Determination of the Exact Particle Radius Distribution for Silica

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Determination of the Exact Particle Radius Distribution for Silica Nanoparticles via Capillary Electrophoresis and Modelling the Electrophoretic Mobility with a Modified Analytic Approximation Anna Fichtner, Alaa Hussein Jalil, and Ute Pyell Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b04543 • Publication Date (Web): 14 Feb 2017 Downloaded from http://pubs.acs.org on February 15, 2017

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Determination of the Exact Particle Radius Distribution for Silica Nanoparticles via Capillary Electrophoresis and Modelling the Electrophoretic Mobility with a Modified Analytic Approximation Anna Fichtner**, Alaa Jalil**, Ute Pyell* University of Marburg, Department of Chemistry, Marburg, Germany **Both authors contributed equally to this work. *Corresponding author: Ute Pyell, University of Marburg, Department of Chemistry, Hans-Meerwein-Straße, D-35032 Marburg, Germany e-mail: [email protected] ABSTRACT Employing aqueous dispersions of amorphous silica nanoparticles of various size, we investigate whether electropherograms recorded from capillary electrophoresis experiments can be converted directly into exact number-based particle radius distributions provided that there is a relaxation effect-based size selectivity of the electrophoretic mobility and provided that the electrokinetic potential ζ of the particles can be regarded to be homogeneous over the surface of the particles and independent of the particle size. The results of this conversion procedure are compared to numberbased particle radius distributions obtained from a large set of transmission electron microscopy data. For this specific example, it is shown that the modified analytic approximation developed by Ohshima adequately describes the mobility-dependent relaxation effect and the electrophoretic mobility of the particle as a function of the reduced hydrodynamic radius and the electrokinetic potential, which is a prerequisite of the presented procedure. Simultaneously, we confirmed that for the given Debye length/particle diameter ratio the electrokinetic surface charge density can be regarded to be size-invariant (including spherical geometry and planar limiting case). It is shown that the accuracy of the results of the developed method is comparable to that gained by a large set of transmission electron microscopy data, which is important when a precise description of the particle size distribution is needed to deduce conclusions regarding the underlying mechanism(s) of particle growth. Those values obtained for the dispersion (width) of the distribution show only a small negative deviation, when compared to the transmission electron microscopy data (4 to 16%).

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KEY WORDS Nanoparticles, silica, capillary electrophoresis, zeta potential, surface charge density, Raleigh scattering, turbidity

INTRODUCTION Nanoparticles of different composition have come into the focus of interest due to their unique properties, which differ from those of bulk materials. Often properties are size dependent, so that for many applications monodisperse nanoparticles having a narrow size distribution are highly desirable. Knowledge of particle growth mechanisms is therefore indispensable to optimize nanoparticle synthesis procedures with regard to mean particle size and particle size distribution, especially if size selection processes are to be avoided [1,2]. Nucleation (denucleation) and growth follow different mechanisms [3]: LaMer mechanism [4], Ostwald ripening [5] (mathematically described by Lifshitz, Slyozov [6] and Wagner [7]), digestive ripening, Finke-Watzky two step mechanism [8], coalescence and oriented attachment [9], and intraparticle growth. Several of these mechanisms might appear combined or can be predominant in different regimes occurring subsequently or overlapping temporally to varying extents [10]. Elucidating the kinetics of these processes requires the development of in situ monitoring methods such as in situ liquid transmission electron microscopy (TEM) [11], which allows to measure the particle size distribution as a function of the reaction time or to observe the growth of single nanoparticles [12]. Another approach is to measure the growth rate of the mean particle radius by a method applicable to a dispersion of nanoparticles, e.g., UV-vis spectrometry, dynamic light scattering (DLS), fluorescence correlation spectroscopy (FCS), or Raman correlation spectroscopy (RCS) [13]. Small angle X-ray


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scattering (SAXS) also allows to monitor in situ the growth rate of the mean particle radius and was employed by several workers to quantify initial steps of nucleation and growth [14]. Simultaneously, the exact knowledge of the final particle size distribution provides additional valuable information about the possible particle growth mechanisms involved, as it can be compared to those particle size distributions, which would be expected from currently available theoretical models [15]. In addition, risk assessment studies regarding (natural and engineered) colloidal nanoparticles require an exact knowledge of size distributions and aggregation degrees. To this background, we examine in this study for aqueous dispersions of colloidal silica (Ludox) nanoparticles whether our method of direct conversion of electropherograms into number-based size distributions described in previous publications [16,17] allows generating particle size distribution functions, which are as precise or even more precise than number-based size distributions generated from transmission electron microscopy (TEM) data. TEM inherently has the disadvantage that (for subsequent investigations) a relatively small sample (few microliters) has to be taken from a larger volume. This sampling step might be associated with a non-quantified sampling error, which could be determined by taking several samples together with a statistical evaluation of the data obtained with the different samples. The high cost of time and effort, which is associated with this approach, however, will impede a quantification of the outlined potential sampling error on a routine basis. In addition, the determination of adequately precise number-based particle size distributions by TEM requires the survey of a minimum of 1000-3000 nanoparticles, which is also inevitably associated with a substantial effort. In contrast to TEM, capillary electrophoresis (CE) works with sample sizes of several mL, of which (in general) several nL are injected for a single run into the separation


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capillary. Consequently, sampling errors due to taking a subsample can be monitored by repeated injections. For charged colloidal nanoparticles, CE permits the precise determination of the mean electrophoretic mobility and of the distribution of the electrophoretic mobility (with regard to an ensemble of nanoparticles) provided that adsorption of the studied particles onto the inner capillary wall can be suppressed, peak distortions due to electric field inhomogeneities can be avoided, and the nanoparticles are sufficiently colloidally stable [18,19]. Likewise CE can be employed to separate and characterize different nanoparticle populations in mixtures [20,21]. CE can also be employed to determine the purity of ferrofluids [22], to probe nanoparticleprotein interactions [23], to study graphene oxide and chemically converted graphene [24], or to characterize nanoparticle-aptamer conjugates [25]. There are several recent review articles on the characterization of nanoparticles by capillary electromigration separation methods [26-28] giving an introduction into theoretical aspects and showing the broad range of applications developed. Beside the determination of the electrophoretic mobility for those nanoparticles, of which the electrophoretic mobility is strongly influenced by the relaxation effect, freezone capillary electrophoresis also allows to gain information on size and shape provided that the electrophoretic mobility can be described adequately as a function of the hydrodynamic radius [29]. Following the classical theory of Overbeek and Booth [30,31] results in the case of a significant relaxation effect in a set of differential equations that cannot be solved analytically, even in the simplest case of a symmetrical 1:1 electrolyte, assuming that the nanoparticles are ideal spheres with a homogeneous surface charge density, and assuming that nanoparticle-nanoparticle and ion-ion interactions can be neglected [32,33]. Solving these equations by numerical calculations [34,35] has shown that the function µep = ƒ(κa) (ζ, Λ°+, Λ°- =


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const.) (µep = electrophoretic mobility, κ = Debye Hückel parameter, a = sphere radius, ζ = electrokinetic potential, Λ°+ = limiting equivalent conductance of the electrolyte cation, Λ°- = limiting equivalent conductance of the electrolyte anion) undergoes a minimum. For a limited range of κa (outside of the minimum range), this function can be approximated to be linear, which simplifies the conversion of electropherograms into size distribution functions [16,17]. These theoretical considerations are supported by the fact that linear functions can be determined experimentally with monodisperse nanoparticle standards of known mean particle size and employed subsequently for the determination of the mean particle radius of nanoparticles by capillary electrophoresis [36,37]. Several approximate analytic expressions have been developed [38-41], which allow to describe the function µep = ƒ(κa, ζ, Λ°+, Λ°-) without recurring to numerical calculations. For gold nanoparticles with different hydrophilic coatings, we have demonstrated that the application of the approximate analytic approach developed by Ohshima [42,43] requires in the case of buffered solutions with cations and anions differing largely in their electrophoretic mobility that the effective ionic drag coefficient is approximated by the ionic drag coefficient of the (strong electrolyte) counter ion [44]. The conversion of electropherograms into size distribution functions after size-selective separation due to the relaxation effect was first suggested by McCann et al. [45] in 1973 based on measurements made with continuous flow electrophoresis. Capillary electrophoresis, however, has the additional advantage of a high separation efficiency. Under ideal conditions, only molecular diffusion contributes to zone broadening. For entities with a high charge and a low diffusion coefficient (e.g. polyelectrolytes, proteins, or nanoparticles) plate numbers > 500.000 can be obtained [46]. Therefore, in CE it is not necessary to deconvolute the particle size distribution obtained from a


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converted electropherogram with regard to inevitable zone broadening, which is in contrast to other size selective separation methods, e.g., asymmetric flow field-flow fractionation (aF-FFF), size-exclusion chromatography (SEC), or hydrodynamic chromatography (HDC). Liu et al. [47] demonstrated for gold nanoparticles prepared under different reaction temperatures that the peak shapes observed in CE directly correspond to size histograms obtained via TEM measurements. Under the conditions of their measurements µep is directly proportional to the mean particle diameter. The excellent matching of CE and TEM data with regard to mean particle size has been confirmed in later studies of the same group [48]. The idea that electropherograms can easily be converted into size distribution curves was pushed forward by Xue et al. [49]. Via DLS, size histograms were obtained for several polystyrene nanoparticle standards (20-125 nm). A calibration plot of migration time vs. particle diameter was constructed by associating the diameters of three cumulative percentiles (5%, 50%, 95%) of the size histogram (obtained via DLS) with the corresponding migration times of three cumulative percentiles (5%, 50%, 95%) of the peaks obtained via CE. The resulting size distributions were corrected for the dependence of the molar absorbance coefficient on the size of the particles. Pyell [16] and Pyell et al. [17] selected a different approach to convert electropherograms directly into size distribution functions. They succeeded in avoiding experimental calibration curves. Their method is based on an exact determination of the electrokinetic potential ζ from measuring the electrophoretic mobility in an electrolyte of known composition. In addition, the mean particle radius has to be determined by a second independent method (e.g. TEM or Taylor dispersion analysis (TDA) [50,51]). Employing the (modified) analytic approximation presented by Ohshima


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[42,43], regression lines were calculated that permitted the direct conversion of recorded electropherograms into particle size distribution functions, with the prerequisite that the particles have to be homogeneous in ζ (resulting from a homogeneity in the electrokinetic surface charge density σζ). For semiconductor/silica core/shell nanocomposites with number mean solid diameters ranging from 15 to 107 nm it was shown that this prerequisite is given (ζ is independent of the particle size) [16]. The overlay of particle size distribution functions calculated from recorded electropherograms with size distribution histograms estimated from TEM micrographs confirmed the validity of the approach. The particle size distribution functions obtained from CE reflect the dispersion of the particle size distribution functions obtained via TEM. The result was shown to be independent of the ionic strength of the background electrolyte. The imprecise TEM data (diagrams calculated from a relatively low number of surveyed particles) and the low signal-to-noise ratio of the recorded electropherograms, however, did not permit a more exact quantitative evaluation of the particle size distributions obtained. A similar method was applied to gold nanoparticles with different hydrophilic coatings [17]. Here CE together with the proposed data evalution method revealed that the analyzed nanoparticle populations had narrow size distributions with a width of 2-4 nm. The hydrodynamic radius distribution of the coated nanoparticles was only slightly broader than the solid particle radius distribution of the gold cores. The particle distribution functions obtained via CE were considerably narrower than those obtained via DLS and data evalution by the CONTIN algorithm. CE in combination with TDA and the proposed data evaluation scheme proved to be an ideal method for the determination of the narrow particle size distributions of the coated particles investigated. It has to be emphasized that the method also permits to determine the


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mean electrophoretic mobility and the electrophoretic mobility distribution, the electrokinetic potential ζ and the electrokinetic surface charge density σζ [44]. Current techniques employed in the characterization and quantification of engineered nanoparticles have been reviewed recently by Lapresta-Fernandez et al. [52]. Following techniques are listed to provide information on morphology, size, and polydispersity characteristics: microscopy techniques (e.g., transmission electron microscopy (TEM) or scanning electron microscopy (SEM)), spectroscopic techniques (e.g., dynamic light scattering (DLS, also termed photon correlation spectroscopy (PCS)), fluorescence correlation spectroscopy (FCS), or small angle X-ray scattering (SAXS)), and chromatography and related techniques including size-exclusion chromatography (SEC), capillary electrophoresis (CE), hydrodynamic chromatography (HDC), and field-flow fractionation (FFF). CE is described as a method which allows the separation of nanoparticles according to differences in size or according to differences in the surface charge density and as a method which permits the evaluation of the mean particle size. Interestingly, in this review article Taylor dispersion analysis (TDA) has not been included into the list of methods described, although it also provides (calibration-free) information on the mean hydrodynamic radius [44,50,51]. It is also remarkable that plate gel electrophoresis (PGE) that emerged into one of the most often used methods for purification and characterization of colloidal nanoparticles [53], was completely overlooked, despite large progress in the theoretical understanding of factors determining migration and separation within the last decade [54,55]. In the present paper we intend to show for aqueous dispersions of amorphous silica (Ludox) nanoparticles of various size that capillary electrophoresis in combination with a method that allows the determination of the number-based mean radius can be a


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powerful technique for the determination of the precise distribution of the electrophoretic mobility and/or for the determination of the precise distribution of the hydrodynamic radius. We also intend to show (with varied temperature and varied ionic strength of the background electrolyte) that CE allows the unbiased determination of the electrokinetic potential ζ and the electrokinetic surface charge density σζ of highly charged colloidal nanoparticles ( ζ ≥ 50 mV), as CE offers a unique simple method of transferring charged nanoparticles from a synthetic mixture of unknown composition into a buffer of exactly known composition. This transfer step is very important, if the calculation of ζ needs to take the relaxation effect into account [17], because under these conditions, the electrophoretic mobility depends not only on the sphere radius a, the electrokinetic potential ζ, and the ionic strength Ι (as in the validity range of the Henry equation), but also on the equivalent limiting conductance of the counter ion Λ°counter with regard to the small buffer ions (mobility-dependent relaxation effect) [34]. The validity of the suggested approach will be tested by comparing size distributions resulting from the CE method with those obtained by a second independent method. As second independent method we have selected TEM because of its known reliability. A precise quantitative method comparison will be enabled by the survey of more than 3500 particles per sample. As with the CE method the recorded signal is the apparent absorbance Aapp (= turbidity τ multiplied with ln(10) and the layer thickness ℓ) caused by Raleigh scattering [56], correction procedures are discussed, which are needed to correct for the dependence of τ on the particle radius. It will be also shown that in the case of a very stable electroosmotic flow velocity the cumulative superposition of subsequently recorded electropherograms allows to improve the signal-to-noise ratio, while avoiding an increase in the particle concentration, which might have the disadvantage of provoking


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nanoparticle-nanoparticle interactions and inhomogeneities of the electric field strength within the separation capillary.


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EXPERIMENTAL

Materials Colloidal dispersions of Ludox TM-40 (SNP22), Ludox HS-30 (SNP12), and Ludox SM30 (SNP7) were supplied by Sigma-Aldrich (Taufkirchen, Germany). The data provided by the manufacturer report for SNP22 a particle concentration (mass fraction) of 40% (w/w), a nominal particle diameter of 22 nm, and a pH of 9.0, for SNP12 a particle concentration (mass fraction) of 30% (w/w), a nominal particle diameter of 12 nm, and a pH of 9.8, and for SNP7 a particle concentration (mass fraction) of 30% (w/w), a nominal particle diameter of 7 nm, and a pH of 10. In all cases Na+ is the counterion of the electrostatically stabilized particles in aqueous dispersion. Borate buffers were prepared by dissolving disodiumtetraborate decahydrate (p.a., Merck, Darmstadt, Germany) in deionized water. MilliQ water (18 MΩcm, Merck MilliPore, Darmstadt, Germany) was used for sample dilution.

Transmission Electron Microscopy TEM measurements were performed on a JEM-3010 UHR (Jeol Ltd., Japan) operating at 200 kV having a lanthanum hexaboride cathode. As detector we used a highresolution CCD camera (Gatan Inc., USA) having a GOS phosphor scintillator. Further data treatment was done with ImageJ and Origin 8.5G (Northampton, MA USA). Samples were prepared by diluting the homogenized delivered nanoparticle dispersion to a mass fraction of 3% (w/w) with MilliQ water. With the intention to remove salt and impurities [57], the SNP22 dispersion was dialyzed two days against MilliQ water by using a regenerated cellulose membrane (Visking dialysis tubing 27/32, MWCO 14kDa,


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SERVA, Heidelberg, Germany). For this sample we did neither determine the final mass fraction nor the pH. After having completed dilution and/or purification we placed one drop of the homogenized diluted sample with the help of a glass Pasteur pipette onto a grid of the type S-160-3 (Plano GmbH, Wetzlar, Germany). Drying was at room temperature in a dust-protected atmosphere.

Dynamic Light Scattering A Zetasizer Nano ZS (Malvern Instruments, Malvern, UK) was used for DLS measurements (non-inverse backscattering at 173°) and further data treatment. Samples were prepared by homogenizing the delivered nanoparticle dispersions by agitating and diluting them to the desired mass fraction with MilliQ water. After this first dilution step the appropriate volume of a solution of 100 mmol L-1 disodiumtetraborate decahydrate (borax) was added and the obtained solution was filled up to the mark of the volumetric flask so that the final concentration of borax is 40 or 30 mmol L-1. The final mass fraction of particles was calculated from the masses of the weighed solutions. From this stock solution further dilutions were prepared by dilution with the corresponding borax solution. All samples were filtered through a nylon membrane filter (0.45 µm, Wicom, Heppenheim, Germany). The temperature of the solution during the measurement was kept at 25 °C or 15 °C.

Capillary electrophoresis and Taylor dispersion analysis All CE and TDA measurements were done with a Beckman (Fullerton, CA, USA) P/ACE MDQ CE system equipped with a UV-absorbance detector. Temperatures of the capillary and the sample tray were kept at 15 °C, 20 °C, or 25 °C by liquid cooling.


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Data were recorded with the Beckman 32 Karat software (v.5.0). Further data treatment (e.g., fitting of data to a Gram-Charlier series of type A, baseline correction, and moment analysis) was done with Origin 8.5 (Northampton, MA, USA). The electrokinetic potential was calculated employing a Matlab (MathWorks, Natick, MA, USA) procedure. Fused silica capillaries (75 µm I.D., 375 µm O.D.) were obtained from Polymicro Technologies, Phoenix, AZ, USA. New capillaries were conditioned by rinsing them first with NaOH solution (0.1 mol L-1) 60 min, water 30 min, and electrolyte 30 min. Between runs the capillaries were rinsed with electrolyte for 5-10 min. Either the total length of the capillary was 395 mm and the length to the detector 292 mm (CE) or the total length of the capillary was 606.5 mm and the length to the detector 505.5 mm (TDA). The electroosmotic mobility was determined using thiourea as a marker. Following mass fractions were employed for the TDA measurements: SNP22, 0.4%; SNP12, 0.9 %, SNP7, 0.9 %.

RESULTS AND DISCUSSION Transmission Electron Microscopy The TEM micrographs (refer to Figure S1, supporting information Part 1) show spheroidal nanoparticles. We therefore estimated the characteristic size (mean diameter) of each nanoparticle as the arithmetic mean of the diameter in the x and in the diameter in the y dimension (two measurements orthogonal to each other). The resulting histograms determined from 1533 (SNP7), 3665 (SNP12) and 3777 (SNP22) particles are presented in Figure 1. For the smallest nanoparticles (SNP7) there is a symmetrical size distribution, which is in accord with a Gaussian function, while for the larger nanoparticles (SNP12) there is some asymmetry (the size distribution is left-


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skewed), which is considerably increased for the largest nanoparticles (SNP22). The existence of an asymmetry was confirmed by drawing the probability plot (refer to Figure 2) that clearly shows two different ranges corresponding to two clearly distinguishable regression lines with different slopes intersecting at 48% (SNP12) or 21% (SNP22). To quantify the observed skewness (= third central moment normalized on σ) we fitted the TEM data to a Gram-Charlier series of type A [58]:

p( x) = y 0 +

−

B σ 2π

e

( x − µ1 )2 2 σ2

 κ3  κ4 H3 (z) + H4 (z)  1 + 3 4 3! σ 4! σ  

with p(x) = probability density, y0 = offset, B = constant, µ1 = first moment, σ2 = second central moment, κ3 = skewness, H3(z) = third Hermite polynomial = z3 – 3z, z = (x µ1)/σ, κ4 = excess, H4(z) = forth Hermite polynomial = z4 - 6 z2 + 3. As can be seen in Figure S2 (supporting information Part 1), the fitted functions excellently describe the experimental data. The results are presented in Table 1. With those data recalculated from Eq. (1) and obtained regression parameters (step width = 0.1 nm, subtraction of base line) it is also possible to perform moment analysis (determination of the (statistical) central moments [59], refer to Section S1, supporting information Part 1), which provides more reliable data regarding skewness and excess (refer to Table 1). The comparison of the parameters given in Table 1 reveals that none of the recorded size distributions can be described correctly with a Gaussian function. While the size distribution function for SNP7 is right-skewed, those for SNP12 and SNP22 are leftskewed. It must be emphasized, however, that the parameter κ3 for SNP7 is associated with a large standard error, so that its significance has to be confirmed. Fitting to a Gram-Charlier series of type A generates significant values for the skewness for SNP12 and SNP22, which are confirmed by moment analysis. Those


(1)

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data generated for the excess (kurtosis) are associated with a large standard error and will not be discussed.

Taylor Dispersion Analysis Following results were obtained by Taylor dispersion analysis (TDA, for experimental details refer to Sections S2+S3, supporting information Part 1): SNP7: 12.60 nm (± 2.77%); SNP12: 14.84 nm (± 4.13%); SNP22: 31.14 nm (± 3.67%); mean particle diameter. The run-to-run repeatability is given in brackets as relative standard deviation for five repeated runs. With SNP22 there is good agreement of the result of TDA with the TEM data, while for SNP12 the results of TDA are considerably lower than those resulting from the corresponding TEM data. For SNP7 we would expect a more drastic negative deviation, while for SNP7 the result of TDA is (in contradiction to our expectations) much higher than the number-based mean radius obtained via TEM. These observation together with a characteristic deviation of the recorded traces (refer to Figure S3, supporting information Part 1) from a cumulative Gaussian function (recorded curves are flatter in the starting and ending regions) points to the presence of a subpopulation of considerably larger particles (agglomerates) in the case of SNP7. The asymmetry of the deviation can be explained by a smaller “elution time” of larger particles than smaller particles due to size exclusion of the particles from the region near the wall [60]. With SNP12 there is only a very small deviation of the recorded trace from the fitted curve in the starting region of the ascending part of the curve, which points to a very small subpopulation of aggregated particles present in the sample. With SNP22 there is no visible deviation of the recorded trace from the fitted curve pointing to the absence of a subpopulation of aggregated particles. It can be also concluded from the


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excellent agreement of the recorded traces with the fitted curves that SNP12 and SNP22 can be characterized to have a monomodal narrow size distribution, which is in agreement with the TEM data. In addition, it can be assumed that under the selected experimental conditions (mass fraction of nanoparticles in the sample = 0.4-0.9%) the measured collective diffusion coefficient Dc is not identical with the diffusion coefficient D0 at infinite dilution due to particle-particle interactions, which would explain the negative deviation from the TEM result obtained for SNP12. This aspect will be discussed in more detail in the next section.

Dynamic Light Scattering Hydrodynamic diameters (mean of number-based size distributions) obtained for infinite dilution (for regression analysis refer to Section S4, supporting information Part 2) are listed in Table 2. These data are remarkably precise (independent of the ionic strength of the dispersant and of the temperature). However, values obtained via the regression method from DLS data are significantly too low compared with the data resulting from the TEM measurements (refer to Table 1, P = 95%). The difference ∆dH between the data obtained by DLS and those obtained by TEM is increasing with dH (∆dH about 10 % of d(mean,TEM)). We ascribe these deviations to the limitations of the employed CONTIN algorithm [61]. Similar problems were reported by [62] who obtained via DLS inconsistent data for coated gold nanoparticles. These authors demonstrated that the hydrodynamic size can be determined alternatively by measuring electrophoretic mobilities over a wide range of electrolyte ionic strength and modelling correctly the influence of the polymeric corona.

Sample analysis by capillary electrophoresis


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As already outlined in our previous work [44], determination of the size distribution function by capillary electrophoresis (CE) requires the selection of experimental conditions that minimize local electric field inhomogeneities (within the separation capillary). Local field inhomogeneities will cause peak broadening and distortion and make a direct conversion of electropherograms into size distributions impossible. Likewise, also adsorption of the particles onto the capillary wall has to be avoided. Adsorption of particles (studied for different adsorption and desorption kinetics and linear and nonlinear adsorption isotherms [63]) onto the capillary wall will result in a drift of the EOF velocity (associated with a drift in migration times), a shift of migration time, sample loss, deterioration of efficiency and asymmetric peak deformation [64]. In the case of negatively charged nanoparticles, adsorptive particle-wall interactions can be effectively suppressed by a negatively charged inner capillary surface. With fused-silica capillaries a high (negative) zeta potential of the inner capillary wall can easily be reached with alkaline aqueous buffers [44], e.g., by dissolving disodiumtetraborate decahydrate (borax) in deionized water: Na2B4O7 • 10 H2O → 2 Na+ + 2 [B(OH)4]– + 2 B(OH)3 + 3 H2O Under these conditions: pH ≈ pKA1 of boric acid and ionic strength Ι = 2 c(borax). As the buffer is generated by the hydrolysis of a salt, there is only one type of counter-ion with precisely known molar concentration, which is identical to the ionic strength of the buffer (Ι = c(Na+)). Electropherograms were recorded for varied ionic strength (Ι = 40-120 mmol L-1, c(borax) = 20-60 mmol L-1) and varied temperature (15-25 °C), while the nanoparticle dispersions were diluted in electrolyte (w = 0.3-0.4 %) to avoid a transient inhomogeneity of the capillary wall zeta-potential (which might induce an additional band broadening) during the electrophoretic run. For optimization of the signal-to-


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noise-ratio a detection wavelength of 200 nm (SNP7 and SNP12) or 214 nm (SNP22) was selected. The method further required to avoid band broadening due to the concentration input function (volume overload) and due to local electric field strength inhomogeneities induced when charge is transported by the nanoparticles (concentration overload). Absence of overload was confirmed by variation of injection conditions (refer to Figure S8, supporting information Part 3). Overload conditions can be identified by a dependence of the peak shape parameters on the amount of substance of analyte or on the volume of sample injected. Signal-to-noise ratio was improved by cumulative superposition of consecutive runs. Figure 3 illustrates the principle for SNP22, c(borax) = 30 mmol L-1, and T = 20 °C. Direct superposition is possible because of the very small variation in migration times. The cumulative superposition of four runs results in a two-fold improvement of the signal-to-noise ratio (see also Figures S9+S10, supporting information Part 3). Drift of the baseline, seen in several electropherograms, stems from a drift in the intensity of the D2-lamp employed for absorbance detection. This drift can be easily corrected and is not inferring the developed data evalution scheme.

Electrophoretic mobilities Electrophoretic mobilities were calculated from the recorded electropherograms assuming a homogeneous electric field strength in the capillary given by the total length of the capillary and the applied voltage (refer to Tables S4+S5a-i, supporting information Part 3). As expected, the electrophoretic mobility is decreased with increasing ionic strength of the buffer and increased with increasing temperature. The relative standard deviation for four consecutive runs is between 0.03 and 0.74%. Data evaluation for measurements repeated at a second day reveals a good reproducibility


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of the method. Small shifts can be ascribed to small day-to-day variations in the electrolyte composition or the temperature.


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Estimation of the zeta potential As already described in the introduction, with nanoparticles having an electrokinetic potential ζ larger than 25 mV, the potential ζ can only be determined from the electrophoretic mobility by numerical calculations [34,35] or via an analytic approximation based on an equation developed by Ohshima [42,43]. The presence of a gel layer on the particle surface (which was postulated in the frame of explaining measured electroviscous effects [65,66]) was regarded to be negligible in accordance with the results presented in [67]. For reasons described in our previous publication with regard to the calculation of ζ for hydrophilic coated gold nanoparticles [44], we employ following equation for the determination of ζ from the determined electrophoretic mobility: µ=

2 2    eζ   eζ  f κ a − m f4 ( κa )   f1 ( κa ) −  ( ) 3 counter     kT   kT   

2εr ε0 ζ 3η

(2)

where εr = relative electric permittivity, ε0 = electric permittivity of vacuum, η = viscosity, κ = inverse thickness of the ionic cloud, a = sphere radius, k = Boltzmann constant, T = absolute temperature, e = elementary electric charge, f1, f3, and f4 are a function of κa and are given by f1 ( κa ) = 1 +

f3 ( κa ) =

f4 ( κa ) =

1

(

)

2 1 + 2.5 /{ κa 1 + 2e− κa } 

(

κa κa + 1.3 e−0.18 κa + 2.5

(

2 κa + 1.2 e

−7.4 κa

+ 4.8

(

)

3

)

(2b)

3

) + 6.02)

9κa κa + 5.2 e−3.9 κa + 5.6

(

8 κa + 1.55 e

−0.32 κa

(2a)

3

Equation (2) corresponds to the equation presented by Ohshima in 2001 [42], where the average dimensionless ionic drag coefficient (m++m-)/2 is replaced by the


(2c)

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dimensionless ionic drag coefficient of the counter ion mcounter [44]. As the silica nanoparticles are negatively charged, the dimensionless ionic drag coefficient of the counter ion mcounter is calculated from the limiting conductance of the cation Λ+0: mcounter =

2εr ε0kTNA 3ηΛ 0+

(3)

where: NA = Avogadro’s number. For 1 < κa < 10 and ζ > 100 mV, the possible relative error of this approximate analytic expression can become very large (∆µ > 50% [16]). It follows for NPs that this equation is only valid if ζ ≤ 100 mV [16]. The sphere radius a was approximated with the radius corresponding to the maximum of the size distribution obtained from the TEM data (refer to Table 1). The Debye-Hückel parameter κ was calculated from: κ=

e 2 NA ∑ z i2c i

(4)

εr ε 0kT

where: zi = charge number (valence) of ith component, ci = molar concentration of ith component. The dimensionless ionic drag coefficients taken into subsequent calculations are in the range of 0.254 to 0.256. With these ionic drag coefficients we calculated electrophoretic mobilities dependent on the reduced sphere radius κa and

ζ (refer to Figure 4). These calculated values were then compared to those experimental values determined by capillary electrophoresis (refer to Tables S4+S5a-I, supporting information Part 3). The superposition of calculated and experimental values (Figure 4) reveals that the experimental electrophoretic mobilities are within the validity range of the analytic approximation (that is ζ < 100 mV). It also reveals that the effect of double layer polarization (the relaxation effect) is important, as in all cases

ζ > 40 mV and there is a distinct minimum of the function µ = ƒ(κa) (for ζ = const.). All data points are close to the calculated minimum of this function. As expected, ζ decreases with increasing ionic strength. For SNP12 and SNP22 the


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resulting ζ potentials are apparently independent of the nanoparticle radius (within the range of the measurement error). However, ζ seems to be considerably lower for SNP7 (refer to Table S7, supporting information Part 3). This apparent deviation can be ascribed to the presence of aggregates in the case of SNP7 as discussed in the previous sections. The need to take the relaxation effect into account was demonstrated recently independently by Hill [68] who calculated in an electrolyte with Na+ as the dominant counterion for particles that are very similar to those studied by us regarding size and electrokinetic surface charge density (sphere radius = 10 nm, electrokinetic surface charge density = 0.04 C m-2) the electrophoretic mobility with varied ionic strength with and without considering ion concentration perturbations (= relaxation terms) employing a software based on the method developed by O’Brien and White [35]. The results depicted in Figure 2b of [68] show the dramatic difference of the results. Hill also concluded that it is the electrophoretic mobility of the dominant counterion that has an influence on the particle drag coefficient in the case of ion concentration perturbations. With the aim to obtain more precise data than those obtainable with a graphical procedure, we applied the iterative scheme described in [44] to calculate ζ from the measured electrophoretic mobility (refer to Table S6, supporting information Part 3) via Eq. (2). This iterative procedure is based on a Matlab script, in which µ is calculated with varied ζ via Eqs. (2+3) at fixed κa and mcounter. The resulting values are listed in Table 3. Comparison of results obtained for two measurement series obtained at different days reveals the high precision of the data. Because ζ is regarded to be independent of the particle radius (which follows from the experimental data shown in Figure 4 and will be discussed in the following section), we have also calculated the


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arithmetic mean of the data obtained for SNP12 and SNP22. With the exception of the bracketed values, ζ is increasing with increasing T. Nanoparticles and capillary wall are composed of the same material: amorphous silica. It is therefore interesting to compare ζ calculated for the nanoparticles via an analytic approximation to those values calculated for ζ at the inner capillary wall/electrolyte interface form the experimentally measured electroosmotic mobilities (refer to Tables S5a-i) via the Smoluchowski equation (refer to Table S7, supporting information Part 3). In all instances we observe the expected decrease in ζ with increasing ionic strength and again an increase in ζ with increasing T. Hence, the data gained from the determination of the electroosmotic mobility confirm the results obtained from the determination of the electrophoretic mobility of the nanoparticles. Listing of the calculated κ and Debye length λD in Table S7 reveals that in the investigated temperature range κ is nearly invariant with T (small increase in κ with increasing T due to a decrease in εr with increasing T). The observed increase in ζ with increasing T is clearly not due to a variation in the Debye length (as in this case we would expect the inverse correlation). It is also not possible to attribute the observed increase in ζ to a decrease in the Bjerrum length λB (because also here there is the influence of a decrease in εr with increasing T), which would have an impact on the condensed fraction of counterions [69]. We can also exclude a shift due to an inaccurate temperature measurement, as our results correspond to those obtained by Evenhuis et al. [70], who calculated ζ from electroosmotic mobility measurements in carefully temperature-controlled fused-silica capillaries corrected for the effects of Joule heating. Within a range of 18.3 to 33.2 °C and employing a 10 mmol L-1 phosphate buffer (pH = 7.21) as electrolyte, the experimentally determined values for ζ increased at 0.39% per °C (3.9% per 10 °C), whereas we observe an


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average increase of 4.3% per 10 °C for c(borax) = 20-50 mmol L-1. We ascribe the observed increase in ζ with increasing T to the fact that the parameter κ is not fully describing the spatial charge density distribution within the diffuse layer (i.e., we are outside the validity range of the linearized Poisson-Boltzmann equation, [the only case] for which κ is an exact descriptor of the spatial charge density distribution [within the diffuse layer]). Additionally, the increase in ζ with increasing T at fixed ionic strength predicts – counterintuitively – an increased colloidal stability with increased T when keeping the ionic strength constant.

Estimation of the surface charge density The electrokinetic potential ζ also gives access to the electrokinetic surface charge density σζ, which is defined here as the surface charge density at the electrokinetic slipping plane (balancing the charge in the diffuse layer beyond the electrokinetic slipping plane) [71] or alternatively as the effective electric charge Qeff normalized on the area of the shear surface (slipping plane) 4πa2 [34]. Under the assumption that the NPs are ideal rigid spheres having a uniform distribution of the charge at the shear surface, σζ can be calculated numerically for given ζ and κa [72]. For a 1:1 electrolyte, σζ can also be calculated employing an approximate empirical formula [72,73]: σζ =

 Qeff ε ε kT  eζ   e ζ  4 = r 0 κ 2 sinh  tanh  +  2 e κa 4πa  2 kT   4 kT  

(5)

For a 1:1 electrolyte, Ohshima et al. [74] derived a more accurate analytic expression:       2ε ε κkT  eζ  1  2 + 1 σζ = r 0 sinh  1 +  e κa    ( κa)2  2 kT   2 e ζ  cosh      4 kT   

   e ζ    8 ln cosh     4 kT         e ζ    sinh2      2 kT    


1/2

(6)

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For a 1:1 electrolyte Loeb et al. [72] compared the results of the numerical treatment with those obtained with Eq. (5). For κa > 0.5 the maximum deviation is only 5% independent of ζ. Ohshima et al. [74] compared the results of Eqs. (5+6) with the exact numerical values obtained by Loeb et al. [72]. The results of both equations include significant errors for κa < 0.5. However, with Eq. (6) the error is smaller than with Eq. (5). A comparison with the experimental values depicted in Figure 4 reveals that we should expect with both Eqs. (5+6) correct values for σζ with a maximum relative error much smaller than 5%. The applicability of Eqs. (5+6) can be extended to buffered electrolytes having weak electrolyte coions and strong electrolyte monovalent cations or anions as counterions [44]. The resulting values for σζ and effective charge number zeff (= Qeff/e) are depicted in Table 4. Results obtained via Eq. (5) do not deviate from those obtained via Eq.(6) (results not shown). It is interesting to note that (within the parameter range selected) σζ is invariant with ionic strength, temperature and particle diameter. With the aim to confirm the validity of the approach used for the calculation of ζ and σζ from electrokinetic data, we also determined σζ for the inner capillary wall of the fusedcapillary employed in the measurements from the electroosmotic mobility data via exact solution of the non-linearized Poisson-Boltzmann equation for the planar limiting case (Gouy-Chapman equation or Grahame equation) [32,75]: σς =

 zeζ 2 εr ε 0 κ k T sinh  ze  2kT

  zeζ   = 8 n εr ε 0 k T sinh     2kT 

where: z = charge number of buffer ions (symmetrical electrolyte, z:z electrolyte), n = number density of buffer ions. The resulting data are also given in Table 4. The comparison shows that there is no significant difference between the results obtained from the nanoparticles investigated and the results obtained for the interface fused


(7)

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silica capillary/buffer, although there is a significant difference for the ζ-potentials at lower ionic strength (refer to Table 3) that will be discussed below. The calculated mean value for the σζ-values obtained for the nanoparticles is -0.0416 C m-2 (± 4.3%), while the calculated mean value for the σζ-values obtained for the capillary is -0.0422 C m-2 (± 1.4%). The remaining difference can be regarded to be insignificant (= within the range of the measurement error). These results indirectly confirm the validity of the modified analytic approximation presented in our previous publication [44]. In the case of a buffered solution with a weak electrolyte co-ion and a strong electrolyte counterion, the effective ionic drag coefficient (refer to Eq. (3)) should be approximated with the ionic drag coefficient of the counterion. Agreement of results for σζ obtained for spherical and for planar geometry also confirms that within the parameter range of this study the results of the employed analytic approximations (Eqs. (2-6)) can be regarded to be associated with a negligible error. Our results are in accordance with those of Evenhuis et al. [70], who reported for fused-silica capillaries that σζ at the inner surface of the capillary was independent of the temperature (temperature range of 18.3 to 33.2 °C). They are also in accordance with the results of Barisik et al. [76], who studied for fixed ionic strength and pH the decrease in the surface charge density σ ζ with an increase in particle size. They report for silica nanoparticles that in the range of λD/dp < 0.2 (which was reached in our investigations) the effect of particle size on surface charge density can be neglected, regardless of the pH and the ionic strength. Interestingly, the small difference in ζ obtained for the nanoparticles and the capillary wall at lower ionic strength (refer to Table 3) can fully be explained by a difference in the diffuse layer specific capacitance, which is curvature-dependant [77].


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By solving numerically the nonlinearized Poisson-Boltzmann equation, Wang and Pilon [77] found for a surface potential of 10 mV, an ionic strength of 10 mmol L-1, T = 25 °C, and εr = 78.5 that the electrode curvature of ultramicroelectrodes has a negligible effect on the predicted diffuse layer specific capacitance only for sphere radii larger than 40 nm. For a sphere radius of 10 nm the predicted diffuse layer specific capacitance is 30% higher than that predicted in the planar limiting case. It must now be taken into consideration that the influence of the curvature is decreasing with increasing the ionic strength and (because of the high ζ ) we are outside the validity range of the linearized Poisson-Boltzmann equation, so that the considerably lower difference in ζ observed by us for the nanoparticles and the capillary wall (refer to Table 3) is in full accordance with the results of the numerical calculations reported by Wang and Pilon. These results support our initial assumption that for highly charged nanoparticles, higher ionic strength and a limited size-range, the parameter ζ can be regarded to be independent of the particle diameter.

Calculation of size distribution functions As the electric field strength within the separation capillary is precisely known and time and space invariant during the electrophoretic run (homogeneous buffer), recorded electropherograms can be easily converted into intensity-weighted electrophoretic mobility distributions (refer to Figure 5). Taking a fixed value for ζ (refer to Table 3) Eq. (2) now permits to calculate functions that relate the electrophoretic mobility µ to the reduced radius κa. In a first step, we calculated for each electropherogram data points of the function µ = c1 + c2 κa (c1 and c2 = constants) by variation of the sphere radius a (at fixed κ and ζ) in a range, in which we expected the particle size distribution. The resulting data points


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for SNP12 and SNP22 are plotted in Figure S11 (supporting information Part 3). From these data points, the constants c1 and c2 were obtained by linear regression. Within the selected ranges for the reduced sphere radius κa the function µ = ƒ(κa) can be approximated with a straight line, although this function is inherently nonlinear (refer to Figure 4). The induced error due to this simplification can be regarded to be negligible (cf., Figure S11, supporting information Part 3) with a maximum error for µ of 10-9 m2 V-1 s-1. The resulting constants c1 and c2 and associated standard errors are listed in Table S8 (supporting information Part 3). For both sizes investigated (SNP12 and SNP22) there is a decrease in c1 and c2 with increasing ionic strength (which reflects the decrease in ζ with increasing ionic strength), while there is an increase in c1 and c2 with increasing temperature (which reflects the increase in ζ with increasing temperature). This comparison shows that from the viewpoint of maximizing sensitivity (here: sensitivity = c2), better measurement conditions are reached with lower ionic strength and higher temperature. An improved accuracy of the method would be reached by calculation of a larger set of data points and fitting to a polynomial function. Size distribution functions can now be obtained by converting the mobility x-axis of the converted electropherograms (refer to Figure 5) into a size coordinate via rearranging the regression function µ = c1 + c2 κa (c1 and c2 = constants) into κa = ƒ(µ) or a = ƒ(µ) with fixed κ. As outlined before, estimation of the size distribution from converted elctropherograms requires the absence of electromigration dispersion due to field strength inhomogeneities. It also requires colloidal stability of the dispersed particles under the conditions of the measurement. Absence of electromigration dispersion was confirmed by variation of the injected volume of sample (refer to Figure S8, supporting information Part 3). Absence of electromigration dispersion can be assumed, if the shape of the recorded trace is independent of the injected amount of particles.


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Influence of electromigration dispersion on the recorded peak will be most pronounced at lowest ionic strength (c(borax) = 20 mmol L-1). Simultaneously, colloidal stability will be lowest at highest ionic strength (c(borax) = 50-60 mmol L-1). A close inspection of the recorded electropherograms depicted in Figure S9 (supporting information Part 3) reveals that in selected cases (at higher ionic strength and lower temperature) distorted, bimodal or right-skewed peaks were obtained. We attribute these distortions to pronounced electrophoretic particle aggregation [78] under conditions of low ζ . Aggregated particles migrate with higher µ (refer to Figure 4). In this context, it is also important to note that at lowest ionic strength (c(borax) = 20 mmol L-1), the estimation of the particle/electrolyte interface ζ potential is associated with the highest imprecision (see Figure 4) due to the nonlinearity of the function µ = ƒ(ζ) under conditions of a strong impact of the relaxation effect. From these considerations we expect best results for c(borax) = 30-40 mmol L-1. A second problem arises from the type of detection employed. A simple conversion of the recorded electropherogram (A = ƒ(t); A = apparent absorbance) into a mobility distribution function (p = const. ⋅A = ƒ(µ); p = probability density) and then into a size distribution function (p = ƒ(d); d = particle diameter) will underestimate the fraction of smaller particles and overestimate the fraction of larger particles. We therefore multiplied the apparent absorbance value of each data point of the initially calculated intensity-weighted size distribution (refer to Figure S12, supporting information Part 4) with a correction factor given by the ratio dmax/d with dmax = d at maximum of the curve. Preliminary experiments with correction factors of the type (dmax)x/dx with x ∈ {1,2,3,4,5,6} resulted in best results for x = 1, so that this type of correction factor was employed in further investigations.


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Following procedure was finally selected for the conversion of recorded electrophorograms (refer to Figures 3 and S9) into size distribution functions: (i) Traces with improved signal-to-noise-ratio were obtained by the cumulative superposition of subsequently recorded electropherograms. (ii) These traces were converted to functions A = ƒ(µ) (A = apparent absorbance, refer to Figure 5) and then to functions A = ƒ(d) (d = particle diameter) by applying the regression lines depicted in Figure S11 (supporting information Part 3). (iii) After smoothing of the obtained traces with a Savitzky-Golay algorithm (500 to 1500 points) and baseline subtraction, the resulting lines were fitted to a Gram-Charlier series of type A (Eq. (1), nonlinear fitting). The obtained regression parameters are listed in Table S11 (supporting information Part 4). (iv) Subsequently, new data points were calculated with a step length of 0.01 nm by employing Eq. (1) and the regression parameters listed in Table S11. After weighing the data points with the correction factor dmax/d, the parameters µ1, σ, κ3, and κ4 (refer to Tables 1 and S10, supporting information Part 4) were calculated by moment analysis (refer to Section S1, supporting information Part 1). Superpositions of resulting size distribution curves with size distribution histograms obtained via TEM are depicted in Figures 6 and S13, supporting information Part 4. Apparently, the particle size distribution functions resulting from the developed procedure give an accurate estimation of the dispersion (width) and represent correctly the observed skewness. Quantitative data for µ1, σ, κ3, and κ4 are listed in Table S10, supporting information Part 4. The parameter σ characterizing the dispersion (width) of the calculated size distribution function was determined with a small negative deviation relative to the TEM data. Moment analysis of the TEM data (refer to Table 1) resulted in σ = 2.40 nm for SNP12 and σ = 3.25 nm for SNP22, while the mean value (taken from all values listed in Table S10) is σ = 2.02 nm (± 0.25 nm) for SNP12 and σ = 3.12 nm (± 0.32 nm) for


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SNP22. The relative standard deviation of this value is 12.3% for SNP12 and 10.3% for SNP22. These values indicate considerable random variations, which can be ascribed to the relatively low signal-to-noise-ratio reached for the recorded electropherograms. The dimensionless parameter κ3, characterizing the skewness of the calculated size distribution function has a very large random variation for SNP12, but is reliably estimated for SNP22 (outliers are marked with brackets, these measurement deviations can be ascribed to electrophoretic particle aggregation discussed above). Corresponding to the left-skewed distribution function derived from the TEM data, κ3 is negative for SNP12 and SNP22. Moment analysis of the TEM data (refer to Table 1) resulted in κ3 = -0.42 for SNP12 and κ3 = -0.57 for SNP22, while the mean value (taken from all values listed in Table S10 without outliers) is κ3 = -0.41 (±0.29) for SNP12 and κ3 = -0.38 (±0.07) for SNP22. The relative standard deviation of this value is 70% for SNP12 and 18% for SNP22. Despite the large random variations, these results clearly indicate a significant negative skewness, which is corroborated by the histograms gained from a large set of transmission electron microscopy data. The parameter κ4, however, which characterizes the excess of a distribution, was determined with very high inaccuracy. It is known, that the determination of κ3 and κ4 by moment analysis requires a high signal-to-noise-ratio [79]. Improvement of precision and accuracy of the developed method requires an improvement of the signal-to-noise-ratio of the recorded electropherograms. To this end, coupling of CE with a more sensitive detection method is highly desirable. Recently, the coupling of CE with evaporative light scattering detection (ELSD) [80] was reported and also the coupling of CE with inductively coupled plasma-mass spectrometry (ICP-MS) [81]. Additionally, the principles of the developed CE-UVD method can be directly transferred to methods based on CE-ELSD or CE-ICP-MS.


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The combination of TEM and CE data clearly reveals a monomodal left-skewed particle size distribution for SNP12 and SNP22. This type of distribution is in accord with the contribution of Ostwald ripening to particle growth following nucleation [5-7,82].


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Investigation of polydisperse and/or multimodal nanoparticle systems Particle growth can result in bimodal size distributions [83]. The accurate size characterization of nanoparticles in complex media often requires the investigation of polydisperse and/or multimodal nanoparticle systems [84]. We therefore prepared samples containing aliquots of the three nanoparticle populations under investigation with the aim to verify that separation by capillary electrophoresis enables the simultaneous determination of the size distributions of the three populations in the mixture. The resulting electropherograms (refer to Figures 7 and S14, supporting information Part 4) clearly reveal that electrophoretic separation results in traces that give access to the simultaneous size distribution determination of three different nanoparticle populations, which differ in the mean size (diameter) by 5 and 14 nm (refer to Table 1). There is no shift in apparent electrophoretic mobilities due to the mixing of different nanoparticle populations (refer to Table S12, supporting information Part 4). While the peak shape for SNP7 (showing clearly a shoulder, refer also to Figure S10, supporting information Part 3) points to the presence of aggregates (bimodal size distribution), which had been confirmed by other methods, the monomodal peak shapes obtained for SNP12 and SNP22 point to the absence of aggregates, which was confirmed by DLS and TDA. In addition, (if calibration is possible) the recorded data for the mixture make it possible to quantify the mass fractions of the different nanoparticle populations within the mixture. If capillary electrophoresis is combined with a selective detection method, the simultaneous characterization and quantification of different nanoparticle populations in a mixture can be performed also in complex media, as was already demonstrated by CE-ICP-MS for two silver nanoparticle populations in a macroemulsion [84] (nanosilver as an antimicrobial agent in a medical product [85]).


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Comparison of results obtained by CE-UVD with those obtained by DLS reveals that CE-UVD can be employed to detect the presence of a bimodal distribution, which would not be detected by DLS (refer to Figure 8), due to the inherent limitations of the employed CONTIN algorithm [61].

CONCLUSIONS The modified analytic approximation introduced by Ohshima [42,44] adequately describes the mobility-dependent relaxation effect and the electrophoretic mobility of the particle as a function of the reduced particle radius and the ζ potential, which was confirmed by the agreement of results obtained for spherical and for planar geometry. If the material to be investigated has (within the range of investigation) a nearly constant size-invariant ζ potential (at fixed dispersion medium composition and given type of counter-ion), the proposed method will allow to determine the parameters mean, dispersion (width), and skewness of any skewed monomodal or multimodal size distribution without the need for calibration or falling back upon microscopic techniques. The method requirement of fixed dispersion medium composition and defined type of counter-ion can be fulfilled easily, as during electrophoretic separation the particles under investigation are electrophoretically transferred from the sample medium into a background electrolyte of exactly known composition. From the electrophoretic data obtained for different ionic strength and temperature, reliable values (relative standard deviation resulting from all measurements ≤ 12%) can be obtained for the dispersion (width) of the distribution showing only a small negative deviation, when compared to the TEM data (-16% for SNP12 and -4% for SNP22). Likewise, the (negative or positive) skewness of the nanoparticle populations


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investigated can be correctly indicated in those cases, where electrophoretic particle aggregation is avoided.

ACKNOWLEDGEMENTS Financial support from the Deutsche Forschungsgemeinschaft (DFG PY 6/11-1) is gratefully acknowledged. AHJ thanks the Iraqi Ministry of Higher Education and Scientific Research (MoHESR) for providing him with a PhD scholarship (via University of Mosul, Iraq). The authors are grateful to Prof. Dr. Bernd Harbrecht, University of Marburg, for helpful discussions regarding particle growth mechanisms. We especially thank Mr. Michael Hellwig (Electron Microscopy and Microanalysis Laboratory, University of Marburg) for his helpful advice and assistance when carrying out the TEM measurements. We also thank Prof. Dr. Wolfgang J. Parak, University of Marburg, for giving us the possibility to measure the DLS data and Dr. Jose Maria Montenegro Martos for helpul assistance with the instrument.

SUPPORTING INFORMATION Part 1 (TEM and TDA): Moment analysis, theoretical background, measurement conditions and data evaluation, validity range of the Taylor-Aris equation, supplementary tables and figures. Part

2 (DLS): Theoretical background,

measurement conditions and data evaluation, supplementary tables and figures. Part 3 (CE): Supplementary tables and figures. Part 4 (Size Distributions): Dependence of τ on the particle radius, supplementary tables and figures.


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FIGURE CAPTIONS Figure 1. Histograms obtained via TEM measurements from (a) SNP22 (3777 particles), (b) SNP12 (3665 particles), and (c) SNP7 (1533 particles) with overlaid fitted Gaussian curves. Figure 2. Probability plot for (a) SNP12 and (b) SNP22 from TEM data. Figure 3. Electropherograms obtained for SNP22: (a) superposition of consecutive runs, (b) cumulative superposition, (c) smoothed cumulative curve (Savitzky-Golay algorithm 500 points), (d) curve fitted to Gram-Charlier series of type A. Experimental conditions: T = 20 °C, total length of capillary = 395 mm, capillary length to detector = 292 mm, inner diameter of fused silica capillary = 75 µm, electrolyte 30 mmol L-1 borax in water (pH = 9.2), voltage 7 kV, sample injection 0.1 psi (6.89 mbar) 6 s, data rate 16 Hz, absorbance detection 214 nm. Figure 4. Calculated electrophoretic mobilities for varied reduced sphere radius κa and varied ζ for (a) T = 15 °C, (b) T = 20 °C and (c) T = 25 °C with superimposed experimental data for different nanoparticle populations (see figure inset) and varied ionic strength (refer to Tables S4 and S5a-i, supporting information Part 3). Figure 5. Intensity-weighted distribution of the electrophoretic mobility µ resulting from the cumulative superposition of four electropherograms obtained for SNP22 recorded at 214 nm (smoothed cumulative curve, Savitzky-Golay algorithm 40 points). For experimental conditions refer to Figure 6. Figure

6.

Superposition

of

size

distribution

functions

calculated

from

electropherograms (black solid line) obtained for (a) SNP12 and (b) SNP22, c(borax) = 30 mmol L-1, T = 25 °C with histograms calculated from TEM data.


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Figure 7. Electropherogram obtained for a mixture of SNP7 (w = 1.2%, w/w), SNP12 (w = 0.6%, w/w), and SNP22 (w = 1.2%, w/w). Experimental conditions: T = 25 °C, total length of capillary = 395 mm, capillary length to detector = 292 mm, inner diameter of fused silica capillary = 75 µm, electrolyte 30 mmol L-1 borax in water (pH = 9.2), voltage 7 kV, sample injection 0.1 psi (6.89 mbar) 6 s, data rate 16 Hz, absorbance detection 200 nm. Cumulative superposition of six consecutive runs, smoothed trace (Savitzky-Golay algorithm 40 points). Figure 8. Number-based probability density plots for mixtures of SNP12 and SNP22 from DLS data dependent on the mass fraction w of nanoparticles in the sample, c(borax) = 40 mmol L-1, T = 25 °C (each data point = mean of five consecutive measurements).


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Figure 1. Histograms obtained via TEM measurements from (a) SNP22 (3777 particles), (b) SNP12 (3665 particles), and (c) SNP7 (1533 particles) with overlaid fitted Gaussian curves.


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99,5 99 98

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t/min

(b)

3

absorbance/mAU

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0 0

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30

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40

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(c)

absorbance/mAU

3

2

0

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20

30

40

t/min

(d)

3

absorbance/mAU

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

2

0

10

20

30

40

t/min

Figure 3. Electropherograms obtained for SNP22: (a) superposition of consecutive runs, (b) cumulative superposition, (c) smoothed cumulative curve (Savitzky-Golay algorithm 500 points), (d) curve fitted to Gram-Charlier series of type A. Experimental conditions: T = 20 °C, total length of capillary = 395 mm, capillary length to detector = 292 mm, inner diameter of fused silica capillary = 75 µm, electrolyte 30 mmol L-1 borax in water (pH = 9.2), voltage 7 kV, sample injection 0.1 psi (6.89 mbar) 6 s, data rate 16 Hz, absorbance detection 214 nm.


Langmuir

- 50 -

4

SNP12 SNP22 70 mV 67.5 mV 65 mV 62.5 mV 60 mV 57.5 mV 55 mV 52.5 mV 50 mV 47.5 mV 45 mV 42.5 mV

-8

2

-1

-1

µ/(-10 m s V )

(a)

3

2 5

10

15

20

25

κa

4

(b)

SNP12 SNP22 70 mV 67.5 mV 65 mV 62.5 mV 60 mV 57.5 mV 55 mV 52.5 mV 50 mV 47.5 mV 45 mV 42.5 mV

3

-8

2

-1

-1

µ/(-10 m s V )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 50 of 60

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10

15

20

κa


25

Page 51 of 60

- 51 -

4

(c)

SNP12 SNP22

70 mV 67.5 mV 65 mV 62.5 mV 60 mV 57.5 mV 55 mV 52.5 mV 50 mV 47.5 mV 45 mV 42.5 mV

-8

2

-1

-1

µ/(-10 m s V )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

3

2 5

10

15

20

25

κa

Figure 4. Calculated electrophoretic mobility µ for varied reduced sphere radius κa and varied ζ for (a) T = 15 °C, (b) T = 20 °C and (c) T = 25 °C with superimposed experimental data for different nanoparticle populations (see figure inset) and varied ionic strength (refer to Tables S4 and S5a-i, supporting information Part 3).


Langmuir

- 52 -

3 absorbance/mAU

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 52 of 60

2

1 1

2 µ/(-10

-8

2

m V

3 -1 -1

s )

Figure 5. Intensity-weighted distribution of the electrophoretic mobility µ resulting from the cumulative superposition of four electropherograms obtained for SNP22 recorded at 214 nm (smoothed cumulative curve, Savitzky-Golay algorithm 40 points). For experimental conditions refer to Figure 3.


Page 53 of 60

- 53 -

relative units

(a)

5

10

15

20

25

d/nm

(b) relative units

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

15

20

25

30

35

40

45

d/nm

Figure 6. Superposition of size distribution functions calculated from electropherograms (black solid line) obtained for (a) SNP12 and (b) SNP22, c(borax) = 30 mmol L-1, T = 25 °C with histograms calculated from TEM data.


Langmuir

- 54 -

9

8

absobance/mAU

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 54 of 60

7

6

5 9

12

15

18

21

24

t/min

Figure 7. Electropherogram obtained for a mixture of SNP7 (w = 1.2%, w/w), SNP12 (w = 0.6%, w/w), and SNP22 (w = 1.2%, w/w). Experimental conditions: T = 25 °C, total length of capillary = 395 mm, capillary length to detector = 292 mm, inner diameter of fused silica capillary = 75 µm, electrolyte 30 mmol L-1 borax in water (pH = 9.2), voltage 7 kV, sample injection 0.1 psi (6.89 mbar) 6 s, data rate 16 Hz, absorbance detection 200 nm. Cumulative superposition of six consecutive runs, smoothed trace (Savitzky-Golay algorithm 40 points).


Page 55 of 60

- 55 -

SNP22 (w = 2%) SNP12 (w = 1.5%) SNP12 (w = 1.5%) + SNP22 (w = 2%) SNP12 (w = 1.5%) + SNP22 (w = 0.15%)

30 25

700

20

600 500 400

15

n u m b er

probability density/%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

300 200

10

100 0 0

10

20

30

40

50

d/nm

5 0 0

10

20

30

40

50

60

dH/nm

Figure 8. Number-based probability density plots for mixtures of SNP12 and SNP22 from DLS data dependent on the mass fraction w of nanoparticles in the sample, c(borax) = 40 mmol L-1, T = 25 °C (each data point = mean of five consecutive measurements) compared to overlayed size histograms (inset) resulting from TEM measurements.


60

Langmuir

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Page 56 of 60

- 56 -

Table 1. Parameters (in nm, if not dimensionless) obtained by different procedures for the size (diameter) distribution functions evaluated from TEM measurements (standard errors in brackets, for symbols refer to Eq. 1). SNP7

SNP12

SNP22

Arithmetic mean

11.17

16.28

29.32

Median

11.10

16.42

29.91

Maximum

11.5

16.5

30.5

Standard deviation

1.87

2.30

3.85

µ1

11.11 (± 0.02)

16.51 (± 0.05)

30.17 (± 0.09)

σ

1.67 (± 0.03)

2.18 (± 0.06)

3.02 (± 0.10)

µ1

11.47 (± 0.18)

15.45 (± 0.16)

29.99 (± 0.17)

σ

1.67 (± 0.03)

2.42 (± 0.08)

3.00 (± 0.08)

κ3

0.49 (± 0.25)

-0.67 (± 0.15)

-0.54 (± 0.11)

κ4

0.27 (± 0.16)

-0.15 (± 0.07)

0.20 (± 0.08)

µ1

11.16

15.77

29.61

σ

1.91

2.40

3.25

κ3

0.25

-0.42

-0.57

κ4

1.68

0.71

0.35

Histograma

Gauß

GCSAb

Moment analysis

a bin-width

= 0.1 nm, b Gram-Charlier series of type A


Page 57 of 60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

- 57 -

Table 2. Number-based mean hydrodynamic diameter dH obtained by DLS from extrapolated reciprocal values. The corresponding regression lines are given in Figure S6, supporting information Part 2.

dH/nm

SE(dH)/nm

R2

SNP12, T = 25 °C, c(borax) = 30 mmol L-1

15.00

0.245

0.92993


14.68

0.055

0.99192


14.64

0.220

0.91513


25.84

0.281

0.94395


25.81

0.276

0.93063

SE = standard error resulting from the standard error of the y-intercept (regression analysis)


Langmuir

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- 58 -

Table 3. Electrokinetic potential ζ calculated from the data listed in Table S4 with the iterative procedure illustrated in Table S6, supporting information Part 3. Buffer concentration, temperature

SNP12 ζ/mV

SNP22 ζ/mV

Meanb ζ/mV

Capillary ζ/mV

c(borax) = 20 mmol L-1, T = 15 °C

-62.42

-60.57

(-61.50)

-67.22

T = 20 °C

-61.53

-60.24

-60.89

-68.46

c(borax) = 20 mmol L-1, T = 25 °C

-61.92

-61.14

-61.53

-69.79

c(borax) = 30 mmol L-1, T = 15 °C

-54.97

-53.93

-54.45

-58.12 c

c(borax) = 30 mmol L-1, T = 20 °C

-55.55

-54.27

-54.91

-58.95 c

c(borax) = 30 mmol L-1, T = 25 °C

-55.67

-55.21

-55.44

-59.64 c

c(borax) = 30 mmol L-1, T = 15 °C

-55.27

-53.79

-54.53

-58.12 c

c(borax) = 30 mmol L-1, T = 20 °C

-54.59

-53.36

(-53.98)

-58.95 c

c(borax) = 30 mmol L-1, T = 25 °C

-55.31

-55.23

-55.27

-59.64 c

c(borax) = 40 mmol L-1, T = 15 °C

-50.43

-50.12

-50.28

-51.98 c

c(borax) = 40 mmol L-1, T = 20 °C

-50.70

-51.03

-50.87

-53.83 c

c(borax) = 40 mmol L-1, T = 25 °C

-51.77

-51.75

-51.76

-55.40 c

c(borax) = 40 mmol L-1, T = 15 °C

-50.46

-49.58

-50.02

-51.98 c

c(borax) = 40 mmol L-1, T = 20 °C

-50.50

-50.59

-50.55

-53.83 c

c(borax) = 40 mmol L-1, T = 25 °C

-51.89

-51.57

-51.73

-55.40 c

c(borax) = 50 mmol L-1, T = 15 °C

--------a

-45.99

-45.99

-47.78

c(borax) = 20 mmol

L-1,

1st series

2nd series

1st series

2nd series

c(borax) = 50 mmol

L-1,

T = 20 °C

-47.44

-46.57

-47.01

-48.83

c(borax) = 50 mmol

L-1,

T = 25 °C

-48.09

-48.17

-48.13

-49.70

c(borax) = 60 mmol L-1, T = 20 °C

--------a

-43.66

-43.66

-45.67

c(borax) = 60 mmol L-1, T = 25 °C

-45.06 -45.02 -45.04 -45.77 c not determined, calculated from value for SNP12 and SNP22, mean values with respect to first and second series a

b


Page 59 of 60

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- 59 -

Table 4. Electrokinetic surface charge density σζ, (obtained via Eq. (6)) and mean effective charge number zeff for the nanoparticles investigated compared to electrokinetic surface charge density σζ of capillary inner wall (obtained via Eq. (7)). Buffer concentration, temperature

SNP12 σζ/(C m-2)

SNP12 zeff

SNP22 σζ/(C m-2)

SNP22 zeff

Capillary σζ/(C m-2)

c(borax) = 20 mmol L-1, T = 15 °C

-0.0429

-229

-0.0389

-710

-0.0430

T = 20 °C

-0.0411

-219

-0.0377

-688

-0.0428

c(borax) = 20 mmol L-1, T = 25 °C

-0.0403

-215

-0.0375

-683

-0.0426

c(borax) = 30 mmol L-1, T = 15 °C

-0.0432

-231

-0.0402

-732

-0.0425 b

c(borax) = 30 mmol L-1, T = 20 °C

-0.0429

-229

-0.0396

-722

-0.0420 b

c(borax) = 30 mmol L-1, T = 25 °C

-0.0419

-223

-0.0394

-719

-0.0413 b

c(borax) = 30 mmol L-1, T = 15 °C

-0.0436

-232

-0.0400

-730

-0.0425 b

c(borax) = 30 mmol L-1, T = 20 °C

-0.0419

-224

-0.0387

-706

-0.0420 b

c(borax) = 30 mmol L-1, T = 25 °C

-0.0415

-221

-0.0395

-719

-0.0413 b

c(borax) = 40 mmol L-1, T = 15 °C

-0.0440

-235

-0.0418

-762

-0.0421 b

c(borax) = 40 mmol L-1, T = 20 °C

-0.0434

-231

-0.0418

-763

-0.0428 b

c(borax) = 40 mmol L-1, T = 25 °C

-0.0434

-232

-0.0415

-757

-0.0431 b

c(borax) = 40 mmol L-1, T = 15 °C

-0.0440

-235

-0.0412

-751

-0.0421 b

c(borax) = 40 mmol L-1, T = 20 °C

-0.0432

-230

-0.0414

-755

-0.0428 b

c(borax) = 40 mmol L-1, T = 25 °C

-0.0435

-232

-0.0413

-753

-0.0431 b

c(borax) = 50 mmol L-1, T = 15 °C

--------a

--------a

-0.0416

-760

-0.0421

c(borax) = 20 mmol

L-1,

1st series

2nd series

1st series

2nd series

c(borax) = 50 mmol

L-1,

T = 20 °C

-0.0442

-236

-0.0414

-755

-0.0420

c(borax) = 50 mmol

L-1,

T = 25 °C

-0.0438

-234

-0.0421

-768

-0.0417

c(borax) = 60 mmol L-1, T = 20 °C

--------a

--------a

-0.0417

-761

-0.0423

c(borax) = 60 mmol L-1, T = 25 °C

-0.0439

-234

-0.0422

-770

-0.0411

a

not determined, bmean values with respect to first and second series


Langmuir

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Table of Contents/Abstract Graphic

3

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0 0

5

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10

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25

30

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25

Determination of the Exact Particle Radius Distribution for Silica

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