Size distribution of nanoparticles by dynamic light scattering. Comparison of Bayesian and Tikhonov inversion methods

Figures & data

Table

B	–	burn-in period (in the Metropolis-Hasting sampling method)
CI	–	Mie scattering coefficient
c1, c2	–	cognitive and social accelerations in a PSO algorithm
	nm	DLS average particle diameter
	nm	mean diameter of a Gaussian PSD
	nm	geometric mean diameter of a log-normal PSD
Di	nm	i-th particle diameter (in the diameter axis of a discrete PSD)
Dmin	nm	minimum diameter (in the diameter axis of a discrete PSD)
Dmax	nm	maximum diameter (in the diameter axis of a discrete PSD)
	nm	(R × 1) vector containing the DLS average particle diameters (at all θr's)
d1, d2	nm	nominal diameters of each mode (in latexes L1 and L2)
fj	–	(j = a, b, 1, 2, L1, L2) discrete number-PSD
f	–	(N × 1) vector containing the ordinates of a discrete number-PSD
Fj	–	(j = 1, 2, L1, L2) discrete weight-PSD
G	–	best position reached by the entire ‘particle’ population in a PSO algorithm
	–	baseline of
	–	second-order autocorrelation function of the scattered light (at θr)
	–	first-order autocorrelation function of the electric field (at θr)
	–	normalized second-order autocorrelation function of the scattered light (at θr)
H	–	(N × N) matrix corresponding to the second-derivative operator
Jf	–	performance index for PSD estimates [Equation (19)]
K1, K2	–	constants in Equations (8)–(10)
k	–	iteration number in Equation (6a)
kB		(= 1.38 × 10^–23) Boltzmann constant
M	–	number of points of the autocorrelation functions
N	–	number of points of the PSD
nm	–	refractive index of the non-absorbing medium
np	–	refractive index of the particles
Q	–	jumping distribution
R	–	number of measurement angles
R1, R2	–	random numbers in the range (0, 1) [Equation (6a)]
T	K	absolute temperature
Vi	–	velocity of the i-th ‘particle’ in a PSO algorithm
w	–	inertia function in a PSO algorithm
Xi	–	i-th ‘particle’ in a PSO algorithm
x1, x2	–	number-fractions for the modes of latex L2
α	–	regularization parameter
β	–	instrumental parameter for DLS measurements [Equation (2)]
Γ0	nm/s	DLS constant [Equation (4)]
γ	–	smoothing parameter [Equation (9)]
ΔD	nm	diameter interval (in the diameter axis of a discrete PSD)
ε	–	Gaussian random sequence (mean = 0; variance = 1) [Equation (18)]
η		medium viscosity
θr	°	detection angle in MDLS
λ	nm	in vacuo laser wavelength
π	–	probability density
σL	–	standard deviation of a log-normal distribution
σG	nm	standard deviation of a Gaussian distribution
τ	nm	decay constant of a decreasing exponential function
τj	s	discrete time delay of

Table

^	estimated value
∼	contaminated with noise
*	convolution product
^T	transposed vector

Figure 1. The ill-conditioned problem. (a) Two arbitrary PSDs. (b) Mie coefficients corresponding to nanometric particles (at 5 angles). (c) Normalized autocorrelation measurements. (d) Derived DLS average diameters.

Figure 2. PSD estimation from MDLS measurements: schematic data treatment paths.

Table 1. Simulated examples.

	m-BI	TI	m-BI	TI
10^1×  (–)	1.56	2.14	0.87/2.62	1.19/9.91
10^1×  (–)	0.32	0.34	0.24/0.64	0.37/2.03
10^1×  (–)	2.41	2.86	1.47/4.16	1.98/12.89
10^1×  (–)	1.00	1.60	0.47/1.35	0.65/5.15
10^3×  (–)	1.86	1.31	1.81	1.34
10^3×  (–)	0.45	0.49	0.33	0.44
10^3×  (–)	3.37	3.03	2.56	2.28
10^3×  (–)	1.25	0.71	1.27	0.62
Note: Performance indexes corresponding to 25 simulations of each analysed example.

Figure 3. Simulated example for the PSD f₁. (a) Average diameters i) calculated from the ‘noisy’ MDLS measurements (dots), and ii) simulated with the estimated PSDs. (b, c) The simulated PSD and its estimates from the TI method (b) and m-BI method (c). (d) Estimated PSD obtained from the m-BI method and the standard deviations calculated from the samples obtained through the Metropolis–Hasting algorithm.

Figure 4. Simulated example for the PSD f₂. (Legends as in Figure 3).

Figure 5. Experimental example for latex L₁. (a) Average diameters i) calculated through the cumulants method 16 (dots), and ii) simulated with the estimated PSDs. (b, c) Comparison of the ‘true’ PSD with its estimates from the TI method (b) and m-BI method (c). (d) Estimated PSD obtained from the m-BI method and the standard deviations calculated from the samples obtained through the Metropolis–Hasting algorithm.

Figure 6. Experimental example for latex L₂. (Legends as in Figure 5).

Table 2. Experimental examples.

	‘true’,			‘true’,
		m-BI	TI		m-BI	TI
(nm)	306	315	315	306	320	320
(nm)	974	971	965	974	965	963
(%)	98.95	99.00	97.40	97.90	98.10	96.80
(%)	1.05	1.00	1.50	2.10	1.90	2.20
10^1× Jf (–)	–	7.7/7.4	7.7/20.7	–	9.8/8.6	9.5/14.6
10^3× JD (–)	–	8.0	7.4	–	4.5	3.5

Koppel, DE, 1972. Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants, J. Chem. Phys. 57 (1972), pp. 4814–4820.

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