Multiangle light scattering

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Multiangle light scattering (MALS) describes a technique for measuring the light scattered by a sample into a plurality of angles. It is used for determining both the absolute molar mass and the average size of molecules in solution, by detecting how they scatter light. A collimated beam 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.[ citation needed ] 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 is used throughout.

Contents

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.

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 [1] [2] 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 [3] was well understood, though not implemented commercially [4] 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 [5] 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 [6] after the quantum mechanical term differential cross section , [7] 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, [8] 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.

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.[ citation needed ] 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 [9] [10] 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 [9] is used for measuring aerosol particles while the latter [10] was used to examine marine organisms such as phytoplankton.

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 Science Spectrum, Inc. [14] patented a flow-through capillary system for a customized bioassay system developed for the USFDA. The first commercial MALS instrument incorporating 8 discrete detectors was delivered to S.C. Johnson and Son, by Wyatt Technology Company, in 1983, [15] followed in 1984 with the sale of the first 15 detector flow instrument (Dawn-F) [16] to AMOCO. By 1988, a three-dimensional configuration was introduced [9] 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. [10] Signals were collected by optical fibers and transmitted to individual photomultipliers. Around December 2001, an instrument was commercialized, which measures 7 scattering angles using a CCD detector (BI-MwA: Brookhaven Instruments Corp, Hotlsville, NY).

The literature associated with measurements made by MALS photometers is extensive. [17] [18] both in reference to batch measurements of particles/molecules and measurements following fractionation by chromatographic means such as size exclusion chromatography [19] (SEC), reversed phase chromatography [20] (RPC), and field flow fractionation [21] (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.

The Rayleigh–Gans approximation is a means of interpreting MALS measurements with the assumption 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 [22] theory is used to interpret the scattering of light by homogeneous spherical particles. The Rayleigh–Gans approximation and the Lorenz–Mie theory produce identical results for homogeneous spheres in the limit as |1 − m| → 0.

Lorenz–Mie theory may be generalized to spherically symmetric particles per reference. [23] More general shapes and structures have been treated by Erma. [24]

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

,

where i(θ) is the scattering function [1] 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

.

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

,   where
,   ,    

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

Applications

The Zimm plot and batch collection

Zimm plot Graphic Explanation of a Zimm Plot.jpg
Zimm plot

MALS is most commonly used for the characterization of mass and size of molecules in solution. Early implementations of MALS such as those discussed by Bruno H. Zimm in his paper “Apparatus and Methods for Measurement and Interpretation of the Angular Variation of Light Scattering; Preliminary Results on Polystyrene Solutions” [1] involved utilizing a single detector rotated about a sample contained within a transparent vessel. MALS measurements from non-flowing samples such as this are commonly referred to as a “batch measurements.” By creating samples at several known low concentrations and detecting scattered light about the sample at varying angles, one can create a Zimm plot [25] by plotting : vs where c is the concentration of the sample and k is a stretch factor used to put kc and into the same numerical range.

When plotted one can extrapolate to both zero angle and zero concentration, and analysis of the plot will give the mean square radius of the sample molecules from the initial slope of the c=0 line and the molar mass of the molecule at the point where both concentration and angle equal zero. Improvements to the Zimm plot, which incorporate all collected data, commonly referred to as a “global fit” have largely replaced the Zimm plot in modern batch analyses. [26]

SEC and flow mode

MALS signals for polystyrene spheres MALS PS Mis 3DPlot MacAstra.jpg
MALS signals for polystyrene spheres

With the advent of size exclusion chromatography (SEC), MALS measurements began to be used in conjunction with an on-line concentration detector to determine absolute molar mass and size of sample fractions eluting from the column, rather than depending on calibration techniques. These flow mode MALS measurements have been extended to other separation techniques such as field flow fractionation, ion exchange chromatography, and reversed-phase chromatography.

The angular dependence of light scattering data is shown below in a figure of mix of polystyrene spheres which was separated by SEC. The two smallest samples (farthest to the right) eluted last and show no angular dependence. The sample, second to the right shows a linear angular variation with the intensity increasing at lower scattering angles. The largest sample, on the left, elutes first and shows non-linear angular variation.

Utility of MALS Measurements

Molar mass and size

BSA Separation and MM distribution MALS BSA separation and MW.jpg
BSA Separation and MM distribution

Coupling MALS with an in-line concentration detector following a sample separation means like SEC permits the calculation of the molar mass of the eluting sample in addition to its root-mean-square radius. The figure below represents a chromatographic separation of BSA aggregates. The 90° light scattering signal from a MALS detector and the molar mass values for each elution slice are shown.

Molecular interactions

As MALS can provide molar mass and size of molecules, it permits study into protein-protein binding, oligomerization and the kinetics of self-assembly, association and dissociation. By comparing the molar mass of a sample to its concentration, one can determine the binding affinity and stoichiometry of interacting molecules.

Branching and molecular conformation

The branching ratio of a polymer relates to the number of branch units in a randomly branched polymer and the number of arms in star-branched polymers and was defined by Zimm and Stockmayer as

Where is the mean square radius of branched and linear macromolecules with identical molar masses. [27] By utilizing MALS in conjunction with a concentration detector as described above, one create a log-log plot of the root-mean-square radius vs molar mass. The slope of this plot yields the branching ratio, g. [28]

In addition to branching, the log-log plot of size vs. molar mass indicates the shape or conformation of a macromolecule. An increase in the slope of the plot indicates a variation in conformation of a polymer from spherical to random coil to linear. Combining the mean-square radius from MALS with the hydrodynamic radius attained from DLS measurements yields the shape factor ρ = , for each macromolecular size fraction.

Other applications

Other MALS applications include nanoparticle sizing, [29] [30] [31] protein aggregation studies, protein-protein interactions, electrophoretic mobility or zeta potential. MALS techniques have been adopted for the study of pharmaceutical drug stability and use in nanomedicine.

Related Research Articles

In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

<span class="mw-page-title-main">Rayleigh scattering</span> Scattering of electromagnetic radiation by particles smaller than the radiations wavelength

Rayleigh scattering, named after the 19th-century British physicist Lord Rayleigh, is the predominantly elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the radiation. For light frequencies well below the resonance frequency of the scattering particle, the amount of scattering is inversely proportional to the fourth power of the wavelength.

<span class="mw-page-title-main">Compton scattering</span> Scattering of photons off charged particles

Compton scattering discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a charged particle, usually an electron. If it results in a decrease in energy of the photon, it is called the Compton effect. Part of the energy of the photon is transferred to the recoiling electron. Inverse Compton scattering occurs when a charged particle transfers part of its energy to a photon.

Circular dichroism (CD) is dichroism involving circularly polarized light, i.e., the differential absorption of left- and right-handed light. Left-hand circular (LHC) and right-hand circular (RHC) polarized light represent two possible spin angular momentum states for a photon, and so circular dichroism is also referred to as dichroism for spin angular momentum. This phenomenon was discovered by Jean-Baptiste Biot, Augustin Fresnel, and Aimé Cotton in the first half of the 19th century. Circular dichroism and circular birefringence are manifestations of optical activity. It is exhibited in the absorption bands of optically active chiral molecules. CD spectroscopy has a wide range of applications in many different fields. Most notably, UV CD is used to investigate the secondary structure of proteins. UV/Vis CD is used to investigate charge-transfer transitions. Near-infrared CD is used to investigate geometric and electronic structure by probing metal d→d transitions. Vibrational circular dichroism, which uses light from the infrared energy region, is used for structural studies of small organic molecules, and most recently proteins and DNA.

<span class="mw-page-title-main">Size-exclusion chromatography</span> Chromatographic method in which dissolved molecules are separated by their size & molecular weight

Size-exclusion chromatography (SEC), also known as molecular sieve chromatography, is a chromatographic method in which molecules in solution are separated by their size, and in some cases molecular weight. It is usually applied to large molecules or macromolecular complexes such as proteins and industrial polymers. Typically, when an aqueous solution is used to transport the sample through the column, the technique is known as gel-filtration chromatography, versus the name gel permeation chromatography, which is used when an organic solvent is used as a mobile phase. The chromatography column is packed with fine, porous beads which are commonly composed of dextran, agarose, or polyacrylamide polymers. The pore sizes of these beads are used to estimate the dimensions of macromolecules. SEC is a widely used polymer characterization method because of its ability to provide good molar mass distribution (Mw) results for polymers.

<span class="mw-page-title-main">Mie scattering</span> Scattering of an electromagnetic wave

The Mie solution to Maxwell's equations describes the scattering of an electromagnetic plane wave by a homogeneous sphere. The solution takes the form of an infinite series of spherical multipole partial waves. It is named after Gustav Mie.

<span class="mw-page-title-main">Time of flight</span> Timing of substance within a medium

Time of flight (ToF) is the measurement of the time taken by an object, particle or wave to travel a distance through a medium. This information can then be used to measure velocity or path length, or as a way to learn about the particle or medium's properties. The traveling object may be detected directly or indirectly.

In physics and chemistry, Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposition of wave fronts scattered by lattice planes, leading to a strict relation between wavelength and scattering angle, or else to the wavevector transfer with respect to the crystal lattice. Such law had initially been formulated for X-rays upon crystals. However, It applies to all sorts of quantum beams, including neutron and electron waves at atomic distances, as well as visible light at artificial periodic microscale lattices.

In polymer chemistry, the molar mass distribution describes the relationship between the number of moles of each polymer species and the molar mass of that species. In linear polymers, the individual polymer chains rarely have exactly the same degree of polymerization and molar mass, and there is always a distribution around an average value. The molar mass distribution of a polymer may be modified by polymer fractionation.

<span class="mw-page-title-main">Powder diffraction</span>

Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is called a powder diffractometer.

Elastic recoil detection analysis (ERDA), also referred to as forward recoil scattering, is an ion beam analysis technique in materials science to obtain elemental concentration depth profiles in thin films. This technique is known by several different names. These names are listed below. In the technique of ERDA, an energetic ion beam is directed at a sample to be characterized and there is an elastic nuclear interaction between the ions of beam and the atoms of the target sample. Such interactions are commonly of Coulomb nature. Depending on the kinetics of the ions, cross section area, and the loss of energy of the ions in the matter, ERDA helps determine the quantification of the elemental analysis. It also provides information about the depth profile of the sample.

<span class="mw-page-title-main">Dynamic light scattering</span> Technique for determining size distribution of particles

Dynamic light scattering (DLS) is a technique in physics that can be used to determine the size distribution profile of small particles in suspension or polymers in solution. In the scope of DLS, temporal fluctuations are usually analyzed using the intensity or photon auto-correlation function. In the time domain analysis, the autocorrelation function (ACF) usually decays starting from zero delay time, and faster dynamics due to smaller particles lead to faster decorrelation of scattered intensity trace. It has been shown that the intensity ACF is the Fourier transform of the power spectrum, and therefore the DLS measurements can be equally well performed in the spectral domain. DLS can also be used to probe the behavior of complex fluids such as concentrated polymer solutions.

Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight Mw of a macromolecule like a polymer or a protein in solution. Measurement of the scattering intensity at many angles allows calculation of the root mean square radius, also called the radius of gyration Rg. By measuring the scattering intensity for many samples of various concentrations, the second virial coefficient, A2, can be calculated.

Electrophoretic light scattering is based on dynamic light scattering. The frequency shift or phase shift of an incident laser beam depends on the dispersed particles mobility. With dynamic light scattering, Brownian motion causes particle motion. With electrophoretic light scattering, oscillating electric field performs this function.

Rutherford backscattering spectrometry (RBS) is an analytical technique used in materials science. Sometimes referred to as high-energy ion scattering (HEIS) spectrometry, RBS is used to determine the structure and composition of materials by measuring the backscattering of a beam of high energy ions (typically protons or alpha particles) impinging on a sample.

Absolute molar mass is a process used to determine the characteristics of molecules.

<span class="mw-page-title-main">Asymmetric flow field flow fractionation</span>

Asymmetrical flow field-flow fractionation (AF4) is most versatile and most widely used sub-technique within the family of field flow fractionation (FFF) methods. AF4 can be used in aqueous and organic solvents and is able to characterize nanoparticles, polymers and proteins. The theory for AF4 was conceived in 1986 and was established in 1987 and first published by Wahlund and Giddings. AF4 is distinct from symmetrical Flow FFF because it contains only one permeable wall so the cross-flow is caused only by the carrier liquid. The cross-flow is induced by the carrier liquid constantly exiting by way of the semi-permeable wall on the bottom of the channel.

Polymerscattering experiments are one of the main scientific methods used in chemistry, physics and other sciences to study the characteristics of polymeric systems: solutions, gels, compounds and more. As in most scattering experiments, it involves subjecting a polymeric sample to incident particles, and studying the characteristics of the scattered particles: angular distribution, intensity polarization and so on. This method is quite simple and straightforward, and does not require special manipulations of the samples which may alter their properties, and hence compromise exact results.

Rayleigh–Gans approximation, also known as Rayleigh–Gans–Debye approximation and Rayleigh–Gans–Born approximation, is an approximate solution to light scattering by optically soft particles. Optical softness implies that the relative refractive index of particle is close to that of the surrounding medium. The approximation holds for particles of arbitrary shape that are relatively small but can be larger than Rayleigh scattering limits.

<span class="mw-page-title-main">Sherman function</span>

The Sherman function describes the dependence of electron-atom scattering events on the spin of the scattered electrons. It was first evaluated theoretically by the physicist Noah Sherman and it allows the measurement of polarization of an electron beam by Mott scattering experiments. A correct evaluation of the Sherman function associated to a particular experimental setup is of vital importance in experiments of spin polarized photoemission spectroscopy, which is an experimental technique which allows to obtain information about the magnetic behaviour of a sample.

References

  1. 1 2 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 (12): 1099–1116. Bibcode:1948JChPh..16.1099Z. doi:10.1063/1.1746740.
  2. ↑ B. A. Brice; M. Halwer & R. Speiser (1950). "Photoelectric light scattering photometer for determining high molecular weights". J. Opt. Soc. Am. 40 (11): 768–778. Bibcode:1950JOSA...40..768B. doi:10.1364/JOSA.40.000768.
  3. ↑ P. J. Wyatt in U.S. Patent 3,624,835 (1971) filed 1968.
  4. ↑ 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. doi: 10.1093/clinchem/21.9.1297 . PMID   1149235.
  5. ↑ 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.
  6. ↑ 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.
  7. ↑ Cf. L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, New York 1955).
  8. ↑ S. Fernbach (1958). "Nuclear Radii as Determined by Scattering of Neutrons". Rev. Mod. Phys. 30 (2): 414–418. Bibcode:1958RvMP...30..414F. doi:10.1103/RevModPhys.30.414.
  9. 1 2 3 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.
  10. 1 2 3 P. J. Wyatt & C. Jackson (1989). "Discrimination of Phytoplankton via Light-Scattering Properties". Limnology & Oceanography. 34 (I): 96–112. Bibcode:1989LimOc..34...96W. doi: 10.4319/lo.1989.34.1.0096 .
  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 (9): 1297–1304. doi: 10.1093/clinchem/21.9.1297 . PMID   1149235.>
  12. ↑ N. George, A. Spindel, J. T. Thomasson in U.S. Patent 3689772A (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. PMID   20221079.
  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. ↑ "Evolution of Wyatt Technology Corp". www.americanlaboratory.com. Retrieved 2017-02-23.
  16. ↑ "museum | about". www.wyatt.com. Retrieved 2017-02-23.
  17. ↑ See, for example Chemical Abstracts
  18. ↑ "MALS Bibliography". www.wyatt.com. Retrieved 2017-02-23.
  19. ↑ 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   978-0-471-20172-4.
  20. ↑ 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.
  21. ↑ M. Schimpf; K. Caldwell; J. C. Giddings, eds. (2000). Field-Flow Fractionation Handbook. Wiley-IEEE. ISBN   978-0-471-18430-0.
  22. ↑ L. V. Lorenz (1890). "Light propagation in and outside a sphere illuminated by plane waves of light". Videnski.Selsk.Skrifter. 6: 1–62.
  23. ↑ 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.Balázs, Louis (1964). "Errata Ibid". Physical Review. 134 (7AB): AB1. Bibcode:1964PhRv..134....1B. doi: 10.1103/physrev.134.ab1.2 .
  24. ↑ V.A. Erma (1968a). "An exact solution for the scattering of electromagnetic waves from conductors of arbitrary shape: I. Case of cylindrical symmetry". Physical Review. 173 (5): 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". Physical Review. 176 (5): 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". Physical Review. 179 (5): 1238–1246. Bibcode:1969PhRv..179.1238E. doi:10.1103/physrev.179.1238.
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  27. ↑ Zimm, Bruno H. (1949). "The Dimensions of Chain Molecules Containing Branches and Rings". J. Chem. Phys. 17 (12): 1301. Bibcode:1949JChPh..17.1301Z. doi:10.1063/1.1747157.
  28. ↑ Podzimek, Stepan (1994). "The Use of GPC Coupled with a Multiangle Laser Light Scattering Photometer for the Characterization of Polymers. On the Determination of Molecular Weight, Size and Branching". Journal of Applied Polymer Science. 54: 91–103. doi:10.1002/app.1994.070540110.
  29. ↑ Waghwani HK, Douglas, T (March 2021). "Cytochrome C with peroxidase-like activity encapsulated inside the small DPS protein nanocage". Journal of Materials Chemistry B. 9 (14): 3168–3179. doi: 10.1039/d1tb00234a . PMID   33885621.
  30. ↑ Waghwani HK, Uchida M, Douglas, T (April 2020). "Virus-Like Particles (VLPs) as a Platform for HierarchicalCompartmentalization". Biomacromolecules. 21 (6): 2060–2072. doi: 10.1021/acs.biomac.0c00030 . PMID   32319761.
  31. ↑ Wang Y, Uchida M, Waghwani HK, Douglas, T (December 2020). "Synthetic Virus-like Particles for Glutathione Biosynthesis". ACS Synthetic Biology. 9 (12): 3298–3310. doi:10.1021/acssynbio.0c00368. PMID   33232156. S2CID   227167991.
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