Data analysis and distribution weighting: FAQ for DLS users

Summary

Dynamic Light Scattering (DLS) is a popular technique for obtaining size distributions of dispersible colloidal particles as well as other nanomaterials. As with all other light scattering based techniques, the intensity of scattered light is strongly skewed towards larger particles, something that must be considered when analyzing samples with two or more major constituents. Given equal populations by number and given that the two particles are composed of the same material, the larger particle will always scatter more light. For this reason, it is logical to treat intensity-weighting as the native distribution weighting. Transformation from intensity- to volume-, surface area-, or number-weighting must be done with extreme caution, especially with multimodal size distributions. When used correctly, this transformation provides valuable information about the relative abundance of different particles; when applied indiscriminately, it can be very misleading.

Introduction

By default, DLS produces intensity-weighted results, where the percent intensity is directly related to the contribution of each peak to the total measured signal. When using this representation, each peak in a bimodal, or otherwise multimodal, distribution is represented in terms of its total contribution to the measured signal. Since it is well known that the amount of light scattered by a given material will vary drastically depending on the size of the particle, it must be understood that two different populations that are of unequal size, and of equal intensity, cannot be equal in number.

The primary optical contrast term in light scattering is derived from the difference in refractive index between sample and solvent, Δn. Light is scattered at the boundary between a particle and surrounding solvent, the same as it would be from a microscopic surface. In addition to this contrast, there is also a prefactor that accounts for size or molecular weight.

The key relationship I ∝d6

This d6 scalar has a major impact on the amount of light scattered by large particles. For small differences in size, this effect is already quite large but, as shown in the table below, it becomes overwhelming when covering more than an order of magnitude in effective size.

d, nmd6Id/I20nm
2064,000,0001X
404,096,000,00064X
3007.3X101411,390,625X
10001 X101815,625,000,000X
Table 1 – Note the outsized impact of the intrinsic d6 term on mixtures of equal number concentration.

While a 40 nm particle will scatter over 60X the light of a 20 nm particle, a single 300 nm particle would scatter over 11 million times more light than a 20 nm particle. When covering more than two orders of magnitude in size, this difference becomes astronomical. 

Conveniently, it is quite rare to prepare samples based on number concentration, so the more common use case for a multimodal size distribution is distinguishing populations that are equal on the basis of mass concentration. Since larger particles by definition have higher molecular weights, these two effects often counteract one another.

For example, assuming two hypothetical silver nanoparticles of 50 nm and 100 nm in size are mixed so that the final concentration of both is 1 mg/mL, then the following would be true:

  • density of silver = 10.5 mg/cm3
  • conc 1 = conc 2 = 1 mg/mL
  • diameter 1 = 50 nm, nanoparticle 1 = 1.5×10 15 particles/mL
  • diameter 2 = 100 nm, nanoparticle 2 = 1.2×10 14 particles/mL
  • volume of particles = 4/3π(0.5 x d)3
  • mass per particle = ( density, mg/cm3 ) X ( volume, cm3 )

As we can see from the above example, the number concentration of the 100 nm nanoparticle would be eight times lower than of the smaller 50 nm nanoparticle. We know from the d6 relationship that each 100 nm particle will scatter 64 times more light, but when we factor in the difference in number concentration, we see that this is effect is nearly compensated for. The result in this hypothetical equal-by-mass case would be a mere 8X net enhancement of the signal from the larger particle, for a twofold increase in diameter.

Preparation of Suspensions of Silica Particles:

Mixtures of several common forms of commercially available silica particles were prepared including LUDOX TM-50, AM-30, and commercially available low-density fumed silica. Despite differences in surface chemistry and macroscale structure, all three materials are silica and thus can be assumed to have very similar refractive indices. Since Δn is the same for all three particles, we only need to consider differences in contrast due to the d6 rule.

LUDOX TM-50 & AM-30 Mixture

[ TM-50 ]: 2.5 mg/mL

Nominal size: 22 nm

Density of TM-50: 1.4 g/cm3


[ AM-30 ]: 1.5 mg/mL

Nominal size: 12 nm

Density of AM-30: 1.21 g/cm3

LUDOX TM-50 & Fumed Silica Mixture

[ TM-50 ]: 2 mg/mL

Nominal size: 22 nm

Density of TM-50: 1.4 g/cm3


[ fumed silica ] : 0.2 mg/mL

Nominal size: 0.2-0.3 μm

Density of fused silica:  0.037 g/cm3

Figure 1 – Comparison of intensity (left) and volume (right) weighted distributions of TM-50 and AM-30.

Examining three measurements of the Ludox TM-50 : AM-30 mixture we can clearly identify two populations in the untransformed, intensity weighted distribution (fig 1). We also note that when applying volume weighting, we suppress the larger of the two peaks, emphasizing the lower abundance of the larger particle.

Figure 2 – Comparison of intensity (left) and volume (right) weighted distributions of TM-50 and fumed silica.

In contrast, when we compare Ludox TM-50 mixed with fumed silica (fig 2), a much less well-defined particle due to its fractal structure, we see a more complex distribution emerge, despite the fact that only two components are present.  This nominally bimodal sample displays several low-abundance data fitting artifacts on an intensity weighted basis, as can be seen around 10 nm and 1000 nm respectively. Despite the low abundance by intensity (left), representing a minor component of the actual measured signal, this artifact becomes amplified when we transform to volume weighting (right).

Figure 3Number weighted distributions of TM-50 and fumed silica.

Conclusions:

The danger in blindly renormalizing a multimodal size distribution is that users can end up ignoring the contributions of the largest components, while overemphasizing the smaller. It is apparent that the effect seen when mixing TM-50 with fumed silica is further compounded by going to number weighting (fig 3), where it appears to be the only peak in the first of three measurements, despite not representing either particle present in the mixture. In extreme cases, this can even mean emphasizing non-existent components.

Recap:

  • Great care is required when converting to number or volume weighting
  • Overfitting when analyzing large numbers of short DLS measurements can yield erroneous distributions
  • Having the ability to resolve three separate populations by DLS is very rare
  • Noise at the lower end of an intensity weighted distribution will dominate a number, volume, or surface area weighted distribution due to the d6 relationship.
Applications: DLS
Instruments: NanoBrook SeriesBI-200SM

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