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# Cross-frontal mode: an alternative methodology for Taylor Dispersion Analysis of monomodal sample

Jérémie Gouyon<sup>a*</sup>, Ariane Boudier<sup>a,b</sup>, Arnaud Pallotta<sup>a,b</sup>, Célia Boukoufi<sup>a,c</sup>, Igor Clarot<sup>a,b</sup>

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

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

<sup>c</sup> Pharmacy Department, University Hospital, 54511 Vandœuvre-lès-Nancy, France

*Corresponding author: jeremie.gouyon@univ-lorraine.fr (ORCID number: 0000-0002-8834-9009), +33 3 72 74 73 10

## Keywords

Capillary, Hydrodynamic radius, Methodology, Taylor Dispersion Analysis

## Abstract

Taylor dispersion analysis (TDA) is a technique dedicated to the determination of the molecular diffusion coefficient (D) of species, using band broadening of an analyte in a laminar flow. Two modes are commonly used to perform TDA: pulse and frontal modes. In each case, a fitting of the signal is required. We propose here a third mode denoted as cross-frontal mode, combining two crossed sample fronts without modification of a classical CE device for the rapid and accurate determination of D of caffeine, reduced glutathione (GSH), insulin from bovine pancreas, bovine serum albumin (BSA) and citrate-capped gold nanoparticles (AuNP). Theoretical aspects and methodology are described, showing a good correlation between the so-called cross-frontal mode and usual frontal mode. Limitations of the techniques are also assessed, and are similar to regular modes while no fitting is required. This new methodology allows improving the sensitivity toward low concentrated sample compared to pulse mode, and an alternative mathematical treatment compared to regular TDA modes.
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# 1. Introduction

Taylor dispersion analysis (TDA) is known as a characterization technique for size determination in the nm range (0.1 – 100 nm)[1]. It has been used for a wide variety of analytes size, from small organic molecules[2] to biomolecules characterization[3] or inorganic nanoparticle size determination[4]. Using capillary electrophoresis (CE) instrument[5], TDA is easily accessible and complementary to other characterization techniques such as Dynamic Light Scattering (DLS) or Transmission Electronic Microscopy (TEM)[6]. Basically, TDA consists in the measurement of the peak broadening of a solute injected in a laminar Poiseuille's flow for sub-nm size range determination. Peak width is, in this particular case, inversely proportional to the diffusion coefficients of the analyte. Such a laminar flow is feasible with unmodified CE devices[7] due to capillary dimensions, pressure control and UV-Visible detector coupling.

Two modes are nowadays commonly used for Taylor dispersion: pulse and frontal modes[8]. The pulse mode consists in the injection of a finite plug of sample before mobilization through the capillary[9] (typically, the plug is inferior to 1% of the volume of the capillary between the inlet and the detection window). The resulting Taylorgrams being described by a Gaussian curve, the peak width is linked to the analyte size. Nonetheless, the sensitivity of the pulse mode is limited (due to low sample volume), which can be counteracted using different types of detectors[10,11], or using the frontal mode. It consists in the continuous injection of the sample[12], leading to a half-wave signal reached after saturation of the capillary with the sample. Compared to pulse mode, frontal mode allows reaching higher sensitivity since there is no limitation in terms of introduced quantity of analyte. Each mode leads to different Taylorgrams profiles, which are summarized in Figure 1.

Mathematical interpretation has already been described in the literature[12,13] for both modes, and is not the object of this paper. Nonetheless, it has to be mentioned that fitting of the signal (either Gaussian or half-wave) using appropriate software is required for the determination of the temporal variance $\sigma^2$ of the elution profile, directly proportional to the size of the analyte (see following review[14] for more details). Specific software and mathematical treatment are required for this operation, which can be unappealing for the non-initiated.
In this paper, we aimed at presenting a third mode, denoted as “cross-frontal mode”, so named because it combines crossed sample fronts. Mathematical demonstration and applications are described for monomodal sample. A comparison with regular frontal mode is also presented, since both modes are comparable in terms of sensitivity and methodology. The main interest of the proposed methodology relies on the absence of fitting required to interpret the data. Similar methodology was applied using capillary zone electrophoresis[15] and is here adapted to TDA application. This method is promising to simplify the interpretation compared to frontal mode while being as simple to perform, since no device modification is required. Demonstration is applied in this paper for analyte size ranging from 0.2 to 5 nm hydrodynamic radius ($r_H$), from small organic molecule to biomolecules. Feasibility of the cross-frontal methodology is thus expressed for a broad range of analytes.

## 2. Material and methods

### 2.1. Materials and reagents

All reagents and solvents were of analytic grade and were used without further purification. Ultrapure deionized water (> 18.2 MΩ cm) was used for the preparation of all solutions. Sodium phosphate monobasic monohydrate $NaH_2PO_4.H_2O$ (for analysis, 99%) was purchased from Merck (Darmstadt, Germany). Sodium phosphate dibasic dodecahydrate $Na_2HPO_4.12H_2O$ (for analysis, 98%) was purchased from Carlo Erba Reagents (Val de Reuil, France). Sodium citrate dihydrate (≥ 99%), sodium borohydride $NaBH_4$ (ReagentPlus®, 99%), gold(III) chloride trihydrate $HAuCl_4.3H_2O$ (ACS reagent, ≥ 49.0 % Au basis), caffeine (SigmaUltra), L-glutathione reduced (GSH, ≥ 98.0 %), Insulin from bovine pancreas (BioReagent, suitable for cell culture), Bovine serum albumin (BSA, 98 % electrophoresis), hydrochloric acid, HCl, and nitric acid, $HNO_3$, were purchased from Sigma Aldrich (St Louis, MO, USA). Solutions preparation, citrate-capped gold nanoparticles (AuNP) synthesis[16,17] and the related characterizations are detailed in Supplementary Information.

### 2.2. Apparatus

Taylor dispersion analyses were performed using a P/ACE MDQ CE system (Beckman Coulter) equipped with a diode array detector (DAD, wavelength ranging from 200 to 600 nm). Measurements were performed at 25 ± 1 °C using a 40 cm bare fused silica capillary (effective length: 30 cm, Polymicro Tech.) with inner diameter of 100 µm.
## 2.3. Cross-frontal TDA experimental methodology

Depending on the analyte, mobilization pressure P and injection time $t_{inj}$ and duration of the experiment were adjusted but the concept remains similar for each tested condition:

(i) The capillary was flushed with water, 0.1 M NaOH, and water.

(ii) the capillary was filled with the sample. Each sample was prepared in a specific solvent, depending on its properties as mentioned in (iii) below (see Supplementary Information for more details).

(iii) the analyte-free solvent, denoted as “empty plug” (a solution of 1.5 mM sodium citrate for citrate-capped AuNP analysis, 20 mM phosphate buffer for caffeine and BSA analysis, 10 mM HCl for insulin analysis or 1 mM HCl for GSH analysis) was injected at P for a duration $t_{inj}$ (P being equal to 0.4, 0.3 or 0.2 psi and $t_{inj}$ varying from 5 to 100 s, depending on the analyte).

(iv) Starting the UV-Visible acquisition (at 514 nm for AuNP[18], 272 nm for caffeine, 214 nm for GSH and 220 nm for BSA and insulin), the empty plug was pushed hydrodynamically with the sample using the same P.

(v) 1.5 mM sodium citrate, 20 mM phosphate buffer, 1 mM HCl or 10 mM HCl (depending on the analyte) were pushed using the same P after measurement of the negative peak.

Figure 2 shows a schematic view of a cross-frontal Taylorgram obtained using the experimental methodology, indicating how the mean solute concentration across the cross-section of the capillary $\bar{C}$ and the concentration of the front $\bar{C}_0$ were measured.

Each measurement was repeated 3 times for each described condition. Experimental conditions are summarized in Supplementary Information section (Table S1). It is to be noted that higher P values were not tested since Taylor's conditions would not be fulfilled (as mentioned in Experimental results section).

# 3. Results and discussion

## 3.1. Theoretical aspects of cross-frontal TDA mode

Considering fundamental aspects of Taylor dispersion, it has been demonstrated[19,20] that the concentration profile (Taylorgram) using frontal mode for monomodal sample is given by eq. (1):

$$ \frac{\bar{c}}{\bar{c}_0} = \frac{1}{2} \pm \frac{1}{2} \operatorname{erf} \left[ \frac{t-t_R}{\sigma\sqrt{2}} \right] (1) $$
where t is time, t_R is the mean residence time, σ is the square root of the temporal variance of the elution profile and erf is the error function. This equation is adequate for symmetric profile.

The new proposed methodology consists in filling the capillary with the sample, followed by injection of variable volumes (for different t_inj) of an “empty plug”. Using same velocity (related to P), the empty plug will then be pushed by re-introducing the sample, resulting in the following theoretical Taylorgram which is the sum of two front Taylorgrams i and j (Figure 3.B):

Considering the continuous laminar flow (Poiseuille's flow) occurring during the experiment and for a precise value of t_inj, both sample profiles (shown by the * on the Figure 3.A) can be described independently using eq. (1). The profile i being centred on t + t_inj/2 and j being centred on t - t_inj/2, eqs. (2) and (3) give mathematical description of profiles i and j.

$$ \frac{\bar{c}_i}{\bar{c}_0} = \frac{1}{2} + \frac{1}{2} \operatorname{erf} \left[ \frac{t - \left( t_R + \frac{t_{inj}}{2} \right)}{\sigma \sqrt{2}} \right] \quad (2) $$

$$ \frac{\bar{c}_j}{\bar{c}_0} = \frac{1}{2} - \frac{1}{2} \operatorname{erf} \left[ \frac{t - \left( t_R - \frac{t_{inj}}{2} \right)}{\sigma \sqrt{2}} \right] \quad (3) $$

The Taylorgram is equal to the sum of both eqs. (2) and (3). At t = t_R, i.e. the intersection of both Taylorgrams i and j (but also the minimum of the peak), the equation can be simplified and give eq. (4):

$$ \frac{\bar{c}}{\bar{c}_0} = 1 - \operatorname{erf} \left[ \frac{t_{inj}}{\sigma 2 \sqrt{2}} \right] \quad (4) $$

It is possible to express the previous equation as follows, giving eq. (5):

$$ \operatorname{erf}^{-1} \left[ 1 - \frac{\bar{c}}{\bar{c}_0} \right] = \frac{t_{inj}}{\sigma 2 \sqrt{2}} \quad (5) $$

For different t_inj, the previous expression will fit a linear curve with a slope equal to 1/(σ2√2). The proposed methodology remains in the possibility to inject variable volumes of empty plug (using variable t_inj) and to plot eq. (5) in order to calculate σ using slope of the linear curve. Using UV-visible detector, the absorbance is directly proportional to the concentration. It can be approximated that C̄ is correlated to absorbance value at t_R and C̄_0 is correlated to absorbance value of the plateau. An illustration of this hypothesis is presented in Figure 3.C.

Fulfilling Taylor's conditions[12], the diffusion coefficient D of the analyte can be determined using eq. (6):
$$D = \frac{r_C^2 t_R}{24 \sigma^2} (6)$$

with r<sub>C</sub>, capillary radius. If no particle interactions occur and analytes are considered as spheres, D can be linked to r<sub>H</sub> using Stokes-Einstein equation (eq. (7)):

$$D = \frac{k_B T}{6 \pi \eta r_H} (7)$$

with k<sub>B</sub> the Boltzmann constant (m<sup>2</sup>.kg.s<sup>-2</sup>.K<sup>-1</sup>), T the temperature (K) and η the viscosity of the medium (Pa.s).

## 3.2. Experimental results

Using cross-frontal TDA methodology for caffeine (194.2 g/mol), GSH (307.3 g/mol, tripeptide), insulin (~ 5 kDa), BSA (~ 66 kDa) and AuNP (inorganic NP) size determination, D and r<sub>H</sub> were obtained and results are reported in Table 1. Peclet (Pe) and dimensionless residence time τ are also indicated for the cross-frontal mode, to insure Taylor's conditions were satisfied[12] (Pe > 69 and τ > 1.4, see Supplementary Information). These conditions are dependent on P and capillary dimensions. Every tested values of P meet the criteria for applicability of TDA modes, and are not supposed to influence size determination.

As summarized, caffeine r<sub>H</sub> was measured using cross-frontal mode and values are comparable with reports from the literature for the lowest tested pressures (0.32 and 0.35 nm for 0.2 and 0.3 psi respectively against 0.32 to 0.49 nm[21–23]). Figure S6 shows the plots for caffeine analysis. These results tend to show the potential of cross-frontal TDA mode for small organic molecules size determination. Concerning the lower value of r<sub>H</sub> obtained at 0.4 psi (0.25 nm), the deviation can be attributed to the low injection time applied, which cannot be fully approximated and undergoes misinterpretation of the final result. It is therefore important to only use the cross-frontal mode with higher t<sub>inj</sub> only (> 10 s), since ramp time to reach the pressure becomes negligible. Results were also calculated using frontal mode, with values close to the cross-frontal mode and the one reported in the literature (0.32 nm for frontal mode and cross-frontal mode at 0.2 psi).

For GSH, values obtained with the cross-frontal mode (0.43, 0.39 and 0.40 nm for 0.2, 0.3 and 0.4 psi) are comparable with reports from the literature (0.36 nm[24]) and with the frontal mode measurement (0.42 nm). Figure S7 shows the plots for GSH analysis. These results tend to show the potential of cross-frontal TDA mode for organic molecules size determination. Moreover, it is the first time TDA was used to measure GSH hydrodynamic radius.
For insulin, values from cross-frontal mode (1.21, 1.16 and 1.09 nm for 0.2, 0.3 and 0.4 psi respectively) are comparable with reports from the literature (1.05 nm[25]) and with the frontal mode measurement (1.04 nm). Figure 4 shows the plots for insulin analysis. These results support the potential of cross-frontal TDA mode for small protein size determination.

Concerning BSA, results on rH are comprised between 5.00 and 3.99 nm (see Figure S8 for cross-frontal Taylorgrams). These values are coherent with previously reported works (see Table 1). In this case, working at P = 0.2 psi, the injection time increased up to 100 s, and gave a deviation from the linear fitting model (visible in Figure S8.B).

Since viscosity of the sample is different than the ‘empty plug’ (due to high concentration of BSA), we suppose that a local change in media properties occurs and leads to such a deviation. Another explanation is based upon adsorption of the protein which will be explained later (see Limits and perspectives). With no consideration for this t<sub>inj</sub>, r<sub>H</sub> becomes equal to 4.13 nm, closer to results obtained using other pressure levels. Still, variations between r<sub>H</sub> from different P are to be mentioned.

For citrate-capped AuNP, results for each tested P are closed (Figure S9) and coherent with DLS measurement: a slightly inferior value is reported for TDA (2.82 nm at 0.3 psi using cross-frontal mode against 3.32 nm using DLS). As already stated in the literature for this type of NP (citrate-capped AuNP), DLS measurements are sensitive to potential aggregates, thus leading systematically to a superior value than with TDA[4] since DLS is an intensity-based method, while TDA is a mass-based method.

Comparing cross-frontal mode with frontal mode, r<sub>H</sub> estimated via TDA also gave similar results (2.74 nm using frontal mode). No significant differences comfort the use of cross-frontal mode as an alternative of frontal mode for inorganic NPs size determination. Moreover, pulse analysis of citrate-capped AuNP is hardly feasible in our case since the sample is not concentrated (< µM). It leads to low signal barely exploitable since injected volume is limited in pulse mode. It is even more convenient to use cross-frontal mode since there is a gain in terms of sensitivity, which makes it a potential alternative for both pulse and frontal modes.

Since results between cross-frontal mode, frontal mode and reported values of the literature are coherent, one can assume that the proposed methodology is adequate.

One advantage of such a methodology relies in the easy data handling, since no fitting is required. Moreover, regular TDA conditions (such as values of P, t<sub>inj</sub>, ...) can be kept,
while non-initiated can start using similar conditions for other molecule/nanoparticle size determination.

## 3.3. Limits and perspectives

Few aspects need to be discussed concerning limitations of the cross-frontal mode, inherent to TDA. It is known that, depending on the operating conditions, measurable size range is variable. In the case of cross-frontal mode, several precisions are required.

For instance, during the development of the methodology, very high injection volume (related to $t_{inj}$) led to low $\bar{C}/\bar{C}_0$ ratio. The function $\text{erf}^{-1}$ being strongly impacted for ratio values close to 0, strong variations of the ratio occur and deeply impact the final results, with high standard deviation. In order to avoid such an effect, $1-(\bar{C}/\bar{C}_0)$ should not exceed 0.96 (with $\text{erf}^{-1}(0.96) = 1.45$) since variation of 0.01 on $1-(\bar{C}/\bar{C}_0)$ leads to a change of more than 0.08 on $\text{erf}^{-1}(1-(\bar{C}/\bar{C}_0)) (> 5 \%$ variability). In other words, preliminary analysis using variable $t_{inj}$ should be performed to determine the maximum $t_{inj}$ to be used for TDA before reaching the specified value. As for frontal or pulse modes, optimization of TDA conditions is required before use.

Moreover, low $t_{inj}$ should be avoided to decrease the probable variation inherent to the CE device (due to ramp time to reach the pressure for instance), as illustrated for caffeine analysis at 0.4 psi. In our case, 10 s minimum was the best option. This value is dependent on the capacity of the apparatus to deliver a stable pressure profile during the injection.

Another aspect which needs to be considered is the sample adsorption. In the case of BSA, no results were indicated for frontal mode (Table 1) since the Taylorgrams were asymmetric and no suitable fittings were obtained for determination of $\sigma$. It suggests that interactions between the protein and the inner surface of the capillary may occur, which is detrimental for TDA application[5], no matter the mode used. Even if results for BSA size determination using cross-frontal mode seem to agree with the literature[26], no conclusion can be adopted since Taylor's conditions are not entirely fulfilled. This adsorption can also explain the variations observed using the cross-frontal mode for BSA characterization. Use of coating (dynamic or covalent/semi-covalent) is suggested, but can be detrimental for detection of the analyte and was not explored here for this reason.
As a prospect, simplification of the interpretation and similarities between cross-frontal and frontal mode results suggest the possibility to consider the cross-frontal mode as an alternative if following conditions are respected:

(i) Usual TDA conditions (on $Pe$ and $\tau$) are respected.

(ii) No adsorption of the sample occurs on the inner capillary wall.

(iii) $t_{inj}$ is sufficiently high to avoid strong variation due to the CE device and misinterpretation (> 10 s). If volume is too important, a decrease in $P$ should be considered.

(iv) $t_{inj}$ is not too high to insure $1-(\bar{C}/\bar{C}_0) < 0.96$.

(v) Sample is monomodal / monodisperse for the equation to be applicable.

Still, cross-frontal mode is more sensitive than pulse mode since the limitation in terms of injected volume of sample is not to be considered (< 1 %). It may allow to measure less concentrated samples or analytes with limited absorption properties, as illustrated with citrate-capped AuNP. Moreover, cross-frontal mode requires no fitting of the signal, in contrary to frontal mode which facilitates data manipulation, even if similar sensitivity is expected. No particular software is necessary to estimate $\sigma$ due to a simplified data treatment, which is more appealing for non-initiated and can makes TDA more accessible.

## 4. Conclusions

To conclude, cross-frontal mode TDA was successfully applied for $r_H$ determination of caffeine (0.32 nm), GSH (0.43 nm), insulin (1.21 nm) and citrate-capped AuNP (2.93 nm), with similar results compared to frontal mode TDA and literature. As well as for the pulse and frontal modes, cross-frontal mode methodology is still limited by several requirements. For instance, adsorption of analytes is incompatible with TDA measurements. Still, this new methodology requires no fitting of the Taylorgram, leading to an interesting simplification of the signal treatment. Moreover, the applicability of the method seems feasible for a wide range of analytes ($r_H$ from 0.3 to 5 nm), as stated in the paper with small organic molecules, biomolecules and inorganic NP measurements. Larger particles and polydisperse / multi-modal samples should also be tested for validation of the methodology.

## Credit authorship contribution statement

Jeremie Gouyon: Conceptualization, Investigation, Methodology, Writing – original draft & editing. Ariane Boudier: Writing - review & editing. Arnaud Pallotta: Writing -
286 review & editing. Celia Boukoufi: Investigation, Writing - review & editing. Igor Clarot:
287 Writing - review & editing.

288 **Declaration of Competing Interest**

289 The authors declare that they have no known competing financial
290 interests or personal relationships that could have appeared to influence
291 the work reported in this paper.

292 **Acknowledgements**

293 This research did not receive any specific grant from funding agencies in the public,
294 commercial, or not-for-profit sectors.

295 **Appendix A. Supplementary material**

296 Supplementary information to this article can be found online.

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<figure><img src="image_1.png" /><figcaption>Figure 1 - Taylor dispersion analysis representation of (A) pulse and (B) frontal mode. [Reprinted, with permission, with modifications from ref [14]. Copyright © 2022 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim]</figcaption></figure>

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<figure><img src="image_2.png" /><figcaption>Figure 2. Schematic view of a cross-frontal Taylorgram. The coloured part indicates the time at which the sample is hydrodynamically pushed, before continuous injection of the solvent in order to decrease the signal to obtain $\bar{c}_0$ and $\bar{c}$.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 3. (A) Schematic capillary profile of the proposed cross-frontal TDA mode and (B) example of resulting Taylorgram centred on tR. Sample (blue) contains the analyte to be measured, and empty plug (clear) is composed of solvent free of analyte. (C) Simulated cross-frontal Taylorgrams using variable tInj for tR = 5 min and σ = 0.3 min and (D) eq. (5) plot for determination of σ, used to calculate diffusion coefficient D and rH.</figcaption></figure>
A

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

B

<figure><img src="image_5.png" /><figcaption>Figure 4. (A) Typical cross-frontal Taylorgrams obtained for insulin using different P and t<sub>inj</sub> and (B) resulting plot for determination of σ (n=3 for each level)</figcaption></figure>

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368

369
<table><caption>Table 1 – Results obtained using cross-frontal mode TDA and comparison with frontal mode TDA and literature data. Values were statistically evaluated by the Student's Test (p < 0.05).</caption><thead><tr><th rowspan="2">Sample</th><th rowspan="2">P (psi)</th><th colspan="3">Cross-frontal mode</th><th rowspan="2">Frontal*</th><th rowspan="2">r<sub>H</sub> th.<br>Ref.</th></tr><tr><th>0.2</th><th>0.3</th><th>0.4</th></tr></thead><tbody><tr><td rowspan="3">Caffeine</td><td>σ (min)</td><td>0.125 ± 0.013</td><td>0.106 ± 0.021</td><td>0.077 ± 0.013</td><td>-</td><td rowspan="3">0.32<sup>a</sup> - 0.491 ±<br>0.008<sup>a</sup><br>[21-23]</td></tr><tr><td>D (10<sup>-10</sup> m<sup>2</sup>.s<sup>-1</sup>)</td><td>6.26 ± 0.66</td><td>5.65 ± 1.11</td><td>8.10 ± 1.38</td><td></td></tr><tr><td>r<sub>H</sub> (nm)</td><td>0.32 ± 0.03</td><td>0.35 ± 0.07</td><td>0.25 ± 0.04</td><td>0.32 ± 0.00</td></tr><tr><td></td><td>τ / Pe</td><td>85.1 / 94</td><td>49.9 / 160</td><td>54.9 / 146</td><td>-</td><td></td></tr><tr><td rowspan="3">GSH</td><td>σ (min)</td><td>0.150 ± 0.005</td><td>0.113 ± 0.007</td><td>0.099 ± 0.004</td><td>-</td><td rowspan="3">0.36<sup>b</sup> - 0.81 ±<br>0.05<sup>c</sup><br>[24,27]</td></tr><tr><td>D (10<sup>-10</sup> m<sup>2</sup>.s<sup>-1</sup>)</td><td>4.64 ± 0.17</td><td>5.16 ± 0.32</td><td>4.98 ± 0.18</td><td></td></tr><tr><td>r<sub>H</sub> (nm)</td><td>0.43 ± 0.02</td><td>0.39 ± 0.02</td><td>0.40 ± 0.01</td><td>0.42 ± 0.00</td></tr><tr><td></td><td>τ / Pe</td><td>66.6 / 120</td><td>47.3 / 169</td><td>33.6 / 238</td><td>-</td><td></td></tr><tr><td rowspan="3">Insulin</td><td>σ (min)</td><td>0.241 ± 0.009</td><td>0.191 ± 0.002</td><td>0.161 ± 0.004</td><td>-</td><td rowspan="3">1.05 ± 0.1<sup>d</sup> - 1.7<sup>a</sup><br>[25,28]</td></tr><tr><td>D (10<sup>-11</sup> m<sup>2</sup>.s<sup>-1</sup>)</td><td>1.66 ± 0.06</td><td>1.72 ± 0.02</td><td>1.84 ± 0.04</td><td></td></tr><tr><td>r<sub>H</sub> (nm)</td><td>1.21 ± 0.04</td><td>1.16 ± 0.01</td><td>1.09 ± 0.03</td><td>1.04 ± 0.01</td></tr><tr><td></td><td>τ / Pe</td><td>22.0 / 364</td><td>15.0 / 535</td><td>12.2 / 654</td><td>-</td><td></td></tr><tr><td rowspan="3">BSA</td><td>σ (min)</td><td>0.494 ± 0.032</td><td>0.367 ± 0.017</td><td>0.307 ± 0.016</td><td>-</td><td rowspan="3">3.3<sup>d</sup> - 4.84 ± 0.03<sup>a</sup><br>[21,26,29]</td></tr><tr><td>D (10<sup>-11</sup> m<sup>2</sup>.s<sup>-1</sup>)</td><td>4.01 ± 0.26</td><td>4.51 ± 0.21</td><td>5.02 ± 0.27</td><td></td></tr><tr><td>r<sub>H</sub> (nm)</td><td>5.00 ± 0.32</td><td>4.44 ± 0.21</td><td>3.99 ± 0.21</td><td>**</td></tr><tr><td></td><td>τ / Pe</td><td>5.4 / 1477</td><td>3.8 / 2114</td><td>3.3 / 2432</td><td>-</td><td></td></tr><tr><td rowspan="3">AuNP</td><td>σ (min)</td><td>0.385 ± 0.021</td><td>0.298 ± 0.010</td><td>0.266 ± 0.017</td><td>-</td><td rowspan="3">3.32<sup>d</sup> (2.9 ± 0.5)<br>[16]</td></tr><tr><td>D (10<sup>-11</sup> m<sup>2</sup>.s<sup>-1</sup>)</td><td>6.84 ± 0.31</td><td>7.10 ± 0.24</td><td>6.54 ± 0.42</td><td></td></tr><tr><td>r<sub>H</sub> (nm)</td><td>2.93 ± 0.14</td><td>2.82 ± 0.10</td><td>3.06 ± 0.20</td><td>2.74 ± 0.10</td></tr><tr><td></td><td>τ / Pe</td><td>9.6 / 833</td><td>6.2 / 1292</td><td>4.2 / 1914</td><td>-</td><td></td></tr></tbody></table>

*Results for frontal analysis were obtained by fitting the experimental curve with eq. (1), see Supplementary Information for more details[30]

**Results for frontal analysis of BSA are not provided since no suitable fitting of the experimental curve were possible, due to asymmetric signals

a Result obtained by TDA / b Result obtained by polarography / c Result obtained by chronoamperometry / d

Result obtained by DLS