Size characterization

Maybe the most basic property of a nanoparticle, besides its material composition, is its size. When dispersed in a solvent, this size is not only the diameter of the hard inorganic particle core but includes its organic ligand shell, possibly other, more complex molecules on the particle surface, and solvent molecules. This yields to an effective, hydrodynamic diameter that is characterized by the diffusion of the particle in its solvent, by a number of different techniques, the diffusion constant can be determined from which then the hydrodynamic diameter is calculated via the Stokes-Einstein relation.

In dynamic light scattering (DLS), the thermal diffusion of particles causes intensity fluctuations of the light scattered by the particles. From the auto- or cross-correlated signal the diffusion constant can be derived, and with appropriate models the particle size distribution. This well-established method is fast and convenient to perform, it has been applied extensively to nanoparticles.[228, 262, 264, 297, 305, 313] However, the scattering cross-section strongly depends on the particle diameter, so large

33 particles contribute with a strong weight to statistics, and the transition from an intensity distribution to a volume or number distribution should to be performed with care. Furthermore, the organic ligand layer is often neglected, as are thicker organic coating shells or biomolecules, yielding to a composite system with different optical properties and their size relation, e.g. in case of small inorganic nanoparticles of given size with possibly several organic layers of varying thickness.

Fluorescence correlation spectroscopy (FCS) is based on the diffusion of fluorescent particles or molecules into and out of the focal volume of a confocal optical microscope. By the confocal setup, only fluorescent light from particles in the focal volume is detected, at high dilution single particles cause intensity fluctuations, of which the dynamics is characterized by the autocorrelation, from which the diffusion constant can be derived. By principle, this method is limited to fluorescent nanoparticles but has been applied with great success to quantum dots of different materials.^[427-429]

Thermophoresis is the effect that small particles move along a temperature gradient, commonly towards to cooler regions as it has been explained by solvation energy.^[430] For particle sizing, a laser can be used to heat a small spot in a flat cuvette, resulting in a static radial temperature gradient in which the particles are depleted. While recording the spatial distribution of the fluorescent particles, the heat gradient is switched off and the nanoparticles diffuse back resulting in an isotropic distribution. The diffusion constant can by derived from the dynamics of the spatial particle distribution, in a way very similar to the technique of fluorescence recovery after photobleaching (FRAP) that has been used to study the lateral diffusion in cell membranes.^[3]

Another related technique is based on single particle tracking, where the diffusion movement of nanoparticles is directly observed under an optical microscope.^[431-433] Analysis of the particle tracks and statistic evaluation allows for calculation of the diffusion coefficient and hence the particle size.

All these techniques are based on free diffusion of the nanoparticles, interacting only with their solvent. A number of other techniques rely on a force field applied to the nanoparticles, or a solid matrix that interacts with the particles in solution.

Analytical ultracentrifugation exploits particle sedimentation for the size characterization.

Sedimentation velocity depends on the different mass densities of particles and medium, and on the particle size. While this method works well for homogenous particles of known density, the density of more complex core/shell nanoparticles strongly depends on the thickness of organic shell, which has to be taken into account for the model applied to derive the particle size.[434, 435] Alternatively, magnetic sedimentation has been applied.^[436]

Size exclusion chromatography (SEC), also known as gel filtration or gel permeation chromatography (GPC), is a technique commonly used for the characterization of polymers and proteins. The separation by particle size takes place in a column packed with a porous gel as matrix, termed stationary phase. The sample is applied under a flow of a solvent, termed mobile phase. While passing the column, small particles can diffuse into the pores of the stationary phase while larger particles are sterically excluded. This results in the retention of small particles that spend more time in diffusing into the pores and out again, while large particles come out first. The stationary phase is characterized by its pore size which allows separation of particles of a certain size range, and size characterization after calibration with standards of known size, e.g. polymers or globular proteins, which allows for an universal calibration based on the particle volume, in contrast to molecular

34 weight.[437, 438] Size exclusion chromatography has been applied to a number of nanoparticles of different materials in both organic solvents[234, 439-442] and aqueous phase,[3, 209, 228, 235, 264, 332, 443, 444] in some cases also in preparative scale. In addition, the separation of nanoparticles of different shapes by SEC has been reported.^[445] Often, this method is limited by the colloidal stability of the particles which can irreversibly aggregate or adsorb to the stationary phase by loss of ligand molecules or reduced electrostatic repulsion in presence of salt. Therefore, additional free ligand molecules or other surfactants are commonly added to the mobile phase.^{[442, 446]}

Electrophoresis is the movement of particles in presence of an electric field, also termed electrokinetic phenomena.^[447] Charged particles are attracted by the oppositely charged electrode, the electric field is mediated by an electrolyte as conductive medium that also causes a hydrodynamic drag on the particles, resulting in a constant velocity in a constant electric field.

Qualitatively, small negatively charged particles migrate faster to the positive electrode than large ones, as do stronger charged particles, while positively charged particles will move towards the negative electrode. Quantitatively, electrophoresis is under certain restrictions described by Henry’s equation, relating electrophoretic mobility to particle size, charge (zeta potential) and ionic strength of the surrounding medium. However, real systems are often found in the intermediate regime where simple approximations such as the Hückel and the Helmholtz-Smoluchowsky limits for low and high ionic strength, respectively, may not be valid, or the shell of ligand molecules is found to strongly influence the mobility.^{[150, 448]}

Free electrophoresis is used for separation of nanoparticles by capillary electrophoresis,^[449-453] where small amounts of samples can be analyzed with high resolution. Gel electrophoresis employs usually poly(acryl amide) or agarose,^[454] a polysaccharide extracted from algae, to form a gel with the conductive medium. The gel avoids flow in the medium, e.g. by convection, and secondly serves as separation matrix by a sieving effect due to the pores.^[455] Agarose gel electrophoresis is commonly used for water-soluble nanoparticles because it allows analysis of a large number of samples in parallel and direct comparison to each other by convenient optical detection, i.e. a camera. In addition, the separation is sometimes superior to other methods (e.g. SEC) allowing for the separation of nanoparticles with a discrete number of attached molecules, as discussed previously.

So far, quantitative analysis of particle size and possibly the conformation of molecules to the nanoparticles has been addressed in different systems, however, the validity of the applied models and approximations should be confirmed carefully.^[456] Colloidal nanoparticles, in particular if decorated with large, soft and complex molecules, are not ideal rigid spheres with a certain homogenous surface charge as commonly assumed, only recently also soft shells have been described theoretically.[457, 458] In real systems, the charge distribution around the particles is not known, and little is known about dynamic effects in presence of an electric field or gel matrix (which might by itself be hard to characterize),^[455] in particular for complex nanoparticle-biomolecule conjugates.^[295] While it is clear that separation is based on both size and charge of the particles, the experimental data can often not be fully interpreted quantitatively^{[150, 453]} or even qualitatively, as in the case for nanoparticles modified with poly(ethylene glycol), a neutral polymer, in which the particles migrate towards the negative electrode, implying a positive net charge (cf. Figure 7, left).

While this effect has been observed by several groups even for methoxy-terminated PEG,[1, 305, 456] no well-supported explanation seems available so far. While gel electrophoresis is mostly applied to nanoparticles for analytical purposes,[4, 350, 448, 459, 460] nanoparticles can be also extracted from the gel

35 for further experiments[1, 293, 294] and also column electrophoresis with continuous elution has been reported.^[461]

Depending on the nanoparticle material, the particle size often directly affects other physical properties that can be used for size characterization. Maybe the most prominent example is optical spectroscopy: The characteristic plasmon peak of gold nanoparticles depends on the nanoparticle size, by its position in the absorption spectrum in terms of energy or wavelength, the particle size can be calculated according to Mie scattering theory.^[462] While the experimental spectra fit well to model calculations for large nanoparticles, the size-dependence of the spectral position of the plasmon absorption decreases for nanoparticles around or smaller than 10 nm in diameter,^[463] which is attributed to increasing surface effects, besides the influence of the refractive index of the surrounding medium.^[464] In Figure 10, absorption spectra of spherical gold nanoparticles with different diameters are displayed. Plasmon resonance is also found for other metal particles like silver, and for rod-shaped particles that exhibit two peaks resulting from a longitudinal and transversal mode. Recently, the absorption of rods and other shapes has been modeled theoretically with excellent agreement to experimental data.^[465] In fluorescent semiconductor nanoparticles, spatial confinement of the wave functions of electron and hole of an exciton results in a size-dependent quantization of the energy levels, and thus size-size-dependent absorption and emission spectra. Being one of the key features of colloidal quantum dots, this effect can also be used to estimate the size of such particles from their optical spectra by means of empirically found calibration curves.^[466] However, for core/shell particles such as CdSe/ZnS there is no such data available, thus commonly the e.g. ZnS shell is neglected which, in fact, is equal to the assumption of a larger, homogenous CdSe particle.

Figure 10: Optical absorption spectra of gold nanoparticles of different size. The spectral position of the plasmon peak changes very little for small particle diameters. The peak width is influenced by the size distribution of the sample in the cuvette, single-particle spectroscopy yields to more narrow peaks. All spectra shown have been normalized.

Together with gravimetric methods, i.e. drying and weighting the mass of a sample after recording the absorption spectrum, and with the particle diameter from TEM, optical absorption spectroscopy is commonly used for concentration measurements, by applying the empirical extinction coefficient to Beer’s law. While this approach works reasonably well, is convenient in application and yields

36 reproducible results, often additional organic shells are neglected (despite in thermogravimetry) which can result in deviations of a few to several ten percent in regard to absolute numbers for concentrations.

In addition, scanning probe microscopy including AFM (atomic force microscopy) have been used for the characterization of nanoparticles immobilized to flat surfaces.[264, 467-469] In combination with fluorescence microscopy, single quantum dots exhibiting differences in quantum yield and the presence of dark quantum dots could be accessed.^[470]

Small-angle x-ray and neutron scattering (SAXS^[471-473] and SANS^[474]) have as well been employed for colloidal nanoparticles, the size and possibly also the core/shell architecture of nanoparticles can be derived by means of a layer model with known parameters.

Figure 11: Nanoparticles saturated with PEG of different molecular weight, “0 g/mol” are plain polymer-coated nanoparticles, hydrodynamic diameter determined by different techniques. Dotted lines are theoretical values consisting of a hypothetical core diameter (10, 15 or 20 nm) plus two times the thickness of the PEG layer that depends on the molecular weight (hydrodynamic diameters for PEG from literature, determined by SEC, data for particles plotted from reference^[3]).

To summarize, there are a number of different techniques with different measurement strategies available that have been successfully applied to the size determination of nanoparticles and to the characterization of particle modification based on the change in size e.g. due to additional molecules or coating layers. However, when absolute values of different methods are compared, often deviations are found that are larger than expected. In Figure 11, diameters of nanoparticles with PEG of different molecular weight is shown, determined by different methods. The observed deviations may be due to the fact that “size” is measured by means of different physical effects, involving possibly different physical properties of the particles, e.g. by interaction of a matrix, or when dry particles vs. particles in solution are compared. Secondly, depending on the method, deviations contribute differently to statistics, as data is collected in completely different ways, i.e. in ensemble or single-particle measurements. Naturally, reliably results are obtained by application and comparison of several complementary methods.[3, 178, 297]

37 Finally, a number of techniques are limited to certain classes of particles, i.e. inorganic particles (TEM) or fluorescent particles (FCS, thermophoresis), while others offer the additional benefit of optional preparative or semi-preparative separation, as found for gel electrophoresis, SEC and ultracentrifugation, in contrast to purely analytical methods.