Food Research International 27 (1994) 195-198

Particle size analysis by dynamic light scattering

F. Ross Hall&t Guelph- Waterloo Program for Graduate Work in Physics, Department of Physics, University of Guelph, Guelph, Ontario,

Canada Nl G 2 WI

Dynamic light scattering is an established technique for measuring the average size and size distribution of particles in a suspension. The technique has the advantage of being fast and non-invasive, but it does require low particle con- centrations. As well, dynamic light scattering results are often open to misinter- pretation if one is unaware of the state of the sample and the method of data analysis. The following discussion reviews some of the basic concepts of dynamic light scattering and outlines some of the pitfalls that are often encoun- tered in data interpretation. A modification of dynamic light scattering, diffus- ing wave spectroscopy, can be used to obtain approximate size information at higher particle concentrations. The fundamentals of this new technique are summarized.

Keywords: dynamic light scattering, particle size analysis, diffusing wave spectro- scopy, moments analysis, exponential sampling.

BASIC PHYSICS SCATTERING

OF DYNAMIC LIGHT

The dynamic light scattering (DLS) technique is based on the scattering of light by diffusing par- ticles. At any instant the suspended particles will have a particular set of positions within the scat- tering volume. The particles scatter the radiation to the detector, but the relative phases of scattered wavelets differ, due to the differing incident phases that they experience at these positions and due to different particle-detector distances. The electric field at the detector is the superposition of the fields due to all the scattered wavelets and will, at time t, have a value E(t). At the time, t + T, which is a very small time later than t, the particles, which are diffusing, will have new positions, slightly removed from those at the earlier time. Superposition of the new slightly shifted wavelets yields a changed electric field at the detector, E(t + T). As time progresses, the electric field and hence the intensity at the detector, will fluctuate as the Brow- nian processes in the sample volume continue.

During the scattering process, the electromag-

Food Research International 0963-9969/94/$07.00 0 1994 Canadian Institute of Food Science and Technology

netic wave can impart or receive energy and mo- mentum from the external and internal motions of the scatterer. The frequency shifts due to Brown- ian motion are so small that the energy difference between incident and scattered photons can be neglected. However, the change in momentum ex- perienced by the photon during the scattering process is a very important parameter in DLS. The mopentum transfer vector or the scattering vector, Q, is defined as,

Q = ;, - zi (1)

where the zs are the wave vectors of the scattered and the incident light. From the geometry of the scattering arrangement (see Fig. l), the magnitude of the scattering vector is,

> Laser beam

Fig. 1. Scattering geometry shywing laser beam, wave vectors (ks), scattering vector (Q) and scattering angle 0.

195

196 F. Ross Hallett

i.0

1 I I I I I

” 30 60 90 120 150

SCATTERING ANGLE (degrees)

160

Fig. 2. Distance scale (Q’) sampled by light (helium-neon laser) as a function of scattering angle.

It is clear from Equation (2) that Q has the dimen- sions of an inverse length. Indeed Q’ sets the length scale that will be probed by the light scattered at angle 0 (see Fig. 2). Thus, light scattered at low angle will contain information on the gross dynamic and structural properties of the scatters, whereas light scattered at high angle contains this informa- Y--- - tion at a finer scale. Figure 3 shows a comparison of scattering techniques and the length scales that they probe. DLS has a greater range than other scattering techniques because it is concerned with the distance (27r/Q) that a particle diffuses in a time interval.

The main problem in data recovery in the DLS experiment is the extraction of quantitative infor- mation from a fluctuating signal. Small rapidly diffusing particles will yield fast fluctuations, whereas larger particles and aggregates general have relatively slow fluctuations. The rate of the fluctuations can be determined through the tech- nique of autocorrelation analysis (Abbiss &

SMALL ANGLE NEUTRON SCATTERING I I I I I

COLD SOURCE LIGHT SCATTERING

SMALL ANGLE X-RAY SCATTERING I I

I I I 100 1000 10000

2n/Q (angstroms)

Fig. 3. Comparison of the particle sizing ranges of several different scattering techniques.

I --- <I(t)%

2 . .

*. *.

0. I 1 ??. . . . .._

*... *...

**.......* .‘...........*

1 ‘.............. <I(t)>2

10 20 30 40 50 60

CHANNEL NUMBER

Fig. 4. Normalized intensity autocorrelation function.

Smart, 1988) from which one can obtain the in- tensity autocorrelation function, C~(T), where r is an instrumental delay time. This function has a T = 0 limit equal to the mean square intensity ~1(t)~> and an asymptotic limit as T approaches infinity equal to the mean intensity squared, i.e. cI(~)>~. The decay time is rouehlv indicative of the tvnical o---J -a r----- fluctuation time of the signal. The normalized form of the intensity autocorrelation function (Fig. 4) can be related to the electric field autocor- relation function through the relationship.

Cl(T) g(2)(7) = - = <I(t)>2

1+ 1 g(‘)(T)1 2 (3)

A biock diagram of a modern DLS spectrometer is shown in Figure 5.

Usually, dynamic light scattering methods are employed to study solutions under conditions where particles are assumed to be small and spherical. Under these conditions, only the relative phases of the scattered waves are of concern and

I

cl Detector

Quantum Photometer

J Fig. 5. Schematic diagram of a DLS spectrometer.

Particle size analysis by dynamic light scattering 197

g(‘)(T) has the form:

g(l)(T) = e-W+

where D is the diffusion coefficient.

(4)

The theoretical function g(‘)(r) approaches a maximum value of unity as T approaches zero. This would also be true experimentally if the pin- holes used to collimate the scattered light were in- finitely small. However, no light would reach the detector. In practice, an experimental compromise is reached, with the effect that the zero delay (T = 0) intercept is ‘a’ where a I 1. In addition, Equation (4) holds only for solutions of small monodisperse particles (all particles are identical in size). The more common situation is one where the solution contains a size distribution of scatterers. In this situation g(l)(r) becomes a summation. Including both these effects, Equation (4) can be written as,

g(i)(r) = a (01e-D,Qz7 + ee-D2Qzr + . . . ) m (5)

= 0 C w,e-D@T i=I

where the wi are weighting factors related to the relative abundance of particles of a size indexed by i and m is the number of sizes. For a continuous distribution of particle sizes, Equation (5) can be replaced by,

g(l)(r) = a [ G (I) eer’& (6)

where I = D@. In principle, a complete Laplace inversion of

Equation 6 would yield the distribution of relax- ation times, G(I), and hence the distribution of particle sizes (through the Stokes-Einstein equa- tion). In general, such an inversion is termed ‘ill conditioned’ because of its mathematical instability and because unattainably high precision in the ex- perimental data is required. Several alternative methods of various complexities have been devel- oped and brief treatments of two of these will be given. The first and most common method of pro- ceeding is called moments analysis or the method of cumulants. A given distribution, such as G(I) can be described in terms of a set of moments about the origin or moments about the mean. The object of moments analysis is to obtain these mo- ments without actually performing the inversion. Specifically one attempts to obtain the first and second moments from which one can obtain the mean value of I and the variance, respectively. In this approach the exponential, eerT from Equation (6) is expanded about the mean value emr7:

_ e-r7 = e-rTe(-(r-r)T)

=e -” i-(r-f)T + (r-r)*T* (r-r)3T3 + 21 - 31 * * * 1 (7)

Substituting Equation (7) into Equation (6) and taking the logarithm yields (Koppel, 1972)

h@‘)(T)] = 1IlU - FT + * T* - E”3 ti + . . . 2! 3!

(8)

The second moment,

is the variance

p2 = (I*) (9)

02” l-42- PI2 = EL2 (10)

since the first moment about the mean (pJ is al- ways zero; o, the standard deviation, is a measure of the width of G(I).

A variety of mathematically more sophisticated procedures have been developed to yield the inver- sion of Equation (6). Most of these approaches are variants of a discrete method in which

VUr = (&T) - 5 % exP(-IJ))* (11)

is minimized with respect to the variables U, and rm. As mentioned earlier, this is notoriously unstable, and if no constraints were applied it is essentially impossible to obtain a unique set of best-fit parameters. One of the more common restraints is to specify, in advance of the fit, the range and values of a ‘trial’ set of I,. In the tech- nique called exponential sampling (Ostrowsky et al., 1981), the I, are themselves exponentially spaced according to the relationship

r m+l = r,exp(>o (12) where x is a constant that sets the initial spacing between the first two Is. The resulting set of OS corresponds to the amplitudes, or relative weights of each of the exponentials in Equation (11). Since each I has a corresponding Y, the amplitudes are usually presented as a histogram. If the data are of sufficient quality (often runs of several hours are necessary), then reliable and reproducible his- tograms of the radius distribution, G(r), can be obtained.

The histograms that are produced from this procedure are intensity-weighted distributions, i.e. the amplitudes represent the amount of light scat- tered by each particle size, r and decay constant I’. In order to obtain number distributions, one must

198 F. Ross Hallett

I-L 0.00 0.02 0.04 0.06 0.08 0.10

RADIUS (micrometres)

Fig. 6. Size distribution of extruded vesicles obtained from DLS.

include the relative scattering ability of each size in the distribution. This can be accomplished by including the relative Rayleigh-Gans-Debye or Mie scattering factors in the summation of Equa- tion (11). Hallett et al. (1991) have shown how this can be accomplished without the introduction of spurious oscillations into the histogram. The in- tensity and the number distribution for a set of vesicles are shown in Figure 6.

Dynamic light scattering requires that the con- centration of scatters be sufficiently dilute that only single scattering takes place. This means that most samples must be diluted to the point where they are optically almost clear. If multiple scatter- ing occurs then multiple phase shifts to the scat- tered light occur, with the result that particle sizes, as determined by moments analysis or by discrete inversion techniques, are too small. As a result standard DLS methods cannot be applied to more concentrated suspensions such as milk. However, a related form of laser light scattering, called di- ffusing wave spectroscopy (DWS), is available and provides estimates of mean particle sizes in con-

centrated suspensions. The apparatus used in DWS is the same as for DLS except that the inci- dent light and the back-scattered light are both transmitted and received by a fibre optic. DWS depends upon multiple scattering being so severe that the incident photons experience a random walk (diffusing wave) in the sample before being finally scattered into the returning fibre optic. Pine et al. (1988) have shown that for back-scatter de- tection from thick samples, the electric field auto- correlation function is

g(l)(r) = exp [-y (67/r,,) ‘1 (13)

In this expression 7. is defined as a characteristic diffusion time

1 70 = -

Dko2 (14)

and k, is the incident wave vector. The constant y must be determined independently by calibration with known size scatterers or by transmission measurements. Once this is done, however, accu- rate estimates of mean sizes have been obtained in a variety of dense scattering media.

REFERENCES

Abbiss, J. B. & Smart, A. E. (1988). Photon correlation tech- niques and applications. OSA Proceedings, 1.

Hallett, F. R., Watton, J. & Krygsman, P. H. (1991). Vesicle sizing: Number distributions by dynamic light scattering. Biophys. J., 59, 357-62.

Koppel, D. E. (1972). Analysis of macromolecular polydis- persity in intensity correlation spectroscopy - Method of cumulants. J. Chem. Phys., 57, 481620.

Ostrowsky, N., Sornette, D., Parker, P. & Pike, E. R. (1981). Exponential sampling method for light scattering polydis- perse analysis. Opt. Acta., 28, 1059-63.

Pine, D. J., Weitz, D. A., Chaikin, P. M. & Herbolzheimer, E. (1988). Diffusing-wave spectroscopy. Phys. Rev. Lett., 60, 11367.