Information Entropy as a Reliable Measure of Nanoparticle Dispersity.

Niamh Mac Fhionnlaoich,Stefan Guldin

doi:10.1021/acs.chemmater.0c00539

Abstract

Nanoparticle size impacts properties vital to applications ranging from drug delivery to diagnostics and catalysis. As such, evaluating nanoparticle size dispersity is of fundamental importance. Conventional approaches, such as standard deviation, usually require the nanoparticle population to follow a known distribution and are ill-equipped to deal with highly poly- or heterodisperse populations. Herein, we propose the use of information entropy as an alternative and assumption-free method for describing nanoparticle size distributions. This measure works equally well for mono-, poly-, and heterodisperse populations and represents an unbiased route to evaluation and optimization of nanoparticle synthesis. We provide intuitive software tools for analysis and supply guidelines for interpretation with respect to known standards.

Highlights

Nanoparticle size impacts properties vital to applications ranging from drug delivery to diagnostics and catalysis
Researchers have largely relied upon standard deviation for evaluating dispersity, but this measurement is only valid when applied to a normal distribution and may provide an insufficient representation of the sample.[6,8,16−19] Standard deviation is often referenced against the mean size to produce the unit-less coefficient of variance (COV), which reflects the relative spread of the given distribution
We propose the use of a modified version of the information entropy equation to accurately evaluate dispersity

Summary

■ METHODS

Figure 1 illustrates an example of the entropy calculation. Here, 35 particles of 7 different possible sizes are sorted into appropriate bins (or intervals) of a histogram. The number of particles in each bin is divided by the total to determine the probability distribution. In order to implement information entropy as a reliable measure of nanoparticle dispersity, three properties are required: (1) A linear relationship between entropy and population dispersity would facilitate interpretation and aid implementation into statistical optimization methods. (2) The entropy needs to be independent of the mean particle size. (3) The data needs to be discretized. Hence, we propose the following modification to the entropy calculation to produce the nanoparticle entropy, E: The response of the information entropy, H, to increasing but equally probably outcomes (i.e., pi = pj≠i) displays a logarithmic trend (Figure S1). Therefore, the exponential, a monotonic function, was added in eq 2. The resulting nanoparticle entropy, E, increases linearly with dispersity. The nanoparticle entropy is independent of the mean particle size, another important characteristic. The same distribution will produce the same entropy regardless of the mean. Further information is found in the SI (Figures S1−S3). Please note that monodispersity criteria are based on the relative deviation from the mean particle size.[29] To this end, a normalized entropy (En) can be obtained via dividing E by the mean. Since the nanoparticle entropy, E, has the same units as the bin width, the normalized nanoparticle entropy, En, follows as being dimensionless. It is important to note that these entropy calculations require discrete data. While the size of nanoparticles is a continuous variable, both the imaging system and analysis method will impose a limit to the exactness of each measurement. This resolution becomes the bin width of a histogram, which effectively presents the nanoparticle distribution as a discrete data set. For an unbiased representation of the nanoparticle dispersity, the bin width must be included in the entropy calculation. The reasoning is as follows: a large bin width will result in fewer bins and therefore a lower entropy; a smaller bin width for the same population will have a larger number of bins and a proportionally larger entropy. By including the bin width in the calculation of the entropy, this variability is avoided (see also Figures S4 and S5). Note that eq 2 assumes the use of a constant bin width. Entropy depends on the sample size and asymptotically approaches the true value with increasing population. While several methods have been proposed to deal with this issue, we implement here the quadratic extrapolation by Strong et al for its simplicity and low computational cost.[30] This process relies on calculating E for the total population of M measurements and two subpopulations comprised of M/2 and M/4 measurements randomly selected from the main. This data is then fitted to eq 3, where x represents the sample size. This method is powerful but requires sufficient data to adequately fit the quadratic. Figure 2 shows the results of the sample size correction for two different populations. For a given sample size, distributions with the characteristics of those shown in Figure 2a,c were randomly generated, and the nanoparticle entropy was calculated with and without sample size correction. This was repeated 100 times. Figure 2b,d shows the mean entropy with standard deviation as a function of the sample size for the respective populations. Without the sample size correction by quadratic extrapolation, at least 500 data points were required for population 1 and 900 for population 2 to achieve an entropy within 15% of the true value. With correction, these reduce to 100 and 150, respectively. On this account, we have developed a reliability index in the accompanying software (Matlab GUI and Excel Macro) to evaluate whether the sample size is sufficiently large for a reliable estimate of the entropy. We note that these sample size requirements are in line with sampling guidelines on conventional approaches.[8] Further details about the sample size correction and the reliability index can be found in the SI. In order to relate the normalized entropy En to established definitions of size uniformity, we have developed evaluation criteria for monodispersity based on definitions for dispersions provided by the National Institute of Standards and Technology (NIST) and guidelines used in nanocluster catalysis.[7,29,31,32] The NIST requires that 90% of the particles must lie within ±5% of the mean for a population to be considered monodisperse.[29] In nanocluster analysis, a population is monodisperse if the standard deviation is ≤5% of the mean and near-monodisperse if it is ≤15% (i.e., COV = 0.05 and 0.15, respectively).[31,32] It is important to note that the https://dx.doi.org/10.1021/acs.chemmater.0c00539 Chem. Mater. 2020, 32, 3701−3706 guidelines from nanocluster analysis assume the population is normally distributed. A linear fit of the data in Figure S3 produces 4.12 σ , μ which corresponds to limits of 0.206 and 0.618 for monodispersity and nearmonodispersity, respectively (Table 1). The NIST standard presupposes no particular distribution. If we assume the distribution described in the NIST is Gaussian, the following must be true: COV = 0.05 . Using the 0.125. However, as the NIST guidelines do not specify a normal distribution, we evaluated the robustness of this limit for non-normal populations. A distribution was designed to maximize entropy in the limits of compliance with the NIST requirements; a schematic describing this shape is shown in Figure 3d. The range of 90% of the population (range90) is set by the NIST requirement, but the total range (rangetotal) remains variant. We have therefore defined a variable r as the ratio of range[90] and rangetotal; this is equal to the number of bins that lie in range[90] divided by the total number of bins. The relationship between En and r is described in eq 4 and plotted in Figure 3. A detailed derivation of this equation can be found in the SI. Figure 3a shows a 3D surface map of eq 4. The majority of the surface exhibits a shallow gradient; overall, 95.9% of all combinations of n90 and n10 result in an En between 0.1 and 0.2 (r = 0.963 and 0.025, respectively). Deviating from a normal En 0.125 0.206 0.618 https://dx.doi.org/10.1021/acs.chemmater.0c00539 Chem. Mater. 2020, 32, 3701−3706 distribution resulted in little change in En. We therefore recommend using the cutoff of En = 0.125, below which populations can be reliably considered as highly monodisperse.

■ RESULTS AND DISCUSSION

■ ACKNOWLEDGMENTS

■ REFERENCES

■ CONCLUSION