By Bruce B. Weiner Ph.D.

### Introduction

The large number of properties that can be determined using various forms of light scattering is impressive: molecular weight, radius of gyration, and second virial coefficient using static light scattering (SLS) on synthetic polymers and natural biopolymers like proteins; diffusion coefficient, hydrodynamic radius, and size distribution using dynamic light scattering (DLS) on polymers, proteins, and other nanoparticles; electrophoretic mobility and zeta potential using electrophoretic light scattering (ELS) or phase analysis light scattering (PALS) on nanoparticles and other colloids. And these are just the routine measurements. With an appropriate sample and the right instrument configuration, SLS can be used to determine the fractal dimension, DLS can be used to determine rotational diffusion, and combinations of the properties determined by DLS and SLS can be used to determine simple shapes. Then there are the indirect, relative or empirical relationships that allow, for example, a molecular weight calculation from a DLS determination of R_{h} (hydrodynamic radius), the technique at the heart of ASEC (absolute size exclusion chromatography).

While the number of useful properties is large, so are the various configurations. Questions arise about what is meant by “absolute” molecular weight and should one, two or multiple angle measurements be made. And if a single angle is used, what are the advantages and disadvantages of back scatter versus ninety degrees or a forward angle. When should ELS be used and when should PALS be used in characterizing the stability of a suspension with electrostatic repulsion?

The purpose of this introduction is to answer these questions and reduce the amount of time it takes to decide on what technique to use in what situation. The author will take an historic approach starting with SLS, DLS and finally the ELS/PALS techniques.

### Static Light Scattering: Peter Debye 1944

**Single-angle SLS Measurements**

That was the year Debye made the first measurements of the weight-average molecular weight, MW, of small polymers. As long as the radius of gyration, R_{g} (a z-average if there is a distribution of chain lengths), is less than about 12 nm, the polymer or protein acts like a point dipole when it comes to scattering the electric field of the light source. A point has no dimensions and size cannot be determined this way. [Size can be determined for such small particles using DLS.] Therefore, any experimentally convenient angle is sufficient. Back angles minimize the effect of any residual and unwanted larger particles (dust, large aggregates or impurities). Small angles maximize the effect of these larger scatters. Thus, choose a back scatter device if you are sure you have no need to know about any larger molecules. Choose a middle angle if you want to verify such molecules exist, or not, in the sample. Choose a low angle if the sample is very clean and you want the effect of extrapolating to zero angle as shown below. This same dictum applies to DLS.

**Multi-angle SLS Measurements**

Within a few years of Debye’s first paper, the interference effect from larger molecules and particles on SLS was developed and with it a means to determine size in the form of R_{g}. Here is the equation:

Where c is the solution concentration, A_{2}, the 2nd virial coefficient is a measure of solution stability (positive means stable, negative means unstable, and slightly negative means metastable), q is the amplitude of the scattering wave vector and is proportional to sin(θ/2) and inversely proportional to the source’s wavelength, λ. The scattering angle is θ.

With this equation a lot is revealed about angular SLS. For example, if the second term in the parenthesis on the right side of the equation is small compared to unity, then there is no angular dependence. A single angle is sufficient. It also means there is no way to determine R_{g}, so if a multiple angle device exists that is convenient, one should use it. Better to show by measurement that R_{g} is too small to determine than to assume it. Still, for lots of common problems like un-aggregated globular proteins, low molecular weight random coils, dendrimers, and highly branched polymers, there is no particular need for a single low angle or even multiple angles.

The second term is small when measurements are made at a single, low angle, or, when measurements are made as a function of angle and extrapolated to zero. It is also small when the molecular size, R_{g}, is much smaller than the wavelength of light in the medium. It is from this latter condition that R_{g} ≤ 12 nm is derived.

The constant K contains the contrast information in the form of the square of the differential refractive index increment, a measure of the difference in refractive index between the macromolecule and the solvent. If this difference is zero, or even if it is small enough, light scattering of any type will not yield good results. Fortunately, this is rarely the case, and when it is, sometimes a different solvent can be chosen. This constant is best determined using a differential refractometer.

**“Absolute” Molecular Weight Using SLS: A Misnomer But With A Good Intention**

Lastly, ∆R is the difference in the Rayleigh ratio of the solution and the solvent. Thus, it is a thermodynamic excess quantity. Since the solvent also scatters light, its effect at each angle measured is determined and subtracted to reveal the scattering due to the polymer or protein. A Rayleigh ratio is determined by measuring the intensity of scattered light using a standard and dividing that into its known ratio. The instrument constant is thus determined and is now independent of which sample is next used. Toluene has emerged as the best standard to use. It is also possible to use a polymer of known molecular weight and determine the instrument constant that way.

Thus, SLS measurements done this way are not absolute; the instrument is always calibrated.

For batch-mode, SLS measurements, an instrument with half a dozen or more angles spread from 30° to 150° offers all the flexibility required for multi-angle SLS and the determination of most R_{g} values. Single angle devices can never be used to determine R_{g}. A two-angle device can always be used to fit a straight line; however, there is no opportunity to minimize the error in the slope from which R_{g} is determined. Three angles affords the minimum number to allow a least squares fit to a straight line and thus the slope. But why settle for the minimum when there are several good multi-angle choices?

**SLS in Flow Mode: A Nearly Ideal SEC/GPC Detector**

The columns in a size exclusion or gel permeation chromatography experiment fractionate the injected macromolecular solution as the pump keeps the eluent flowing at a constant rate. With an SLS detector, the intensity and Rayleigh ratio are determined. Add a concentration detector (RI or UV device, rarely an ELSD (polysaccharides)), and Debye’s equation can be used to determine M_{W}, if and only if the excess scattering is high enough. Thus, light scattering is of little use in HPLC (molecular weights too low).

Prior to the introduction of on-line, SLS devices, the retention of polymers on the columns was calibrated using standards. With the introduction of viscometers, some “universality” was added to the calibration method. Still, one was calibrating the columns. With SLS, one is calibrating the light scattering instrument not the columns. It is in this sense that the misnomer “absolute molecular weight” became fashionable.

If standards exist, it is less expensive to use the old-fashioned, standard calibration technique. If standards do not exist, one can use a viscometer and several standards to show the columns are operating exclusively in the size exclusion mode. But doing a thorough calibration this way is time consuming (at least three different chemistries, each with several molecular weights to cover each decade of interest). SLS is just simpler since only the instrument is calibrated.

As in batch-mode, if a multi-angle device is available use it. If R_{g} is large enough it can be determined. Unlike the batch-mode method where the injection of several concentrations allows the determination of A_{2}, in flow-mode there is just the one concentration at any instant. Fortunately, at the concentrations of interest, the term 2A_{2}c is usually much smaller than 1/M_{W} and can be ignored. When it cannot, and this rarely happens, approximate values of A_{2} can be used and the small correction applied.

**Dynamic Light Scattering: Herman Cummins 1964**

That was the year Cummins made the first measurements of the time-dependence of the scattered light to determine the translational diffusion coefficient, D_{T}, of a narrow suspension of polystyrene latex spheres of hydrodynamic diameter d_{h}. Due to the Brownian motion of the colloidal-sized particles, the fluctuations in the scattered intensity contain information on their diffusive motion. While Cummins, his colleagues, and a few other groups around the world were exploring this new form of light scattering using spectrum analyzers to look at the spread in the frequency of light scattered, within a few years every one was using digital autocorrelation. In this signal processing technique, the number of photons registered in a small sampling or bin time is multiplied with the number arriving at some delayed time. The full power of analyzing scattered photons using a digital correlator reduced dramatically the laser power required, the experiment duration, and the losses involved with analogue techniques.

For monodisperse spheres, the correlation function is a single exponential decay riding on top of a baseline. There is nothing to calibrate as there is in SLS. Thus, DLS is absolute. Furthermore, molecules and nanoparticles too small to show an angular dependence still diffuse. Thus, they too can be measured. Examples include tetrapropylammonium bromide with R_{h} = 0.5 nm (modestly large salt ion: size calculable from bond lengths and bond angles), lysozyme with R_{h} = 1.9 nm (globular protein), vitamin E TPGS with R_{h} = 5.4 nm (micelle), and many more.

There are times when multiple angles are needed: To determine if a particular mode is due to rotational (no angular dependence) or translational (q² dependence) motion; to determine if the relative intensity in multiple modes varies as expected with the intensity as a function of angle (follows Mie theory, requires particle refractive index), and many more reasons having to do with complex fluid behavior and dynamic structure factors.

However, for particle sizing, a single angle is often sufficient. But which angle? The same arguments apply here that applied for SLS: A high back angle minimizes the contribution from larger particles (good sometimes; bad other times); a middle angle like 90° picks up contributions from a variety of different sizes; and if dust were not a serious problem, scattering at low angles would be ideal (it isn’t, generally).

Like SLS, there are batch-mode cells for DLS from just a few microliters to one or two milliliters. One picks a cell based on convenience for the measurement at hand and what the instrument manufacturer offers for the particular configuration.

**Combining SLS and DLS: ASEC, How to Characterize Protein Aggregation**

Recently, DLS has been used on-line in SEC-type experiments. Here, since the SLS signal is not calibrated, this is truly an “absolute” measurement. So what is measured? The intensity and R_{h} are measured vs elution volume (or elution time). The relative area under any intensity peak gives the percent by intensity in that peak. If the peak is narrow, as it will be if it represents, for example, a globular protein not stuck to another one, then the R_{h} measured using DLS can be used to calculate an empirical molecular weight. This follows from one or another form of the Mark-Houwink-Sakurada equations such as R_{h} = 0.307M^(0.428), a formula that applies to globular proteins. For un-aggregated proteins ranging from 10 kDa to 670 kDa, a plot of log(R_{h}) vs log(M) gave a straight line from which the constants in the MHS equation were determined. Other equations exist for other geometries. MHS equations occur frequently when working with intrinsic viscosity of polymers.

Though it is an empirical equation, and the error bars high (±10%), it does make it easy to spot which peaks correspond to singlets, doublets, triplets and other aggregates. Once the molecular weight is known, dividing that into the relative intensity yields the relative concentration. From this information, it can be determined how badly the protein has aggregated. Aggregates can dramatically reduce bioactivity.

### Determining Stability Using Zeta Potential and Light Scattering

For charged particles placed between electrodes of opposite sign, there develops an electrophoretic velocity in addition to diffusion. Scattered light is affected. There is a Doppler shift in the frequency (ELS) and also a shift in the phase of the scattered light (PALS). Both can be measured relative to a reference beam and thus the sign of the charge density surrounding and moving with the particle can be determined. The concept of shear plane is invoked. Particle (including single molecules, micelles, micro-emulsions and many other types of nanoparticles) and tightly bound charges (natural or additive: wetting agents, surfactants, dispersing agents) and counterions that move with the particle determine its hydrodynamic size (R_{h}).

The frequency or phase shift yields an electrophoretic velocity from which an electrophoretic mobility is calculated. For many practical cases, this is further reduced to an electrostatic potential called the zeta potential. Its absolute magnitude determines stability and indicates what has actually attached to the particle surface.

Since phase measurements are so much more sensitive than frequency measurements, the PALS technique is preferred when electrophoretic effects are small such as in nonpolar solvents or relatively concentrated salt solutions. Otherwise, the ELS technique is sufficient for most simple measurements in aqueous suspensions at relatively lower salt concentrations.

### Summary

Molecular weight, size –two types, R_{g} and R_{h}–, size distribution, stability measures –two types, A_{2} and zeta potential, fractal dimensions (no space here for details, sorry), and more all available by looking at the intensity, the fluctuations, the frequency, and the phase of scattered light. Is there a more powerful set of techniques to characterize polymers, proteins, and nanoparticles.

P. Debye, “Light Scattering in Solutions”, Journal of Applied Physics, Vol. 15, pp. 338-342, (1944).

B. Zimm, “The Scattering of Light and Radial Distribution Function of High Polymer Solutions”, Journal of Chemical Physics, Vol. 16, No. 12, pp. 1093-1099, (1948).

H.Z. Cummins, N. Knable, & Y.Yeh, “Observations of Diffusion Broadening of Rayleigh Scattered Light”, Physics Review Letters, Vol. 12, pp. 150-153, (1964).

B.R. Ware & W.H. Flygare, “The simultaneous measurement of the electrophoretic mobility and diffusion coefficient in Bovine Serum albumin solutions by light scattering”, Chem. Phys. Lett., Vol. 12, p. 81, (1971).

J.F. Miller, K. Schatzel, & B. Vincent, “The Determination of Very Small Electrophoretic Mobilities in Polar and Nonpolar Colloidal Suspensions Using Phase Analysis Light Scattering”, Journal of Colloid and Interface Science, Vol. 143, No. 2, (1991)

Special thanks to Dr. Dave Fairhurst, Colloid Consultants Ltd. for his invaluable suggestions.

©September 2011

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