Multiangle light scattering

Multiangle light scattering (MALS) is a technique for determining, independently, the absolute molar mass and the average size of particles in solution, by detecting how they scatter light. Collimated light from a laser source is most often used, in which case the technique can be referred to as multiangle laser light scattering (MALLS). The insertion of the word “laser” was intended to reassure those used to making light scattering measurements with conventional light sources such as Hg-arc lamps that low angle measurements could now be made. Until the advent of lasers and their associated fine beams of narrow width, the width of conventional light beams used to make such measurements prevented data collection at smaller scattering angles. In recent years, since all commercial light scattering instrumentation use laser sources, this need to mention the light source has been dropped and the term MALS used throughout.

The "multi-angle" term refers to the detection of scattered light at different discrete angles as measured, for example, by a single detector moved over a range that includes the particular angles selected or an array of detectors fixed at specific angular locations. A discussion of the physical phenomenon related to this static light scattering, including some applications, data analysis methods and graphical representations associated therewith are presented.

A MALS measurement requires a set of ancillary elements. Most important among them is a collimated or focused light beam (usually from a laser source producing a collimated beam of monochromatic light) that illuminates a region of the sample. In modern instruments, the beam is generally plane-polarized perpendicular to the plane of measurement, though other polarizations may be used especially when studying anisotropic particles. Earlier measurements, before the introduction of lasers, were performed using focused, though unpolarized, light beams from sources such as Hg-arc lamps. Another required element is an optical cell to hold the sample being measured. Alternatively, cells incorporating means to permit measurement of flowing samples may be employed. If single-particles scattering properties are to be measured, a means to introduce such particles one-at-a-time through the light beam at a point generally equidistant from the surrounding detectors must be provided.

Although most MALS-based measurements are performed in a plane containing a set of detectors usually equidistantly placed from a centrally located sample through which the illuminating beam passes, three-dimensional versions^[1][2] also have been developed wherein the detectors lie on the surface of a sphere with the sample controlled to pass through its center where it intersects the path of the incident light beam passing along a diameter of the sphere. The former framework^[1] is used for measuring aerosol particles while the latter^[2] was used to examine marine organisms such as phytoplankton.

Background

The measurement of scattered light from an illuminated sample forms the basis of the so-called classical light scattering measurement. Historically, such measurements were made using a single detector^[3][4] rotated in an arc about the illuminated sample. The first commercial instrument (formally called a “scattered photometer”) was the Brice-Phoenix light scattering photometer introduced in the mid 1950s and followed by the Sofica photometer introduced in the late 1960s.

Measurements were generally expressed as scattered intensities or scattered irradiance. Since the collection of data was made as the detector was placed at different locations on the arc, each position corresponding to a different scattering angle, the concept of placing a separate detector at each angular location of interest^[5] was well understood, though not implemented commercially^[6] until the late 1970s. Multiple detectors having different quantum efficiency have different response and hence needs to be normalized in this scheme. An interesting system based upon the use of high speed film was developed by Brunsting and Mullaney^[7] in 1974. It permitted the entire range of scattered intensities to be recorded on the film with a subsequent densitometer scan providing the relative scattered intensities. The then-conventional use of a single detector rotated about an illuminated sample with intensities collected at specific angles was called differential light scattering^[8] after the quantum mechanical term differential cross section,^[9] σ(θ) expressed in milli-barns/steradian. Differential cross section measurements were commonly made, for example, to study the structure of the atomic nucleus by scattering from them nucleons,^[10] such as neutrons. It is important to distinguish between differential light scattering and dynamic light scattering, both of which are referred to by the initials DLS. The latter refers to a technique that is quite different, measuring the fluctuation of scattered light due to constructive and destructive interference, the frequency being linked to the thermal motion, Brownian motion of the molecules or particles in solution or suspension.

The traditional differential light scattering measurement was virtually identical to the currently used MALS technique. Although the MALS technique generally collects multiplexed data sequentially from the outputs of a set of discrete detectors, the earlier differential light scattering measurement also collected data sequentially as a single detector was moved from one collection angle to the next. The MALS implementation is of course much faster, but the same types of data are collected and are interpreted in the same manner. The two terms thus refer to the same concept. For differential light scattering measurements, the light scattering photometer has a single detector whereas the MALS light scattering photometer generally has a plurality of detectors.

Another type of MALS device was developed in 1974 by Salzmann et al.^[11] based on a light pattern detector invented by George et al.^[12] for Litton Systems Inc. in 1971. The Litton detector was developed for sampling the light energy distribution in the rear focal-plane of a spherical lens for sampling geometric relationships and the spectral density distribution of objects recorded on film transparencies.

The application of the Litton detector by Salzman et al. provided measurement at 32 small scattering angles between 0° and 30°, and averaging over a broad range of azimuthal angles as the most important angles are the forward angles for static light scattering. By 1980, Bartholi et al.^[13] had developed a new approach to measuring the scattering at discrete scattering angles by using an elliptical reflector to permit measurement at 30 polar angles over the range 2.5° ≤ θ ≤ 177.5° with a resolution of 2.1°.

The commercialization of multiangle systems began in 1977 when Phillips^[14] introduced a flow-through capillary surrounded by 8 discrete detectors for a customized bioassay system developed for the USFDA. His implementation was commercialized in 1984 with the introduction of a 15 detector instrument (Dawn-F: Wyatt Technology Corporation, Santa Barbara, CA). By 1985, a three-dimensional configuration was introduced^[1] specifically to measure the scattering properties of single aerosol particles. At about the same time, the underwater device was built to measure the scattered light properties of single phytoplankton.^[2] Signals were collected by optical fibers and transmitted to individual photomultipliers. A more recent instrument was commercialized, which measures 7 scattering angles using a CCD detector (BI-MwA: Brookhaven Instruments Corp, Hotlsville, NY).

The commercial introduction of conventional light scattering instrumentation in the early 1970s that incorporated a laser light source^[15] was the impetus for the development of a new class of photometers. In 1972, Beckman Instruments, recognizing that the use of a laser source would permit scattered light measurements to be made at very small angles (for example for determining the weight average molar mass of a sample following the method of Zimm^[3]), introduced their low angle laser light scattering instrument, developed by Wilber Kaye^[16][17][18] and his colleagues, at the American Chemical Society’s 1972 National meeting in Los Angeles. They referred to their instrument as a LALLS (Low-Angle Laser Light Scattering) photometer with the word “laser” added, as discussed earlier, to emphasize the ability to measure at low angles since the very narrow beam produced by a laser would permit measurement at correspondingly smaller scattering angles than achievable using conventional light sources. The only LALS system currently available is commercialized by Viscotek/Malvern. By 1992, the first commercial light scattering photometer providing a plurality of discrete detectors was introduced. Although the term DLS (differential light scattering) was initially associated with such multiangle detectors (the term is now most commonly used to refer to “dynamic light scattering”), following Beckman’s popularization of the term “LALLS”, MALLS became a common descriptive. The laser reference was finally dropped and MALS has survived. Two angle devices have appeared with the descriptor DALS (dual angle light scattering) or TALS. The classic 90° detection angle is referred to as right-angle light scattering, RALS.

The literature associated with measurements made by MALS photometers is extensive.^[19] both in reference to batch measurements of particles/molecules and measurements following fractionation by chromatographic means such as size exclusion chromatography^[20] (SEC), reversed phase chromatography^[21] (RPC), and field flow fractionation^[22] (FFF).

Theory

The interpretation of scattering measurements made at the multiangular locations relies upon some knowledge of the a priori properties of the particles or molecules measured. The scattering characteristics of different classes of such scatterers may be interpreted best by application of an appropriate theory. For example, the following theories are most often applied.

Rayleigh scattering is the simplest and describes elastic scattering of light or other electromagnetic radiation by objects much smaller than the incident wavelength. This type of scattering is responsible for the blue color of the sky during the day and is inversely proportional to the fourth power of wavelength.

Rayleigh-Gans scattering is generalization of Rayleigh scattering, assumes that the scattering particles have a refractive index, n₁, very close to the refractive index of the surrounding medium, n₀. If we set m = n₁/n₀ and assume that |m - 1| << 1, then such particles may be considered as composed of very small elements, each of which may be represented as a Rayleigh-scattering particle. Thus each small element of the larger particle is assumed to scatter independently of any other.

Lorenz-Mie^[23] scattering refers to the scattering of light by spherical particles. Rayleigh-Gans scattering and Lorenz-Mie scattering produce identical results for spheres in the limit as |1 − m| → 0.

Lorenz-Mie scattering may be generalized to spherically symmetric particles per reference.^[24] More general shapes and structures have been treated by Erma.^[25]

Scattering data is usually represented in terms of the so-called excess Rayleigh ratio defined as the Rayleigh ratio of the solution or single particle event from which is subtracted the Rayleigh ratio of the carrier fluid itself and other background contributions, if any. The Rayleigh Ratio measured at a detector lying at an angle θ and subtending a solid angle ΔΩ is defined as the intensity of light per unit solid angle per unit incident intensity, I₀, per unit illuminated scattering volume ΔV. The scattering volume ΔV from which scattered light reaches the detector is determined by the detector’s field of view generally restricted by apertures, lenses and stops. Consider now a MALS measurement made in a plane from a suspension of N identical particles/molecules per ml illuminated by a fine beam of light produced by a laser. Assuming that the light is polarized perpendicular to the plane of the detectors. The scattered light intensity measured by the detector at angle θ in excess of that scattered by the suspending fluid would be

${\displaystyle I(\theta )={\frac {I_{0}N\Delta V}{(kr)^{2}}}i(\theta )}$,

where i(θ) is the scattering function^[3] of a single particle, k = 2πn₀/λ₀, n₀ is the refractive index of the suspending fluid, and λ₀ is the vacuum wavelength of the incident light. The excess Rayleigh ratio, R(θ), is then given by

${\displaystyle R(\theta )={\frac {I(\theta )r^{2}}{I_{0}\Delta V}}=Ni(\theta )/k^{2}}$.

Even for a simple homogeneous sphere of radius a whose refractive index, n, is very nearly the same as the refractive index "n₀" of the suspending fluid, i.e. Rayleigh-Gans approximation, the scattering function in the scattering plane is the relatively complex quantity

${\displaystyle i(\theta )={\frac {k^{2}V^{2}\left|{m-1}\right|^{2}}{4\pi ^{2}}}G^{2}\left({2ka\sin {\frac {\theta }{2}}}\right)}$,   where

${\displaystyle G(\xi )={\frac {3}{\xi ^{2}}}(\sin \xi -\xi \cos \xi )}$,   ${\displaystyle k={\frac {2\pi n_{0}}{\lambda _{0}}}}$,    ${\displaystyle V={\frac {4}{3}}\pi a^{3}}$

and λ₀ is the wavelength of the incident light in vacuum.

References

1. P. J. Wyatt; Y. J. Chang; C. Jackson; R. G. Parker; et al. (1988). "Aerosol Particle Analyzer". Applied Optics. 27 (2): 217–221. Bibcode:1988ApOpt..27..217W. doi:10.1364/AO.27.000217. PMID 20523583.
2. P. J. Wyatt & C. Jackson (1989). "Discrimination of Phytoplankton via Light-Scattering Properties". Limnology & Oceanography. 34 (I): 96–112. doi:10.4319/lo.1989.34.1.0096.
3. B. A. Zimm (1948). "Apparatus and methods for measurement and interpretation of the angular variation of light scattering; preliminary results on polystyrene solutions". J. Chem. Phys. 16: 1099–1116. Bibcode:1948JChPh..16.1099Z. doi:10.1063/1.1746740.
4. B. A. Brice; M. Halwer & R. Speiser (1950). "Photoelectric light scattering photometer for determining high molecular weights". J. Opt. Soc. Am. 40: 768–778. doi:10.1364/JOSA.40.000768.
5. P. J. Wyatt in U.S. Patent 3,624,835 (1971) filed 1968.
6. G. C. Salzmann; J. M. Crowell; C. A. Goad; K. M. Hansen; et al. (1975). "A Flow-System Multiangle Light-Scattering Instrument for Cell Characterization". Clinical Chemistry. 21 (9): 1297–1304. PMID 1149235.
7. A. Brunsting & P. F. Mullaney (1974). "Differential Light Scattering from Spherical Mammalian Cells". Biophys. J. 14 (6): 439–453. Bibcode:1974BpJ....14..439B. doi:10.1016/S0006-3495(74)85925-4. PMC 1334522. PMID 4134589.
8. P. J. Wyatt (1968). "Differential Light Scattering: A Physical Method for Identifying Living Bacterial Cells". Applied Optics. 7 (10): 1879–1896. Bibcode:1968ApOpt...7.1879W. doi:10.1364/AO.7.001879. PMID 20068905.
9. Cf. L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, New York 1955).
10. S. Fernbach (1958). "Nuclear Radii as Determined by Scattering of Neutrons". Revs. Modern Phys. 30: 414–418. Bibcode:1958RvMP...30..414F. doi:10.1103/RevModPhys.30.414.
11. G. C. Salzmann; J. M. Crowell; C. A. Goad; K. M. Hansen; et al. (1975). "A Flow-System Multiangle Light-Scattering Instrument for Cell Characterization". Clinical Chemistry. 21: 1297–1304.>
12. N. George, A. Spindel, J. T. Thomasson in U.S. Patent 3,689,772A(1972) filed 1971.
13. M. Bartholdi; G. C. Salzman; R. D. Hiebert & M. Kerker (1980). "Differential light scattering photometer for rapid analysis of single particles in flow". Applied Optics. 19 (10): 1573–1581. Bibcode:1980ApOpt..19.1573B. doi:10.1364/AO.19.001573.
14. L. V. Maldarelli, D. T. Phillips, W. L. Proctor, P. J. Wyatt, and T. C. Urquhart, Programmable action sampler system, U.S. Patent 4,140,018 (1979) filed 1977.
15. D. T. Phillips, D. T. (August 15, 1971). "Evolution of a light scattering photometer". BioScience. American Institute of Biological Sciences. 21 (16): 865–867. doi:10.2307/1295813. ISSN 0006-3568. JSTOR 1295813.
16. W. Kaye; Grossman, D. M.; Jacobs, A. L.; Michaelis, A. F.; Salstiz, B. (1973). "Low-angle laser light scattering". Anal. Chem. 45: 221A–225A. doi:10.1021/ac60325a068.
17. W. Kaye & A. J. Havlik (1973). "Low-angle laser light scattering—Absolute Calibration". Applied Optics. 12 (3): 541–550. Bibcode:1973ApOpt..12..541K. doi:10.1364/AO.12.000541. PMID 20125341.
18. W. Kaye & J. B. McDaniel (1974). "Low-angle laser light scattering—Rayleigh factors and depolarization ratios". Applied Optics. 13 (8): 1934–1937. Bibcode:1974ApOpt..13.1934K. doi:10.1364/AO.13.001934. PMID 20134598.
19. See, for example Chemical Abstracts
20. A. M. Striegel; W. W. Yau; J. J. Kirkland & D. D. Bly (2009). Modern Size-Exclusion Liquid Chromatography: Practice of Gel Permeation and Gel Filtration Chromatography. John Wiley and Sons. ISBN 0-471-20172-3.
21. I. V. Astafieva; G. A. Eberlein; Y. J. Wang (1996). "Absolute on-line molecular mass analysis of basic fibroblast growth factor and its multimers by reversed-phase liquid chromatography with multi-angle laser light scattering detection". Journal of Chromatography A. 740 (2): 215–229. doi:10.1016/0021-9673(96)00134-3. PMID 8765649.
22. M. Schimpf; K. Caldwell; J. C. Giddings, eds. (2000). Field-Flow Fractionation Handbook. Wiley-IEEE. ISBN 0-471-18430-6.
23. L. V. Lorenz (1890). Videnski.Selsk.Skrifter. 6: 1–62. Missing or empty |title= (help)
24. P. J. Wyatt (1962). "Scattering of Electromagnetic Plane Waves from Inhomogeneous Spherically Symmetric Objects". Physical Review. 127 (5): 1837–1843. Bibcode:1962PhRv..127.1837W. doi:10.1103/PhysRev.127.1837."Errata Ibid". The Phys.Rev. 134 (7AB): AB1. 1964. Bibcode:1964PhRv..134....1B. doi:10.1103/physrev.134.ab1.2.
25. V.A. Erma (1968a). "An exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape: I. Case of cylindrical symmetry.". The Phys.Rev. 173: 1243–1257. Bibcode:1968PhRv..173.1243E. doi:10.1103/physrev.173.1243.V.A. Erma (1968b). "Exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape: II. General case.". The Phys.Rev. 176: 1544–1553. Bibcode:1968PhRv..176.1544E. doi:10.1103/physrev.176.1544.V.A. Erma (1969). "Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape: III. Obstacles with arbitrary electromagnetic properties.". The Phys.Rev. 179: 1238–1246. Bibcode:1969PhRv..179.1238E. doi:10.1103/physrev.179.1238.