The determination of the molar mass of soft nanoparticles is essential to estimate their (molar) concentration in dispersions. Measuring this quantity by conventional methods, however, often proves challenging. We describe a facile approach to determine the molar mass of (soft) nanoparticles via counting their number per volume using a widefield fluorescence microscope. The method is exemplified on a microgel dispersion as a model system, while it is applicable to other types of stainable nanoparticles. For this, covalent labeling or modification of the nanoparticles is not required. The dispersion is simply mixed with a Nile Red solution in a defined ratio and measured in an optical fluorescence microscope accessible to most researchers in the field.
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Introduction
The molar mass M is a fundamental quantity in chemistry and many methods are available for the determination of M of (macro)molecules. For soft nanoparticles, knowledge of the molar mass allows for a calculation, or in case of polydisperse systems at least an estimation, of the number of particles from their weight, which is important since in most cases solutions of soft nanoparticles are prepared by weighing nanoparticle powders and dispersing them in a known volume. Knowing the number of microgels in a certain volume for example is crucial when using them to stabilize droplets in liquid-liquid systems, e.g., for extraction processes or emulsion stabilization, as the occupancy of the droplet’s interface strongly influences their behavior [1‐3]. Furthermore, the topography of a layer of ultra-low crosslinked microgels depends on the concentration/compression of the microgels. Hence, it is fundamental to estimate the number of microgels on the surface [4]. Also the flow properties of colloidal suspensions depend on the particle concentration [5, 6].
Determination of the molar mass of soft nanoparticles can be achieved via static light scattering (SLS) with Zimm plot analysis [7‐10] or via viscosimetry yielding the volume fraction of the microgels that enables the calculation of the molar mass by knowledge of the microgel size [10‐12]. Analytical Ultracentrifugation is also able to yield the molar mass, requiring knowledge about further properties of the particles namely the partial specific volume which is the inverse of their density and/or their anisotropy. These properties have to be determined in an additional run in the analytical ultracentrifuge or by using further analytical tools like a density oscillation tube in case of particle density [13, 14]. If particles exhibit the attribute of absorbing light, it is possible to apply absorption spectroscopy and dynamic light scattering to yield the number density of the particles and therefore the molar mass, as has been described in the literature for gold nanoparticles coated with a polymer shell [15]. Moreover, quartz crystal microbalance coupled with absorption spectroscopy gives access to accurate molar concentrations of nanoparticle suspensions [16]. Other techniques for measuring molar concentrations are based on scattering, plasma generation or electric resistance when particles pass through an electric field, but often need calibration with a standard solution [17]. All these methods are quite sophisticated, but also require elaborate analysis. Further, optical methods based on counting objects by their scattering or after a fluorescence labeling procedure are available [18]. In the former case, the scattering contrast is often limited for small objects with high water content. Fluorescence labeling on the other hand is an additional procedure that might also affect the properties of the particles since functional groups have to be introduced or reacted with the fluorescence label. This can change interactions between particles and might even cause aggregation. For particles with low scattering contrast, a suitable and universal labeling procedure to enable detection via fluorescence would be of significant advantage. A suitable dye for this purpose is Nile Red. Its solvatochromy ascertains that the aqueous solution remains colorless whereas more hydrophobic regions/objects such as particles become fluorescent [19]. This property has already been exploited to label microplastics [20, 21]. Also soft particles, such as microgels can be stained with Nile Red [22, 23].
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Here, we present a straightforward microscopy method to determine the molar mass of soft particles, which can be basically applied at any microscope. One outstanding feature is the facile preparation procedure with an “in situ” labeling. A dispersion/solution of soft particles in water is mixed with freshly prepared aqueous Nile Red solution. As sketched in Fig. 1, images of microgels are taken and the microgels close to the focal plane are counted. For the detection of the fluorescence signal of the microgels readily available software, such as the ImageJ plugin ThunderSTORM [24], can be applied. The number of particles per volume equals the particle number concentration (\(C=\frac{N}{V}\)). Division of the known mass concentration \(\gamma\) by the determined particle concentration C and multiplication with Avogadro’s number \(N_A\) yields the molar mass of the particles: \(M=\frac{\gamma }{C}N_A\).
Fig. 1
Principle to determine the molar mass of soft particles: Labeled particles are excited and detected for example on a camera. Particles sufficiently close to the focal plane (black cylinder in the center) of the objective can be localized. The diameter of the field of view and the distance for the focal plane in which particles can be still localized defines the observation volume \(V_{\text {obs}}\)
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Materials and methods
Microgel synthesis and characterization
The microgels were provided by the DWI – Leibniz-Institute for Interactive Materialials. The poly(N-vinylcaprolactam) (PVCL) based microgels with poly(glycidyl methacrylate) (PGMA) core were synthesized using batch free radical precipitation polymerization [25]. N-Vinylcaprolactam (VCL, Sigma-Aldrich, Darmstadt, Germany, 98%) was distilled and recrystallized from hexane before use. Glycidyl methacrylate (GMA, Sigma-Aldrich, Darmstadt, Germany, 97%) was purified via column chromatography on basic aluminum oxide. VCL (10.62 mmol, 90 mol%), GMA (1.18 mmol, 10 mol%) and N,N\('\)-methylenebisacrylamide (BIS, 0.354 mmol, 3 mol%, Sigma-Aldrich, Darmstadt, Germany, 99%) were dissolved in HPLC grade water (163 mL, VWR Chemicals). The solution was purged with nitrogen for 1 h at \(70\,^{\circ }\text{C}\). Subsequently, a solution consisting of 2,2’-azobis(2-methylpropionamidine) dihydrochloride (AMPA, 0.094 mM, 0.8 mol%, Sigma-Aldrich, Darmstadt, Germany, 97%) and HPLC grade water (5 mL) was added to initiate the polymerization. The mixture was stirred for 2 h at \(70\,^{\circ }\text{C}\) to obtain the microgels which were subsequently dialyzed against deionized water (Molecular weight cut-off (MWCO): 12,000–14,000 Da) for 5 days.
The Raman spectrum of the microgels was measured on a Bruker FS 100/S Raman Spectrometer (Bruker Corporation, Billerica, MA) and is shown in Fig. S4 in the Supporting Information. For the measurement, the microgel was lyophilized and pressed into an aluminum pan. A Nd:YAG laser (\(\lambda =1064~\tt nm\)) with an energy of 200 mW was used. The measurement was performed at a wavenumber between 400 and 3500 cm\(^{-1}\) with a resolution of 4 cm\(^{-1}\), and 1000 scans. The GMA content was determined using a calibration curve based on a previous publication [26].
Sample preparation
For microscopy measurements, borosilicate glass high precision coverslips #1.5H (24 \(\times\) 50 mm and (\(170 \pm 5 \, \mu \textrm{m}\)) thickness; Paul Marienfeld) were used. For cleaning, they were washed with a 1% Hellmanex III® solution (Hellma Analytics) and left in it overnight. Afterwards they were rinsed with water and dried in a drying cabinet at \(55\,^{\circ }\text{C}\) overnight. For both steps, ultra pure water produced by Astacus2 system (membraPure) with an electric conductivity of max. 0.06 \(\mu\)S cm\(^{-1}\) was used. In the last step, the coverslips were cleaned using a plasma cleaner (Femto; Diener electronic) for 15 min running with oxygen 2.5 (Westfalen).
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From the freeze dried microgels a dilution series with the following concentrations was prepared: 0.1 g L\(^{-1}\), 0.05 g L\(^{-1}\), 0.01 g L\(^{-1}\), 0.005 g L\(^{-1}\), 0.001 g L\(^{-1}\) using demineralized water (electric conductivity > 0.7 \(\mu\)S cm\(^{-1}\)) and lyophilized microgels for preparing the stock solution. The aqueous Nile Red solution was freshly prepared by mixing 20 \(\mu\)L of a \(1 \times 10^{-4}\) M Nile Red solution in methanol with 180 \(\mu\)L of water (LiChrosolv® water for chromatography; Merck). To label the microgels, 20 \(\mu\)L of the microgel dispersion and 20 \(\mu\)L of the aqueous Nile Red solution were mixed. Therefore, the mass concentration of the microgel dispersion was halved, which was considered in our analysis. For the microscopic measurements SecureSeal™ imaging spacer (9 mm diameter, 0.12 mm thickness; Sigma-Aldrich) were placed onto the coverslip, 10 \(\mu\)L of the microgel-Nile Red mixture were added and everything was closed with a round coverslip (borosilicate glass, 22 mm diameter, 130–160 \(\mu\)m thickness; Thermo Scientific). For calibration 10 \(\mu\)L of a microgel-Nile Red mixture with a mass concentration of 0.05 g L\(^{-1}\) was spin coated onto a coverslip (4000 rpm for 40 s) and wetted with 10 \(\mu\)L of water. For static light scattering (SLS) and dynamic light scattering (DLS) measurements, the microgel dispersions were filtered using 1.2 \(\mu\)m syringe filters (CHROMAFIL®XTRA PET-120/25, Macherey-Nagel GmbH & Co. KG).
Microscopy measurements
Fluorescence microscopy measurements were performed at a Nikon Eclipse Ti-E inverted microscope with the possibility of performing both widefield fluorescence microscopy and spinning disc confocal microscopy in two different paths (VisiScope Spinning-Disc-DC Confocal System and VisiTIRF system; Visitron Systems GmbH). For collection of light a 60x/1.2 w water immersion objective (UPlanSapo; Olympus) was applied connected with an adapter (PLE152; Thorlabs) to the Nikon microscope. For moving the sample, a piezo table (SmarAct) was utilized. A single mode diode laser with an excitation wavelength of 488 nm (200 mW max., iBeam smart 488-S-HP, Toptica) was used for excitation. The laser light was first coupled into a glass fiber using a \(\mu\)Aligna system (TEM Messtechnik) and coupled into the microscope via the VisiTIRF system, which allows for switching between TIRF and widefield illumination. The latter one was used in our case. To separate excitation and fluorescence light a quadline dichroic (ZT405/488/561/640rpcv2-UF2; Chroma) and a quadband emission filter (ZET405/488/561/640mv2; Chroma) were applied. The fluorescence light was imaged onto a CMOS camera (Prime 95B CMOS; Teledyne Photometrics). The final pixel size was 163.6 nm. The exposure time was set to 20 ms and the laser power density to about \(I\approx 2 \times 10^{6} ~\textrm{ W~ m}^{-2}\). In case of the tracking experiment the measurement was performed approx. 37 \(\mu\)m above the coverslip surface with an power density of about \(1.3 \times 10^{6} ~\textrm{ W ~m}^{-2}\), measuring a microgel-Nile Red mixture with \(\gamma =0.05\) g L\(^{-1}\). For tracking, the maximum step length (maximum distance of two consecutive positions to be considered as a step) was set to 10 px, the join radius (to interpret two localizations within this distance as one) was set to 0 px, the minimum track length to 7 positions and the maximum number of frames between two positions (in case a microgel vanishes out of focus) was set to 5 frames. For MSD analysis the integration time was set to 128 ms, read out from the raw data as time between two consecutive frames.
Dynamic light scattering measurements
DLS measurements were performed on a goniometer setup with a HeNe laser (\(\lambda\) = 632.8 nm) from ALV-Laser Vertriebsgesellschaft mbH (Hessen, Germany) equipped with an ALV/LSE 5003 multiple tau correlator. The samples were measured at angles from \(30^{\circ }\) to \(150^{\circ }\) under temperature control in a thermal bath at \(T=20\,^{\circ }\text{C}\). The intensity time correlation functions were analyzed with the second order cumulant method [27] yielding the q-dependent decay rate \(\Gamma =Dq^2\), where \(q=4\pi n/\lambda \sin (\theta /2)\) is the length of the scattering vector with the refractive index of the sample n and the scattering angle \(\theta\). The diffusion coefficient D was obtained by a linear fit of \(\Gamma\) against \(q^2\), and then gave the hydrodynamic radius \(r_H\) by the Stokes-Einstein equation \(r_H=k_BT/(6\pi \eta D)\) with the Boltzmann constant \(k_B\) and the viscosity of the solvent \(\eta\).
Static light scattering measurements
Static light scattering measurements were performed on a FICA instrument (SLS-Systemtechnik, Denzlingen, Germany) at \(20~^{\circ }\text{C}\), using a laser with a wavelength of \(\lambda =640\) nm. Toluene (analytical reagent, VWR) was used as a calibration standard for absolute scale. The measurements were performed after a resting time of at least 30 min for each sample, for equilibration. Quartz cuvettes with 20 mm diameter (Hellma Analytics) were used as sample containers. The scattered intensity expressed as the Rayleigh ratio R was extrapolated to \(q=0\) and \(c=0\) in a Zimm-Guinier plot according to the following equation:
with the concentration c in g/mL, the radius of gyration \(R_\textrm{G}\) in nm, the weight-averaged molecular weight \(M_\textrm{W}\) in g/mol, the second virial coefficient \(A_2\) in \(\mathrm {ml\ mol\ g^{-2}}\), and the optical constant \(K=4\pi ^2n_\textrm{s}^2/(N_\textrm{A}\lambda ^4) (\textrm{d}n/\textrm{d}c)^2\) with Avogadro’s constant \(N_\textrm{A}\), the refractive index of the solvent \(n_\textrm{s}\) and the refractive index increment \((\textrm{d}n/\textrm{d}c)\) that expresses the contrast of the microgel to the solvent. The Guinier approximation was chosen for the form factor of the microgel because it describes the scattering of spherical objects to higher q-values than the classical Zimm approximation. The analyzed intensities were ensured to lie in the Guinier regime \(qR_\textrm{G}<1.3\) for spherical objects [28].
Results and discussion
Molar mass determination
To calculate the molar mass of the microgels, first the number of microgels per volume has to be determined. The lateral area of the field of view is well-known whereas an observable z-range has to be appropriately determined. For this, we used a calibration by recording z-stacks of microgels which were adsorbed to the coverslip by spin-coating to a defined layer and, subsequently, wetted with a drop of water. Three representative images for \(z=-1~\mu\)m, \(z=0~\mu\)m and \(z=+1~\mu\)m are shown in Fig. 2. To estimate the width of the detectable z-range, we used the full width at half maximum (FWHM) of the Gaussian fit of the distribution of detected microgels plotted against the z-position, which is \(\text {FWHM}= 0.97~\mu\)m. The calibration is optimally performed with the particles of interest that are immobilized on a surface, for example by spin-coating or drop-casting. It can, however also be performed with fluorescent beads.
Fig. 2
Number of detected microgels in dependence of the focal distance from the coverslip measured with widefield fluorescence microscopy
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In order to obtain more reliable statistics, we not only concentrated on one z-plane when measuring the samples for M determination, but measured an entire z-stack with 0.5 \(\mu\)m step size from the coverslip surface up to 40 \(\mu\)m into the solution. This allowed surface effects to be assessed, which would be important if particles were strongly adsorbed to the coverslip surface. Essential for the measurements is a suitable choice of the camera integration time (20 ms in our case) which has to be set short enough to be able to localize single particles as they are diffusing inside the solution and avoid “smearing”. We localized the respective signals of single microgels using the ImageJ Plugin ThunderSTORM [24]. Representative images and single microgel localizations (marked with yellow circles) for some consecutive images with 0.5 \(\mu\)m difference in z are displayed in Fig. 3. It also illustrates that defocused molecules far from the focal plane are not detected. The histogram in Fig. 3 outlines the statistical fluctuation of the number of microgels in different planes. Consequently, it is important to account for such statistical effects in the analysis.
Fig. 3
Example of a z-scan measured via widefield fluorescence microscopy at different z-heights above the coverslip surface for a mass concentration of 0.005 g/L. The localized microgels are marked as yellow circles and the number of detected microgels at different foci is shown in the histogram. The histogram bars obtained from images in the five different focus layers are color-coded with the border color around the images. The scan was performed with a piezo-table with a precision in the low nanometer range
Fig. 4
Number of microgels in dependence of the height of the z-scan above the coverslip surface measured via widefield fluorescence microscopy, for mass concentrations \(\gamma\) of a–b 0.0005 g L\(^{-1}\), c–d 0.0025 g L\(^{-1}\), and e–h 0.005 g L\(^{-1}\). For (a)–(f), the coverslips were used approx. 1–2 weeks after the cleaning procedure with oxygen plasma. g–h presents measurements with \(\gamma =\) 0.005 g L\(^{-1}\) using coverslip surface stored for two month after plasma-cleaning. Apparently this additional storage time caused stronger adsorption of the microgels to the surface whereas the concentration in solution remained bascially unchanged compared to (e) and (f). For analysis we used the values obtained for heights of 20 to 40 \(\mu\)m above the coverslip surface as no surface effects are expected
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Apart from statistics, it is also important to consider surface effects that can have a significant impact on the molar mass evaluation. Strong attraction between particles and the surface can result in irreversible binding and, thus, a depletion of particles close to the interface. With the direct visualization of z-stacks as described here, such potential sources of error and the resulting misinterpretations can be minimized. Additionally, as outlined below, we varied the concentration of microgels. Typical distributions of single microgels in different distances from the surface at \(z=0\) to 40 \(\mu\)m away from the surface are shown in Fig. 4. Panels (a) to (f) show rather homogeneous concentration profiles through the z-scan with no significant adsorption on the coverslip surface. In contrast, a significant amount of microgel adsorption can be observed in (g) and (h). Prior to the latter measurements, the plasma-cleaned coverslips were stored for 2 months, whereas in the former cases (a) to (f) they were stored only for 1–2 weeks. It is known that plasma-cleaning induces a more hydrophilic surface on glass, which becomes less hydrophilic after several days [29, 30]. Previous studies have shown that microgels tend to reduce the contact area with a surface for more hydrophilic surfaces [31, 32]. As the hydrophilicity of the glass surface is lost after a few days, the microgels may tend to adsorb stronger to coverslips that were plasma cleaned 2 months prior to measurement. However, even if some microgels adhere to the coverslip surface, the number within the bulk solution is not greatly affected, as can be seen by comparing (e) and (f) with (g) and (h).
We prepared dispersions with three different microgel concentrations and recorded series of microscopy images with different z-focus (z-stacks). The number of microgels was counted for each focus (see Fig. 4a–f) and histogrammed for the different concentrations as shown in Section S1 of the Supporting Information. To avoid surface effects in our analysis, we used the numbers obtained for the distances between 20 \(\mu\)m and 40 \(\mu\)m above the coverslip surface and fitted the distribution with a Poisson function (see Fig. S1 of the Supporting Information). From the Poisson fits we obtained the expectation values and standard deviations, which in the case of a Poisson distribution is simply the square root of the expectation value. We present the results of this analysis for three different microgels concentrations in Fig. 5. The data points were fitted with a linear function, the slope of which is the inverse of the molar mass. If effects of adsorption to the interface or aggregation occur, this can be directly observed in the measured image series and as a deviation of the linear behavior of the curve. Indeed, in our case, aggregates were observed at higher concentrations \(\ge\) 0.025 g L\(^{-1}\) (see Fig. 6a) and therefore these measurements were excluded from the analysis. The possibility to observe aggregation is a significant advantage of our novel techniques with respect to existing methods to determine the molar mass.
Fig. 5
Mean number of microgels per volume in dependence of the mass concentration \(\gamma\), measured via widefield fluorescence microscopy. The linear fit is visualized as solid line and the error limits of the slope as dashed lines
Fig. 6
Widefield fluorescence microscopy images at the coverslip surface from samples with mass concentrations of a\(\gamma =\) 0.05 g L\(^{-1}\) and b\(\gamma =\) 0.005 g L\(^{-1}\). In case of the sample with \(\gamma =\) 0.05 g L\(^{-1}\) aggregation is visible. Aggregates are marked with blue circles
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Applying linear fits (with zero y-intercept) to the data obtained yielded a slope of \((4.3 \pm 0.3) \times 10^{13}~ \textrm{g}^{-1}\). Multiplying the inverse of the slope by Avogadro’s number gives the molar mass, which is \((1.4 \pm 0.1) \times 10^{10} \text { g mol}^{-1}\).
We compared this result with the molar mass obtained by static light scattering [7‐10]. For analysis of the SLS data, the refractive index increment for PGMA (\((\textrm{d}n/\textrm{d}c)_\textrm{PGMA}=\)0.0931 mL g\(^{-1}\)) [33] as well as for PVCL (\((\textrm{d}n/\textrm{d}c)_\textrm{PVCL}=\) 0.186 mL g\(^{-1}\)) [34] were taken from the literature. Weighting the refractive index increments by the weight fraction of the polymers gives the refractive index increment of the copolymerized microgel \((\textrm{d}n/\textrm{d}c)_{\mu \textrm{G}}=w_\textrm{PVCL}\cdot (\textrm{d}n/\textrm{d}c)_{\textrm{PVCL}}+w_\textrm{PGMA}\cdot (\textrm{d}n/\textrm{d}c)_{\textrm{PGMA}}=\) 0.177 mL g\(^{-1}\) [35]. For the SLS measurements, microgel concentrations needed to be sufficiently high to produce good statistics of the scattered intensity and were chosen as 0.04 g L\(^{-1}\), 0.06 g L\(^{-1}\) and 0.08 g L\(^{-1}\). Extrapolations of the scattered intensity to \(0^{\circ }\) and 0 g L\(^{-1}\) were done in a Zimm-Guinier plot (see Fig. 7) yielding a weight-averaged molar mass \(M_w = (1.9 \pm 0.2) \times 10^{8} \text { g mol}^{-1}\), a second virial coefficient \(A_2=(-3.0 \pm 0.4) \times 10^{-4}\) mL mol g\(^{-2}\), and a radius of gyration \(R_G = (189 \pm 2)\) nm. Here, \(M_w\) is almost two orders of magnitude lower than the number averaged molar mass \(M_n\) determined with our microscopy method. We ascribe this rather large discrepancy to the effects of aggregation, sedimentation and filtering of the samples in SLS. Filtering may decrease the microgel concentration in the sample, especially when aggregated microgel clusters exceed the pore size of the filter. Accordingly, unfiltered microgel samples produce higher scattering intensities compared to filtered samples, as shown in Section S2 in the supporting information. Furthermore, aggregated microgels may sediment and thus deplete from the scattering volume. Both filtering and sedimentation can thus lead to an overestimation of the microgel concentration and an underestimation of the molar mass in SLS. The negative second virial coefficient indicates attractive forces between the microgels, which explains their aggregation in this concentration range as also shown in Fig. 6a.
Fig. 7
Zimm-Guinier plot of SLS data of the microgel dispersion at 20 \(^{\circ }\)C at concentrations of 0.04 g L\(^{-1}\), 0.06 g L\(^{-1}\), 0.08 g L\(^{-1}\). The lines are extrapolations to \(q=0\) and \(c=0\)
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We suggest that our method based on fluorescence microscopy yields a more reliable molar mass than SLS. The microscopy method offers high sensitivity at low microgel mass concentrations and allows to exclude aggregation by simple observation. It should be noted that the statistics has to be carefully taken into account, especially when low concentrations and, thus, low particle numbers are analyzed. Also, the possible interaction with the glass coverslip has to be considered. If particles stick to the surface, the number N of microgels counted will be lower and the calculated molar mass will be overestimated. Overall, our method provides a quick and easy way to obtain the size range of M which is, for example, crucial for applying microgels to stabilize liquid-liquid interfaces [1].
Single particle tracking experiments
To confirm that the fluorescent signals of the Nile Red-labeled microgels arise from individual microgels and not from aggregates, single particle tracking (SPT) experiments of diffusing microgels (37 \(\mu\)m above the coverslip surface) were performed and analyzed using a MATLAB routine developed in our research group, applying a cost function in space and time to connect the individual positions to tracks [36]. From the obtained tracks the diffusion coefficients were calculated by performing linear fits to the mean squared displacement (MSD) plotted against the lag time, using the second to the fifth time points according to Ernst et al. [37]. The representation of the MSD data plotted against the lag time for all tracks is presented in Fig. 8a indicating free diffusion. The distribution of the diffusion coefficients, weighted by the length of the track, and the distribution of the hydrodynamic radius calculated from them are depicted in Fig. 8b and c. A mean diffusion coefficient of \(D=8.7 \times 10^{-13} \textrm{ m}^{2} \textrm{s}^{-1}\) and a hydrodynamic radius of \(r_{\textrm{h}}=\) 247 nm were obtained. This is consistent with the results of dynamic light scattering (DLS), which provided \(D=(8.61 \pm 0.05) \times 10^{-13} \textrm{ m}^{2} \textrm{s}^{-1}\) and \(r_{\textrm{h}}=\) (\(249 \pm 2\)) nm (see Section S3 of the Supporting Information), i.e., the self-diffusion of individual microgels is tracked rather than the one of aggregates. As SPT also indicates the size distribution of the microgels, it shows the presence of polydispersity in the sample. This offers an approximation of the distribution of the molar mass obtained using the single colloid counting method presented. Additionally, we have showcased various applications of this colloid labeling technique.
Fig. 8
Representation of the mean squared displacement plotted against the lag time (a), as well as distributions of the diffusion coefficient (b) and the hydrodynamic radius (c)
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Estimation of the amount of bound water
Microgels are polymer networks swollen by a solvent [38]. The amount of bound water therefore is of great interest. Using the experiments and results described above, we can estimate the water content as follows. Dividing the molar mass of about \(M = 1.4 \times 10^{10} \text { g mol}^{-1}\) by the volume of the microgel (\(V_\textrm{swollen}=6.31 \times 10^{7} \text { nm}^{3}\)), calculated from the hydrodynamic radius rh = 247 nm obtained from the tracking experiment, we get a value of \(M/V_\textrm{swollen}=222 \text { g mol}^{-1} \text { nm}^{-3}{}\) as molar mass per volume. Taking the molar mass and molar ratios of the monomers into account (\(M_\textrm{VCL}=139.19 \text{ g mol}^{-1}{}\), \(M_\textrm{GMA}=142.15 \text { g mol}^{-1}{}\); \(M_\textrm{monomers}=0.9\cdot M_\textrm{VCL}+0.1\cdot M_\textrm{GMA}=139.49 \text { g mol}^{-1}{}\)), a number concentration of \(C=1.59 \text { nm}^{-3}\) of monomer units inside the microgel is obtained, and from this Ntot = 1 × 108 monomer units inside the whole microgel by multiplying C with the volume of the microgel (Vswollen). The volume of the “dry” microgel without any water is obtained by dividing the mass (\(m=M/N_\textrm{A}{}\)) by the density of the microgel. The latter one is calculated as \(\rho _{\mu \textrm{G}}=0.9\cdot \rho _\textrm{PVCL}+0.1\cdot \rho _\textrm{PGMA}=1.226 \text { g mL}^{-1}{}\) considering the densities of PVCL (\(\rho _\textrm{PVCL}=1.23 \text { g mL}^{-1}{}\)) [39] and PGMA (\(\rho _\textrm{PGMA}=1.19 \text { g mL}^{-1}{}\))[33], giving \(V_\textrm{dry}=1.90 \times 10^{7} \text { nm}^{3}{}\) corresponding to a theoretical radius of \(r_\textrm{dry}=165 \text{ nm}\). The water content is calculated as \(V_{\mathrm{H}_{2}\mathrm{O}}/V_\textrm{swollen}=(V_\textrm{swollen}-V_\textrm{dry})/V_\textrm{swollen}=0.70\). Thus, the water content can be estimated as 70% for the swollen microgel. As we used the hydrodynamic radius to calculate \(V_\textrm{swollen}{}\) this value might be overestimated. However, a value, in the range of 70% to 90% is common for microgels such as a PNIPAM (poly(\(N\)-isopropylacrylamide) microgel dispersed in water [40]. Therefore, despite some rough assumptions, our method seems to give a quite reasonable result for the calculation of the water content and for the molar mass included in the calculation.
Conclusions
In conclusion, the method presented in our work is capable of providing the molar mass of microgels or other types of colloids. It is simple and does not require any special equipment other than a fluorescence microscope. Sample preparation is straightforward, requiring only the mixing of a Nile Red solution with the colloidal system to be studied. Combined with single particle tracking, the method even provides information on possible polydispersity. We have avoided the potential misinterpretation of microgels adsorbed at the coverslip interface by recording z-scans over several micrometers, and errors due to aggregation of microgels by performing a dilution series and ensuring the observation of individual microgels by single particle tracking.
Our method enables the avoidance of commonly encountered experimental challenges, including aggregation and sedimentation, which can negatively impact conventional methods such as SLS and viscosimetry that are frequently employed to determine molar mass.
Acknowledgements
We thank Anish Gulati for his introduction to the SLS setup and his help with the measurements.
Declarations
Ethical approval
Not applicable.
Conflict of interest
The authors declare no competing interests.
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