extinction efficiencies. Similarly, it can be shown that a positive κ _eff can be obtained from

κ_{eff} = (\frac{- q_{2} \pm \sqrt{{(2 q_{1} κ + q_{2})}^{2} + 4 q_{1} (Q_{Mie} (n, κ) - Q_{RDG} (n, κ))}}{2 q_{1}})

subject to the following constraints

{(2 q_{1} κ + q_{2})}^{2} \geq 4 q_{1} (Q_{Mie} (n, κ) - Q_{RDG} (n, κ))

and

\sqrt{{(2 q_{1} κ + q_{2})}^{2} + 4 q_{1} (Q_{Mie} (n, κ) - Q_{RDG} (n, κ))} \geq q_{2}

It is evident that, although effective values for the optical properties can be obtained, the approach is not satisfactory; the effective properties are also functions of the particle size, they have to be evaluated at every wavelength, and the optical property values calculated through Eqs. (26)–(29) do not have any physical meaning. In addition, the estimation of either n _eff (λ) or κ _eff (λ) conditional upon κ(λ) or n (λ) will not necessarily satisfy the Kramers-Kronig transforms.

5. Discussion

Prior to this study, a comparative evaluation of the Rayleigh-Debye-Gans approximation relative to Mie theory had not been studied for particle sizes relevant to biological systems over the UV-Vis spectral region (200–900nm). The importance of this study lies on the potential use of multiwavelength spectrophotometric measurements for the characterization of biological particles. Of particular interest is the region where relevant biological chromophores absorb (190–400nm); it is primarily in this region where strict adherence to the RDG constraints cannot be ensured. The simulation studies reported herein lead to the conclusion that there are significant differences in the spectroscopy behavior of homogeneous spherical particles when their transmission spectra are predicted with Rayleigh-Debye-Gans and with Mie theory. In these studies, the optical properties relevant to the study of industrial and biological particles have been used. The disagreement between RDG and Mie theories is most severe when absorption is present and when the particle size becomes larger than the wavelength. The effect of changes in the refractive indices has been explored as a means to extend the range of validity of RDG over the complete UV-Vis wavelength range. In studying the effects of changes in the refractive indices the expectation was to bring Rayleigh-Debye-Gans into better agreement with Mie theory for larger particles and for particles containing strong chromophoric groups. It has been demonstrated that it is not possible to adequately, or realistically, compensate for the differences between Mie and RDG through the use of hypochromicity models and/or effective refractive indices. The simplicity of the Rayleigh-Debye-Gans approximation for the prediction of shape and orientation, and its potential application to the characterization of complex biological particles, continues to justify the efforts to improve its accuracy and precision. Although the compensation approaches reported herein have shown not to be effective over a broad wavelength range, the reformulation of RDG in terms of a hybrid theory shows considerable promise. The results from such formulation will be reported shortly.

Appendix

Fig. 19. Optical properties of AgBr as function of wavelength
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Fig. 20. Optical properties of AgCl as function of wavelength
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Fig. 21. Optical properties of Soft bodies as function of wavelength
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Fig. 22. Optical properties of Hemoglobin as function of wavelength
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Fig. 23. Optical properties of Water as function of wavelength
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Acknowledgments

This work has been supported in part by the National Science Foundation through the Particle Engineering Research Center located at the University of Florida.

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Light Source Wavelength	Particle Concentration	Particle Density
200–900 nm	1E-4 g/cc	1 g/cc