Taylor dispersion analysis of metallic-based
nanoparticles – A short review

Jérémie Gouyon, Ariane Boudier, Fatima Barakat, Arnaud Pallotta, Igor

Clarot

► To cite this version:

Jérémie Gouyon, Ariane Boudier, Fatima Barakat, Arnaud Pallotta, Igor Clarot. Taylor disper-
sion analysis of metallic-based nanoparticles – A short review. Electrophoresis, In press, 43 (23-24),
pp.2377-2391. 10.1002/elps.202200184. hal-03851273

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https://hal.univ-lorraine.fr/hal-03851273v1

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Distributed under a Creative Commons Attribution 4.0 International License
1 Taylor dispersion analysis of metallic-based nanoparticles

2 – a short review

3 *Jeremie Gouyon*<sup>a*</sup>, *Ariane Boudier*<sup>a,b</sup>, *Fatima Barakat*<sup>a</sup>, *Arnaud Pallotta*<sup>a,b</sup>, *Igor

4 Clarot<sup>a,b</sup>

5

6 <sup>a</sup>Université de Lorraine, CITHEFOR, F-54000 Nancy, France

7 <sup>b</sup>Nanocontrol, F-54000 Nancy, France

8

9 *Correspondence should be addressed to the following author:

10 Jeremie Gouyon (PhD)

11 CITHEFOR, Université de Lorraine

12 F-54000 Nancy

13 jeremie.gouyon@univ-lorraine.fr (ORCID number : 0000-0002-8834-9009)

14

15 **Keywords:** Capillary, Hydrodynamic radius, Metallic nanoparticles, Quality control,

16 Taylor Dispersion Analysis

17 **Abbreviations:**

AgNPs Silver nanoparticles

AuNPs Gold nanoparticles

CRLI Constrained Regularized Linear Inversion

dH Hydrodynamic diameter

DLS Dynamic Light Scattering

FCS Fetal Calf Serum

MUA 11-MercaptoUndecanoic Acid

NTA Nanoparticles Tracking Analysis

PDI Polydispersity index

PLMA Poly(LaurylMethAcrylate)

PMA Poly(isobutylene-alt-Maleic Anhydride)-graft-dodecyl

PTMAEMA Poly((2-(MethAcryloyloxy)Ethyl)phosphonic Acid)

QDs Quantum Dots

rH Hydrodynamic radius

SPIONs Superparamagnetic iron oxide nanoparticles

TDA Taylor Dispersion Analysis

Zinc mono substituted carboxyphenoxy phthalocyanine-Glutathione-

ZnMCPPc CdTe/CdS/ZnS

ZnTCPPc Zinc tetra substituted carboxyphenoxy phthalocyanine- Glutathione -
CdTe/CdS/ZnS
19

**Abstract**

Taylor Dispersion Analysis (TDA) is an interesting tool for nanoparticle (NP) size determination, feasible using simple capillary electrophoresis (CE) apparatus. Based upon the radial diffusion of analytes upon a laminar stream, the diffusion coefficient of species is easily estimable. Moreover, TDA is generally more adequate than conventional Dynamic Light Scattering (DLS) methodologies since it is less dependent on the polydispersity of the sample, leading to accurate measurement and reliable results. This review provides every paper mentioning the use of TDA for metallic-based NPs size determination. Diverse strategies for the detection of metallic NPs (like UV-Visible and inductively coupled plasma-mass spectrometry -ICP-MS- for instance) and interpretation of the Taylorgrams are discussed. Based upon the literature, advices on future prospects are also indicated, especially for comparison of TDA results with other classical techniques.
# 1. Introduction

Size matters when it comes to nanoparticles (NPs). To be more accurate, core size, hydrodynamic diameter ($d_h$, including core size and the electric dipole layer to its surface, comprising solvents and stabiliser) value and polydispersity index (PDI) strongly impact their physicochemical properties, from their bioavailability [1] to their colloidal stability [2]. For instance, cytotoxicity largely depends on the polydispersity of the NPs [3], which can be detrimental for therapeutic purposes. The use of experimental apparatus for NPs size measurement has become a critical aspect for NPs characterisation, with a wide variety of techniques available nowadays, and reviewed in the literature [4]. Transmission Electron Microscopy (TEM), NPs Tracking Analysis (NTA), Dynamic Light Scattering (DLS), UV-Visible spectroscopy, Field-Flow Fractionation, single-particle inductively coupled plasma mass spectrometry (sp-ICP-MS) or analytical ultracentrifugation are well-known examples described in various cases, from liposome [5] to ultra-small (1-3 nm range) [6] NPs characterisation. For the most popular of them, advantages and drawbacks are reported in **Table 1**, justifying the orthogonality requirements to exhaustively characterise NPs.

<table><caption>**Table 1 - Advantages and drawbacks of the most commonly used techniques for NPs size / polydispersity determination according to the literature [7–14]**</caption><thead><tr><td>Technique</td><td>Advantages</td><td>Drawbacks</td></tr></thead><tbody><tr><td>TEM</td><td>Morphology, core size / NPs size dispersity determination shape, aggregation state</td><td>No information about d<sub>h</sub><br/>Large deviation for smallest size (&lt;10 nm)<br/>Not representative, sample preparation and user interpretation bias<br/>Expensive</td></tr><tr><td>DLS</td><td>d<sub>h</sub> determination, size distribution and dispersity, zeta potential speed and easy to use</td><td>Difficulties for complex media<br/>Sensitive to aggregates<br/>Bias toward larger objects in complex media<br/>Large deviation for smallest size (&lt;10 nm) and for fluorescent sample<br/>Essential knowledge about NPs nature are required</td></tr><tr><td>NTA</td><td>d<sub>h</sub> determination, size distribution speed and easy to use</td><td>Large deviation for smallest size (&lt;10 nm)</td></tr></tbody></table>
<table><tr><td rowspan="3"><b>TDA</b></td><td><b>Single/double window mode:</b> d<sub>n</sub><br>monodisperse determination<br>Compatible with several detectors<br>(UV-Visible, LIF, ICP-MS...)<br>Can be considered as a separative<br>method<br>Medium viscosity determination<br>possible</td><td>Experimental conditions to be<br>controlled precisely<br>Difficulties for complex media<br>(depending on the detector)</td></tr><tr><td><b>Double window mode:</b> counteract<br>the influence of technical aspects<br>(pressure ramp, injection...) on the<br>peak variance</td><td>Device modification required</td></tr></table>

NPs can be classified according to their size, polydispersity, but also their chemical composition: some are constituted of polymeric materials or macrostructural organic arrangements (such as liposomes), some will be constituted of inorganic core like mesoporous silica covered with organic moieties [15]. Others will also contain metallic or semimetallic elements, from mono-atomic complexes to metallic core in colloidal suspension. The latter – designated as metallic-based NPs – are currently gaining much attention thanks to their versatility [16]. They are tuneable (size, shape, charge) and generally characterised by surface plasmon resonance properties. They are particularly useful in bio-imaging thanks to their relative biocompatibility, and also for cancer immunotherapy in which metallic-based NPs can act as adjuvant, for instance [17]. Nowadays, metallic-based NPs and associated nanostructures are developed for a wide variety of applications, from catalysis to sensor technology [18]. Nonetheless, metallic-based NPs stability and toxicity need to be monitored, indirectly through their characterisation, including their size and PDI determination. Aforementioned size determination techniques are obviously reported in the associated literature, and the most commonly adopted. But the synthesis routes of metallic-based NPs or their use in biological media could lead to complex analytical conditions, which can be incompatible with the selected methodology. One technique can overcome some disadvantages associated to the size determination of metallic-based NPs: Taylor Dispersion Analysis (TDA). Surprisingly, TDA is almost not mentioned in the moderately recent review of S. Mourdikoudis et al. [4] among previously mentioned characterisation techniques. This can be explained by the low - but increasing - number
72 of publications dedicated to the use of TDA for NPs (20 in 2001 against 70 in 2021,
73 see Figure 1).

74

<figure><img src="image_2.png" /></figure>

75

<caption>76 Figure 1. Number of publications related to TDA as a function of time (results
77 obtained on WebOfScience, using “Taylor dispersion analysis” keywords for
78 Title / Topic)</caption>

79 TDA is a non-destructive method adapted to d<sub>n</sub> determination [19], ranging from Å to
80 µm, thus including the NP range (1 to 100 nm [20]). A recent review by M.R. Moser
81 and C.A. Baker [21], intending to bring a better comprehension of the technique and
82 making it accessible to a wider community, gave a guide for TDA users. A particular
83 focus was realised on practical considerations and mathematical treatment of the
84 generated signals using pulse mode on capillary electrophoresis (CE) device,
85 especially (originated from coupling with different detector types, such as UV-Visible
86 or Laser induced fluorescence). Briefly, to perform TDA in pulse mode, a solute is
87 introduced and hydrodynamically mobilised (via pressure difference, ΔP) through a
88 capillary filled with a solution (buffer for example); the diffusion occurring under a
89 laminar flow leads to a Gaussian distribution as a function of time S(t), resulting in a
90 Taylorsgram (Figure 2A) which can be fitted with equation 1 [22]:

$$
91 S(t) = y_0 + \frac{A}{\sigma\sqrt{2\pi}} \exp\left(-\frac{1}{2}\frac{(t-t_0)^2}{\sigma^2}\right) (1)
$$

92 with t the experimental time, t₀ the mean residence time of the solute, y₀ the baseline
93 intensity, A a constant which depends on NPs concentration and detection sensitivity,
and $\sigma$ the mean temporal variance of the fitted Gaussian distribution. Using appropriate software, it is possible to approximate the signal with a theoretical curve, thus estimating $\sigma$ value.

<figure><img src="image_3.png" /><figcaption>Figure 2. Principle of TDA in (A) pulse mode and Gaussian curve parameters resulting from the fitting defined in equation 1 and (B) frontal mode and frontal curve resulting from the fitting defined in equation 4. Adapted from [21].</figcaption></figure>

The following formula (equation 2) can be used for the determination of the diffusion coefficient of the solute D if the dimensionless residence time $\tau$ is superior to 1.4 and the Peclet number Pe is superior to 69 (defined as the 'relative rates of mass transfer along the axis of the capillary due to convection and diffusion') [23].

$$D = \frac{r_c^2}{24\sigma^2} t_0 \quad (2), \text{ with } \tau = \frac{Dt_0}{r_c^2} \text{ and } Pe = \frac{ur_c}{D}$$

with $r_c$, capillary radius and $u$, the linear flow rate. The previously mentioned criteria are based on previous work [23] and are generally adopted. It has been demonstrated that these conditions lead to a determination of D with a relative error $\epsilon$ of 1%. Then, Stokes-Einstein's equation leads to hydrodynamic radius $r_h$ estimation (equation 3):
110 $$D = \frac{k_B T}{6\pi\eta r_H} \quad (3)$$

111 with k_B, the Boltzmann's constant, T the temperature and η the viscosity of the media
112 inside the capillary.

113 In the frontal mode, the sample is continuously introduced in the capillary, leading to a
114 Taylorsgram described with a frontal curve as a function of time (Figure 2B), which can
115 be fitted with equation 4 [23].

116 $$\frac{\bar{c}}{c_0} = \frac{1}{2} \pm \frac{1}{2} \operatorname{erf} \left[ \frac{(t-t_0)}{\sigma\sqrt{2}} \right] \quad (4)$$

117 with $\bar{c}$ the mean solute concentration across the section of the capillary, proportional
118 to the signal at a given time and $\bar{c}_0$ the concentration of the front, typically proportional
119 to the signal of the front. One of the interests of the frontal mode compared to pulse
120 mode is the lower relative standard deviation obtained for the determination of diffusion
121 coefficient. This is attributed to a lower number of pressure changes steps, which can
122 be detrimental toward the Taylor-Aris regime required for TDA.

123 In this review, we focus on the use of TDA using both modes for size determination of
124 specific metallic-based NPs, or associated nanoobjects like nanoclusters,
125 SuperParamagnetic Iron Oxide Nanoparticles (SPIONs) or even Quantum Dots (QDs).

126 We aim at listing every publication mentioning the technique for such a precise
127 application, whether it is for quality control or for deep fundamental study. Summarised
128 in Table 2, applications, TDA conditions and main results are compared to regular
129 characterisation techniques such as DLS or TEM, even if the considered techniques
130 can be complementary for a wide range of NP size determination. The following
131 paragraphs will expose the possibilities for full characterisation of NPs, in comparison
132 with regular techniques and, regarding the reported results, advices for future
133 development and democratisation of TDA in laboratories, for academics, industrial and
134 regulatory purposes. TDA conditions are reported to give a quick overview of possible
135 setups used for analysis with corresponded r_H range. The studied media are also
136 reported since their composition affects the interactions between nanoparticles and
137 capillary walls, which need to be negligible to remain under Taylor-Aris regime.
<table><caption>Table 2 – Reported application of TDA in metallic-based NPs size determination</caption><thead><tr><th>Application</th><th>Nanoparticle type</th><th>Ref. technique / r<sub>H</sub> size</th><th>TDA r<sub>H</sub> size det.</th><th>Detection technique</th><th>Studied media</th><th>TDA conditions</th><th>r<sub>H</sub> range** (nm) (ε=1%)</th><th>r<sub>H</sub> range** (nm) (ε=3%)</th><th>Ref.</th></tr></thead><tbody><tr><td rowspan="2">Theoretical comparison</td><td>PVP-coated AgNPs</td><td>TEM (mean core radius):<br>r<sub>core</sub> = 14 ± 1,9 nm</td><td>r<sub>H</sub> ≈ 23 nm</td><td>UV-Visible<br>(2 det.wind.)<br>400 nm</td><td rowspan="2">MilliQ water</td><td>L<sub>tot</sub> = 145 cm ; i.d.<br>74.5 µm ; L<sub>det</sub> = 37 /<br>72 cm ; V<sub>inj</sub> = 140 nL<br>(2.2 %) ; ΔP = 90<br>mbar</td><td rowspan="2">[0.3 – 14.4]</td><td rowspan="2">[0.2 – 43.3]</td><td rowspan="2">[9]</td></tr><tr><td>PEGylated AuNPs<br>with carboxylic acid<br>(COOH) end group</td><td>TEM (mean core radius):<br>r<sub>core</sub> = 21.5 ± 4.9 nm</td><td>r<sub>H</sub> ≈ 19 nm</td><td>UV-Visible<br>(2 det.wind.)<br>520 nm</td></tr><tr><td rowspan="6">Proof of concept<br>for small NPs</td><td rowspan="3">AuNPs</td><td>DLS : a. r<sub>H</sub> = -<br>b. r<sub>H</sub> = 13.5 ± 2 nm<br>c. r<sub>H</sub> = 22.5 ± 0.1 nm</td><td>a. r<sub>H</sub> = 1.7 ± 0.1<br>nm</td><td rowspan="3">UV-Visible<br>(2 det.wind.)<br>520 nm</td><td rowspan="6">0.001% (wt/V)<br>Tween®20 in<br>MilliQwater<br>(to prevent<br>adsorption)</td><td rowspan="6">L<sub>tot</sub> = 130 cm ; i.d.<br>74.5 µm ; L<sub>det</sub> = 45 /<br>85 cm ; V<sub>inj</sub> = 100 -<br>200 nL (1.8 - 3.5 %) ;<br>ΔP = 70 mbar</td><td rowspan="6">[0.4 – 20.2]</td><td rowspan="6">[0.3 – 60.6]</td><td rowspan="6">[10]</td></tr><tr><td>TEM : a. r<sub>H</sub> = 2.3 ±<br>0.6 nm</td><td>b. r<sub>H</sub> = 11.5 ± 0.5<br>nm</td></tr><tr><td>b. r<sub>H</sub> = 7.5 ± 1.0 nm<br>c. r<sub>H</sub> = 28 ± 5 nm</td><td>c. r<sub>H</sub> = 33.5 ± 2.0<br>nm</td></tr><tr><td rowspan="3">SPIONs</td><td>DLS : a. r<sub>H</sub> = 27 ± 1<br>nm</td><td>a. r<sub>H</sub> = 7.5 ± 0.5<br>nm</td><td rowspan="3">UV-Visible<br>(2 det.wind.)<br>280 nm</td><td rowspan="3"></td><td rowspan="3"></td><td rowspan="3"></td><td rowspan="3"></td><td rowspan="3"></td></tr><tr><td>b. r<sub>H</sub> = 13.5 ± 0.2 nm</td><td>b. r<sub>H</sub> = 16.5 ± 1.0<br>nm</td></tr><tr><td>TEM : a. r<sub>core</sub> = 6.5 ±<br>1.0 nm<br>b. r<sub>core</sub> = 12.0 ± 1.5<br>nm</td><td></td></tr><tr><td rowspan="2">Theoretical<br>comparison</td><td rowspan="2">Citrate stabilised<br>SPIONs<br>(2 batchs)</td><td>TEM : a. r<sub>core</sub> = 6.0 ±<br>0.7 nm</td><td>a. r<sub>H</sub> = 6.0 ± 0.2<br>nm or 4.6 ± 0.2<br>nm</td><td rowspan="2">UV-Visible<br>(2 det.wind.)<br>520 nm</td><td rowspan="2">MilliQ water</td><td rowspan="2">L<sub>tot</sub> = 145.5 cm ; i.d.<br>74.5 µm ; L<sub>det</sub> = 72.5 /<br>104.5 cm ; V<sub>inj</sub> = 112<br>nL (1.8 %) ; ΔP = 90<br>mbar</td><td rowspan="2">[0.4 – 28.3]</td><td rowspan="2">[0.2 – 85.0]</td><td rowspan="2">[11]</td></tr><tr><td>b. r<sub>core</sub> = 9.9 ± 0.6<br>nm</td><td>b. r<sub>H</sub> = 9.4 ± 0.7<br>nm or 9.3 ± 0.5<br>nm</td></tr><tr><td rowspan="5">Model (related to<br>biomedical<br>application for<br>spions and Au) +<br>Industrial app.<br>(wood<br>impregnation Cu<br>/ pigment TiO2 /<br>cosmetic ZnO)</td><td>Citrate stabilised<br>AuNPs</td><td>TEM : r<sub>core</sub> = 28.0 ±<br>4.7 nm</td><td>r<sub>H</sub> = 33.8 nm</td><td>UV-Visible<br>(2 det.wind.)<br>520 nm</td><td rowspan="5">MilliQwater (+0.001%<br>(wt/V) Tween®20<br>for SPIONs / Au)</td><td rowspan="5">L<sub>tot</sub> = 130 cm ; i.d.<br>74.5 µm ; L<sub>det</sub> = 45 /<br>85 cm ; V<sub>inj</sub> = - (- %) ;<br>ΔP = 70 mbar</td><td rowspan="5">[0.4 – 20.2]</td><td rowspan="5">[0.3 – 60.6]</td><td rowspan="5">[13]</td></tr><tr><td>SPIONs</td><td>TEM : r<sub>core</sub> = 6.7 ±<br>1.1 nm</td><td>r<sub>H</sub> = 7.6 nm</td><td>UV-Visible<br>(2 det.wind.)<br>520 nm</td></tr><tr><td>ZnO-NPs</td><td>TEM : r<sub>core</sub> = 28.5 ±<br>18.1 nm</td><td>r<sub>H</sub> = 28.8 nm</td><td>UV-Visible<br>(2 det.wind.)<br>280 nm</td></tr><tr><td>TiO2-NPs</td><td>TEM : r<sub>core</sub> = 117.5 ±<br>101.4 nm</td><td>r<sub>H</sub> = 50.0 nm</td><td>UV-Visible<br>(2 det.wind.)<br>280 nm</td></tr><tr><td>Cu-NPs</td><td>TEM : r<sub>core</sub> = 27.9 ±<br>26.8 nm</td><td>r<sub>H</sub> = 33.1 nm</td><td>UV-Visible<br>(2 det.wind.)<br>400 nm</td></tr></tbody></table>
<table><tr><td>Theranostic agent in biological media</td><td>Gd chelated polysiloxane nanoparticle</td><td>DLS : r<sub>H</sub> = 2.8 ± 0.4 nm</td><td>r<sub>H</sub> = 2.3 ± 0.2 nm</td><td>UV 200 nm &amp; ICP-MS</td><td>10 mM Tris + 150 mM NaCl buffer pH 7.4</td><td>L<sub>tot</sub> = 70 cm ; i.d. 75 µm ; L<sub>det</sub> = 21.6 cm / outlet ; V<sub>inj</sub> = 8 nL (0.2 %) ; ΔP = 50 mbar</td><td>[0.3 – 7.1]</td><td>[0.2 – 21.4]</td><td>[22]</td></tr><tr><td>Colloïdal stability prediction</td><td>SPIONs</td><td>DLS : r<sub>H</sub> = 7.3 / 9.3 / 16 nm (sub.pop from 1 batch)</td><td>r<sub>H</sub> = 5.2 / 7.3 / 11.7 nm (sub.pop from 1 batch)</td><td>UV-Visible (1 det.wind.) Front concentration profile 254 nm</td><td>10 mM HCl</td><td>L<sub>tot</sub> = 35 / 60 / 100 cm ; i.d. 50 µm ; L<sub>det</sub> = 26.5 / 52.5 / 91.5 cm ; ΔP = 20 - 50 mbar (frontal)</td><td>[0.5 – 30.5]*</td><td>[0.3 – 91.4]*</td><td>[23]</td></tr><tr><td>Theoretical comparison</td><td>Highly polydisperse Citrate-capped AgNPs</td><td>TEM (no data available)</td><td>r<sub>H</sub> ≈ 16 nm</td><td>UV-Visible (2 det.wind.) 400 nm</td><td>MilliQ water</td><td>L<sub>tot</sub> = 145 cm ; i.d. 74.5 µm ; L<sub>det</sub> = 37 / 72 cm ; V<sub>inj</sub> = 32 nL (1.0 %) ; ΔP = 90 mbar</td><td>[0.3 – 7.1]</td><td>[0.2 – 43.3]</td><td>[24]</td></tr><tr><td>Photodynamic therapy (nanotherapy)</td><td>Phtalocyanine conjugated QDs a. ZnTCPPc b. ZnMCPPc</td><td>DLS : a. r<sub>H</sub> = 13.6 ± 7.0 b. r<sub>H</sub> = 67.1 ± 32.9 nm</td><td>a. r<sub>H</sub> = 13.5 ± 3.9 nm b. r<sub>H</sub> = 64.1 ± 7.7 nm</td><td>IF/UV-Visible (2 det.wind.) Front concentration profile 480 nm ex. / 630 nm em. / 680 nm abs.</td><td>phosphate buffer pH 7.4 I.S. 15 mM (and other media)</td><td>L<sub>tot</sub> = 45 cm ; i.d. 50 µm ; L<sub>det</sub> = 21 (IF) / 36.5 (UV) cm ; ΔP = 70 mbar</td><td>[1.2 – 37.5] (IF) / [1.2 – 65.3] (UV)</td><td>[0.7 – 112.6] (IF) / [0.7 – 195.8] (UV)</td><td>[25]</td></tr><tr><td>Colloïdal stability prediction</td><td>Hydrophilic surface coated AuNPs a. PMA-AuNPs b. PEG-2kDa-PMA-AuNPs c. PEG-10kDa-PMA-AuNPs d. MUA-AuNPs e. PTMAEMA-stat-PLMA-AuNPs PEG-10kDa-PMA-AuNPs MUA-AuNPs PTMAEMA-stat-PLMA-AuNPs Carboxylic-acid modified fluorescently labelled SPIONs silica core-shell (NP*-COOH): a. (NP*-COOH) without aptamer b. NP*-COOH with aptamer 1/150</td><td>DLS (cumulant method) a. r<sub>H</sub> = 26.80 ± 0.09 nm b. r<sub>H</sub> = 22.46 ± 0.11 nm c. r<sub>H</sub> = 33.74 ± 0.14 nm d. r<sub>H</sub> = 19.59 ± 0.09 nm e. r<sub>H</sub> = 11.15 ± 0.14 nm</td><td>a. r<sub>H</sub> = 9.22 ± 0.10 nm b. r<sub>H</sub> = 12.81 ± 0.13 nm c. r<sub>H</sub> = 14.44 ± 0.19 nm d. r<sub>H</sub> = 8.08 ± 0.08 nm e. r<sub>H</sub> = 5.98 ± 0.04 nm</td><td>UV-Visible (1 det.wind.) Front concentration profile 280 nm</td><td>5 mM disodiumtetraborate decahydrate</td><td>L<sub>tot</sub> = 61.5 cm ; i.d. 75 µm ; L<sub>det</sub> = 51.4 cm ; V<sub>inj</sub> = 24 nL (0.9 %) ; ΔP = 70 mbar</td><td>[1.0 – 53.9]</td><td>[0.6 – 161.8]</td><td>[14,26]</td></tr><tr><td>Cancer diagnosis and therapy</td><td>DLS : a. r<sub>H</sub> = 32 nm b. - c. -</td><td>a. r<sub>H</sub> = 34.5 ± 2.5 nm b. r<sub>H</sub> = 39.5 nm c. r<sub>H</sub> = 41.5 nm</td><td>UV-Visible (1 det.wind.) Front concentration profile 254 nm</td><td>MOPS buffer pH 7.4 I.S. 100 mM</td><td></td><td>L<sub>tot</sub> = 35 cm ; i.d. 50 µm ; L<sub>det</sub> = 26.5 cm ; ΔP = 50 mbar (frontal)</td><td>[0.5 – 30.5]</td><td>[0.3 – 91.4]</td><td>[27]</td></tr></table>
c. NP*-COOH with aptamer 1/200

<table><tbody><tr><td rowspan="2">Nanoparticle-based biocatalyst</td><td><b>AuNPs</b><br/>a. Citrate stabilised<br/><b>AuNPs</b><br/>b. Pepsin-AuNPs*<br/>*other pepsin concentration were tested</td><td>DLS: a. r<sub>H</sub> = 22.1 ± 0.2 nm<br/>b. r<sub>H</sub> = 39.9 ± 2.9 nm<br/>TEM: a. r<sub>core</sub> ≈ 15 nm<br/>b. -</td><td>a. r<sub>H</sub> = 17.8 ± 0.2 nm<br/>b. r<sub>H</sub> = 21.5 ± 0.3 nm</td><td>UV-Visible (2 det.wind.) 214 nm</td><td>Ultrapure water</td><td>L<sub>tot</sub> = 130 cm; i.d. 75 µm; L<sub>det</sub> = 44 / 85 cm; V<sub>inj</sub> = 40 nL (0.7 %); ΔP = 140 mbar</td><td>[0.2 – 9.6]</td><td>[0.1 – 28.9]</td><td>[28]</td></tr><tr><td>Carboxylated magnetic beads (commercial SPIONs)</td><td>-</td><td>r<sub>H</sub> = 100 ± 0.15 nm (given by the manufacturer)</td><td>UV-Visible (1 det.wind.) 200 nm</td><td>Water</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Effective charge determination</td><td>CeO<sub>2</sub></td><td>-</td><td>r<sub>H</sub> = 16.05 ± 0.48 nm</td><td>UV-Visible (1 det.wind.) 254 nm</td><td>Betaine-Cl<sup>-</sup> pH 2<br/>I.S. 10 mM</td><td>L<sub>tot</sub> = 33.5 cm; i.d. 50 µm; L<sub>det</sub> = 25 cm; V<sub>inj</sub> = 2.6 nL (0.4 %); ΔP = 50 mbar</td><td>[0.5 – 20.0]</td><td>[0.3 – 59.9]</td><td>[29]</td></tr><tr><td>TiO<sub>2</sub></td><td>-</td><td>r<sub>H</sub> = 13.71 ± 0.55 nm</td><td>UV-Visible (1 det.wind.) 254 nm</td><td>Betaine-Cl<sup>-</sup> pH 2<br/>I.S. 10 mM</td><td></td><td></td><td></td><td></td></tr><tr><td>Gd speciation</td><td><b>AGulX</b> ultrasmall nanoparticles (gadolinium-chelated polysiloxane nano-Particles)</td><td>-</td><td>r<sub>H</sub> = 2.8 ± 0.1 nm (bimodal deconvolution)<br/>r<sub>H</sub> = 2.7 ± 0.1 nm (CRLI)</td><td>ICP-MS</td><td>10 mM Tris + 150 mM NaCl buffer pH 7.4</td><td>L<sub>tot</sub> = 70 cm; i.d. 75 µm; L<sub>det</sub> = outlet; V<sub>inj</sub> = 7 nL (0.2 %); ΔP = 50 mbar</td><td>[0.3 – 23.1]</td><td>[0.2 – 69.2]</td><td>[30]</td></tr><tr><td rowspan="2">Protein corona formation on Au-Si-NPs</td><td><b>AuSiNPs</b><br/>a. protein-free<br/>b. with BSA<br/>c. with fetuin from FET</td><td>DLS: a. r<sub>H</sub> = 30 ± 2 nm<br/>TEM: a. r<sub>core</sub> = 28.5 ± 1.5 nm</td><td>a. r<sub>H</sub> = 30.0 ± 0.7 nm<br/>b. r<sub>H</sub> = 36.8 ± 0.7 nm<br/>c. r<sub>H</sub> = 35 ± 1 nm</td><td>ICP-MS</td><td>10 mM phosphate buffer pH 7.4</td><td>L<sub>tot</sub> = 70 cm; i.d. 50 µm; L<sub>det</sub> = outlet; V<sub>inj</sub> = 2.9 nL (0.2 %); ΔP = 50 mbar</td><td>[1.1 – 116.8]</td><td>[0.6 – 350.5]</td><td>[31]</td></tr></tbody></table>

Note: PVP = PolyVinylPyrrolidone; PMA = Poly(isobutylene-alt-Maleic Anhydride)-graft-dodecyl; MUA = 11-MercaptoUndecanoic Acid; PTMAEMA = Poly((2-MethAcryloyloxy)Ethyl)phosphonic Acid; PLMA = Poly(LaurylMethAcrylate); NP* = fluorescently labelled NPs; I.S. = Ionic Strength

Different samples in the same article are identified by letters (a, b, c...)

<footnote>* estimated by considering L<sub>tot</sub> = 35 cm, L<sub>det</sub> = 26.5 cm; ΔP = 50 mbar</footnote>

<footnote>**r<sub>H</sub> range are estimated using TDA conditions as described in the corresponding paper and : for ε = 1%, Pe > 69 and τ > 1.4 and for ε = 3%, Pe > 40 and τ > 1.2 using equations 2 and 3</footnote>
140

## 2. Capillary device for TDA applications

141

### 2.1. Single window mode

142

Capillary Electrophoresis is a separative technique based upon the application of an electric field in a capillary filled with electrolyte buffer. Due to the reduced inner capillary diameter (down to 5 µm) and pumping system adapted to relatively low laminar flow control, CE apparatus is well-suited for TDA, even though some other devices can be used as it will be presented. Previously mentioned criteria on τ and Pe values can be easily satisfied according to the analyte size and capillary dimensions (such as capillary length or inner diameter) for both pulse and frontal mode. Another advantage of CE is the inline coupling with UV-Visible detector, making it easy to monitor most of the NPs since they generally possess optical properties, from deep UV to visible absorbance range (Figure 3A).

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<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

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<caption>**Figure 3.** Scheme of TDA with CE device in (A) single window mode and (B) double window mode.</caption>

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Historically, the first use of CE for TDA application was reported in 1994 by M.S. Bello *et al.* [32], with a proof of concept based on the determination of the diffusion coefficient of small organic molecules and proteins. A single UV-Visible detector was used inline and the results were in good agreement with the tabulated values of the literature. The technique has been presented as an accessible tool for bioanalytical laboratories since relatively regular devices are required (compared to DLS devices, back at the time) but
also for a quick determination and a low sample consumption, which is quite
advantageous when analysing biological samples. It was only in 2008 that the first use
of CE device for TDA of metallic-based NPs was reported, with the characterisation of
Maghemite particles (or SPIONs) in a single window mode (also denoted as single-
point detection configuration) [23]. Careful considerations were taken using TDA as
described in other papers, such as the effect of a flow ramp while mobilising the fluid
through the capillary [33], or an absence of interaction between the particles and the
capillary inner wall [34]. Didodecyldimethylammonium bromide was used as dynamic
coating reagent between each run to limit particle-capillary inner wall interactions and
size-exclusion effect (SPIONS being cationic and uncoated silica capillary being
anionic at neutral pH), ensuring that the mean axial velocity of the NPs is equivalent to
that of the electrolyte.

Performing TDA in hydrochloric acid (HCl) background electrolyte (BGE) from a front
concentration profile (using frontal mode) on three sorted size populations (below 10
nm diameter), F. d'Orlyé et al. [23] successfully determined the SPIONs $d_H$, comprised
between 10.4 and 23.4 nm (Table 2). It is noteworthy that TDA results tend to be
superior to particle mean solid radii estimated by magnetic measurements, attributed
to hydration effects.

They also pointed out DLS results for comparison, with overestimated $r_H$ of around
30%, which is inherent to the technique. Indeed, DLS intensity is proportional to the
square of the particle volume/mass, implying that the largest particles will strongly
impact the results and potentially lead to an overestimation of the real NPs size (Figure
4). Since TDA is measured with a UV-Visible detector, the signal is proportional to the
mass concentration, which is less impacted by the polydispersity, unlike DLS
measurements. The main interest of this study was to examine colloidal stability
through estimation of the diffusion coefficient, discerning interaction and aggregation
phenomena, while limiting the impact of partial aggregation on the size determination.
Showing the existence of repulsive interactions between SPIONs, the influence of
electrolyte ionic strength has been highlighted. Addition of sodium chloride to the BGE
to tune ionic strength allowed to decrease NPs interactions, which accounted for the
determination of the Brownian diffusion coefficient of a single sphere. Thus, BGE
composition is of utmost importance in the determination of NPs size since D is
dependent on NPs interactions.
<figure><img src="image_6.png" /><figcaption>Figure 4. DLS hydrodynamic diameter as a function of the TDA measurements reported from various papers; some NPs may be constituted of aggregates [10,22,23,25–28]. dH,DLS and dH,TDA refer to dH measured by DLS and TDA, respectively.</figcaption></figure>

The same team used a similar strategy to characterise silica core-shell SPIONs, chemically modified with an aptamer for miniaturised molecular diagnostic tool development [27]. Since silica core shell SPIONs were doomed to be characterised by Capillary Zone Electrophoresis (CZE) separation, TDA was naturally chosen to complete NPs characterisation. They demonstrated that associated devices are helpful in the development of biocompatible tools, with determination of the grafting efficiency and prediction of the conditions in which SPIONs will aggregate. They evidenced the successful grafting of synthetic oligonucleotides on maghemite, with an evolution of dn from 69 nm to up to 83 nm, with a colloidal suspension and no apparent aggregation. In this case, both DLS and TDA gave close results, which can be attributed to a low PDI (less than 0.1 before grafting). No additional details were given about experimental conditions, except the use of a BGE composed of 3-(N-morpholino)propanesulfonic acid (MOPS) buffer (I.S. = 100 mM, pH 7.4). A near-neutral pH was selected since CZE measurements were shorter, and insuring no aggregation for both SPIONs and aptamer grafted SPIONs.

In a more anecdotal way, TDA coupled with UV-visible detection was also used for inorganic NPs size estimation such as CeO2 and TiO2 [29], but sizes were not compared to any other regular technique; it testifies the confidence of the team,
working actively with TDA for the determination of d<sub>H</sub> (see Table 2 for reference [29]),
on the development of TDA for the analysis of mixtures for instance [35]. Analyses
were performed in an acidic BGE (betaine/chloride pH 2) to prevent CeO<sub>2</sub> and TiO<sub>2</sub>
precipitation, but also for further CZE experiments.

In the same manner, U. Pyell et al. [14,26] used TDA to characterise five different gold
NPs (AuNPs), with a negatively charged surface obtained after coating with
amphiphilic polymer or polyethyleneglycol (PEG). Since interactions between NPs and
inner wall of the capillary are detrimental for TDA and CZE experiments, an alkaline
aqueous BGE (pH ~ 9) was chosen (5 mM borate buffer) to yield a negatively charged
inner capillary surface, since AuNPs were negatively charged and thus suppressing
potential interactions. With AuNPs of hydrodynamic radii determined by TDA (r<sub>H,TDA</sub>)
ranging from 5.98 to 15.44 nm, (for Au-core of about 2 nm), they confronted those
results with DLS measurements. Actually, they did compare TDA appreciation with the
number-weighted means hydrodynamic radii (r<sub>H,DLS,N</sub>) obtained by employing the
CONTIN algorithm on the DLS acquisition [36] for interpretation. It is important to
highlight that depending on the DLS size characterising parameters used for
comparison with r<sub>H,TDA</sub>, there could be an important deviation. Careful attention should
be paid to the exact DLS size characterising parameters and mathematical approach
reported in the literature, which is not systematically indicated in experimental papers
according to our reading (Figure 5).

<figure><img src="image_7.png" /><figcaption>Figure 5. DLS treatment types and graphical representation of intensity, volume and number distribution. DLS acquisitions are first treated with the CONTIN algorithm, which results in intensity distribution graphic. Since DLS signals are proportional to the NPs radius power of 6, the conversion in volume distribution</figcaption></figure>
is done by dividing to radius to power of 3, while number distribution approximation is divided by the radius to the power of 6, resulting in variable radius value when comparing the different distribution. One may consider each data when reporting rH values, or should at least indicates the distribution type used in the report. In this example, 2 subpopulations of 4.0 and 20.0 nm respectively (σ = 0.4 nm for each population) are represented with an initial ratio of 1:15625 in intensity distribution. Since there is a ratio 1:5 between both NPs size, the volume distribution gives a ratio 1:(15625/5³) = 1:125, and 1:(15625/5⁶) = 1:1 for number distribution. Mean radius values in this representation are equal to 20.1 ± 0,4, 19.9 ± 1,5 and 12.0 ± 8,0 nm for each distribution, respectively, which justifies the requirement of DLS information, when reported in the literature.

For instance, CONTIN algorithm is well suited for polydisperse sample analysis, while the cumulant method is generally commonly adopted for monomodal population. As concluded by U. Pyell et al. [14,26], using CE device coupled with a UV detector, TDA allows the estimation of number-weighted and volume-weighted mean hydrodynamic radii.

Moreover, TDA evidenced a subpopulation for two different AuNPs, possibly attributed to agglomerates as highlighted with DLS rH,N estimation (Table 2, for references [14,26]). This was demonstrated by a deviation of the fitting curve (based on a Gaussian cumulative function, using pulse mode) from the experimental Taylorsgram, which is interpreted here by the presence of larger AuNPs in the media.

Finally, G. Ramírez-García et al. worked on the characterisation of functionalised QDs, using TDA to measure the apparent equivalent sphere diameter of the metallic-based NPs, with coupling to either UV-Visible and/or fluorescence detectors [25]. Frontal mode was used with adequate equations from previously mentioned paper [23], with mathematical models being very well-fitted to the Taylorsgram. Size but also surface charge and stability of the QDs were concomitantly estimated by CZE, with a particular interest to the comparison between TDA and DLS results. No interaction between the capillary and the NPs was highlighted, since no variation of the retention time occurred. QDs consisted of zinc mono substituted carboxyphenoxy phthalocyanine-Glutathione-CdTe/CdS/ZnS (ZnMCPPc-QDs) and zinc tetra substituted carboxyphenoxy phthalocyanine-Glutathione-CdTe/CdS/ZnS (ZnTCPPc-QDs). The colloidal stability of ZnTCPPc-QDs in low ionic strength was evidenced since TDA measurements
indicated a dH of around 27 nm, while ZnMCPPc-QDs was measured at around 130 nm. This was explained by the presence of aggregates due to lower electrostatic repulsions since the latter contains only one carboxyl group. In this case, TDA gave lower standard deviation value than DLS, since QDs aggregates strongly impact the latter, but also since alteration of the refractive index is due to the fluorescence properties of the NPs and by the incident laser light absorption (Table 2, for reference [25]). In this situation, TDA with CE apparatus clearly presents the advantage of filling the inherent inconveniences originated from the DLS methodology, while working in physiologically relevant condition.

Another interesting coupling with TDA is the use of ICP-MS as detector in order to increase the specificity since UV-Visible detector or DLS measurements are sensitive to matrix interferences. Single particle ICP-MS can be used to determine metallic content of a suspension [37], allowing analysis in complex media. This methodology can be used alone to determine NP size, but is not adapted to hybrid NPs (composed of multiple metals) since intensity of the signal depends on the metallic composition. Coupling of ICP-MS with TDA allows to characterise a wider range of sizes and compositions of NPs. Experimental conditions were assessed using previous work of H. Cottet's team [38,39], in which they expressed a range of operating mobilising pressure based upon a relative error on the determination of D of 3% instead of 1% (translated here by following prerequisite : Pe > 40 and τ > 1.2). Using a gadolinium-containing theragnostic NP for TDA coupled with ICP-MS, L. Labied et al. [22] measured a hydrodynamic radius of 2.2 nm in pulse mode, accurate with DLS measurements. They also highlighted the existence of a slight polydispersity since Taylorgrams presented slight deviation from the Gaussian fitting model. One of the advantages of using ICP-MS with TDA instead of DLS is the lower concentration of sample required for proper detection (down to 0.05 g.L⁻¹, 200 times lower), but also the multi-elemental character of the detector, allowing for multiplex assay. The same team also worked on the size determination of a commercially available gadolinium-chelated polysiloxane NPs (AGuIX) [30], using a bimodal fitting in comparison with a constrained regularised linear inversion (CRLI) method. They were able to discriminate two subpopulations with only few nanometers differences, one comprising Gadolinium-containing ultrasmall NPs (5.4 ± 0.2 nm), and a second comprising its fragments (1.8 ± 1.6 nm), which is hardly feasible with DLS.
309 They also worked on the protein corona formation on gold silica NPs (AuSiNPs) [31], 310 arguing that ICP-MS allows to detect metallic content selectively in complex biological 311 media. Studying the adsorption of proteins (bovine serum albumin, BSA and fetuin 312 from fetal calf serum, FCS), the team showed the formation of protein layers on the 313 surface of the AuSiNPs ($r_H = 30.0 \pm 0.7$ nm) with an increase to up to 36.8 $\pm$ 0.7 nm 314 (12.5 $mg_{BSA} \cdot mL^{-1}$) and $35 \pm 1$ nm (15 $mg_{FCS} \cdot mL^{-1}$) respectively. The change in $r_H$ 315 suggests the formation of a protein monolayer in the described conditions. In the case 316 of FCS incubation, higher concentrations were tested (up to 30 $mg_{FCS} \cdot mL^{-1}$), resulting 317 in probable NPs aggregation of protein multilayer. The reversibility of the protein 318 corona formation was also studied, showing that BSA corona stability is influenced by 319 incubation time of AuSiNPs with the protein, with a reversible corona, if incubation time 320 is too short (2 h). For longer incubation time tested in the study (4 h), an irreversible 321 corona was evidenced through TDA measurements, explained since irreversible 322 adsorption of BSA requires structural changes in the adsorbed BSA, occurring for 323 longer incubation time. In the case of fetuin corona, a hard corona was evidenced with 324 the formation of a monolayer upon the surface of the NPs, and that NPs or protein 325 aggregation occurring in concentrated protein-containing media are reversible. 326

327 As mentioned in this paragraph, one of the main applications of TDA in single window 328 mode remains in the size estimation but also in the determination of the colloidal 329 stability, with the advantage of being more precise than DLS. Another advantage is 330 that the use of CE apparatus is generally concomitant to CZE analysis, which allows 331 charge determination through measurement of the electrophoretic mobility [29], while 332 working with low sample volume. Coupling with ICP-MS allows better selectivity 333 towards metals constituting the NPs, which has been presented as an interesting mode 334 for study in complex media. 335

## 2.2. Double window mode

335 To counteract the influence of the injected volume and pressure ramp on the peak 336 variance and elution time, a two-detection window mode (also denoted as dual-point 337 detection configuration, Figure 3B) was first proposed by A.J.S. Chapman and D.M. 338 Goodall [40]. It consists in using two detection points during the same analysis to 339 highlight the influences of the analytical running conditions on the results. J. Chamieh 340 et al. [41,42] demonstrated that for low injection volume ($< 1\%$ capillary volume till the 341 detection window), there is no significant difference between $d_H$ estimated with a
single-window and a double-window approach. Still, the use of the latter is preferable since there is no need for approximation, and it increases the sensitivity of the detector (with higher injected volume for instance), at least for monodisperse NPs. The conception of a double window apparatus is not simple, which is why it is not systematically used for TDA. It requires changes of the regular apparatus, with in-house development adaptation, even if several strategies have been presented in the literature as illustrated in the following section. In the case of double-window mode, the Taylor-Aris equation is modified as follows (equation 5):

$$D = \frac{r_c^2}{24(\sigma_2^2 - \sigma_1^2)}(t_2 - t_1) (5)$$

with $t_1$ and $t_2$ the mean residence time of the solute to reach the first and second window respectively, and $\sigma_1$ and $\sigma_2$ the mean temporal variance of the fitted Gaussian distribution at the first and second window respectively, as illustrated in Figure 3B.

One use of this technique for AuNPs characterisation has been presented by M. Höldrich et al. [28], with analysis of pepsin-coated AuNPs to study the mechanism of formation of protein corona, as potential use for nanobiocatalyst with attachment of enzyme on metallic NPs. In this study, TDA was employed on a Viscosizer device since DLS results were questionable, due to the sensitivity of the method towards bigger NPs, responsible for probable particle distribution distortion. This can lead to undistinguishable $r_H$ between pepsin-coated AuNPs at different pepsin incubation concentration. It has to be noted that TDA conditions were not discussed, especially concerning the boundary conditions in which TDA is applicable ($\tau$ and $Pe$). Results showed that TDA was useful and sufficiently resolved to distinguish uncoated AuNPs (citrate-capped, $r_H = 17.8 \pm 0.2$ nm) and pepsin-coated AuNPs (up to $21.5 \pm 0.3$ nm), in agreement with DLS results and allowing to distinguish $r_H$ for different pepsin concentration level incubating with AuNPs. It is noteworthy that when analysing citrate-capped AuNPs, Taylorgrams with peaks corresponding to a bimodal distribution were evidenced. Deconvolution of the signal were done with two overlapping Gaussian signals, one attributed to free citrate in solution and another to citrate-capped AuNPs. Pepsin grafting occurs through ligand exchange mechanism by monolayer formation. The binding isotherm built from $r_H$ of AuNPs as a function of the logarithm of the pepsin concentration led to an effective dissociation constant of 30.7 µM. Data from TDA measurements were used for estimation of the amount of bound pepsin, in the same
374 order with the one determined by Lowry's assay for an intermediate studied pepsin
375 concentration incubation (1 mg/mL, 89200 against 51745 pepsin/AuNPs respectively).

376 The dual detection window mode has been extensively used by S. Balog et al. [9–11,13,24] for several applications in metallic-based NPs, from AuNPs to SPIONs TDA
377 characterisation with a particular interest in theoretical predictions for a better
378 modelling to estimation of dH, especially concerning polydisperse samples.
379

380 A first study aimed at comparing TDA results for the characterisation of diverse NPs
381 with sizes obtained by DLS and TEM [10]. Citrate-capped AuNPs (dH between 3.3 and
382 67 nm TDA) and SPIONs (dH between 15 and 33 nm TDA) showed higher results than
383 TEM measurements in most cases (except for lower sizes of AuNPs). This is attributed
384 to the fact that TDA is used for dH determination, TEM resulting in a core-dried state
385 which implies a lower value than in aqueous media. Deviation from DLS, however, was
386 attributed to the presence of larger NPs and/or aggregates, as discussed previously in
387 this paper. This tends to be overestimated when using light scattering and justifies the
388 use of TDA instead of DLS. It has to be pointed out that authors highlighted the
389 determination of two different sizes with DLS, one based on cumulant analysis, another
390 on CONTIN analysis [43]. Careful attention was already given to the reader to interpret
391 these characteristics. Another important note is that the lower tested sizes (for AuNPs)
392 gave results with TDA in good agreement with TEM, in contrary to DLS which tends to
393 be problematic when NPs became too small (below 2 nm here). So TDA can be used
394 for small size determination with low sample consumption, oppositely to DLS.

395 A second study [13] aimed at developing a new model describing the polydispersity
396 affecting the Taylorsgram in pulse mode, which results in a non-Gaussian curve since
397 it consists in an overlay of multiple Gaussian curves. By considering the size-
398 dependent optical extinction μ(r, λ) of a particle at the detection wavelength, they
399 introduced the probability density distribution of the particle radius in the Taylorsgram
400 expression, P(r). To illustrate their model, two metallic-based NPs were tested and
401 accurately measured by considering that optical extinction is dominated by absorption.
402 Citrate-capped AuNPs and SPIONs apparent radii were measured at 33.8 nm and 7.6
403 nm experimentally, against 30.4 nm and 7.2 nm predicted with TEM results,
404 respectively (Table 2). This tends to demonstrate the validity of the methodology and
405 the equations used to describe the results. Industrial samples of ZnO, TiO₂ and Cu
406 NPs were also tested according to the same process, resulting in non-Gaussian
407 Taylorsgrams and/or multimodal and asymmetric distribution. Large polydispersities
408 were expected for the samples, which were testified by TEM measurements. But due
409 to low signal-to-noise ratio, TDA cannot provide clear information about population
410 fraction resulting from aggregates and agglomerates. In the case of CuNPs, matrix
411 interferences (molecules or subnanometer particulates) were also responsible for
412 overlapping a narrow peak to the Taylorsgram, but were not further discussed.

413 A third study focused on the possibility to improve the dynamic range of TDA [11];
414 actually, proper particle size determination with TDA is operational with only a valid
415 combination of parameters, comprising capillary inner diameter size, mobilisation
416 pressure, injected volume and so on. This can be particularly tricky when working with
417 weakly absorbing colloids (UV-Visible), but also for routine analysis when comparing
418 different NPs sizes. To counteract this inconvenient and to extend the use of TDA for
419 a wide dynamic range for a given apparatus, P. Lemal et al. considered TDA as a linear
420 time-invariant system [11]. A new equation describing the signal, suited either for mono
421 or moderately polydisperse NPs, has been proposed and used for SPIONs
422 characterisation with sample volume of around 1.8 % of the total capillary volume. TDA
423 results gave better estimation on the basis of the linear time-invariant system with 6.0
424 ± 0.2 nm against 4.8 ± 0.2 nm with regular consideration, for 6.0 ± 0.7 nm determined
425 with TEM. Larger NPs were correctly estimated with both models (9.4 ± 0.7 nm and
426 9.3 ± 0.5 nm for linear time-invariant system and regular consideration respectfully
427 against 9.9 ± 0.6 nm with TEM), highlighting particle size impact on the accuracy of the
428 method. The development of a new equation for the description of the Taylorsgram was
429 needed to extend the dynamic range of size estimation while using a single set of
430 parameters. As written, "for a given range of particle sizes, one may always find a valid
431 combination of operation parameters", which is not always compatible with routine
432 analysis or for the uninitiated. The new equation allows for the determination of uniform
433 and moderately polydisperse particles with arbitrary injection and detection volumes,
434 which is interesting for an increase in sensitivity towards NPs with low absorbance
435 properties.

436 Using similar approach, P. Taladriz-Blanco et al. [9] described, with a mathematical
437 model, the Taylorsgram originating from polymer coated AuNPs and AgNPs, with good
438 accuracy between theoretical approach and experimental results. TDA gave sizes
439 reported in Table 2 (around 20 nm each), but were not discussed since it was not the
440 purpose of the paper. However, higher values were obtained with TDA approach than
441 TEM results, which is in good agreement with previously reported remarks on similar metallic NPs.
442

443 Finally, the team worked on another mathematical model, reliable for highly
444 polydisperse AgNPs suspension [24] and describing the limit of resolution of TDA.
445 Actually, resolution of TDA was explored, i.e. the capacity to resolve the difference
446 between rH and rH ± δr. Temperature, viscosity, capillary radius or peak width were
447 identified as directly related to the “precision” of the measurement. The regular model
448 was used to estimate AgNPs size (around 16 nm), using TEM to validate the high
449 polydispersity of the sample (no data available). The main interest of the study was not
450 only to present the analysis of a degradable metallic NPs as a function of time (with
451 evolution of the rH as a function of time). It was also to use a new mathematical
452 description of the polydispersity, with good agreement with the prediction. However,
453 the low signal-to-noise ratio did not allow for high resolution, owing to differentiate
454 particle size and comparing experimental results and theoretical expectation with good
455 accuracy.

3. Conclusion
---

457 TDA has been proven as a promising characterisation tool for metallic-based NPs size
458 determination, and easily implementable in laboratories since CE apparatus are
459 perfectly equipped for this application. UV detector in single or double window mode
460 have been extensively used for various metallic-based NPs characterisation (Au, Ag,
461 Fe, ...). Other couplings are still being optimised. ICP-MS, for instance, allows for a
462 better specificity toward metals, reducing the interferences due to matrix component,
463 promising for complex media study such as for protein corona formation. CZE coupling
464 is also feasible, and coherent since no significant modification of the CE apparatus is
465 required.

466 TDA allows for the determination of monodisperse NPs population, since signals
467 originated from NPs are directly proportional to the mass of the metal. In the case of
468 polydisperse NPs, theoretical studies are still in progress, with mathematical
469 approximations requiring a sensitive detection method to increase the resolution and
470 thus the exact PDI of a sample. Nonetheless, numerous teams are nowadays using
471 TDA in both pulse and frontal mode as a potential routine tool and are confident enough
472 to conclude about NPs size using only this methodology. Using both single and double
473 window mode (the latter being less accessible since it requires customisable device),
474 metallic-based NPs size has been screened from sub-NPs to larger inorganic objects
475 (rH ranging from 2.3 to 128.2 nm), making it a suitable alternative to DLS
476 measurements.

477 **Conflict of interest**

478 The authors have declared no conflict of interest.

479 **Author contributions**

480 Conceptualisation, J.G., I.C. and A.B.; writing—original draft preparation, J.G.;
481 writing—review and editing, J.G., A.B., F.B., A.P. and I.C. All authors have read and
482 agreed to the published version of the manuscript.

483 **Data Availability Statement**

484 Data sharing not applicable to this article as no datasets were generated or analysed
485 during the current study.

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