Rayleigh-Gans-Debye Approximation Research Articles

On the Discrepancy between Modeled and Measured Mass Absorption Cross Sections of Light Absorbing Carbon Aerosols

Recent modeling studies based on the Rayleigh-Debye-Gans (RDG) approximation have revealed a discrepancy between modeled and measured mass absorption cross sections (MAC) for atmospheric light absorbing carbon (LAC) aerosols. One plausible explanation is that this discrepancy is due to errors introduced by neglecting electromagnetic interactions among monomers in LAC aggregates within the RDG approximation. Here we compute MAC by use of numerically exact solutions to Maxwell's equations and investigate the sensitivity of the results to a variation in the aggregates’ physical properties and refractive index. The results do confirm that approximate methods can introduce large errors in the results for the optical properties. However, these errors alone cannot explain the discrepancy between measured and modeled values of MAC. An agreement between observations and theoretical results can only be attained when assuming a fairly high value of the real and imaginary parts of the refractive index along the void-...

Effect of aggregation on the absorption cross-section of fractal soot aggregates and its impact on LII modelling

This study concerns the effect of particle aggregation on laser heating rate of soot aggregates in laser-induced incandescence. Three aggregate absorption models were investigated: the Rayleigh–Debye–Gans approximation, the electrostatics approximation, and the numerically exact generalized multi-sphere Mie-solution method. Fractal aggregates containing 5–893 primary particles of 30 nm in diameter were generated numerically using a combined particle-cluster and cluster-cluster aggregation algorithm with specific fractal parameters typical of soot. The primary particle size parameters considered are 0.089, 0.177, and 0.354. The Rayleigh–Debye–Gans approximation neglects the effect of particle aggregation on absorption; so it underestimates the aggregate absorption cross-section area by approximately 10%, depending on the aggregate size and primary particle size parameter. The electrostatics approximation is somewhat better than the Rayleigh–Debye–Gans approximation, but cannot account for the effect of primary particle size parameter. The aggregate absorption submodel affects the calculated soot temperature in laser-induced incandescence mainly in the low laser fluence regime. At high laser fluences, the effect diminishes due to the enhanced importance of soot sublimation cooling and neglect of aggregation effect in the sublimation in the present numerical model of laser-induced incandescence.

Complex refractive index of nonspherical particles in the visible near infrared region--application to Bacillus subtilis spores

A method is presented for the estimation of optical constants in the ultraviolet-visible-near-infrared (UV-Vis-NIR) region of nonspherical particles in a suspension at concentrations where multiple scattering is significant. The optical constants are obtained by an inversion technique using the adding-doubling method to solve the radiative transfer equation in combination with the single scattering theories for modelling scattering by nonspherical particles. Two methods for describing scattering by single scattering are considered: the T-matrix method and the approximate but computationally simpler Rayleigh-Gans-Debye (RGD) approximation. The method is then applied to obtain the optical constants of Bacillus subtilis spores in the wavelength region 400-1200 nm. It is found that the optical constants obtained using the RGD approximation matches those obtained using the T-matrix method to within experimental error.

On the validity of Rayleigh–Gans–Debye type approximations in red blood cell size determination

The validity of the Rayleigh-Gans-Debye (RGD) approximation for the size determination of a soft prolate spheroidal scatterer has been examined. Calculations have been carried out for the volume range 30–100 µm3 and for aspect ratios ranging from 1.1 to 4.0. Particular attention has been paid to size determination of a red blood cell (RBC) in a flow cytometric arrangement. The applicability of the RGD approximation has been compared with its two modified forms and also with the WKB approximation.

Light scattering and absorption by fractal-like carbonaceous chain aggregates: comparison of theories and experiment

This study compares the optical coefficients of size-selected soot particles measured at a wavelength of 870 nm with those predicted by three theories, namely, Rayleigh-Debye-Gans (RDG) approximation, volume-equivalent Mie theory, and integral equation formulation for scattering (IEFS). Soot particles, produced by a premixed ethene flame, were size-selected using two differential mobility analyzers in series, and their scattering and absorption coefficients were measured with nephelometry and photoacoustic spectroscopy. Scanning electron microscopy and image processing techniques were used for the parameterization of the structural properties of the fractal-like soot aggregates. The aggregate structural parameters were used to evaluate the predictions of the optical coefficients based on the three light-scattering and absorption theories. Our results show that the RDG approximation agrees within 10% with the experimental results and the exact electromagnetic calculations of the IEFS theory. Volume-equivalent Mie theory overpredicts the experimental scattering coefficient by a factor of approximately 3.2. The optical coefficients predicted by the RDG approximation showed pronounced sensitivity to changes in monomer mean diameter, the count median diameter of the aggregates, and the geometric standard deviation of the aggregate number size distribution.

Light-scattering form factors of asymmetric particle dimers from heteroaggregation experiments

Measurements of form factors of asymmetric particle dimers composed of oppositely charged polystyrene latex particles are presented. These measurements are based on time-resolved static and dynamic light scattering on dilute aggregating aqueous suspensions. The experimental form factors are compared with independent calculations based on the superposition T-matrix method and Rayleigh-Debye-Gans (RDG) approximation. While the RDG approximation is found to be reliable only up to particle diameters of about 250 nm, the superposition T-matrix method is very accurate for all types of dimers investigated. The present results show clearly the appropriateness of the superposition T-matrix method to estimate the optical properties of colloidal particles in the micrometer range reliably.

Forward light scattering to characterise structure of flocs composed of large particles

Abstract Forward light scattering has been a popular technique for measuring the fractal dimension of aggregated or flocculated particles. The method relies on the measurement of the slope of scattered intensity versus scattering vector ( q ) on a log–log plot. In the Rayleigh–Gans–Debye approximation, the negative of the slope has the value of the mass fractal dimension of the aggregates. In this work, simulation results were used to test the idea that Fraunhofer diffraction can be used to describe scattering from fractal aggregates. Polystyrene latex of diameter 4.9 μm was aggregated in a forward light scattering instrument, and the fractal dimension was determined independently by the volume obscuration method. Comparison of the experimental data with simulations showed poor agreement. It is suggested that multiple scattering within the aggregate structure, ignored in the usual Fraunhofer formulation, is significant.

Evaluation of on-line high-performance size-exclusion chromatography, differential refractometry, and multi-angle laser light scattering analysis for the monitoring of the oligomeric state of human immunodeficiency virus vaccine protein antigen

Chiron has developed a novel mutant form of the human immunodeficiency virus (HIV) envelop protein, o-gp140, that is currently entering Human Phase 1 clinical trials for testing as a prophylactic HIV vaccine. The o-gp140 protein is oligomeric and the quaternary structure is thought to play an important role in its activity as an antigen. As o-gp140 proceeds through the clinical trial process and prior to marketing approval, analytical methods that are able to demonstrate manufacturing consistency with respect to degree of oligomerization will need to be developed and validated. On-line high-performance size-exclusion chromatography, differential refractometry, and multi-angle laser light scattering analysis (HPSEC–RI–MALLS), a method commonly used to obtain the molar mass of macromolecules based on the Rayleigh–Gans–Debye approximation, was evaluated for this purpose. The results obtained demonstrated intra- and inter-day precisions to be 0.9 and 3.6% R.S.D., respectively. Accuracy was found to be equal to, or better than, 11% when comparing the known molar masses of test proteins to that of the molar masses determined by the method. Additionally, the method compared favorably to orthogonal native polyacrylamide gel electrophoresis and ultracentrifugation analyses. R-factor analysis was used to demonstrate that HPSEC–RI–MALLS is capable of discriminating compositional differences between o-gp140 test lots. Based on the data presented, HPSEC–RI–MALLS may be a suitable manufacturing control method.

Scattering Behavior of Restructured Aggregates: A Simulation Study

Simulations of diffusion-limited cluster–cluster aggregation (DLCA) with no restructuring, full restructuring, and partial restructuring have been performed. The scattering patterns produced from these aggregates have been simulated using the Rayleigh–Gans–Debye approximation. Pure DLCA aggregates produced a scattering pattern with the slope of the fractal region being about −1.8. In contrast, the slope of the fractal region of the scattering pattern for fully restructured aggregates was about −2.1, indicating an increase in fractal dimension. Partial restructuring at large length scales produced an upward turn in the scattering pattern at low qro, while at high qro the fractal section of the pure DLCA aggregate was retained. This last result was expected and is consistent with the results and postulations of several other workers. This simulation shows that the type of scattering pattern often obtained from orthokinetic or sheared aggregation can be produced by restructuring of aggregates at large length scales.

Light-scattering by optically soft randomly oriented spheroids

In the framework of the Rayleigh–Gans–Debye approximation and anomalous diffraction approaches, the light scattering characteristics of randomly oriented spheroids have been investigated. It has been proved that the system of randomly oriented spheroids is equivalent to the system of polydisperse spherical particles that have the same values of volume and surface area as nonspherical particles. The power law size distribution meeting these requirements has been obtained.

Polymerization of Rod-Like Macromolecular Monomers Studied by Stopped-Flow, Multiangle Light Scattering: Set-Up, Data Processing, and Application to Fibrin Formation

Many biological supramolecular structures are formed by polymerization of macromolecular monomers. Light scattering techniques can provide structural information from such systems, if suitable procedures are used to collect the data and then to extract the relevant parameters. We present an experimental set-up in which a commercial multiangle laser light scattering photometer is linked to a stopped-flow mixer, allowing, in principle, the time-resolved extrapolation of the weight-average molecular weight M w and of the z-average square radius of gyration 〈R g 2〉 z of the polymers from Zimm-like plots. However, if elongated structures are formed as the polymerization proceeds, curved plots rapidly arise, from which M w and 〈R g 2〉 z cannot be recovered by linear fitting. To verify the correctness of a polynomial fitting procedure, polydisperse collections of rod-like or worm-like particles of different lengths, generated at various stages during bifunctional polycondensations of rod-like macromolecular monomers, were considered. Then, the angular dependence of their time-averaged scattered intensity was calculated in the Rayleigh–Gans–Debye approximation, with random and systematic noise also added to the data. For relatively narrow size distributions, a third-degree polynomial fitting gave satisfactory results across a broad range of conversion degrees, yielding M w and 〈R g 2〉 z values within 2% and no greater than 10–20%, respectively, of the calculated values. When more broad size distributions were analyzed, the procedure still performed well for semiflexible polymers, but started to seriously underestimate both M w and 〈R g 2〉 z when rigid rod-like particles were analyzed, even at relatively low conversion degrees. The data were also analyzed in the framework of the Casassa approximation, from which the mass per unit length of the polymers can be derived. These procedures were applied to a set of data taken on the early stages of the thrombin-catalyzed polymerization of fibrinogen, a rod-like macromolecule ∼50 nm long. The polymers, grown in the absence of Ca 2+ by rate-limiting amounts of thrombin, appeared to be characterized by a much broader size distribution than the one expected for a classical Flory bifunctional polycondensation, and they seem to behave as relatively flexible worm-like double-stranded chains. Evidence for the formation of fibrinogen–fibrin monomer complexes is also inferred from the time dependence of the mass/length ratio. However, our data are also compatible with the presence of limited amounts of single-stranded structures in the very early stages, either as a secondary, less populated pathway, or as transient intermediates to the classical double-stranded fibrils.

Electrooptic effects in dilute suspensions of bacterial cells and fractal aggregates

The relative change in optical density (orientational turbidimetric effect) of dilute suspensions of bacterial cells is studied theoretically and experimentally. Optical properties of bacterial cells are modeled by homogeneous spheroids using the T-matrix method. Calculated differential spectra of the steady-state turbidimetric effect, as well as its electric field dependence at fixed wavelengths, are shown to be in good agreement with experimental measurements for dilute suspensions of Escherichia coli and Bacillus polymyxa cells. The theoretically predicted unusual non-exponential decay of optical density (after removal of the orienting field) has been illustrated by an experimental electrooptical example. By using the Rayleigh–Debye–Gans approximation, we have also computed orientational effects for dilute suspensions of 3D-lattice ballistic and diffusion-limited fractal aggregates built from optically soft small spheres. We have found that the turbidimetric effect is determined solely by the gyration radius diffraction parameter and is essentially independent of the cluster mass. In terms of orientational effects, the effective structural anisotropy of the random fractal aggregates is close to the anisotropy of the homogeneous spheroidal model with the axial ratio 2 for small clusters and reduces to 1.5 for larger clusters.

Theoretical description of the absorbance versus time curve in a homocoagulation process.

We have developed a simple theory within the Rayleigh–Gans–Debye (RGD) approximation to follow the homocoagulation process by a turbidity measurement. We have considered all the contributions in the aggregate from particle–particle pairs, with none, one and two particles between them and a von Smoluchowski process. It has been possible to obtain a theoretical description of the experimental absorbance vs. time curves for the first minute in most general cases. Due to the dependence on the RGD approximation, the smaller the particle is, the better agreement between experimental data and theoretical calculations is obtained. We have used this model to develop a technique to calculate kinetic constants from experimental turbidity measurements from homocoagulation of polystyrene latexes. Great agreement has been found between kinetic constants obtained by this method and others by fitting the initial slope. In this new method we can adjust up to 60 s of experimental data (for the particle concentrations we have used), to be compared with 4 or 5 s that can be used in adjusting the initial slope in the classical turbidimetric technique.

Phosphatidylcholine Vesicle Diameter, Molecular Weight and Wall Thickness Determined by Static Light Scattering

Phosphatidylcholine vesicles in the diameter range, 40 to 115 nm scatter light as isotropic hollow spheres within the Rayleigh-Gans-Debye (RGD) approximation. A more complex model (i.e., the exact or Lorenz-Mie theory) is not required for accurate interpretation of the static light scattering data. The scattered light intensity measurements also allow for the determination of the vesicle weight-average molecular weight through a Zimm plot analysis. This information, in combination with the vesicle size as determined within the RGD approximation, indicates that the vesicle wall thickness is close to 3.6 nm as previously determined by X-ray diffraction methods. Thus, static light scattering techniques may provide excellent estimates of surfactant vesicle wall thicknesses.

Kinetics of heterocoagulation. Part 4.—Evaluation of absolute coagulation rate constants using a classical light scattering technique

The rate of coagulation in systems of identical and unlike colloids in aqueous media has been studied by a simultaneous multi-angle static light scattering technique. The mixed dispersions consisted of polystyrene latex particles of different radii and of opposite sign of charge. A procedure based on the Rayleigh–Gans–Debye approximation permits one to estimate the absolute rate constant either for homo- or hetero-coagulation. The absolute coagulation rate constants were compared to those calculated theoretically, using the Smoluchowski theory. In homosystems, the fast coagulation rate corresponded to ca. 55–60% of the Smoluchowski value. In the heterosystem, the coagulation rates exceeded the calculated values based on the diffusion-controlled process at a low ionic strength. An increase in the electrolyte content led to a decrease of the coagulation rate, which at high concentrations corresponded to ca. 85% of the Smoluchowski value.

Scattering of Light from Cylindrical Particles: Coupled Dipole Method Calculations and the Range of Validity of the Rayleigh-Gans-Debye Approximation

Abstract We calculate the single particle scattering properties of cylindrical particles with the iterative version of the Coupled Dipole Method (CDM), and using these results we discuss the validity regime of the Rayleigh-Gans-Debye (RGD) theory. The CDM includes, contrary to RGD scattering, the higher order intraparticle multiple scattering contributions, which become more important when the dimensions of the rods increase and/or the refractive index difference with the solvent increases. The range of validity of the RGD approximation is assessed from the deviations from the CDM, which becomes exact in the limit of infinitely small distances between the subunits. Systematic calculations are performed for rods with a diameter smaller than one-tenth of and a length smaller than two times the wavelength of the light in the solvent, and a refractive index relative to the solvent that lies between 1 and 1.5.

Small‐angle scattering of polarized light. I. comparison of theoretical predictions for isotropic and anisotropic spheres

AbstractThis paper describes small angle light scattering (SALS) calculations and compares the results for isotropic and anisotropic spheres. For isotropic spheres the exact Mie theory is compared with the Rayleigh‐Gans‐Debye (RGD) and the anomalous diffraction (AD) approximations in order to study the validity of the approximations. For anisotropic spheres the RGD is compared with the AD approximation. A study of SALS in the RGD approximation shows that the first‐order intensity maximum corresponding to a lobe of the four‐leaf clover pattern is not only a function of the sphere diameter but also depends on the isotropic and the anisotropic refractive index of the sphere. The practical implications of this are discussed.

Modified Rayleigh-Gans-Debye approximations applied to sizing transparent homogeneous long fibres of intermediate size

Modified Rayleigh-Gans-Debye (RGD) approximations have been used as approximations to exact theory for fibre size measurements from the positions of the minima in the scattering curve. Size parameters in the range 2.0<or=x<or=20.0 are of interest. Percentage errors in size parameters obtained using the RGD approximation and modified RGD approximations have been tabulated for relative refractive indices m=1.05 and m=1.15.

General theory of dynamic light scattering from cylindrically symmetric particles with translational-rotational coupling

The general solution to the scattered electric field time autocorrelation function for a dilute system of cylindrically symmetric, optically anisotropic particles undergoing coupled translational-rotational diffusion is presented. The solution is exact within the Rayleigh–Gans–Debye approximation. The scattered time autocorrelation function is an infinite series of decaying exponentials with time constants τ−1Ms =q2D+[s(s+1)+wMs(γ)−γ/3]Θ⊥, (M=0,1,2; s=0,1,2, ⋅ ⋅ ), containing both translational and rotational diffusion coefficients and spheroidal harmonic eigenvalues. The latter depend on the coupling parameter γ=q2 (ΔD)/(Θ⊥), where ΔD is the anisotropy in the translational diffusion coefficient. The dynamical terms are weighted by structure factors that depend on scattering geometry, the particle geometry, and the coupling parameter. These structure factors are given for arbitrary geometries and polarization of the radiation. In general, the weights for two to six exponentials will be significant depending on the scattering geometry and particle size. The domain of applicability of the theory is also discussed.

Detection of small polydispersities by photon correlation spectroscopy

We consider dilute suspensions of homogeneous polydisperse spherical particles for which the Rayleigh–Gans–Debye (RGD) approximation is valid. For two model particle size distributions we calculate the dependence on scattering vector Q of the average scattered intensity I(Q) and the effective diffusion coefficient De(Q) obtained from the first cumulant measured by photon-correlation spectroscopy. If the mean particle radius R̄ is large enough (≳170 nm) that the intensity form factor P(QR̄) shows at least one minimum in the accessible range of Q, we find that De(Q) exhibits a characteristic variation with Q which is very sensitive to the sample polydispersity. Under favorable conditions it should be possible to measure polydispersities (standard deviation/mean size) as small as 0.01. These theoretical considerations are supported, at least qualitatively, by experiment. We also discuss briefly the effect of relaxing the RGD approximation and the implications of this work for the more common PCS probes of polydispersity such as Laplace transformation of the light scattering correlation function.