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Taylor Dispersion of Polydisperse Nanoclusters and Nanoparticles: Modelling, Simulation and Analysis Sandor Balog Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.7b04476 • Publication Date (Web): 08 Mar 2018 Downloaded from http://pubs.acs.org on March 10, 2018
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Analytical Chemistry
Taylor Dispersion of Polydisperse Nanoclusters and Nanoparticles: Modelling, Simulation and Analysis Sandor Balog University of Fribourg, Chemin des Verdiers 4, 1700 Fribourg, Switzerland ABSTRACT: The dimensions of ultrasmall inorganic nanoparticles (US-NPs) is in the heart of the design of diagnostic and therapeutic efficacy; yet its accurate measurement is challenging for most experimental techniques. We show here how to design and analyze Taylor dispersion experiments to characterize the two most sought-after parameters describing size distributions: the number-averaged mean size and polydispersity index. To demonstrate the power of the method, we simulated and analyzed taylograms corresponding to gold US-NPs distributed normally. By using simulation and including experimental noise, we had the advantage that the true values describing size distribution were known exactly, and thus, we were able test the absolute accuracy of our analysis and its robustness against noise. Theory and computational experiments were found in very good agreement, providing a significant step in the analysis of ultrasmall inorganic nanoparticles and Taylor dispersion experiments.
The dimensions of ultrasmall inorganic nanoparticles (USNPs) are halfway between molecules, proteins and classical nanoparticles, and thus, show significant potential for biomedical applications.1-3 While size distribution is a key parameter governing diagnostic and therapeutic efficacy, its accurate measurement is challenging. It requires a spatial resolution on the sub-nanomater length scale and a nondestructive probe in situ under native conditions and possibly even in the presence of a biological/physiological environment.4,5 While size polydispersity is an issue6 that also affects bio-distribution7-10 and cytotoxicity,11 experimental techniques dedicated to particle size analysis—including transmission electron microscopy (TEM), dynamic light scattering (DLS), nanoparticle tracking analysis (NTA), X-ray diffraction (XRD), and small-angle Xray scattering (SAXS)12—frequently struggle to determine correctly the number-averaged distribution of size.13 To determine the number-averaged mean and polydispersity index of US-NPs, we present an analysis based on the technique of Taylor dispersion.14-16 Previously, Taylor dispersion was successfully used for characterizing the distribution of the diffusion coefficients of polydisperse polymers, namely, poly(styrenesulfonate) standards and their heir bimodal or trimodal mixtures.17-19 The instrumentation of Taylor dispersion is well established, suits very well to particles with a diameter on the 0.1-10 nm range, and compared to X-ray techniques and electron microscopy, is highly affordable. Furthermore, given the dimensions and operational principles, it matches perfectly the needs of opto-microfluidic lab-on-achip—in situ and real-time—analytical systems.20 Taylor dispersion relies on a steady laminar flow driven by a pressure gradient in a microfluidic channel, which is most frequently formed by a cylindrical capillary tube. After injection, the parabolic velocity profile of the flow disperses the initially homogenous band of the particles—frequently referred to as the ‘analyte’—and the advection creates a concentration gradient at the front and back of the band, which induces a spon-
taneous net transport of particles via translational diffusion. Particles at the front of the band migrate towards the capillary walls, and particles in the back of the band migrate towards the capillary center, and hence, the band broadens.21 The rate of band-broadening of the analyte is therefore a function of the shear rate and the translational diffusion coefficient of the particles. The diffusion coefficient () is determined via the so-called ‘taylogram’, which is the record of the temporal evolution of the optical absorbance of the flow at a given distance from the injection point. The absorbance is a dimensionless quantity, defined as = − where is the optical transmission at a wavelength that is usually chosen from the UV-Vis range. From the self-diffusion coefficient, the hydrodynamic radius is determined via the StokesEinstein equation.22-24 The taylogram corresponding to uniform particles of hydrodynamic radius r can be written as14-16 (1)
∙ ∙
= ∙ ⁄ e
where is the amplitude, =
! " #$ %
, & the capillary radius, '
the viscosity of the fluid, the temperature, and () the Boltzmann constant, and = */, the so-called residence time defined by the distance between detection and injection points (* = *-./ − *012 ) and the mean velocity of the flow (,) averaged over the cross section of the capillary. Equation 1 is essentially a time-dependent Gaussian function, whose width increases with the residence time and particle size. As shown eslewehere,21 Equation 1 can be adapted to polydisperse particles by considering that the optical extinction 34, is sizedependent (2)
6 =
=
7 68∙9:,8∙;< /,8 -8 =
7 68∙9:,8 -8
.
>4 is the probability density distribution of the particle radius, that is, >4 × @4 quantifies the probability that the 1
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radius is found in @4 interval about 4. Equation 2 is of general validity that expresses that the presence of each particle in the taylogram is weighted by its optical extinction. In the case of ‘discrete’ binary, ternary, etc. mixtures of respectively uniform particles, the ‘modes’ may be clearly distinguishable in the taylogram, and it is straightforward to fit a multi-modal Gaussian. However, when the size distribution is continuous over a finite range, this simplicity vanishes, and Equation 2 constitutes an inverse problem represented by an inhomogeneous Fredholm integral equation of the first kind with a Gaussian kernel. Consequently, to obtain the full form of >4 from 6 is profoundly difficult. In consequence, the question we ask is whether we could B analyze 6 to quantify the mean radius, µ ≡ 7 >4 4 @4 ,
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3; ≫ 3F , and the power-law relationship thus remains valid. Using this relationship, we adopt the approach presented elsewhere21 to express Equation 2 as a linear combination of weighted Gaussian functions (5)
6 =
where tion.
e4 ] f
〈8 b 〉
=
7 68 8 b ;< /,8 -8
B
=
7 68 8 b -8
∙ 7 >4 4 ] e
is the *
/g
∙ ∙
=
@4
raw moment of the particle size distribu-
B
and polydispersity index, C ≡ ∙ D7 4 − µ" ∙ >4 @4 . It is µ
obvious that to determine the values of these two unknown variables from 6 , we need to find two independent relationships between µ and C, and set up and solve a system of equations. Accordingly, we need to describe two independent and measurable quantities of 6 that are both functions of µ and C. We begin with by considering that the extinction coefficient is the sum of the absorption and scattering coefficients: 34, = 3; 4, E 3F 4, . US-NPs belongs to the realm of Rayleigh particles, where 4 ≪ λ and |I| ∙ 4 ≪ λ, and I4, = J4, E K (4, is the complex refractive index. In the case of Rayleigh particles, the dipole approximation is fully adequate to describe extinction, and both the absorption and scattering coefficients can be factored in two terms: (3)
3; ∝
where (4)
8M :
∙ N; and 3F ∝
N; 4, = − QI R and
NF 4, = V
8O :P
∙ NF
S8,:
S8,: T"
S8,:
S8,: T"
U
"
V .
Given that gold assumes a particular interest in biomedical applications,25,26 here we restrict our analysis to gold particles, and for the calculations we use a wavelength that is commonly used in photodynamic therapy (λ = 600 nm).27 Both is an arbitrary choice that embody all the essential aspects of our technique, nonetheless they do not in any way imply the limits of the technique. Figure 1a shows the real and imaginary parts of the refractive index of gold at this wavelength.28 The dependence of J on 4 below 10 nm radius is evident, while ( approaches its bulk value already at 5 nm. In the case of conventional nanoparticles, I is practically constant and independent of the particle size. The consequence is that on the range of US-NPs—unlike in the case of conventional NPs— the factors N; 4, and NF 4, are not constant but size dependent (Figure 1b). In the case of conventional gold NPs, absorption and scattering is respectively proportional to the third and sixth power of the size. Our calculations, however, show that for gold US-NPs N; ∝ 4 .\ (Figure 1c), and accordingly, the scaling exponent of the extinction coefficient is smaller than three, that is, 34, 600nm ∝ 4 ] where * = 3 − 0.6 = 2.4. The 95% confidence interval of * is 2.38-2.42. For gold US-NPs, absorption dominates extinction because
Figure 1. a) The real (J) and imaginary part (() of the refractive index of gold particles at λ = 600 nm.28 They approach their bulk value beyond 10 nm and 5 nm, respectively. The dashed lines are guides to the eye. b) The dependence of N; and NF (Equation 4) on the radius of gold particles dispersed in water. The dashed lines are guides to the eye. c) Fi shown on the range of US-NPs as a function of radius. The solid lines is a power function obtained by non-linear regression. Coefficient of determination is 0.997 (adjusted R2). d) The extinction coefficient on the range of gold USNPs as a function of radius. The solid lines is a power function obtained by non-linear regression. Coefficient of determination is 0.999 (adjusted R2).
We recall that our goal is to describe two independently measurable quantities of 6 that are both functions of µ and C. We will use first the concept of statistical moments to describe the shape of 6 .21,29,30 It is shown elsewhere21 that the mean and the variance of 6 are B
k l8 bmn o
(6)
j/ ≡ 7 ∙ 6 @ = E
(7)
p/ ≡ 7 − j/ " ∙ 6 @ = ∙
and B
" e8 b f "
l8 bmn o e8 b f
.
Solving Equation 6 and 7 determines and an apparent radius equal to 4 ≡ e4 ]T f⁄e4 ] f.21 When C q 0, 4 is larger than µ, and in fact, increases with polydispersity. At moderate polydispersity (C r 0.25) this apparent radius can be expressed simply by the mean and polydispersity index 2
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Analytical Chemistry (8) 4 = e4 ]T f⁄e4 ] f ≅ µ ∙ 1 E * ∙ C " . Equation 8 is the first of the two relationships we were interested in. To establish the second one, we express √ ∙ , 4 via its Talyor series around , and approximate it by a quadratic function (9)
√ ∙ , 4w/ ≅ ∙ ∙ x1 −
// k∙8∙/
y.
Then we substitute Equation 9 into Equation 5 (10)
√ ∙ 6 |/ ≅ ∙ B
7 >4 4 ] x1 − and obtain that (11)
// k∙8∙/
y @4
√ ∙ 6 w/ ≅ ∙ |1 −
=
7 68 8 b z ∙ ∙ { -8 = 7 68 8 b -8
// 〈 b 〉
k∙ bn ∙/ 〈 〉
/
= 〈8 b〉 ∙
}.
Thus, the quadratic profile of the taylogram around its center determines another apparent radius equal to 4" ≡ e4 ] f⁄e4 ] f. Similarly to 4 , at moderate polydispersity 4" can be also expressed simply via the mean and polydispersity index (12) 4" = e4 ] f⁄e4 ] f ≅ µ ∙ 1 E * − 1 ∙ C " .
In case of uniform particles 4 = 4" = µ, but 4 q 4" when C q 0 because 4 − 4" ≅ µ ∙ C " . Thus, Equation 8 and 12 provide the system of equations we wanted to establish. The solution is (13)
µ = 4 − * ∙ 4 − 4" and C = x
8n
8n 8
n
− *y .
To demonstrate this approach, we simulated and analyzed taylograms corresponding to polydisperse gold US-NPs distributed normally. Using simulation, we had the advantage that the true values describing the size distribution are known exactly, and thus, we were able test the absolute accuracy of our analysis. To integrate the most common phenomenon corresponding to experimental techniques, we included noise. The realistic level of noise was calculated from basic principles, taking into the account the quantized nature of light and the fundamentals of photon detectors and UV-Vis spectroscopy. To measure the transmission of the analyte and to determine its absorbance, the intensity of the transmitted light must be measured. Measuring the intensity of light is never instantaneous, but involves detecting photons over a time interval ~ q 0. Detecting photons is, however, intrinsically random, and the consequence is that the number of photons detected during τ is a random variable. In other words, if we measure the absorbance of a sample ten times under the very same conditions, we tend to obtain ten different results. This is because even if the intensity of the illumination is completely stable, the probability density of the photon counts follow a Poisson distribution.31 This randomness is an inherent property of classical linear spectroscopy and referred to as shot noise. Accordingly, measuring transmission ( ) and absorbance () is also probabilistic, and thus noisy. Due to the relatively fine temporal resolution, this stochastic character becomes relevant to Taylor dispersion where a long continuous sequence of short and single measurements is required to resolve the dynamics of the band dispersion, while measuring the transmission of the solvent background may be a single and considerably longer measurement.
By starting from the Poisson distribution of the photon counts, and by applying the rule of transforming random variables, we calculated the probability density of the absorbance when the intensity of illumination is precisely known and the influence of particle number density fluctuations32 is negligible: (14) > = J10 ∙ e∙% ∙ ∙ /
where = ∙ 10; , = α ∙ Q ∙ ~ is the number of the photons illuminating the flow during ~ integration time, α a detectorand wavelength-specific constant, Q the intensity of the illuminating light reaching the flow, the value of the transmission, and the gamma function. It can be shown that when q 100, the signal-to-noise ratio () of —defined as the ratio of the mean and standard deviation of >—is (15) = − ∙ 2 E 2 ∙ √ ∙ . The mean value of the absorbance is equal to − , which is in fact the ‘true’ value one would always measure in the absence of shot noise. When q 0.5, Equation 15 simplifies:
(16) ≅ √ ∙ 1 − . Therefore, the expected value of is not a constant over the course of a measurement. The smaller the transmission the better the signal, which can be improved by a) increasing the concentration of the analyte, b) increasing the capillary diameter, c) increasing the detection area, d) increasing the intensity of the illuminating lamp, and e) increasing the integration time. The concentration of the analyte must be chosen so that the Lambert-Beer law is valid and inter-particle interactions are negligible and collective-diffusion is vanishing. To resolve the taylogram, sufficient resolution in both space and time is required, which define the upper limits for the integration time and the area of observation. When increasing the power of the illuminating lamp, one must ensure that the optical detector response remains linear. The linear range of UV-Vis detectors can be quite high, up to a counting rate of 10 MHz.33 When we simulated the taylograms, we took into account these fundamental properties of UV-Vis spectroscopy. For each computational experiment, the mean radius and polydispersity index were drawn randomly and independently from two uniform distributions: 0.5 r r 2.5 nm and (0.1 r C r 0.25. The shape of the particle size distribution >4 was modelled as Gaussian. The noiseless taylogram was created via Equation 2 with a temporal resolution of 0.5s, by using the power scaling between optical extinction and radius shown in Figure 1d. The minimum transmission value was limited to 0.8. The distance between the injection and detection points was set to 50 cm, the radius of the capillary (&) to 50 µm, the flow velocity (,) to 1 mm/s, and the temperature to 25 °C. The fluid we used in the simulation was water. Each taylogram was thus fully realistic and fulfilled all the vital experimental conditions.14-16 The Reynolds number was 0.11, and the axial diffusion was negligible compared to convection, that is, the inequality 69 r , ∙ &/ (Peclet number) is satisfied by the mean particle size. Also, the residence time was much larger than the characteristic (dimensionless) diffusion time on a distance equal to the capillary radius, that is, the inequality 1.4 r = 4 ∙ /& " was also satisfied. Furthermore, we used the fact that single-detection analysis is accurate when the injected volume (~ 10 J) is less than 1% of the total capillary volume till the detection window.34 Independently of 3
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one another, noise was included for each point of the taylogram, and the ‘true’ absorbance value was replaced by a single variable draw randomly from the density distribution of the absorbance (Equation 14). We expected that the level of noise influences the absolute accuracy of the analysis of determining and C. To depict this dependence, we used nine increasing values of signal-to-noise ratio defined at the center of the peak. The integration time, and thus, the temporal resolution was set to ~ = 0.5, and the intensity of illumination (/~) were systematically increased (Figure 2a).
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As anticipated, the expected accuracy is better when SN is higher, and compared to polydispersity, measuring the mean value is always more accurate (Figure 2b). This is not surprising when considering that noise may obscure subtle features of the taylograms that differentiate uniform and polydisperse particles. Nonetheless, the expectable accuracy is already within 10% if the signal-to-noise ratio is not less than 100. Although it is beyond the scope of this work, the accuracy may be improved further by filtering and smoothing the noise in taylograms. To conclude, our technique provides a simple yet reliable analysis of Taylor dispersion experiments to determine the mean and polydipeirsty index of US-NPs. We took into account that the refractive index of US-NPs may exhibit strong dependence on the size, and demonstrated its relevance by calculating the extinction of gold particles. We established straightforward relationships between two independent and measurable ‘shape’ parameters of a taylogram, and used them to characterize particle size distribution. In this work particle size distribution was treated as normal, but the concept is fully appropriate for other parametric distributions as well. We treated exhaustively the dependence of the expectable accuracy on noise, and theory and computational experiments were found to be in very good agreement. Our technique is nontrivial and—to the best of our knowledge—new to Taylor dispersion, and thus, provides a significant improvement in the analysis of Taylor dispersion experiments addressing polydisperse nanoclusters and nanoparticles.
ASSOCIATED CONTENT Supporting Information Figure 2. a) A taylogram with different levels of noise. The baselines are shifted for the sake of clarity. Intensity (count rates, / ~): 31.3 kHz, 45 kHz, 125 kHz, 281 kHz, 500 kHz, 1.125 MHz, 2 MHz, 3.13 MHz, and 4.5 MHz, from top to bottom. The quantum efficiency of photon detectors at = 600 nm is approximately 50%,33 and thus, these numbers are ensured by any standard lamp. b) The absolute value of the relative accuracy of determining the mean value (µ) and polydispersity index (C) as a function of the signal-to-noise ratio. (Equation 17). ∆µ and ∆ are the averages of 500 measurements, and the signal-to-noise ratio was calculated at
= 0.8 and ~ = 0.5 via Equation 16. Supporting Information files show representative taylograms and list the individual results of the analyses.
For the analyses, we draw randomly 500 -C pairs as described above, representing 500 different particle size distributions. For each pair, the experiment was repeated 100 times, and each time with a new noise configurations. The analyses—described by Equation 8, 12 and 13—were performed without either filtering, smoothing or conditioning the taylogram. For each analysis, the absolute value of the relative accuracy of determining the mean and polydispersity index was quantified by (17)
∆µ = V
µ µ µ
V and ∆ = V
V
where µ and C are the true values, and µ and CF are the values obtained from the analysis of the computational experiments.
Supporting Information (PDF) shows representative taylograms for each of the 500 experiments performed at three SN values (Taylograms) and lists the results of the analyses (Absolute accuracy). SN values are 50 (0.125MHz), 100 (0.5MHz), and 300 (4.5MHz). Bookmark notation: 31_0.62_0.09 indicates experiment number_µ_C.
AUTHOR INFORMATION Corresponding Author
[email protected] ACKNOWLEDGMENT The author is grateful for the financial support of the Adolphe Merkle Foundation, the University of Fribourg, and the Swiss National Science Foundation through the National Centre of Competence in Research Bio-Inspired Materials
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