Cationic Polyelectrolyte Adsorption onto Anionic Nanoparticles Analyzed with Frequency‐Domain Scanning Fluorescence Correlation Spectroscopy

Abstract

An improved small‐circle scanning fluorescence correlation spectroscopy (sFCS) technique is introduced by combining acousto‐optical laser scanning with fitting the autocorrelation function in the frequency domain. The technique is validated using both simulation and experimental data on various fluorescent nanoparticles, including polystyrene beads, CdSe/CdS quantum dots, and lipid nanoparticles. Then, the sFCS method is used to investigate the adsorption of in‐house synthesized poly(2‐guanidinoethyl methacrylate) (PGUMA) polymers on polystyrene beads as a model system for polymer‐coated particles in biomedical and gene delivery applications. Using the particle diffusion and illumination beam waist values obtained from our sFCS analysis, regions of polymer concentrations are identified where polymer‐particle complexes remain stable. An increase in hydrodynamic size is also observed with the molecular mass of the adsorbed polymer. Beyond quantifying polymer‐particle stability and hydrodynamic size, the sFCS technique offers the advantage of not requiring a time‐consuming calibration step for the measurement volume, unlike standard FCS.

A small‐circle scanning fluorescence correlation spectroscopy (sFCS) technique is presented using acousto‐optical laser scanning and frequency‐domain fitting. Validated with simulations and experiments, sFCS investigates cationic polyelectrolyte adsorption onto anionic nanoparticles, revealing stable complexes and size increases with polymer mass. Unlike standard FCS, sFCS eliminates calibration steps, offering a streamlined approach for analyzing polymer‐coated particles in biomedical applications.

1. Introduction

Fluorescence correlation spectroscopy (FCS) has been around for over half a century and is nowadays routinely used in a diverse range of investigations, from photo‐physics to conformational dynamics and from diffusion measurements of fluorophores in bulk media to multi‐parameter measurements of fluorescent proteins in living cells.^{[ 1 ]} The concept of FCS was introduced by Magde, Elson, and Webb in 1972^{[ 2 ]} as a means to measure the reaction kinetics of the reversible binding between ethidium bromide and DNA. The measurement was performed on a traditional fluorescence microscope with a large beam waist and low collection efficiency which made the experiment challenging. However, the advent and widespread use of innovations in optical microscopy, such as confocal and two‐photon microscopes, significantly boosted the signal‐to‐noise ratio. These advancements opened up a wide range of new applications in single‐molecule fields, leading to numerous adaptations of the methodology. Some examples of these adaptations include: dual‐color fluorescence cross‐correlation spectroscopy (FCCS),^{[ 3 ]} Förster resonance energy transfer (FRET‐FCS),^{[ 4 ]} stimulated emission depletion (STED‐FCS),^{[ 5} , ^{6 ]} total internal reflection (TIR‐FCS),^{[ 7 ]} single plane illumination (SPIM‐FCS),^{[ 8 ]} label‐free Inverse (iFCS),^{[ 9 ]} label‐free ultraviolet (UV‐FCS),^{[ 10 ]} photon counting histogram (PCH),^{[ 11 ]} computational methods^{[ 12} , ^{13 ]} and scanning FCS (sFCS).^{[ 14} , ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ , ^{19 ]}

In its most basic form, the working principle of FCS is based on the analysis of a fluctuating intensity signal acquired by exciting a small volume in a sample containing diffusing fluorophores. As fluorophores diffuse through the excited volume, they emit fluorescent photons resulting in the fluctuations of the intensity signal. Consequently, the amplitude and duration of these fluctuations depend on the number of excited fluorophores and the average time for fluorophores to move through the measurement volume (which is a combination of illumination and detection volume^{[ 20} , ^{21 ]}). By knowing the shape of the measurement volume, the average amplitude and time of fluctuations can be converted into the concentration and diffusion of the fluorophore.

However, determining the dimensions of the aforementioned measurement volume can be challenging due to the dependence of the illumination and detection volume shape on multiple factors, making direct volume measurements difficult.^{[ 22 ]} Therefore a calibration measurement is often performed using a known diffusion standard dye or particle which is then used to establish the measurement volume. Particularly in biological media such as the surface of a membrane, the interior of a living cell, or when the microscope is part of a multi‐purpose optical setup, this calibration procedure can be undesirable or even impossible under the same circumstances.^{[ 23 ]}

Therefore, to circumvent the volume calibration procedure, methods have been developed that incorporate spatial information into the acquired fluorescence fluctuation signal. One way of adding spatial information is through using multiple measurement volumes like in dual‐focus FCS^{[ 24 ]} or by using a detector with an inherent distance calibration (pixel pitch of a camera system) like multi‐scale FCS^{[ 25 ]} and imaging FCS.^{[ 26 ]} Another way to avoid volume calibration is by moving or scanning the illumination volume along a calibrated path (offline calibration of a beam deflector). Methods such as: large circle‐,^{[ 17 ]} line‐,^{[ 18} , ^{27 ]} raster scan‐,^{[ 19 ]} and small circle‐^{[ 14} , ¹⁵ , ¹⁶ , ^{23 ]} FCS scan the illumination beam along a distinct path, each having their advantages and disadvantages.^{[ 1 ]}

For bulk medium diffusion measurements, small‐circle sFCS is particularly advantageous due to its time‐invariant autocorrelation function similar to that of traditional static volume FCS. This allows small‐circle sFCS to be analyzed using a standard autocorrelation model, as demonstrated by Berland et al. ^{[ 14 ]} Later, Skinner et al. ^{[ 15 ]} implemented a position sensitivity to not only measure the diffusion but also the drift of a particle, while the group of Schwille proved that the spatial information could be used to fit the measurement or confocal volume parameters in bulk^{[ 16 ]} and on a lipid membrane.^{[ 23 ]} Despite its robustness, small circle sFCS has been utilized in only a small amount of works such as the previously mentioned studies and most recently as a beam waist calibration procedure for a different non‐scanning FCS methodology.^{[ 28} , ^{29 ]} However, its ability to simultaneously measure particle diffusion, particle concentration, and the measurement volume, makes it particularly useful to enhance the understanding of a wide range of applications and particle systems.

An example of a particle system that has gained a lot of interest is the study of charged polymer (or polyelectrolyte) adsorption onto oppositely charged particles. Since polymer‐coated particles can be tailored toward their application by finding the most suitable polymer, they are increasingly being investigated for applications such as gene delivery, drug delivery, imaging, and biosensing.^{[ 30} , ³¹ , ³² , ^{33 ]} Consequently, the adsorption process and resulting polymer‐particle complex must be well understood and characterized.

In the past two decades several methods have been proposed to analyze this problem such as dilute multi‐angle time‐resolved DLS,^{[ 34} , ^{35 ]} single particle electrophoresis (SPE),^{[ 36} , ^{37 ]} particle tracking and bulk electrophoresis,^{[ 38} , ^{39 ]} small‐angle neutron scattering (SANS),^{[ 40 ]} and many more.^{[ 41 ]} However, often one or multiple measurement parameters for these methodologies are far from the potential use case. Indeed, DLS suffers from particle stability issues requiring low sample concentrations, SANS and EM are destructive while SPE requires a low ionic strength environment. FCS on the other hand has been proven to work in these environments, as demonstrated in adsorption studies performed on protein‐particle complexes using traditional static FCS.^{[ 42} , ⁴³ , ^{44 ]}

In this work, we present an improved approach to small‐circle sFCS (from now on referred to as sFCS) by utilizing an acousto‐optical laser scanning method for increased flexibility on the hardware side, and by fitting the autocorrelation function in the frequency domain for improved accuracy on the analysis side. We demonstrate, with a theoretical model and simulations, that the variation of the obtained diffusion coefficient and beam waist can be reduced using frequency domain fitting. Additionally, this new approach is evaluated using experimental data of fluorescent polystyrene (PS) particles, CdSe/CdS quantum dots (QDs) with a thick shell structure^{[ 45 ]} and DiO‐labeled lipid nanoparticles^{[ 46 ]} (LNPs). The measured particle sizes are in the range of 15–220 nm, representing the size range most optimal for drug delivery^{[ 32 ]} and transfection^{[ 47 ]} applications. Finally, we apply the optimized sFCS technique to assess the adsorption of the in‐house synthesized polymer poly(2‐guanidinoethyl methacrylate) (PGUMA) with low (18.7 kg/mol) and high (40.2 kg/mol) molar mass to 28 and 51 nm PS beads, as a model system for polymer‐coated particles destined for cellular uptake. Analysis of the sFCS data is used to determine the polymer‐particle complex stability at different polymer concentrations and to measure the increase in hydrodynamic radius for polymer‐particle complexes in the stable high‐polymer‐concentration regime.

2. Theory

Just like regular FCS, scanning FCS also obtains information on the suspension under test from the variation in the fluorescence intensity signal denoted as F(t). This fluorescence signal F(t) is created by illuminating a dilute particle‐ or molecule suspension with a tightly focused laser beam, which is scanned along a circular path with radius R and frequency f. As fluorescent particles or molecules diffuse in and out of the dynamic illumination field they emit fluorescent photons that are detected by a sensitive photodetector. The average time a particle remains in the observation volume and thus emits light is also called the residence time τ _D . This characteristic residence time τ _D depends on both the average diffusion coefficient D and the shape of the static confocal volume which is defined by the radial‐ and axial beam waist denoted w ₀ and z ₀. Additionally, by scanning the illumination, the emitted light will consist of short bursts with a time scale much smaller than τ _D and a temporal shape dependent on the confocal volume, average diffusion, scanning radius, and scanning frequency (assuming a point approximation for the fluorescent particles). Consequently, by analyzing the autocorrelation G _S (τ) of the fluorescence fluctuation signal δF(t) a meaningful analysis of the average diffusion and the confocal volume can be obtained at the same time. However, contrary to the traditional static FCS measurements, this means that sFCS requires no additional beam calibration except for a one‐time calibration of scanning radius R.

Figure  1 comprehensively illustrates how the autocorrelation model for a sFCS measurement is obtained. First of all, the location of all particles is described by a so‐called local concentration function $C\left(\vec{r},t\right)$. This function is zero everywhere except for the specific instances of position $\vec{r}$ and time t where a particle is present. Therefore the local concentration function $C\left(\vec{r},t\right)$ must satisfy the diffusion equation and fulfill the stationary‐ and ergodic properties^{[ 20} , ^{48 ]} shown below.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{\partial C\left(\vec{r},t\right)}{\partial t}& =D{\nabla }^{2}C\left(\vec{r},t\right)\hfill \\ \hfill {⟨C\left(\overrightarrow{r_0},t\right)⟩}_t& ={⟨C\left(\vec{r},t_0\right)⟩}_{\vec{r}}\to \text{ergodicity}\hfill \\ & ={⟨C\left(\vec{r},0\right)⟩}_{\vec{r}}\to \text{stationary}\hfill \end{array}\end{array}$$ (1)

Figure 1.

The working principle of sFCS. a) Illustration of a sampled particle propagation through the illumination area. b) Illustration of the confocal volume. c) Detected fluorescent signal of the particle propagation in a). d) Resulting autocorrelation of the fluorescence fluctuation with the dashed line representing the autocorrelation function of a static confocal volume and dashed‐dotted line representing the cross‐correlation of two confocal volumes separated by a distance 2R.

Furthermore, the combined illumination‐ and detection path is approximated by a time dependent confocal volume function $P\left(\vec{r},t\right)$ which can be described as a Gaussian function in each direction (also called the quasi‐cylindrical profile^{[ 20 ]}) as follows:

$$\begin{array}{c}P\left(\vec{r},t\right)=P_0{e}^{-\frac{2\left(x^2(t)+y^2(t)\right)}{w_0^2}}{e}^{-\frac{2z^2}{z_0^2}}\\ x(t)=R\cos \left(\omega t\right),y(t)=R\cos \left(\omega t\right)\end{array}$$ (2)

with P ₀ the peak intensity, w ₀ the radial beam waist (which will be called the beam waist from now on), R the scanning radius, ω = 2πf the angular scanning frequency and z ₀ the axial beam waist. Though more accurate models of the confocal volume have been investigated by Berland et al. ^{[ 20 ]} and Rigler et al. ^{[ 21 ]} by taking into account both the full point spread function of the detection optics and replacing the illumination by a Gaussian–Lorentzian intensity function, the approximation described in equation (2) remains relevant. This is particularly true when the aperture of the back focal plane is not severely under or overfilled and the confocal pinhole is larger than the beam waist^{[ 49 ]} as is the case for our experiments.

Now, by combining the concentration function $C\left(\vec{r},t\right)$ and confocal volume function $P\left(\vec{r},t\right)$ through multiplication and spatial integration, the fluorescence signal F(t) can be obtained. Subsequently, subtracting the fluorescence signal F(t) by its expected value 〈F(t)〉 the fluorescence fluctuation signal δF(t) can be found which finally leads to the calculation of the autocorrelation function G _s (τ). In the case of a circular scanning confocal volume defined by equations (2) and the local concentration function $C\left(\vec{r},t\right)$ defined by Equation (1), Berland et al. ^{[ 14 ]} first showed that the autocorrelation function G _s (τ) can be split in a multiplication of two time‐independent functions (meaning no change with t). These two functions are the static/no‐scanning FCS autocorrelation function denoted G(τ) and the so‐called scan envelope function S(τ):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G_s(\tau )& =\stackrel{G\left(\tau \right)}{\overbrace{\frac{1}{2\sqrt{2}N}\frac{1}{\left(1+\frac{\tau }{{\tau }_D}\right)\sqrt{1+\frac{\tau }{a^2{\tau }_D}}}}}\underset{S\left(\tau \right)}{\underbrace{\exp \left(-\frac{4{\rho }^2{\sin }^2\left(\frac{\omega \tau }{2}\right)}{1+\frac{\tau }{{\tau }_D}}\right)}}\hfill \end{array}\end{array}$$ (3)

with ${\tau }_D=w_0^2/4D$ the aforementioned residence time, N as a scaled average amount of particles in the confocal volume, a as the ratio between the radial‐ and axial beam waist and ρ the ratio between the scanning radius R and the radial beam waist w ₀. As illustrated in Figure 1d), the autocorrelation function G _s (τ) is bounded by the static autocorrelation G(τ) at the top (for τ = n/f where n is a natural number) and the cross‐correlation of a dual‐focus FCS at opposite sides of the scanned circle G _x (τ) at the bottom (for τ = (1/2 + n)/f where n is a natural number). This shows a clear addition of spatial information on the autocorrelation as compared to the traditional FCS methodology.

Moreover, in earlier work by Skinner et al. ^{[ 15 ]} the autocorrelation function G _s (τ) was further expanded with a drift component which combined with a position‐sensitive scanning (synchronized photon acquisition and beam scanning) allowed for the extraction of a lateral drift velocity vector. However, the work by Petrásêk and Schwille^{[ 16 ]} on the other hand, first showed that the spatial information contained within the autocorrelation G _s (τ) could be used to extract both the sample diffusion D and confocal volume parameters w ₀ and a if the scanning radius R is known.

Indeed, as shown in Figure  2a both scanning‐ and static autocorrelation models are illustrated for several particle hydrodynamic radii r _h (top) and beam waist sizes w ₀ (bottom). As can be observed from the static autocorrelation models (dashed lines) both changes in r _h and w ₀ shift the residence time τ _D while keeping the overall shape of the autocorrelation curve similar in a qualitative way. Contrary to this conservation of the curve shape, the scanning autocorrelation models (full lines in Figure 2a) show distinct behavior for both parameter sweeps. By increasing the hydrodynamic radius of the diffusing species r _h , the amplitude of the local maxima increases for higher τ values while the minima mostly stay the same except for a shift of the cross‐correlation bump (not clearly visible for the given values). On the other hand if the beam waist w ₀ increases, both the local maxima and minima shift to higher amplitudes. This distinct behavior is also evaluated quantitatively by assessing the square difference for a range of r _h (or diffusion coefficients D) and w ₀ values in a so‐called χ²‐map^{[ 23 ]} (see Section S1, Supporting Information) as shown in Figure 2c. For a diffusion coefficient D ≈ 20 µ m² s⁻¹ (r _h = 12.5 nm) and a beam waist w ₀ = 400 nm, the square difference maps are plotted for the static‐ and scanning FCS autocorrelation models (utilizing a multiple‐tau sampling scheme^{[ 16} , ^{50 ]}) in the top and middle subfigures of Figure 2c) respectively. For the static autocorrelation G(τ), a correlation between D and w ₀ is clearly visible, which verifies the qualitative assessment previously stated. This strong correlation between the two parameters also illustrates the need for calibration before fitting. For the scanning autocorrelation model G _s (τ), the correlation between D and w ₀ is much less prominent and a clear global minimum is present in the true values. This global minimum illustrates the possibility of fitting both D and w ₀ simultaneously without calibration.

Figure 2.

Comparison of the time‐ and frequency domain autocorrelation models. a) Model autocorrelation curves for non‐scanning (dashed) and scanning FCS measurements (full) for multiple r _h (top) and w ₀ (bottom). The other parameters used for these curves are f = 2 kHz, R = 600 nm and a = 5. b) Similar plot as a), but for frequency domain. c) Normalized χ²‐maps with set values of: f = 2 kHz, r = 12.5 nm, w ₀ = 400 nm, R = 600 nm and a = 5.

However, by evaluating the scanning autocorrelation models from Figure 2a in the frequency domain (amplitude of the Fast Fourier Transformed (FFT) autocorrelation models without multiple‐tau sampling), the curves illustrated in Figure 2b are obtained. The amplitude of the Fourier transformed autocorrelation denoted as G _s (f) shows an overall decreasing function with peaks at the scanning frequency harmonics, preceded by local minima. Similar to the time domain evaluated autocorrelation models, different behavior occurs for changes in the particle hydrodynamic radius r _h as compared to the beam waist w ₀. By increasing the particle hydrodynamic radius the peaks (local maxima) and valleys (local minima) of G _s (f) will respectively undergo an increase and a decrease while enlarging the beam waist will increase the low frequency harmonics, decrease the high frequency harmonics, and shift the valleys to higher frequencies. This stronger distinction in behavior between r _h and w ₀ is also visible in the χ²‐map on the bottom of Figure 2c. The χ²‐map is calculated from the square difference of the natural logarithm of G _s (f) to scale the peak and valleys into the same order of magnitude as both contain essential information. Similar to G _s (τ), a global minimum is present in the set values. Though, in comparison to the χ²‐map of the time domain evaluated model, the correlation between D and w ₀ is reduced even further which should lead to an improved fitting in the frequency domain.

Note, in an earlier statement it was mentioned that a, the ratio between the radial‐ and axial beam waist, could also be extracted from the scanning autocorrelation model, though a is not included in the parameter sweep of Figure 2. This is because for values of a larger than 5, its importance diminishes.^{[ 51 ]} In fact, for a ≈ 10 a 2D approximation of the confocal volume (cylindrical volume) is almost an exact model^{[ 51 ]} (see Section S1, Supporting Information). Therefore, especially in the case of an oil‐immersion objective (which is used in our experiments) where the confocal volume drastically increases in the axial direction due to spherical abberations^{[ 22} , ⁴⁹ , ⁵¹ , ⁵² , ⁵³ , ^{54 ]} the fitting of a is omitted and set to a high enough value to reduce its importance. In our case a is fixed to a value of 5 for all experiments and simulations, which is an underestimation and therefore a non‐optimal value. However, given the use of an oil‐immersion objective, the aberrations will far outweigh the contribution of a to potential discrepancies between the data and the proposed model.

2.1. Simulation

To verify the improved fitting precision of the autocorrelation function in the frequency domain G _s (f) over its time domain counterpart G _s (τ), a simulation is implemented. Instead of simulating the fluorescence signal in a probabilistic way or adding noise to the autocorrelation models,^{[ 16 ]}, here the motion of individual particles is simulated, similar to the work of Dix et al. and Wohland et al. ^{[ 55} , ^{56 ]} The simulation algorithm of a single ‘experiment’ consists of the following four steps.

In the first step, a so‐called generation/simulation region is defined in which the simulation will take place. Along with this initialization of this region, the starting position of all particles is determined using a uniform probability distribution, which is bounded by the size of said generation region. In all simulations, the generation region is a cube with an edge size d _s which is at least one order of magnitude larger than R (the scanning radius). This allows for particles to fully diffuse in and out of the scanning region, similar to a real experiment.

In the second step, once the starting locations are determined for all particles, the individual Brownian trajectories are generated using a Gaussian process with a zero mean and a standard deviation $\sqrt{2D\mathrm{\Delta }t}$, with D the diffusion coefficient and Δt simulation time step. This means we consider an ensemble of particles all having the same size. To make sure the particle concentration is uniform over the entire simulated experiment, a periodic boundary condition is defined. If a particle diffuses out of the simulation region it will automatically appear on the opposite face of the cube to continue its trajectory. Additionally, the concentration c of the particle suspension using this method can easily be calculated as $c=\frac{C}{d_s^3}$ with C the amount of particles inside the simulation region.

In the third step, a time‐dependent confocal volume function is defined according to Equation (2). For every particle, a random brightness I (Gaussian probability density function) is assigned and fixed over its entire Brownian trajectory. In this work, we do not take into account the photon statistics using Poissonian emission probabilities, since we are only interested in bright particles. With the particle brightness assigned, the particle trajectories and confocal volume function are combined to extract the photon count signal assigned to each particle. The summation of all photon count signals then results in the final fluorescence signal.

Lastly, in the fourth step, the analysis of the simulated photon count is executed in the same way as for real experimental data. This data analysis is described in the materials and methods section.

Figure  3 illustrates the fitting results of 25 simulations over a set of hydrodynamic radius r _h and beam waist w ₀ combinations each consisting of 250 “experiments” with parameters similar to the real experiments performed in the results section. The simulation parameters are as follows: a simulation time step of 5 µ s, duration of 10 s, scanning radius R of 400 nm, particle brightness I with an average value of 30 counts per sample period and standard deviation of 5 counts per sample period, a Poissonian background count with average value of 4 counts per bin, scanning frequency f of 2 kHz and the number of particles per simulation volume C of 100 (average particles inside the confocal volume is ≈0.1 for a 300 nm beam waist and a = 5).

Figure 3.

Comparison of the time‐ and frequency domain fitting for simulated data. a) Scatter plot of fitting results in both frequency domain (dark) and time domain (light) for 25 simulations with different hydrodynamic radius r _h and beam waist w ₀ combinations (GT). All point clouds consist of N points each representing a single “experiment.” b) Fitted standard deviation of r _h as a function of the average of r _h for all point clouds in a). c) Fitted standard deviation of w ₀ as a function of the average of w ₀ for all point clouds in (a).

In Figure 3a, each data point in the scatter plot represents the fitting of a single simulated experiment (i.e., 10 s of data). The ground truth is highlighted by the crosses while the time‐ and frequency domain fitted results are illustrated by the light green‐ and dark purple color dots respectively. Overall the frequency domain fitted results show a greater precision which is clearly visible from the cumulative histograms in Figure 3a. However, for some combinations of the r _h and w ₀ the frequency domain fitted results show a bias for both parameters. Furthermore, by analyzing the individual distributions by means of Bayesian inference with a bivariate Gaussian as a prior, the results are obtained for the average and standard deviations of r _h and w ₀ as shown in Figure 3b,c. Indeed, the standard deviation of both the hydrodynamic radii and beam waist are in most cases smaller and grow with the average value which is to be expected. The increased bias of the frequency domain fitted results is also clearly visible, especially for the largest particles. Additionally, as was mentioned earlier, the χ²‐maps also showed a stronger correlation between D (or r _h ) and w ₀ for the time domain fitting. Though not represented here, the average Pearson correlation coefficient of all 25 simulations was approximately 0.35 for the time domain fitting and 0.22 for the frequency domain. On the whole, the difference in the fitting results of the simulated data shows an incremental advantage in favor of the frequency domain fitting especially for the measurement of larger particles and beam waists.

3. Results and Discussion

3.1. Analysis and Fitting of the Fluorescence Signal

Unlike a mirror‐based scanning microscope, which is most common for sFCS,^{[ 16} , ¹⁸ , ²³ , ²⁷ , ^{57 ]} our optical setup contains two orthogonal acousto‐optical deflectors (AOD) to deflect the laser beam before entering the microscope objective. Since the AODs deflect the laser beam through the interaction of electromagnetic‐ and acoustic waves within a piezoelectric crystal, they do not suffer from resonant behavior as is the case for an inertial deflector. The scanning characteristics of AODs are therefore independent of the scanning frequency and radius. In other words, this means that the power of the illumination beam going to the microscope objective is independent of the scanning parameters (see Section S2, Supporting Information). Therefore, in our approach, changing between different scanning parameters only requires a single scanning radius‐to‐AOD voltage calibration curve (see Section S5, Supporting Information) and not a full radius‐and‐frequency‐to‐modulation voltage sweep as is the case for an inertial scanning system.^{[ 16 ]}This flexibility allows for fine adjustment of the scanning parameters based on the sample brightness and diffusion during the experiment. Indeed, as a faster scanning rate results in a shorter illumination time of the particles within the confocal volume, a quick and easy adjustment of the scanning frequency can become necessary during the measurement of a particle suspension with unknown brightness. Additionally, this adaptability allows for fine adjustment of the scanning parameters to their optimal values as described in the work of Petrášek and Schwille^{[ 16 ]} since the optimal scanning radius is dependent on the scanning frequency.

As can be seen in Figure  4a, the resulting photon count traces (for a duration of 5 s) are shown for 51 nm PS particles under different scanning frequencies with fixed scanning radius, while Figure 4d shows 28 nm PS particles under different scanning radii with fixed scanning frequency. Figure 4b,e zoom in on these respective photon count traces, illustrating that both the circle scanning frequency and radius have a strong influence on the pulse shape of the photon count signal, and illustrating the additional information added to the fluorescence signal by scanning the confocal volume as compared to the traditional static approach (not shown here). Since the individual traces, with different scanning parameters were performed on the same respective particle suspensions, their autocorrelations can be fitted together to a single diffusion D and beam waist w ₀.

Figure 4.

Measured fluorescence signals and frequency domain model verification. a) Fluorescence signals (5 s) of 51 nm PS beads for different scanning frequencies. b) 5 ms snippets of fluorescence signal in a). c) Simultaneously fitted autocorrelations of a) in the frequency domain. d) Fluorescence signals (5 s) of 28 nm PS beads for different scanning radii. e) 5 ms snippets of fluorescence signal in d). f) Simultaneously fitted autocorrelations of d) in the frequency domain.

Fitting multiple autocorrelations at once is obtained by concatenating the individual Fourier transforms over a frequency range (for example between 0 and 10 times the scanning frequency) and fitting this with a similar concatenation of the Fourier transformed model (see Section S6, Supporting Information). In Figure 4c,f the following values are respectively obtained for the diffusion coefficient D and beam waist w ₀ through simultaneous fitting: D = 8.4 µ m² s⁻¹ and w ₀ = 350 nm for 51 nm PS, and D = 18.1 µ m² s⁻¹ and w ₀ = 350 nm for 28 nm PS. These values are close to the expected values of the diffusion and beam waist (see following section) and the results show an accurate fit to the Fourier‐transformed model for different scanning parameters. The same photon count traces were also fitted individually in both the frequency and time domain yielding similar values (see Section S6, Supporting Information).

3.2. Diffusion Coefficient Measurement

To compare the autocorrelation fitting in the time‐ and frequency domain, we analyzed the diffusion coefficient and beam waist for a set of PS nanoparticles and three functional particles, namely DiO labeled lipid nanoparticles (LNPs) carrying unlabeled fLuc‐mRNA^{[ 46 ]} and two different sizes of CdSe/CdS QDs with a thick shell structure.^{[ 45 ]} To perform this set of measurements, two different lasers were used. A 450 nm blue laser was used for the excitation of the LNPs, QDs, and PS particles (106 nm and 220 nm diameter), while a green laser of 532 nm was used for the excitation of a different set of PS particles (28 nm, 51 nm, and 190 nm diameter).

These sFCS measurements are illustrated in Figure  5 , with Figure 5a–c representing results fitted in the frequency domain and Figure 5d–f representing fitting results in the time domain. Every point in the scatter plots represents approximately 30 s of individually fitted photon count data (i.e. no simultaneous fitting of multiple scanning parameters), following the data processing and fitting procedure as described in the materials and methods. Note that each point cloud contains measurements at different scanning frequencies (1, 2, and 5 kHz) and scanning radii (210–780 nm) with each parameter combination having the same prevalence. For both fitting in the time and frequency domain, all diffusion coefficient distributions associated with different particles are easily distinguishable and the fitted beam waist gives similar values, from the smaller QDs to the bigger 220 nm PS particles. Qualitatively the frequency domain fitting produces a narrower range of beam waist w ₀ values compared to the time domain fitting. The distributions of the fitted diffusion coefficients D are mostly narrower with frequency domain fitting, but with some exceptions.

Figure 5.

Comparison of time‐ and frequency domain fitting on measured data. a–c) Scatter plot of fitting results in the frequency domain for respectively 532 nm illuminated particles, 450 nm illuminated particles, and multiple ethylene glycol/water mixtures. d–f) Same as (a–c) with fitting in the time domain.

In Table  1 , a summary is presented of the data in Figure 5a,b,d,e along with the expected values of the diffusion coefficients for the different particles (i.e., manufacturer specifications (M), dynamic light scattering (DLS), and electron microscopy (EM)). Across almost all particle samples the expected diffusion coefficient D _E and the fitted diffusion coefficient in the time and frequency domain, respectively denoted as D _T and D _F , lay well within their standard deviation. The LNPs on the contrary show a stronger deviation from their expected values. The LNPs were characterized using DLS, which has a strong bias toward the strongest scattering particles, i.e., the largest particles in the suspension, causing an underestimation of the expected diffusion coefficient, which can explain the higher diffusion coefficient for both D _T and D _F . For the QDs on the other hand, the expected diffusion coefficient D _E is somewhat smaller than the nominal size (as indicated by the sample name). However, using an automated size calculation on a TEM image of a limited sample and taking into account the stabilizing surfactant coating (see Section S7, Supporting Information) the expected diffusion coefficient in Table 1 was obtained for both QD samples. Moreover, the Auger recombination rate and exciton blinking are strongly suppressed in the QDs used in this work due to their large shell structure. Therefore these QDs show little to no blinking as reported by Christodoulou et al. ^{[ 45 ]}

Table 1.

Summary of measurements values in Figure 5a,b,d,e. The sample names describe the particle type (PS: Polystyrene, QD: Quantum Dot, LNP: Lipid Nanoparticle), if applicable the nominal diameter, and the excitation color (G: 532 nm and B: 450 nm). The expected diffusion coefficients D _E are marked with the characterization method (M: Manufacturer specs, EM: Electron microscopy (see Section S7, Supporting Information), DLS: Dynamic Light Scattering). D _F and w _0F represent the fitting results in frequency domain and D _T and w _0T the time domain. The error margin ±σ is the standard deviation. The concentration of all samples are given in the Supporting Information (see Section S11).

Sample	D E [µ m2 s−1]	D F [µ m2 s−1]	D T [µ m2 s−1]	w 0F [nm]	w 0T [nm]
PS 28 nm (G)	16.2 (M)	18.4 ± 4.7	17.6 ± 3.1	340 ± 30	340 ± 30
PS 51 nm (G)	8.9 (M)	9.7 ± 2.2	11.1 ± 2.8	350 ± 30	370 ± 50
PS 190 nm (G)	2.4 (M)	2.6 ± 0.5	3.3 ± 0.7	350 ± 40	370 ± 70
QD 15 nm (B)	13.1 (EM)	12.8 ± 1.9	13.1 ± 2.4	360 ± 50	420 ± 40
QD 25 nm (B)	9.7 (EM)	10.3 ± 1.4	9.5 ± 2.0	350 ± 50	380 ± 70
PS 106 nm (B)	4.3 (M)	4.5 ± 0.9	4.5 ± 0.6	360 ± 20	360 ± 20
LNP (B)	3.3 (DLS)	4.5 ± 0.9	5.4 ± 0.9	350 ± 30	350 ± 50
PS 220 nm (B)	2.1 (M)	2.4 ± 0.7	2.4 ± 0.6	360 ± 20	350 ± 30

As expected from the simulation results (see Figures 2 and 3), the obtained beam waist values indeed benefit from using frequency domain fitting for all particles. All measurements combined give a beam waist in the frequency domain of w _0F = 350 ± 30 nm for the 532 nm laser and w _0F = 350 ± 40 nm for the 450 nm laser, while the time domain fitting gives respectively w _0T = 360 ± 50 nm and w _0T = 370 ± 50 nm. Compared to the values extracted from the glass/water interface images provided in the Supporting Information (See Section S5), these fitted beam waists are on average smaller, but still within the error margins. For most samples the standard deviation of the diffusion coefficient when fitting in the frequency domain is also lower than in the time domain, which is in line with expectations from the simulation results (see Figures 2 and 3). However, the effect is quite small, and for 3 out of 8 samples the standard deviation actually increased. Theses deviation from the simulated expectations could be due to several reasons which can influence the fitting stability. First and foremost the simulation does not take into account the physical dimensions of the particles being simulated (point approximation) which can cause a change of the observed residence time.^{[ 58 ]} Second, a non‐ideal confocal volume as a result of spherical abberations due to the use of an oil immersion objective^{[ 51 ]} and third the inherent filtering of the autocorrelation functions using a the multiple‐tau sampling scheme by reducing the weight of longer term correlations and noise.

To assess the change in diffusion due to increasing viscosity η without altering the particles under test, 51 nm PS particles were suspended in three different ethylene glycol (EG)/water mixtures and measured. The measurements in Figure 5c,f are expressed as a relative diffusion coefficient $\epsilon (v)=\frac{D(0)}{D(v)}$, with D(0) being the diffusion coefficient in pure water and D(v) the diffusion coefficient for each respective EG‐weight ratio v (EG/(EG+water)). These measurements are compared to a relative viscosity model developed by Bohne et al. ^{[ 59 ]} For both the time domain and frequency domain fitting, the average diffusion coefficient decreases with increasing EG‐weight ratios, as expected. Although the average relative diffusion coefficient ϵ(v) at the highest EG‐weight ratio of v = 0.816 (equivalent to a volume ratio of 0.800) deviates from the relative viscosity model, the absolute diffusion coefficients for both the time and frequency domain fits fall within each other's standard deviations. Additionally, the fitted beam waist w ₀ for the combined measurements was w ₀ = 350 ± 30 nm for frequency domain fitting and w ₀ = 360 ± 50 nm for time domain fitting. These averages align well with the values in Table 1, indicating that changes in the suspension medium refractive index (from 1.33 for water and 1.43 for pure EG^{[ 60 ]}) does not cause an observable beam waist variation in our case (measurements between 7.50 µ m to 12.50 µ m removed from the coverslip using an immersion oil with a refractive index of approximately 1.52 and an under filled back focal aperture^{[ 51} , ^{53 ]}). Changes in the confocal volume elongations (axial beam waist) are however difficult to observe as its influence is rather limited as mentioned earlier.

3.3. Stability of Cationic Polymers/Anionic Particle Complexes

We now apply the sFCS technique to investigate the adsorption of charged polymers onto oppositely charged particles. More specifically, we examine the adsorption of the cationic polymer PGUMA onto anionic PS particles, serving as a model system for polymer‐coated particles destined for biomedical applications.^{[ 30 ]} Similar to previous works on protein adsorption using static FCS,^{[ 42} , ^{44 ]} the stability and hydrodynamic radius of the PGUMA/PS complexes is investigated. However, since sFCS is used for this investigation, both the fitted (in the frequency domain) beam waist and particle diffusion can be used to extract information on the PGUMA/PS complex behaviour.

As a starting point, it is necessary to estimate the required PGUMA concentration to achieve complete surface coverage of the PS particles (the 51 nm PS particles) from a charge perspective. This implies that at minimum a charge balance must be established in the PGUMA/PS suspension for the PS particles to be considered fully coated. Therefore the charge density of both the PS surface and the PGUMA chain needs to be measured. Assuming that the PS particles are perfectly spherical and hard, the average amount of charges on their surface can be estimated by measuring both the electrophoretic mobility µ and diffusion coefficient D. Using the diffusion coefficient D = 9.7 ± 2.2 µ m² s⁻¹ from Table 1 for the 51 nm PS particles and their electrophoretic mobility µ = −4.3 ± 0.6 × 10⁻⁸ m² V⁻¹s⁻¹ (which was measured using a Zetasizer (see Section S8, Supporting Information)), the amount of surface charges per PS particle q ⁻ is estimated to be 114 ± 40 e⁻ (with e⁻ =−1.602 × 10⁻¹⁹ C), which equals a surface charge density of s = −2.4 ± 0.9 mC m⁻² (see Section S8, Supporting Information). Even though the charge per particle based on the bulk electrophoretic mobility is only a rough estimate, both arguments can be made to consider it an over‐^{[ 61} , ^{62 ]} and underestimate.^{[ 63 ]} Nonetheless, it provides an order of magnitude which serves as a good starting point.

As previously mentioned, to saturate the PS particles, the negative charge concentration Q ⁻ of the PS particles needs to be countered by the positive charge concentration Q ⁺ of the polyelectrolyte, requiring at least a charge balance $\overline{Q}=\frac{Q^{+}}{Q^{-}}=1$. From the measurements in the previous subsection, an appropriate dilution of the 51 nm stock solution (2.5% weight/volume) for sFCS was found to be × 10⁻⁴, resulting in a particle concentration of approximately c = 4.0 × 10¹⁰ particles mL⁻¹ or 65 pm, which combined with the above estimates of surface charges per particle q ⁻ leads to a negative charge concentration of Q ⁻ = q ⁻ · c = 7.4 ± 3.0 nm. Similar to previous work by Van Daele et al. and Amer Cid et al.,^{[ 36} , ^{37 ]} the medium in which the PS particles and PGUMA are suspended has a pH value close to 7 which means that the PGUMA molecules can be approximated as fully protonated and therefore have an estimated mass per charge of M ≈ 205 g mol⁻¹ for high guanylation degrees^{[ 37 ]} (confirmed by ¹H NMR see Section S9, Supporting Information). Hence, to find the mass of PGUMA m _PGUMA that needs to be added to the PS particle dilution for a desired charge ratio $\overline{Q}$, one simply needs to multiply the mass per charge M with the desired charge ratio $\overline{Q}$ and the negative charge concentration Q ⁻ or $m_{PGUMA}=M·\overline{Q}·Q^{-}$. This means that for 1 mL of appropriately diluted 51 nm PS particles (approximately m _PS = 2.5 µ g) a charge balance is reached by adding approximately m _PGUMA = 1.5 ng of PGUMA.

In Figure  6a,b, the fitted diffusion coefficients D and beam waists w ₀ are shown for 12 different PGUMA concentrations (or charge balances), consisting of 37 photon count signals per data point with a duration of 60 s or more per photon count signal. The PGUMA concentration ranges from zero (bare PS particles) to very high concentrations of 100 µ g mL⁻¹. From the smallest to the largest PGUMA content a decrease in the average diffusion coefficient is observed. A drop of D is expected but only occurs after a PGUMA concentration of at least 30 ng mL⁻¹ ($\overline{Q}\approx 20$) has been reached. Along with this decrease in the average diffusion coefficient, an increase of the standard deviation by a factor of two is observed, indicating a change in the fitting stability, which could indicate an increase in the sample polydispersity and potentially aggregation. This reduced fitting precision is also noticeable in Figure 6b) when looking at the fitted beam waist. Indeed, while no change in the average value of the beam waist is expected for a higher polydispersity (i.e., without aggregation), an increase is observed along with an increased standard deviation for the same intermediate PGUMA concentrations (between 40 and 100 ng mL⁻¹). Therefore, this indicates that the reduced fitting precision for the intermediate PGUMA concentration is most likely due to an increasing formation of aggregates. In such cases FCS measurements tend to deviate from the analytical model and become difficult to reproduce, resulting in a bad fit.^{[ 42} , ^{64 ]}

Figure 6.

PGUMA/PS stability analysis using photon counting and sFCS. a,b) Fitted diffusion coefficient and beam waist as a function of PGUMA concentration. c) Physical interpretation of aggregation in low charge ratio and stabilization in high charge ratio. d,e) Fluorescence signals of respectively low and high charge ratios with their respective photon count histograms to the right. These fluorescence signals are related to the respectively labeled points in subfigures a) and b). PGUMA molecules used in this Figure have a number average molecular mass M _n of 31.7 kg mol⁻¹ and polydispersity of 2.5 (based on the size exclusion chromatography (SEC) of PAEMA).

As illustrated in Figure 6c, when the PGUMA and PS particle solutions are combined, strong adsorption takes place due to their opposite charges. Hence, if the amount of negative and positive charges are in balance, the PS/PGUMA complex becomes on average neutrally charged.^{[ 36 ]} Since the stability of these colloidal PS particles stems from their strong electrostatic repulsion, their reduced overall charge will cause stronger affinity to form aggregates with one another,^{[ 65} , ⁶⁶ , ⁶⁷ , ^{68 ]} especially so when heterogeneous attraction and bridging occurs between the particles.^{[ 69} , ⁷⁰ , ^{71 ]} However, if the PGUMA concentration is high enough, combining it with the PS particles results in a stable suspension. In this case, the anionic particles adsorb enough cationic polymers to not only neutralize the PS particle surface charge but also make them positively charged. In turn, this makes the PS/PGUMA complexes again more stable through electrostatic repulsion and steric hindrance, a phenomenon well‐known for the adsorption of polyelectrolyte (charged polymer) on oppositely charged colloids.^{[ 36} , ⁴² , ⁶⁵ , ⁶⁶ , ⁶⁷ , ⁶⁸ , ⁶⁹ , ⁷⁰ , ⁷² , ^{73 ]} This is confirmed by comparing the measurements with the highest PGUMA‐ and no PGUMA content. Namely, a similar beam waist and standard deviations are observed for both w ₀ and D while the average diffusion coefficient dropped from D = 10.5 ± 1.6 µ m² s⁻¹ to D = 7.2 ± 0.9 µ m² s⁻¹ indicating a stable PS/PGUMA complexation.

Moreover, in Figure 6d,e this process is further confirmed by analyzing the photon count signals themselves. As can be seen for the photon count signal and the photon count histogram next to it in Figure 6d, strong bursts of photons are detected which saturate the single photon counting module (SPCM) (saturation is reached between 60 and 70 counts per bin), while the bare PS is situated well below this saturation count. Throughout all the measurements with intermediate PGUMA concentration (measurements with larger standard deviation), similar bursts can be found. This increased brightness along with a longer average residence time implies that indeed the unstable fitting is caused by an increased presence of aggregates. In Figure 6e, photon count signals are shown for high PGUMA concentrations of 100 µ g mL⁻¹. In this case the photon count histograms of the bare PS‐ and PS/PGUMA particles are very similar, which indicates no significantly brighter particles entering the confocal volume, reflecting the higher precision of the fitting outcomes as compared to an unstable suspension.

For the 51 nm PS particles aggregation started at approximately $\overline{Q}=20$ and stabilization was reached at much higher ratios (the exact ratio was not determined). Note, however, that this charge balance is based on the bulk electrophoretic mobility of the PS particles and can therefore deviate significantly. Moreover, in earlier work reported by Amer Cid et al., PGUMA of similar molar masses were adsorbed on much larger PS particles with a diameter of 1 µ m, which only reached charge neutralization (directly measured using single particle electrophoresis^{[ 36 ]}) at an equivalently calculated charge balance of $\overline{Q}>20000$. This indicates that both size and concentration of PS play an important role in the required charge balance for stable complex formation. Additionally, the ionic strength, polymer type, and even microchannel materials can play a big role^{[ 36} , ⁴² , ⁶⁵ , ⁶⁶ , ⁶⁷ , ⁶⁸ , ⁶⁹ , ⁷⁰ , ⁷² , ^{73 ]} in the complex formation through electrostatic adsorption.

3.4. Size of Cationic Polymers/Anionic Particle Complexes

With the PS/PGUMA complexes stabilized at high PGUMA concentration, an estimate of their hydrodynamic radius when fully saturated can be made using sFCS. The performed experiments are presented in Figure  7a. PS particles with diameters of 51 and 28 nm are mixed (and incubated for at least 15 min) with a high concentration PGUMA (100 µ g mL⁻¹) of molar masses M _n = 18.7 kg mol⁻¹ (LMM) and M _n = 40.2 kg mol⁻¹ (HMM) (more information see Table in Section S9, Supporting Information). For each data set, 44 photon count signals of approximately 90 s are recorded across four different microfluidic devices (11 photon count signals per microfluidic device) and individually fitted to obtain the data represented in Figure 7b,c. The hydrodynamic radii are calculated from the fitted diffusion coefficients using the Stokes–Einstein equation with a viscosity of η = 0.954 mPas for water at 22 °C.^{[ 74 ]}

Figure 7.

Analysis of hydrodynamic radius increase (bare PS vs. PGUMA/PS) as a function of PGUMA MM. a) Set of performed PGUMA/NP measurements. b,c) Fitting results for respectively 28 nm PS and 51 nm PS particles. Radius is calculated from the fitted diffusion coefficient d) Averaged fitting results of (b) and (c) with error bars representing the standard deviations.

From the scatter plots in Figure 7b,c it is clear that the fitted beam waist is similar to the ones found in Table 1. For the 28 nm particles an average beam waist of w ₀ = 350 ± 50 nm is fitted and for the 51 nm particles w ₀ = 350 ± 40 nm. As was mentioned in the previous subsection, this indicates that no aggregates were present in the fitted data set. Additionally, an increase of the hydrodynamic radius r _h is observed from the histograms for both the 28 and 51 nm PS/PGUMA complexes when compared to their respective bare PS particles. The average values and standard deviations of these histograms are summarized in Figure 7d along with the average hydrodynamic radius increase Δr _h . For both the 28 and 51 nm particles the hydrodynamic radius increase has a similar absolute value namely and average radius increase of about 3 nm (Assuming higher measurement precision we obtain Δr _h = 2.7 nm for 28 nm PS and Δr _h = 3.3 nm for 51 nm PS) is observed for the adsorption of LMM PGUMA, while an average increase of approximately 7 nm (Δr _h = 6.9 nm for 28 nm PS and Δr _h = 7.8 nm for 51 nm PS) is measured for the adsorption of HMM PGUMA.

This observed scaling of hydrodynamic radius increase with respect to the number averaged molar mass M _n and therefore the polymer's chain length is somewhat surprising when compared to prior research by other groups. Earlier experimental results obtained using a dilute multi‐angle DLS measurement by Hierrezuelo et al. ^{[ 67 ]} and Seyrek et al. ^{[ 35 ]} with similar MM, but different polymers (PSS, anionic) and particles (amidine latex particles 200 nm, cationic), show very little hydrodynamic radius scaling with the polymer chain length for polymer‐particle complexes in solution with low ionic strength (smaller than 10 mm according to Hierrezuelo et al. ^{[ 67 ]}). Additionally, previous simulation studies^{[ 75} , ⁷⁶ , ⁷⁷ , ⁷⁸ , ⁷⁹ , ⁸⁰ , ⁸¹ , ⁸² , ⁸³ , ⁸⁴ , ^{85 ]} of oppositely charged polymer‐particle complexes at similar or smaller polymer chain lengths show that highly charged polymers (as is the case for PGUMA) in low ionic strength media are usually adsorbed in a tightly wrapped fashion to form a thin layer with a thickness of several Kuhn lengths (one or multiple monomers) independent of the total polymer chain length.

However, from the previously mentioned theoretical and simulated studies available in the literature, it is evident that the conformation of adsorbed polymers is influenced by a wide range of factors, including polymer, particle, and solution properties. Some of these factors may explain the unexpectedly large hydrodynamic radius observed. In our case, the PS particle have a lower charge density, approximately 0.02 e⁻ nm⁻² (or 0.25 e⁻ nm⁻¹ when considering a uniformly spaced charge distribution and e⁺ = ‐ e⁻) compared to the PGUMA chains which exhibit a high charge density between 4.0 e⁺ nm⁻¹ (average monomer spacing) and 1.7 e⁺ nm⁻¹ (average protonated guanidine group spacing) assuming 100% guanylation (see Section S9, Supporting Information). In this scenario, the polymers will adsorb onto the surface only at locations where opposite charges are present. In the regions between these adsorption sites, where the surface is neutrally charged, the polymer's conformation will be primarily influenced by the repulsive Coulomb forces between similar charges along the polymer chain. As a result, the polymer will undergo only partial deformation, i.e., stretching or elongating along the surface of the particle,^{[ 76} , ⁸⁰ , ^{82 ]} while also wrapping around the nanoparticle.^{[ 81 ]} This leads to the formation of loops and tails of the polymer on the surface. Additionally, by working at a high polymer concentration (relative to the NPs) and low ionic strength, a dense polymer coverage will be present on the NP surface causing a strong charge inversion. In this regime, the inter‐chain coulomb repulsion can also promote the formation of loops and tails^{[ 79} , ^{82 ]} which do scale with the chain length. However even in these regimes where the formation of loops and tails are promoted, the scaling should be less than linear since a highly charged polymer does not obtain a rod‐like shape in low ionic strength media, but rather a shape between a self‐avoiding chain and rod.^{[ 76} , ^{81 ]} Moreover, from particle tracking experiments performed by Doan and Adachi^{[ 39 ]} on larger polymers and particles (particle size close to 1 µ m), it was shown that combining a high concentration of polymer with oppositely charged NPs, an initial overloading is observed. After the charge inversion has taken place, the adsorbed polymers will adjust their conformation and potentially desorb until an equilibrium is reached leaving behind a thin polymer layer that is densely packed. This could indicate that our measurements were performed before conformational equilibrium was obtained.

Additionally it is worth noting that the measurement spread of these measurements is large as compared to the simulations, but of the same proportion as the measurements in Table 1. In comparison to other FCS modalities^{[ 42} , ⁴⁴ , ^{86 ]} in previous studies on protein adsorption, smaller measurement spread would be expected though these methods average over much larger time intervals, use water immersion objectives with a small confocal volumes, filter between autocorrelations which do not fit well to the analytical models and perform rigorous sample preparations.

The same PS/PGUMA complexes were also measured with a standard DLS measurement performed on a Zetasizer. In Figure  8a,b, the probability density functions (PDF) of the sFCS measurements (histograms in Figure 7b,c) are shown together with the ones obtained through DLS (intensity normalized). As was mentioned earlier, the DLS measurements tend to bias larger particle signals which is also evident from the data on the bare 51 and 28 nm PS particles. The numerical values are summarized in Table  2 . From the data in Table 2, it is clear that the average hydrodynamic increase has a similar trend for both the sFCS and DLS methodology, except for the 51 nm PS/HMM PGUMA complex. In the latter case, some aggregates might be present in the solution causing a bias toward higher values of the DLS measurements. Additionally, different normalizations of the DLS data such as volume and or number normalizations can filter out this bias. However, since our measurements were performed with an excess of PGUMA different normalizations will show a bias toward free PGUMA (see Section S10, Supporting Information).

Figure 8.

Comparison of DLS and sFCS measurement of PGUMA/PS a,b) Probability density comparison of the obtained results (normalized histograms of Figure 7b,c) with the DLS intensity normalized measurements.

Table 2.

Summary of the sFCS and DLS data shown in Figure 8a,b. For the sFCS measurements the average hydrodynamic diameter d _h is represented along side the standard deviation ${\sigma }_{d_h}$. The DLS measurements show the intensity averaged hydrodynamic diameter d _h and the polydispersity index Đ.

	sFCS ($d_h\pm {\sigma }_{d_h}$ [nm])			DLS (d h [nm]; Đ [a.u.])
Sample	Bare	LMM	HMM	Bare	LMM	HMM
PS 28 nm	24 ± 6	30 ± 12	38 ± 10	34; 0.19	37; 0.18	48; 0.34
PS 51 nm	49 ± 14	55 ± 12	65 ± 9	53; 0.07	59; 0.11	106; 0.29

4. Conclusion

In this study, we presented an enhanced scanning fluorescence correlation spectroscopy (sFCS) technique by utilizing acousto‐optical laser scanning and fitting the autocorrelation function in the frequency domain. The introduction of frequency‐domain fitting reduces the variation in beam waist estimation. Although simulations predicted a reduction in the standard deviation of the diffusion coefficient due to frequency‐domain fitting, this effect was not prominent in the experimental results. The robustness of our sFCS method was validated by applying this technique to various particle types, including polystyrene beads, quantum dots, and lipid nanoparticles, without the need for confocal volume calibration. This lack of calibration, combined with the increased flexibility of inertial‐free scanning, offers advantages such as reduced measurement time compared to traditional methods.

Using our sFCS methodology, the adsorption of poly(2‐guanidinoethyl methacrylate) (PGUMA) onto polystyrene (PS) particles was investigated. The results provide valuable insights into the stability of the PGUMA/PS complex under varying PGUMA concentrations, based on both the analysis of fitted parameters and photon statistics. Additionally, the scaling of the hydrodynamic radius as a function of PS particle size and average PGUMA chain size was examined. The concentration of PGUMA required to stabilize the polymer/particle complex was higher than expected, even though aggregation formation began to occur within the same order of magnitude as estimated from a basic charge balance calculation. For the hydrodynamic radius of fully saturated complexes, the increase in hydrodynamic radius was larger than observed in earlier experimental reports, although various experimental parameters can influence polymer adsorption outcomes.

Future work could expand this study of polyelectrolyte adsorption onto oppositely charged particles by exploring a broader range of experimental parameters, such as particle concentration, an extended analysis of the hydrodynamic radius in relation to particle and polymer size ratios, the influence of suspension medium pH on adsorption and hydrodynamic radius, temporal measurements of hydrodynamic radius, particle surface charge effects, and extending the study to polymers with different charge densities. Additionally, the sFCS methodology could be further enhanced by incorporating synchronized scanning and photon binning^{[ 15 ]} to enable drift measurements under the influence of an external electric field, thus adding an electrophoretic dimension to the study of polymer‐particle complexes. And, also across a wide range of fields beyond the application of polymer adsorption the presented sFCS methodology may be used as a versatile tool for characterizing complex particle systems.

5. Experimental Section

Laser‐Scanning Microscopy

The laser‐scanning microscope (see Section S2, Supporting Information) is as described in a previous work.^{[ 87 ]} In short, the optical setup consists of a custom‐built fluorescence microscope using a 100x oil immersion objective (CFI Plan Fluor 100X Oil, Nikon) and a single photon counting module (SPCM) (SPCM‐AQRH‐15, Excelitas Technologies) combined with several optical filters for highly sensitive fluorescence signal detection and background suppression. The laser beam is deflected using two orthogonal acousto‐optic deflectors (AODs) (MT110‐A1.5‐ VIS, AA Optoelectronic) to obtain 2D laser scanning.

Particle Synthesis—Synthesis of Quantum Dots

Materials: Cadmium oxide (CdO) (99.99%, Aldrich), n‐octadecylphosphonic acid (ODPA) (97%, Plasma chem), Tetradecylphosphonic acid (TDPA) (97%, Plasma chem), trioctylphosphine oxide (TOPO) (Merck), trioctylphosphine (TOP) (97%, Strem chemicals), Selenium powder (Se) (99.5%, Aldrich) and solvents: hexane, toluene, methanol (ChemLab Analytical). Preparation of 0.8 m TOP‐Se: prepared inside a glove box under an inert atmosphere by dissolving Se powder (89.5 mg) in TOP (3 mL) at room temperature. Preparation of 0.5 m TOP‐S: prepared under an inert atmosphere by dissolving S (512 mg) in TOP (16 mL) and ODE (16 mL) at 120 °C for 20 min.

Preparation of 0.5 m Cd(ol)₂: CdO (2.568 g, 20.06 mmol), oleic acid (20 mL), and ODE (20 mL) were added to a 50 mL three‐necked flask. The mixture was degassed at 120 °C for 30 min, followed by heating the mixture to 280 °C under an inert atmosphere for CdO to dissolve, and then left for 10–15 minutes at 210 °C to obtain a colorless solution. Then, the reaction mixture was again degassed for 30 min at 120 °C to remove the water droplets.

Synthesis of 2 batches of colloidal CdSe QDs: CdSe core‐only quantum dots were synthesized analogous to the procedure of Carbone et. al.^{[ 88 ]} using an in situ prepared Cadmium precursor and TOP‐Se as the Se source. A 50 mL three‐neck flask was charged with CdO (117 mg, 0.914 mmol), TDPA (84 mg, 0.301 mmol), ODPA (450 mg, 1.345 mmol) for batch 1, or ODPA (485 mg, 1.450 mmol) for batch 2, along with TOPO (6 g, 0.015 mmol) and flushed with nitrogen. The mixture was then heated to 100 °C and stirred until the TOPO dissolved. TOP (3 mL) was added and the mixture was degassed 140 °C for 1 h. After degassing, the reaction mixture was put under an inert atmosphere and heated to 300 °C to completely dissolve CdO, and heated further to 370 °C. 0.8 m TOP‐Se (1 mL) was added and the mixture was allowed to react for 35 s for batch 1, or 26 s for batch 2, before being rapidly cooled to 100 °C using a water bath. Toluene (10 mL) was added; the mixture was cooled further and transferred to a falcon tube. Methanol (20 mL) was added and the mixture was centrifuged for 4 min at 5000 rpm to precipitate the dots. For purification, the dots were dissolved in hexane, precipitated in methanol, and centrifuged at 5000 rpm for 4 min. This was done twice. The final solution was stored in hexane (60 mL), at –4 °C. Growth of CdS shell on top of CdSe core‐only QDs: CdS shelling procedure was performed in accordance with the work of Christodoulou et. al.^{[ 45 ]} To synthesize a final structure of 15 nm core‐shell QDs a 50 mL three neck flask was charged with 8.8 nmol of previously synthesized batch 2. CdSe cores (10 nmol of batch 1 in case of 25 nm QDs), and ODE (10 mL). The solution was degassed for 20 minutes at 100 °C and subsequently heated up to 300 °C (320 °C in case of 25 nm QDs). Meanwhile, in a separate vial, 0.5 m TOP‐S (2 mL), ODE (2 mL), TOP (1 mL), 0.5 m Cd(ol)₂ (2 mL) and oleic acid (1 mL) was mixed. This solution was then injected into the solution comprising CdSe cores in the flask under nitrogen flow at a rate of 2 mL/h (1.4 nmol in case of 25 nm QDs) by means of a syringe pump. Aliquots were then taken at specific time intervals. When the injection was completed, the reaction was left to cool down to room temperature. The solution was transferred to a falcon tube and isopropanol (25 mL) was added in order the precipitate the quantum dots. The dots were then centrifuged for 10 min at 5000 rpm. The supernatant was discarded and the precipitate re‐dispersed in hexane. The precipitation and resuspension were done twice. The final solution was stored in hexane, at –4 °C. For all experiments the QDs were resuspended in dodecane with the addition of polyisobutylene succinic anhydride (PIBSA) (Dovermulse H1000 (Dover ChenicalCorporation)) (0.005 wt%) for increased stability at low concentrations.

Particle Synthesis—Synthesis and Characterization of Cationic Polymers

The polymers were synthesized and consecutively modified using a previously described protocol.^{[ 36} , ³⁷ , ^{89 ]} In short, the monomer 2‐aminoethyl methacrylate (AEMA, ⩾ 95 %, Polysciences) was dissolved at a final concentration of 0.5 m in double distilled water at pH 4 (acidified using HCl (37 %, Chem‐Lab)). Subsequently, ammonium persulfate (⩾ 98 %, Sigma‐Aldrich) was added as the initiator at a monomer‐to‐initiator molar ratio of 5 and 25 to obtain polymers with low and high molar mass, respectively. This solution was then flushed with an inert gas for 30 min and consecutively heated to 70 °C for 24 h to perform the free radical polymerization of AEMA to poly(2‐aminoethyl methacrylate) (PAEMA). The polymer was isolated by dialysis against double distilled water at pH 4 (acidified using HCl) with 3500 g mol⁻¹ cut‐off membranes (SpectraPor) for 24 h, followed by freeze‐drying. To guanylate the primary amines of PAEMA and obtain poly(2‐guanidinoethyl methacrylate) (PGUMA), the polymer was first dissolved in double distilled water and reacted with 1 equivalent of both 1‐H‐pyrazole‐1‐carboxamidine hydrochloride (99 %, Sigma‐Aldrich) and triethylamine (⩾ 99.5 %, Sigma‐Aldrich) for 24 h at room temperature. Again, purification was performed by dialyzing against double distilled water at pH 4 (acidified using HCl) with 3500 g mol⁻¹ cut‐off membranes for 24 h, followed by freeze‐drying. Confirmation of the chemical structure and quantification of the degree of guanylation was obtained by proton nuclear magnetic resonance (¹H NMR) spectroscopy. The polymers were dissolved in deuterium oxide (99.90 % + 0.1 % 3‐(trimethylsilyl)propionic‐2,2,3,3‐d4 acid sodium salt, Eurisotop) at a concentration of around 10 mg mL⁻¹. All NMR spectra were recorded using a Bruker Avance II Ultrashield 400 MHz NMR spectrometer and data analysis was performed using the Mestrenova Software. Size exclusion chromatography (SEC) was used to determine the number‐average and weight‐average molar mass (M _n and M _w , respectively) of PAEMA, as well as its dispersity (Đ). The setup consisted of a Waters 600 controller equipped with an isocratic pump, Waters 610 fluid unit, Rheodyne injection unit with 20 µ L loop, two Shodex OHpak SB‐806 m HQ columns, a Waters 410 differential refractometer and Waters 996 photodiode array detector. The samples were dissolved at a concentration of around 10 mg mL⁻¹ in an aqueous buffer with 3 % m/v acetonitrile (99.5 %, Chem‐Lab) and 4 % m/v sodium dihydrogen phosphate (⩾ 98 %, Chem‐Lab), which was also used as the mobile phase. Mono‐disperse dextran standards were used to obtain a calibration curve. The Empower2 software was used to perform the data analysis. The results of this analysis are provided in the supplementary information (see Section S9, Supporting Information).

Particle Synthesis—Synthesis of Lipid Nanoparticles

Lipid nanoparticles (LNPs) were formulated via microfluidic mixing using the NanoAssemblr Spark™ system (Precision Nanosystems, Vancouver, Canada) at a nitrogen‐to‐phosphate ratio (N/P) of 6. Here, one part ethanol phase (containing lipids) was mixed with two parts aqueous phase (containing mRNA). The ethanol phase comprised the ionizable lipid C12–200, DSPC (1,2‐distearoyl‐sn‐glycero‐3‐phosphocholine), cholesterol and DMG‐PEG2000 (1,2‐ dimyristoyl‐rac‐glycero‐3‐methoxypolyethylene glycol‐2000) (Avanti Polar Lipids) at a molar ratio of 50:10:38.5:1.5, respectively. Additionally, 1% of lipophilic dye DiO (3,3′‐ dioctadecyloxacarbocyanine perchlorate) (Invitrogen) was added to this phase, to visualize the particles during the measurements. The aqueous phase contained FLuc mRNA (TriLink biotechnologies, L7202) dissolved in 50 mm acetate buffer (pH 4) at 0.3125 mg mL⁻¹. The LNPs were stored in Tris buffer (pH 7.4) at 4 °C until use. The LNPs were diluted in Phosphate‐buffered saline (PBS).

sFCS Measurement Procedure

After a suitable concentration of particles was established, a microfluidic device was placed on top of the piezo stage and carefully filled with the particle suspension using a pipette. By temporarily removing the long pass filter in front of the pinhole, the excitation laser was made visible, such that the bottom of the microfluidic channel could be identified from its strong back reflection. With the bottom of the microfluidic channel in focus, the long pass filter was placed back, such that fluorescent particles could be observed using the microscope's CMOS camera. Once all liquid flow had come to a rest (for cell heights smaller than 50 µ m this ccould take up to a minute for our in‐house microfluidic channels), the piezo stage was moved down by about 10 µ m (moving the focal plane inside the microchannel) and the measurement were started.

DLS and ELS Measurement Procedure

Dynamic‐ and electrophoretic light scattering (DLS and ELS) measurements were performed using a Malvern Zetasizer Nano‐ZS device equipped with a He‐Ne laser. DLS measurements were performed on LNPs, PS 28 nm, PS 51 nm, and PS/PGUMA complexes while the ELS measurements were only performed on the 51 and 28 nm PS particles. For most size measurements, samples were measured in disposable cuvettes while the ELS measurements were performed in folded capillary cells. The dilutions of all samples were made within 24 h of performing DLS and ELS measurements and could deviate within an order of magnitude from the dilutions made for sFCS. However, the PGUMA/PS complexes were measured (DLS) at the same concentrations as the performed sFCS experiments. Similar incubation times were maintained and multiple measurements were performed to assure sample stability. The sFCS and DLS measurements of PGUMA/PS complex were performed several weeks apart.

Microfluidic Devices

All experiments were performed in one of two types of microfluidic devices as shown in the supplementary information (see Section S4, Supporting Information). The first microfluidic device type is fabricated in‐house using a similar method as described in a previous work.^{[ 87 ]} The second is a commercial pre‐coated microfluidic device (coated #1.5 µ‐Slide VI 0.4, Ibidi) for particles that are prone to adsorption to the microfluidic channel walls, such as polymer‐coated PS particles. Bare PS particles were measured in both in‐house and commercial microfluidic channels to assess the potential influence of the sample background.

Statistical Analysis

All measurement data is captured by the FPGA, and through the use of FIFO buffers it is written to the host PC in a Technical Data Management Streaming (TDMS) file. The photon count signals are captured with a 5 µ s binning window for all experiments. Additionally, to reduce the effects of background photons on the Fourier transform, a threshold photon count is defined. Based on the photon count histogram (see Section S3, Supporting Information), this threshold value was determined for each experiment as multiple particles were used with widely‐ranging brightness. Intervals with a photon count below the threshold are then set to zero. A visual representation of the data analysis procedure can be found in the supplementary information (see Section S3, Supporting Information). The time domain autocorrelations were calculated using a multiple‐tau sampling scheme and fitted to a similarly sampled model. The frequency domain autocorrelation was calculated from time domain autocorrelation (without multiple‐tau sampling) for several segments and averaged. Fitting in the frequency domain was performed on natural logarithm of the FFT amplitude and to avoid infinite/very high values in the valleys (local minima) of the FFT models a finite but small value was added to both the model and calculated FFT. All data is presented using the average value over all measurements with the standard deviation as the error, the sample size is shown in each figure since these are different for most particles. All processing and data representation was implemented in Python.

Conflict of Interest

The authors declare no conflict of interest.

Supporting information

Supporting Information

Oorlynck L., Van Daele L., Myslovska A., et al. “Cationic Polyelectrolyte Adsorption onto Anionic Nanoparticles Analyzed with Frequency‐Domain Scanning Fluorescence Correlation Spectroscopy.” Small Methods 9, no. 9 (2025): 2401985. 10.1002/smtd.202401985

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.