Comprehensive Characterization of Nano- and Microparticles by In-Situ Visualization of Particle Movement Using Advanced Sedimentation Techniques

Abstract

The state of a suspension is crucial with regard to processing pathways, functionality and performance of the end product. In the past decade, substantial progress has been made in designing highly specialized and functionalized particles. In current particle technology, besides classic particle properties such as particle size distribution, shape and density, surface properties play an essential role for processing, product specification and use. For example, in medical therapy, analytical diagnostic applications, as well as in separation processing and harvesting of high-valued materials, magnetic micro- and nanoparticles play an increasing role. In addition to traditional parameters such as size, the particle magnetization has to be quantified here.

Sedimentation techniques have been used for hundreds of years to determine the geometrical characteristics of dispersed particles. Numerous national and international standards regarding these techniques have been published. Mainly due to the fast growing market share of laser scattering techniques over the past two decades, most customers these days are not aware of some advantageous features of particle characterization via a first-principle fractionating approach such as sedimentation. This is unfortunate as sedimentation techniques have made huge technological leaps forward regarding electronics, sensors and computing abilities.

This paper aims to give a short review about different cumulative and incremental sedimentation approaches to measure the particle size distribution. It focuses mainly on the in-situ visualization (STEP-Technology^®) of particle migration in gravitational and centrifugal fields. It describes the basics of the new multi-sample measuring approaches to quantify the separation kinetics by spatial and time-resolved particle concentration over the entire sample height. Based on these data, the sedimentation velocity and particle size distribution are elucidated and estimates of accuracy, precision and experimental uncertainties are discussed. Multi-wavelength approaches, correction of higher concentration, and the influence of rheological behavior of continuous phase will also be discussed. Applications beyond the traditional scope of sedimentation analysis are presented. This concerns the in-situ determination of hydrodynamic particle density and of magnetophoretic velocity distributions for magnetic particulate objects.

1. Introduction

Particles were ubiquitous on our and on other planets all across the galaxy long before mankind. Regarding particle separation methods, winnowing grain in ancient Egypt is perhaps one of the oldest approaches to separating/classifying particles by their mass. Sir George G. Stokes’ work about the resistance of a particle moving relative to a viscous incompressible liquid marks the milestone of “modern” sedimentation analysis. He linearized the corresponding general equation of motion, arrived at the time-dependent equation of creeping motion and applied it to the frictional damping of a spherical pendulum bob (Stokes, 1851). The obtained relationship between the resistance and the movement of such a spherical body at constant slow velocity is today known as Stokes’ law and forms the basis of all sedimentation techniques. Since the days of Stokes, additional other techniques were developed to measure the particle size. The ISO Technical Committee ISO/TC24/SC4 “Particle Characterization” alone counts 13 active working groups dealing with numerous different physical principles to determine particle size distributions (ISO, 2017). There are plenty of articles and numerous books dealing with size measuring techniques and approaches focusing on the needs of academia, regulatory administration or industry. We restrict ourselves to referencing several books of the last 10 to 20 years (Leschonski, 1988), (Allen, 1999), (Babick, 2016), (Masuda et al., 2006), (Merkus et al., 2014), (Schärtl, 2007). It should be emphasized that most commercial sizing analyzers feature high repeatability. But the reproducibility of these techniques depends in large part on representative sampling, sample size and sample preparation (e.g. (Sommer, 1986), (ISO 14887:2000), (Jillavenkatesa et al., 2001)).

General principles

Any measurand depending directly or implicitly on particle size can be used for size determination. Due to the various dependencies, developed different measuring techniques “see” a particle with different eyes. Particle sizing techniques may be grouped based on particle characteristics such as geometrical properties, mobility in external force fields, and signals due to interaction responses with externally applied fields. Most straightforward is the visual analysis of geometrical parameters by light, scanning electron or transmission electron microscopy. These techniques have the advantage that they can evaluate each individual particle. But even for this relatively simple technique, the primary measuring information is not the size but the 2D contour of the particle. It depends on the orientation of the particle with regard to the imaging focus plane. Image analysis of falling particles reduces orientation effects, but is limited to particles of micrometer scale. Other single particle analyzing techniques are based on counting principles. Particle detection is made by electrical or optical means. Well known is the Coulter Counter principle (electrical detection, (Coulter, 1953)) famous for blood cell counting. These instruments allow precise size and count determinations for narrow particle size distributions. Special single particle optical sizing detectors (White, 2002) enlarge the dynamic range and are widely used for tail detections of droplet size distributions of emulsion (USP-788, 2012), (USP-729, 2016). Measuring techniques with hydrodynamic focusing allow for an especially high size resolution (Shuler et al., 1972). Lichtenfeld was able to discriminate between 600 nm and 610 nm particles as well as to measure changes in thickness of individual layers of about 0.5 nm of LBL-coated particles by means of an optical detection system (Buske et al., 1980), (Lichtenfeld et al., 2004). Due to the latest EU regulations regarding the classification of nanomaterials (European Union, 2011), (Linsinger et al., 2012) improved methods are needed for providing direct number-weighted size distributions for nanoparticles.

Today, most laboratories employ ensemble methods such as static (SLS) or dynamic (DLS) light scattering as their standard analysis methods, (Allen, 1999), (Schärtl, 2007), (Merkus, 2009), (ISO 13320:2009), (ISO 22412:2017). Classic SLS analysis is based on the Fraunhofer diffraction theory, stating that the intensity of light forward-scattered by a particle bigger than the wavelength is directly proportional to the particle size (Ward-Smith et al., 2013). Size distribution is specified volume-weighted despite the fact that primary measurement data is area-weighted. Fraunhofer-based size analysis should only be used for micron-sized particles. For submicron particles, the Mie scattering theory has to be employed (Mie, 1908), (van de Hulst, 1981). However, since most real-world particles are non-spherical, any analytical results should be viewed with caution. Furthermore, the results are influenced by the deconvolution algorithms used by individual manufacturers. For additional details, refer to (Kuchenbecker et al., 2012), (Kelly et al., 2013).

In contrast, DLS is based on the temporal fluctuation of scattered laser light (speckle pattern) reflecting temporal fluctuations of the particle position due to Brownian motion (Schärtl, 2007). The autocorrelation function allows determination of the mean travel distance, and based on this, the diffusion coefficients of particles. This information can be converted by the Stokes-Einstein relation to a hydrodynamic equivalent spherical particle size distribution. Obtained distributions are intensity-weighted. The dynamic range is from about 1 nm to 1 μm. Review in nanoparticle sizing using DLS were very recently given by (Zhou et al., 2017).

Particle tracking analysis (PTA) is another emerging technique based on tracking the stochastic movement (trajectories) of individual dispersed particles due to Brownian motion (diffusion) by light scattering microscopy (ISO 19430:2016). Similar to DLS, hydrodynamic spherical equivalent diameters are obtained. Distribution is number-weighted and covers a size range from about 50 nm to 1 μm. Estimation of particle concentration can be given but has to be treated with care (Filipe et al., 2010), (Chenouard et al., 2014).

Finally, acoustic spectroscopy is another measurement principle to obtain particle size information based on interaction with externally applied fields. It is based on measuring the attenuation of ultrasound at a range of MHz frequencies. The attenuation at these different frequencies (raw data) is used for calculating a particle size distribution. In contrast to optical methods, the material parameters and acoustic properties such as sound speed, density and heat conductivity have to be known to come up with a particle size. Particle sizes can be measured in the range of 10 nm to 3 mm. In contrast to, e.g. SLS techniques, the volume concentration of particles should meet a given critical concentration. On the other hand, developed theories take into account particle-particle interactions, and particle size distributions can be extracted from attenuation spectra in concentrated systems (ISO 20998-1:2006), (Dukhin et al., 2010), (Zhou et al., 2017).

In our review, we aim to give an overview regarding different sedimentation approaches to measuring particle size distribution (ISO 13317-3:2001), (ISO 13318-2:2007), (ISO 13318-3:2004). In addition, we would like to point out to the reader the analytical ultracentrifugation (AUC) method. Numerous publications regarding particle characterization (see, e.g. (Mächtle et al., 2006), (Planken et al., 2010), (Uchiyama et al., 2016)) were published over nearly the last 100 years since its invention by (Svedberg et al., 1924).

Classic sedimentation techniques can be classified according to particle distribution at the beginning (line start or homogeneous sedimentation), to the measuring principle (incremental/differential or cumulative/integral) and to the driving force field (earth gravity or high gravity by centrifugation) (e.g. (Allen, 1999), (ISO 13317-1:2001), (ISO 13318-1:2001)).

This article focuses on the most recent and advanced sedimentation techniques. In particular on one using STEP-Technology^®, which allows the direct and in-situ observation of particle migration in gravitational and centrifugal fields. The primary experimental data obtained by this technique are terminal particle sedimentation velocities (first-principle measurement) obtained from the kinetics of spatial concentration changes within the sample over time due to gravity fields. Particle velocity distributions allow detailed characterization of dispersed particles such as size and/or density modalities, and broadness and kind of distribution. Based on experimental particle velocities, which demand neither prior assumptions nor particle and liquid material constants, the size distribution of particles can be derived via Stokes’ law (Stokes, 1851). It results in an extinction-based size distribution according to the transmission measurement principle. Volume (number)-based particle size distribution calculation demands optical particle properties such as the refractive particle index and particle shape. The Mie theory derived for spherical particles (Mie, 1908), (van de Hulst, 1981) is commonly used for taking into account the refractive contrast between particles and liquid. With regard to volume- and number-weighted particle size calculations based on extinction-weighted distribution, we restrict ourselves to a short overview, as this part is fairly standardized (ISO series 13317 and 13318).

There are numerous references and books dealing with various aspects of particle characterization but in general they all center on particle sizing. The third part of this paper reports on the extension of the traditional scope of sedimentation size analysis towards particle characterization in a more general sense. This includes in-situ determination of the density of particles dispersed into liquids (ISO 18747-1:2018), magnetization and magnet responsiveness of particles.

While this article is about the current state of sedimentation research, it centers somewhat on our own research and experience.

2. In-situ visualization of particle migration—STEP-Technology^®

Conventional cumulative sedimentation techniques record the amount of settled particles. The reader may think of the weight-balance approach, where the weight of settled particles is recorded on a special tray (ISO 13317-4:2014). Another example is the time-dependent detection of the hydrostatic pressure at a given position in the lower part of the cell (Bickert, 1997). On the other hand, incremental methods determine the change of particle concentration in a small measuring zone of known position by optical or X-ray measuring systems. The latter techniques provide concentration data of the sample only from a small fixed volume at a time t. Scanning sensors (Mächtle et al., 2006) typical for AUC, provide more information but do not allow for the capture of fast concentration profile changes.

Initially, particle settling was analyzed at gravity (e.g. Ladal photosedimentometer, Paar Lumosed). An early centrifugal field application was invented by (Whitby, 1955). Known is the start line incremental disc rotor technique and was first used by (Kaye, 1962), and today is marketed by CPS Instruments Inc., Stuart, Florida, and Brookhaven Instruments Corp., Holtsville, New York. Homogeneous incremental methods were also available using cuvette-based photocentrifuges up to the turn of the last century from Japanese manufacturers such as Horiba Scientific Ltd. (e.g. CAPA 500), and Shimadzu Corp. (e.g. SA-CP3). All of these techniques pass a more or less parallel, focused light beam (ROI about 0.2 mm up to 1 mm) through the sample and measure at a single given position p the changes of transmission over time due to settling or creaming of particles (Fig. 1a, b). These techniques therefore provide the concentration only at a distinct small fixed data-point at a time t_i in the sample. Scanning sensors are typical for AUC and provide more information but do not allow recording an entire concentration profile over sample height at selected points of time.

Fig. 1

Schematic drawing (^© by LUM GmbH, Berlin) of homogeneously distributed particles in a sample cell at the start of the experiment (t = 0) and the corresponding constant particle concentration over the entire height of the sample (a). Classic techniques monitor changes of particle concentration due to sedimentation or creaming at a small volume with no information of spatial concentration (b). STEP-Technology^® quantifies the concentration profile over the entire sample at selected points of time (c).

As Fig. 1c displays, more comprehensive information can be obtained illuminating the entire sample by means of a narrow line beam of parallel light I₀ and recording the transmitted light intensity I_t instantaneously for any position p and any set time t by multiple sensors. This measuring technique allows the user to record concentration changes across the entire sample height due to the migration of particles or droplets driven by earth gravity or centrifugal fields, respectively, as well as, e.g. magnetic fields (Mykhaylyk et al., 2015).

Fig. 2 displays the main features of patented STEP-Technology^® (Space- and Time-resolved Extinction Profiles) schematically in more detail as implemented in the multi-sample analytical photocentrifuge LUMiSizer^® (AC-L, LUM GmbH, Germany). A linearly shaped parallel NIR or VIS light beam passes through the optical cells placed horizontally into one of the 12 channels of a disk rotor. The channel positions at the rotor are indicated by pale areas and numbering 11, 12, 1 and 2 in Fig. 2. Particles scatter the light, resulting in an attenuation of the forward beam. This forward intensity is recorded during centrifugation in real time as a function of the radial position p over the entire sample height with a space resolution of 13.7 μm. In the case of, e.g. a gravity sedimentometer LUMiReader^® (GS-L, LUM GmbH, Germany), it amounts to 8.7 μm.

Fig. 2

Principle of measuring space- and time-resolved extinction profiles (displayed as transmission profiles) implemented in an analytical photocentrifuge (© by LUM GmbH, Berlin). (a) Transmission profile of an unseparated sample at t₀. (b) Transmission profile (green line) at t₁. Dotted lines indicate the situation at t₀ and t₂ for comparison. (1) light source(s); (2) parallel NIR or VIS light; (3) sample cell; (4) multi-sample rotor with indicated channel numbers 11, 12, 1 and 2; (5) line detector

In this way, space-resolved intensity profiles are obtained at adjustable time steps, monitoring concentration changes over time due to the sedimentation or creaming of dispersed particles. The recorded intensity at every position I(r_i, t_j) is normalized by I₀(r_i) measured with no cells (empty rotor) as is usually done in a photometer. The change of transmission and extinction profiles during centrifugation are calculated according to Eq. 1

  

$$\begin{array}{l}\frac{I(r_i,\hspace{0.17em}{t}_j)}{I_0(r_i)}=T(r_i,\hspace{0.17em}{t}_j)\\ =\text{exp}(-E(r_i,\hspace{0.17em}{t}_j\left)\right)\\ =\text{exp}(-A_{\text{v}}·\phi (r_i,\hspace{0.17em}{t}_j)·L)\end{array}$$(1)

and displayed by the software package SEPView^® as fingerprints of transmission or extinction (Fig. 3a, b).

Fig. 3

Progress of separation over time visualized and recorded by STEP-Technology^®. Fingerprints of particle separation can be quantified based on transmission (a), extinction (b) or clarification (c). The first recorded profile (red) displays the initial particle distribution of the sample just after centrifugation start (< 1 s). The green profiles visualize the final distribution of particles over sample height (position r) due to separation.

At the position of about 107 mm (distance from the center of revolution), the meniscus (filling height of sample) is identified as well as the bottom of the sample cell (130 mm). As indicated, the first red profile corresponds to the first recorded measurement, e.g. one second after start. The sharp decline (a, c) and increase (b) of green profiles at 107.4 mm indicates the interface between suspension and sediment at the end of the separation process (here by centrifugation). Clarification is another visualization approach. In this case, the initial transmission I(r_i, t₀) is subtracted from any later recorded transmission profile I(r_i, t_j) and the incremental change is displayed (Fig. 3c). For details refer to (Detloff et al., 2014). Visualization of the separation process by the change of transmission profiles at wavelengths of visible light gives similar information to an observation by the naked eye. Extinction profiles, on the other hand, are directly related to the corresponding concentration alterations within the sample according to Lambert-Beer-law (Eq. 1, 3^rd line, right part). Fig. 4 displays experimentally determined extinction for β-Methyl Orange solutions of different concentration (points) determined by the AC-L and a calibrated UV/VIS spectrophotometer Lambda 2 (Perkin Elmer). The correlation is very satisfying. The slight deviation from the ideal slope of unity is explained by the spectral line half-width of used LED as light source of AC-L which amounts to 25 nm compared to the monochromatic source (470 nm +/− 3 nm) of the calibrated spectrophotometer.

Fig. 4

Extinction of different concentrations of β-Methyl Orange (0.014 g/ml–0.142 g/ml, optical glass cells) measured by a calibrated photometer (y-axis) and an AC-L (x-axis). Points: Experimental data of different concentrations; Line: Linear fit.

The response (extinction) of the sensor systems of AC-L was also validated for a concentration range of more than 3 decades of different dispersed phases as shown in Fig. 5 (Detloff, 2004). The theoretical extinction values were calculated by the Lambert-Beer law combined with a semi-empirical approach to account for multiple light scattering at higher particle volume concentrations. An error function was determined which is subtracted from the extinction values of the Lambert-Beer law. This function does not depend directly on the particle size x. The information about the differences in the particle sizes are delivered by the volumetric scattering cross-section A_V calculated by the Mie Theory. Only one set of parameters in the empirical part is required to calculate the extinction. The parameters were determined by fitting to experimental data (Detloff et al., 2007).

Fig. 5

Natural extinction of different silica and latex suspensions (280 nm to 1550 nm particle size, volume fractions from 0.014 % to 14.9 %) measured by an AC-L (x-axis) and theoretically calculated (y-axis) by the Lambert-Beer law extended by a semi-empirical approach for multiple scattering (Detloff et al., 2007).

The light beam, penetrating the suspension or emulsion, interacts with both the particles (dispersed phase) and the continuous phase. The NIR-range of 850 nm–900 nm is advantageous to minimize the influence of optical properties of the continuous phase. For these wavelengths, most liquids and solutions are transparent independent of their color at visible light. Fig. 6 displays the transmission changes monitored by STEP-Technology^® at NIR (top) during centrifugation over the entire dispersion sample made of an organic pigment. No separation is detectable by the naked eye at visual light (photo middle). In contrast, recorded concentration changes could be also visualized by photographic means (Fig. 6, bottom), if the sample was illuminated in a special set-up with light in the NIR- range. Originally, this approach was invented to quantify the settling of red particles (red blood cells) in a continuous phase (blood plasma) of different color due to medication or surgery (Lerche et al., 1988), (Lerche et al., 1992).

Fig. 6

Separation kinetics of a pigment recorded by STEP-Technology (NIR) and visual appearance of the sample cell at the end of centrifugation (last green profile) photographed by visible light (middle) and by a special NIR illumination set-up (bottom), respectively. Note that the sample cell is lying horizontally in the rotor with the applied centrifugal acceleration being from left to right.

Visible light on the other hand, e.g. blue wavelengths, has the advantage that it increases the sensitiveness towards much lower particle concentrations and nanoparticles as shown in Fig. 7. Scattering of non-adsorbing particles decreases with decreasing particle radius by (x/2)⁶ according to Rayleigh, and correspondingly for NIR wavelengths, the dispersion sample is more transparent compared to blue light as indicated by the first profile (red) of each fingerprint. Accordingly, the total transmission change amounts to 45 % and 75 %, respectively, for complete separation of particles (indicated by green profiles).

Fig. 7

Fingerprints of the separation behavior of the same silica nanoparticle suspension quantified by transmission changes recorded by NIR (λ = 870 nm) or blue light (λ = 470 nm) illumination. Arrow lengths indicate the difference between transmission of the initial sample and the particle-free supernatant, respectively.

Eq. 1 obeys the linearity between extinction and particle concentration. In other words, multiple scattering of particles at higher particle concentration should be avoided. The applicability of the Lambert-Beer law regarding particle concentration can be broadened with no sample dilution by employing cells with shorter optical path (virtual dilution). As Fig. 8 verifies, using sample cells of an optical path length as small as 1 mm allows working at 10-times higher volume concentrations with no sample dilution compared to 10-mm cells. (Luigjes et al., 2012), (Erne, 2012) reported on super thin made glass cells of 50 μm path length allowing the analysis of magnetic iron oxide particles dispersed in oleic acid-decalin of concentrations up to 300 g/l and measurement of osmotic pressures of strongly adsorbing and very concentrated dispersions.

Fig. 8

Maximum concentration of β (Methyl Orange) accurately measured (linear range of detector) by AC-L using optical glass cells with path lengths from 0.1 cm to 1 cm.

Typical dispersion fingerprints—patterns of transmission profiles—visualizing the behavior of different types of suspensions and emulsions at original concentration (no dilution) are shown in Fig. 9. Sample fingerprints allow classification of the different types of dispersions based on their separation behavior over time. SEPView^® provides the quantitative and direct evaluation of fingerprints by different analysis tools. The instability or separation index (Detloff et al., 2014), integral transmission and front tracking (Lerche et al. 2001), (Sobisch et al., 2010) or first derivative (Wiese, 2010) allow application-specific comparison and ranking of differently designed formulations and products and identification of destabilization phenomena as well as prediction of shelf life (Badolato et al., 2008), (Lerche et al., 2011), (Takeda, 2012), (Staudinger et al., 2014), (Lerche et al., 2014), (Cai et al., 2018). Nowadays, the use of AC-L gains also large interest in characterizing highly concentrated dispersions, especially in the field of centrifugal filtration and sediment consolidation (Usher et al., 2013), (Cao et al., 2014), (Loginov et al., 2017a, 2017b) as well as mechanical properties (rigidity, deformability) of e.g. hydrocolloid beads (Schuldt et al., 2018).

Fig. 9

Typical fingerprints of different types of dispersions recorded by an analytical space-resolving photocentrifuge (AC-L): (a) monomodal monodisperse suspension, (b) tetramodal monodisperse suspension (c) polydisperse suspension (d) particle-particle interaction, flocculated suspension (e) emulsion (f) suspoemulsion (flotation and sedimentation).

On the other hand, particle characteristics such as size, density and magnetic responsiveness can be quantified, upon which we focus in this paper.

3. Particle sizing in line with ISO 13317 and 13318

3.1 Particle velocity distribution

Any experimental set-up to analyze particle size must measure the terminal (stationary) velocity of particles (solid particles or droplets) due to gravity or centrifugal fields, respectively. If the density of dispersed particles is higher than that of the continuous phase, they settle. They cream or float in the opposite case. Conventionally, sedimentation velocity is determined by measuring the required time t for particles to settle from the filling height (meniscus) a fixed distance h (gravity sedimentation) or to a fixed radial position r (analytical centrifugation), respectively. This is the typical procedure in the case of line start centrifugal sedimentation techniques (Constant Position Approach, CPA) based on the time course of registered intensity of the emergent optical beam (ISO 13318-1:2001). Alternatively, the particle velocity can be determined by the “Constant Time approach (CTA)” (Detloff et al., 2006), (ISO 13318-2:2007) measuring the traveled distance of particles in a given time t by spatial resolving techniques.

  

$$v(r,\hspace{0.17em}t)=\frac{r-r_{\text{m}}}{t}$$(2)

Here, v denotes the particle velocity and r_m the position where the particles start to sediment/cream (sedimentation = position of the meniscus; creaming = position of the cell bottom). For a suspension of monodisperse particles, all profiles, independent of measurement time t_i, have a vertical transmission profile (horizontally placed sample cell) and therefore all particles have moved the same distance h = r – r_m (Fig. 9a). Accordingly, a unique velocity is obtained. For polydisperse particle systems, the profiles deviate from the vertical shape and become more and more skewed with increasing time t_i, displaying the different traveling distances of the smaller and larger particles (Fig. 9c). The resolution of the particle velocity distribution improves with increasing sedimentation time (larger distances). It should be emphasized that the light intensity is merely recorded to determine the distance the particles move over a set time interval and is not used in itself to determine the particle size. This is in contrast to scattering methods, e.g. to laser diffraction, which uses the light intensity patterns to deduce the particle size based on applied algorithms.

In any case, the experimental determination of the particle velocity is a first-principle approach—measurands are time and distance and no assumptions have to be made, e.g. such as density, viscosity, refractive index, shape, and empirical models or fitting parameters. High accuracy (precision, trueness) as well as traceability can be achieved for the physical quantity “sedimentation velocity v_st”, dividing the basic quantities settling distance h for a particle of size x and settling time t.

The hydrodynamics for a sphere slowly settling in a liquid due to gravity and the corresponding frictional coefficient were first described by (Stokes, 1851), and Eq. 3 was derived based on these considerations called Stokes law.

  

$$v=\frac{h}{t}=\frac{\left({\rho }_{\text{p}}-{\rho }_1\right)·x^2·g}{18·\eta }=K·x^2·g$$(3)

Fig. 10 displays as an example velocity distribution for a suspension consisting of a mixture of four quasi-monomodal PMMA particles (reference material (RM), same weight concentration) obtained by GS-L. Calculation of the sedimentation velocity was based on detection of the extinction changes at constant positions of 38.0 mm, 41.5 mm and 45.0 mm and thereafter averaged. The velocity of particles of the nominal size 15 and 20 μm can be differentiated easily (left two peaks).

Fig. 10

Particle velocity distribution of a suspension made from batches of 4 monodisperse PMMA reference particles (nominal size indicated, LUMiReader PSA, 120 profiles, Δt = 20 s, ϑ = 30°C, λ = 870 nm). Particles were supplied by the company Dr. Lerche KG, Berlin, Germany.

The uncertainty of velocity measurement depends mainly on time resolution of the measuring system and, on the other hand, on distance resolution and precision of meniscus detection (r_m). Time resolutions that are technically easy to reach are in the order of less than a second (e.g. LUMiReader^® PSA, 250 ms). The physical space resolution of sensors of GS-L was determined by calibrated mechanical references to be 8.7 μm and the expanded measurement uncertainty was determined for k = 2 to be 4.63 % (Rodriguez, 2017).

In a centrifugal field, the applied local gravity depends on angular velocity and the distance of the considered particle from the center of revolution r (Eq. 4)

  

$$a={(2·\mathrm{\pi }·n)}^2·r={\omega }^2·r$$(4)

Where n equals the frequency of revolution (1/s = RPM/60s), ω equals the angular speed (radians/s) and r is the distance from the axis of rotation to be inserted in meters. According to (Svedberg et al., 1924), the earth acceleration g in Stokes’ law (Eq. 3) is replaced by centrifugal acceleration a. Particle motion in this frame is a/g = RCA (relative centrifugal acceleration, also called separation factor G or Froude number F) times faster than Stokes’ velocity under earth gravity g. Therefore, small nanoparticles and even molecules can be analyzed by high-speed AUC (Schuck et al., 2002), (Mächtle et al., 2006), (Mehn et al., 2017). Svedberg also introduced the so-called sedimentation coefficient:

  

$$s=\frac{v}{{\omega }^2·r}=\frac{\mathrm{\pi }·x^3·\left({\rho }_{\text{p}}-{\rho }_1\right)}{6·f}$$(5)

where f describes the translational frictional coefficient well known from hydrodynamic theory (Ungarish, 1993), (Happel et al., 1983).

Svedberg’s simplified approach of constant RCA fits well, if particles move only short distances within the measurement time. But in general, acceleration and therefore particle velocity depends also on the position within the sample cell (Eq. 4). Therefore, the velocity of dispersed particles increase as they move radially outward from the center of rotation. This can be seen in the dispersion fingerprint profiles where the distance between consecutive profiles taken at the same time interval increases (see Fig. 9a, monodisperse particles). To take this into account, acceleration and therefore particle velocity v has to be expressed in a differential way. Eq. 3 becomes

  

$$v(t)=\frac{\text{d}{r}_p}{\text{d}t}=K·x^2·{\omega }^2·r$$(6)

After integration we get:

  

$$t(r)=\frac{1}{K·x^2·{\omega }^2}·\text{ln}\left(\frac{r}{r_{\text{m}}}\right)$$(7)

Fig. 11 displays extinction-based velocity distributions of a suspension consisting of a mixture of 4 quasi-monomodal silica particles (RM, same weight concentration) obtained by the AC-L (Uhl, 2015). The uncertainty of velocity measurement depends on the time resolution of the measuring system and on the distance resolution and precision of meniscus detection (r_m). Time resolution scales with applied RPM and amounts of 0.1 s (4000 RPM) to 0.3 s (200 RPM) for a typical analytical photocentrifuge (AC-L type). Space resolution is independent of RPM and was determined to be 13.9 μm by means of a calibrated physical reference. Determination of the meniscus position was evaluated by 3 different approaches (SEPView algorithm, direct boundary model (Walter et al., 2015b)) for 2-mm and 10-mm cells and 4 different RPMs of the rotor. The standard deviation for each approach (n = 65) was below 0.2 % (Boldt, 2017). The expanded measurement uncertainty was determined for k = 2 to be 4.29 % in the case of default settings by the manufacturer (Rodriguez, 2017).

Fig. 11

Particle velocity distributions for a 4 modal monodisperse silica particle suspension (nominal size indicated, analytical photocentrifuge LUMiSizer, RCA ramp, ϑ = 25°C, λ = 470 nm). RMs were supplied by the company Dr. Lerche KG, Berlin, Germany.

The terminal sedimentation velocity demands a time-independent centrifugal acceleration. In other words, the final rotor speed has to be reached in a short time compared to the expected necessary sedimentation time and should not fluctuate. The above-described AC-L instruments reach a final RCA of 1000 at 8 s and of 2300 at 15 s and the RPM variation is below 0.75 % (RPM = 200) and 0.1% (RPM = 4000), respectively. To take into account the acceleration kinetics, the quantitative data obtained should be plotted against the so-called running time integral “∫ω²dt” (Mächtle et al., 2006).

The velocity distribution is of great importance for the classification and separation of polydisperse fine particulate suspensions. Of practical interest is the flow of the main fraction as well as the particle size distribution of the fines in the centrate as a function of the residence time in a process unit (Leung, 2004). The composition of centrate can be simulated with velocity distributions of the sedimenting particles obtained by a laboratory analytical photocentrifuge using the “Constant Time Approach” (CTA) (Detloff et al., 2012). From Fig. 12 it is obvious that with increasing time of centrifugation, the amount of coarser particles within the centrate decreases and that the distribution becomes narrower. The velocities for smaller particle fractions remain constant.

Fig. 12

Particle velocity distribution for an aqueous technical latex suspension within the centrate in dependency of the time of centrifugation (AC-L, RCA = 2325, (Detloff et al., 2012)).

As the velocity determination is a primary method, the terminal sedimentation velocity can be determined experimentally for any particle shape, particle concentration, rheological behavior of liquid, laminar flow condition, etc.. But simulations or calculations based on Eq. 3 and Eq. 5 are only valid for dispersions of spherical particles of low concentration and settling in an unbounded Newtonian liquid at low Reynolds numbers.

Shape of particles

The geometry of particles can be taken into account by the so-called Stokes shape factor. As any non-spherical particle shape has a larger surface area than a sphere with the same volume, frictional coefficient f will consequently differ along with its terminal sedimentation velocity (Eq. 5). It was shown that sphericity can be used to predict the average resistance (Pettyjohn et al., 1948), (Happel et al., 1983). The sedimentation shape factor equals the ratio of corresponding sedimentation velocities. It is one for a sphere and smaller than one for non-spherical isotropic particles. For a cube, for example, it amounts to 0.921 according to the above authors. In the case of non-isotropic, non-spherical particles, the settling velocity depends in addition on particle orientation (Happel et al., 1983). The resistance of rod-like objects as, e.g. MWCNT were found to depend on the cross-section multiplied by the square root of its length in the case of disk centrifuge experiments (Nadler et al., 2008).

(Hogg, 2015) described in detail a spheroid model to account for the shape influence regarding different sizing methods. For different basic geometries, Fig. 13 displays the settling (Stokes shape) factor in dependence of different height-to-diameter (cross-section) ratios (Happel et al., 1983).

Fig. 13

Settling factor for cylinders, rectangular parallelepipeds, and spheroids calculated in dependence of the height-diameter ratio (Happel et al., 1983).

The above considerations apply for rigid particles. In the case of deformable ones, the particle geometry and orientation depend on the flow conditions and may change depending on the applied shear stress. In general, shear-induced deformation decreases frictional resistance and enhances sedimentation. Pioneering work for emulsion droplets was published by (Taylor, 1932), relating the shear deformation to the ratio of inner viscosity of the droplet and the viscosity of the continuous phase (Kaur et al., 2010).

Lerche showed experimentally that rigidified human red blood cells settle at a slower rate compared to physiologically deforming ones (Lerche et al., 2001). The basics of this phenomenon are similar to the shear thinning effects of deformable particles and droplets (Pal, 2000).

There are also other definitions and approaches to take into account geometrical shapes (Arakawa et al., 1985), (Leith, 1987), (Luerkens, 1991), (Hölzer, 2007).

High particle concentration

Stokes’ law was derived for a single particle settling in an unbounded (infinite) liquid. These requirements are never fulfilled by sedimentation methods. Let us first discuss concentration effects. In general, particles dispersed in liquid are separated by finite distances and mutually affect each other resulting in disturbed flow path trajectories around particles. This hydrodynamic interaction between particles reduces the settling velocity compared to their Stokes velocity. It is obvious that hydrodynamic hindrance depends on the number of particles dispersed in a given volume. In addition, hindrance depends on particle size distribution for the simple reason that the mean distance between two particles narrows with decreasing size of the dispersed phase for a given volume concentration. The retardation can be quantified by the so-called hindrance function H(φ), defined as the ratio of sedimentation velocity as a function of the volume concentration v(φ) to the Stokes velocity v of a single particle under otherwise identical settling conditions. Both (Steinour, 1944) and (Kynch, 1952) introduced the basic idea for the first time to describe sedimentation kinetics for virtually any particle volume concentration. The hindrance function is frequently used as given in Eq. 8,

  

$$H(\phi )=\frac{v}{v_0}=\frac{{(1-\phi )}^2}{{\eta }_{\text{rel}}}$$(8)

where φ is the initial volume concentration of the dispersed phase and η_rel is the relative viscosity of the dispersion. Numerous empirical, semi-empirical and analytical models have been proposed since the pioneering work of Kynch (Fig. 14). The generic approach of Eq. 8 allows taking into account any apparent viscosity-concentration relationship η(φ) for dispersions, (Quemada, 1977, 1978a, b), (Bullard et al., 2009), (Brouwers, 2010).

Fig. 14

Overview of the hydrodynamic interaction between particles in dependence of the volume concentration described by differently proposed hindrance functions.

Maude and Whitmore (Maude et al., 1958) proposed a two-parameter approach (Eq. 9) to describe hindered settling, where β depends on size, shape and Re.

  

$$H(\phi )={(1-\phi )}^{\beta }$$(9)

This relationship is very often used for solid micronized particles, and with β = 4.65, it is the well-known Richardson-Zaki-approach (Richardson et al., 1954). Eq. 9 does not take into account the experimental fact that separation velocities for suspensions go towards zero at volume concentrations of between 0.64–0.72. It also does not consider polydispersity. Michaels and Bolger (Michaelis et al., 1962) introduced a three-parameter approach for the relative viscosity of Eq. 8:

  

$${\eta }_{\text{rel}}(\phi )={\left(1-\frac{\phi }{{\phi }_{\text{max}}}\right)}^{-n}$$(10)

The introduction of maximum packing concentration φ_max enables accounting for particle shape and particle size distribution but not for particle size. Different exponents were proposed. (Quemada, 1977, 1978a, b) set n = 2. Eq. 10 is also known as the Krieger-Dougherty equation in the rheology community with a factorized exponent n = [η]φ_max, where [η] is the intrinsic viscosity (Krieger et al., 1959). In the case of deformable particles such as blood cells or oil droplets, intrinsic viscosity [η] quantifies the cell or droplet deformability (Dintenfass, 1980).

Fig. 15 shows as an example of the velocity distributions for an aqueous technical latex suspension at different volume concentrations. It can be seen that with increasing concentration, the distributions are shifted to smaller velocities. This occurs due to the increasing hindrance. The distance between the particles is smaller at higher concentrations and so the hydrodynamic interactions increase and result in slower settling.

Fig. 15

Particle velocity cumulative distribution for an aqueous technical latex suspension (x = 100 nm–170 nm, density 1230 kg/m³) in dependence of the mass ratio (AC-L data, RCA = 2325) (Detloff et al., 2012).

If experimentally determined sedimentation velocity values are plotted against the volume concentration, the functional dependence of the hindrance function can be obtained easily by curve-fitting (Detloff et al., 2007). Fig. 16 shows the particle sedimentation velocity dependence of monodisperse spherical silica particles on the volume concentration. The results show that the general shape of the hindrance function is independent of the particle size. Hindrance diminishes for particle volume concentrations lower than about 0.005. Interestingly, the exponent of the power law (Eq. 9) increases with decreasing size, and becomes more strongly pronounced in the lower nanometer size range. In addition to the particle-particle separation distance, both the immobilized water at the particle surface as well as the ionic double layer becomes increasingly important for smaller particle sizes (Salinas-Salas et al., 2007). In other words, the effective or apparent particle volume is larger than the geometrical one. On the other hand, it influences also the effective density of the particle (Hinderliter et al., 2010), (DeLoid et al., 2013). The hindrance also depends on particle shape. (He et al., 2010) demonstrated that for iso-volumetric platelet-like particles having different shapes, the hindrance function also depends on the aspect ratio. A single functional relationship over the whole concentration range for spherical particles was found. In contrast, the behavior of platelet-like-shaped particles show two different functional relationships with regard to volume concentration. Both slopes and critical onset concentration depend on the platelet shape aspect ratio.

Fig. 16

Hindrance function in dependence of the size of monodisperse particles (see symbols) determined by means of an analytical centrifuge (LUMiFuge). Settling velocities (symbols) v were determined from the slope of the position of the interphase supernatant/dispersion versus time by Front Tracking. The given exponents were obtained by best-fitted curves (lines) based on Eq. 9.

Furthermore, the polydispersity of the dispersed phase has to be taken into account with respect to the hindrance function. The mean particle-particle separation distance depends on the size distribution of the dispersion. This is especially important since most applications involve broad distributions. The idea is supported by rheological references regarding the relative viscosity of dispersions made of a mixture of different-sized particles, e.g. (Horn et al., 2000).

Fig. 17 compares the hindrance of a suspension of monodisperse silica particles (0.18 μm) in dependence of the volume concentration (red line and red symbols) and binary suspensions containing 500-μm particles of increasing ratios (2/1 to 1/2) at the same total volume concentration. Hindrance is most pronounced for the small particles and decreases with increasing ratio of the large ones. Correspondingly, the exponent of Eq. 9 decreases. For a given total volume concentration, again, the hindrance for the monodisperse suspension of small particles (0.18 μm) is greater than for mixtures with the larger ones. This effect is also shown for the polydisperse technical latex dispersion (Fig. 15). The hindrance function determined based on different velocity quantiles showed that the exponent of Eq. 9 amounts to 6.5 and 8.2 for v₁₀ and v₉₀, respectively.

Fig. 17

Hindrance functions of binary mixtures (ratios as indicated) of differently sized monodisperse particles in dependence of the total volume concentration. Other details as Fig. 16.

Wall effects

In addition to the hydrodynamic particle-particle interaction, under real conditions, we also have to deal with a “bounded” liquid. Stokes’ law did not account for the influence of cell walls (bounded liquid), where fluid streamlines are disturbed. Any container will influence the flow pattern around a settling particle causing increased drag for spheres. This applies to walls as well as to the top and bottom. In the early part of the last century, (Lorentz, 1907) and (Ladenburg, 1907) analyzed this phenomenon for single-ball viscometers. A correction term was added to the friction force taking into account the ratio of particle size x to the diameter of the circular cylinder-shaped container multiplied by a k-factor of 2.1044 (rectangular cross-section deviates only slightly). In the case of a 500 nm particle and a measuring cell of 1 mm optical path, the real settling velocity is about 0.11 % lower compared to Stokes velocity. If the size-container ratio amounts 1:100, the error is up to about 2 %. Regarding the influence of the bottom, the measurement position (e.g. for CPA) should be 50 particle diameters above the bottom to account for a friction error of less than 0.6 % (Allen, 1999). As shown recently, cylindrical particles may orient with regard to the wall and settle in narrow cylindrical cells even faster (Lau et al., 2010).

Reynolds number

It should be further underlined that Stokes’ law (Eq. 3) was derived assuming the steady low-speed motion of a rigid spherical particle through a Newtonian liquid. A laminar flow condition is classified by the dimensionless Reynolds number defined as the characteristic flow speed multiplied by the characteristic flow length divided by the liquid kinematic viscosity. It reads (Re = ρ_l·v·x/η = v·x/κ). The numerator characterizes the flow, whereas the denominator characterizes the liquid. Re increases with continuous phase density ρ_p and size x, faster sedimentation velocity v and lower liquid viscosity η. Re should be lower than Re < 0.25 for the application of Stokes’ law (laminar Stokes regime). This critical Re would be reached if a particle of 1 μm in size settles in water at 20°C as fast as 0.25 m/s. In general, the drag coefficient decreases with increasing Re. If Re equals 0.25, the experimental terminal particle velocity will be about 3.5 % higher than that calculated by Stokes’ law (Allen, 1999). Note that for particle sizing, this number is reduced by its square root (Eq. 12), and therefore amounts to 1.4 %.

In the case of high-gravity settling (centrifugation), the weight force and associated flow pattern around moving particles depends not only on particle mass and rheological behavior of the liquid but also on RCA, and the Re numbers may reach values above 0.25 (Stokes regime). For higher particle velocities (Re > 0.25), laminar flow ceases and two symmetrical recirculating eddies emerge near the rear stagnation point. For a Reynolds number at about Re = 130, these eddies already occupy a space larger than the particle and finally the flow around the particle becomes unstable and turbulent. Stokes’ law underestimates the drag coefficient (C_D = 24/Re) as the Reynolds number increases, and the measured real velocity will be larger than that estimated for the Stokes regime. There are numerous papers dealing with extensions of Stokes’ law for Re numbers up to turbulent flow, e.g. (Allen, 1999).

Non-Newtonian liquids

Eq. 3 and Eq. 5 hold only for slow particle movement in a Newtonian liquid. This means that shear viscosity does not vary with shear rate, and viscosity is a constant with respect to the time of shearing, respectively. Particle velocity is inversely proportional to the dynamic viscosity of the continuous phase and, in the case of analytical centrifugation, it scales linearly with centrifugal acceleration (Lerche et al., 2006), (Lerche et al., 2011), (ISO/TR 13097:2013). As far as that goes, sedimentation is very comparable to a vertical falling ball viscometer. Typical Newtonian liquids are those containing compounds of low molecular weight, as sugar in food, or low concentrations of dissolved polymers (e.g. starch, pectin, xanthan). Dynamic viscosities increase linearly with low solute concentration (Einstein equation) to become increasingly sloped upwards at higher concentrations. For Newtonian continuous phases, this concentration-dependent increase in viscosity can be taken into account according to Eq. 5.

Fig. 18 displays “normalized” sedimentation velocities of monodisperse polystyrene particles dispersed in different concentrated sucrose solutions normalized for dynamic viscosity and solution density (Werner, 1966). Despite the absolute solution viscosities differing by a factor of 5.4, a master curve is obtained after “normalization” of experimentally determined velocities by Stokes’ law with respect to continuous phase density and dynamic viscosity, respectively.

Fig. 18

Normalized sedimentation velocities of polystyrene particles (0.03 % m/m, 1.1 μm) in water and solutions of increasing mass fraction of sucrose up to 28.9 % (AC-L, ϑ = 4°C, RCA = 580).

Nowadays, research and dispersion formulators are dealing more and more with rheologically tuned continuous phases that exhibit non-Newtonian or gel-like behavior (Schulz et al., 1991), (Kuentz et al., 2003), (Arabi et al., 2016). Under such circumstances, dynamic viscosity is not a constant anymore and has to be described by means of a shear-stress versus shear-rate relationship (for details refer to (Macosko, 1994)). The dependence is non-linear and/or does not have its origin at zero. Another complication is often the time-dependent rheological behavior as a result of structural changes of the continuous phase due to shearing. Shear-thinning, the decrease in viscosity with increasing shear rates, is the most common relevant behavior with respect to sedimentation (Quemada, 1977, 1978a, b). The question arises as to how the translational frictional coefficient f (Eq. 5) is influenced? During sedimentation, particles are driven by gravity fields through a stagnant liquid creating streamlines and shear rate gradients in the immediate vicinity of each particle. The qualitative and quantitative characteristics of this flow field in the neighborhood depend primarily on the velocity of the moving particle, as well as on the rheological behavior of the continuous phase. In addition, the flow pattern is influenced by particle shape and orientation of the settling particles. There are numerous papers dealing theoretically with the gravity settling of solid particles in rheologically well-defined continuous phases, e.g. (Deshpande et al., 2010). But there is no theoretical model to predict structural changes of the continuous phase due to forced particle movement or to deduce an apparent viscosity to be inserted into Eq. 3 and Eq. 5.

Interestingly, the apparent viscosity can be experimentally determined by monodisperse tracer particles. To this end, continuous phases of anionic polyacrylamide solutions (PAA, molecular mass 7 million Dalton, charge density 32 %, Praestol 2540 Ashland) of concentrations up to mass ratios of 0.1 % were prepared and the apparent viscosity in dependence of the shear rate was determined by means of a rheometer (LOW SHEAR 40, Contraves AG, Switzerland). The apparent viscosity rises with PAA concentration and, on the other hand, decreases with increasing shear rates (shear thinning behavior). Monodisperse silica tracer particles of about 1 μm in size were dispersed in these solutions and the settling velocities (CPA) of primary tracer particles were determined for increasing RCA of 33, 270, 740 and 1900. The apparent viscosities of the PAA solutions of different concentrations were calculated by Eq. 5 based on experimentally obtained velocities (first-principle approach), densities of the two phases and particle size. Concentration-dependent shear thinning behavior was detected with increasing centrifugal force. To compare the shear thinning behavior measured by a Couette rheometer and the sedimentation approach (multi-ball viscometer), the maximal shear rates created by gravity-driven falling balls (particles) in its neighborhood were approximated according to (Guyon et al., 2001) by:

  

$${\dot{\gamma }}_{\text{max}}=\frac{3}{2}·\frac{v}{x}$$(11)

Fig. 19 displays the results of both approaches. There is a very reasonable coincidence. Higher maximal shear rates result in a more pronounced shear thinning. Results can be interpreted in a way that the settling of polydisperse particles in a polymer solution (e.g. rheological thickener application) results in a broadening of the velocity distribution compared to Stokes’ law prediction assuming a shear-rate-independent viscosity (Newtonian liquid).

Fig. 19

Shear-dependent apparent viscosity of PAA solutions of different concentrations determined traditionally by a Low Shear 40 rheometer:

and based on Stokes’ law by sedimentation velocity of monodisperse tracer particles (“falling multi-ball viscometer”):

In-situ visualization and quantification of the movement of tracer particles of known size can also be applied to study the time-dependent structuring and aging phenomena of polymeric liquids. Using a model polymer solution system, the sedimentation velocity of monosized silica particles was measured in continuous phases of increasing pectin concentrations and under different centrifugal forces (Sobisch et al., 2018).

Fig. 20 clearly demonstrates that the settling velocity of tracer particles slows down by about 2 decades with increasing pectin mass fraction up to 3 % (RCA = 2300). The separation behavior of silica particles in pectin solutions (non-Newtonian continuous phases) does not depend linearly on RCA. In contrast, shear thinning-like behavior is observed and this behavior becomes more pronounced with increasing pectin concentration.

Fig. 20

Sedimentation velocity of silica tracer particles in dependence of the pectin concentration and RCA. Velocity determination by the slope of interface supernatant/suspension (front) tracking (threshold 10 % transmission). Data by AC-L, 2-mm cells, ϑ = 20°C, mass fraction of 1-μm silica particles 5 % (Sobisch et al., 2018).

To investigate the aging of the pectin solution in situ, solutions of different pectin concentration, using silica tracers with a mass fraction of 2 %, were filled into sample vials immediately after preparation, and aged for up to 29 days at 4°C. Vials analyzed for structural changes after a set aging period (see indications in Fig. 20) were equilibrated at room temperature for about 30 minutes, carefully placed into the analytical centrifuge LUMiSizer, and then analyzed for any movement of the primary silica particles. In this way, the process of structure build-up can be investigated with no distortion by sampling quasi in-situ. Fig. 21 displays the settling of the tracer particles for the freshly prepared sample (day 0, top). The red line represents the first transmission profile, green the last one taken. As can be seen, the area between transmission profiles for a sedimentation time of 230 minutes (quantifying clarification) decreases with increasing pectin concentration and aging. Interestingly, pectin solutions of 3 % and 3.5 % exhibit about the same aging alteration after 15 days and 1 day, respectively. No particle movement could be detected at RCA = 2300 after 29 days (3 %) and 15 days (3.5 %), respectively. This indicates that some kind of structure building occurred, resulting in a motion decrease of the tracer particles with aging time. The analytical approach in this case tracks the particle movement directed by gravity fields, in contrast to particle tracking methods based on Brownian motion (Schuster et al., 2015).

Fig. 21

Probing pectin gels at different concentrations and aging times by tracking the settling velocity of monodisperse tracer particles (AC-L, RCA = 2300, 230 min., 2-mm cells, ϑ = 20°C). Only the first (red) and last (green) transmission profiles are displayed.

Fig. 22 summarizes possible relationships between particle velocity and relative centrifugal/relative centrifugal force acceleration with regard to the rheological behavior of the continuous phase, as well as structural alterations due to shear stress acting on moving particles. In accordance with Stokes’ law, the particle velocity is proportional to the centrifugal acceleration or RCA for a Newtonian continuous phase (line 1). In contrast, if the continuous phase exhibits shear thinning (line 2) or shear thickening (line 3), the particle velocity increases non-linearly. On the other hand, non-linearity may indicate “changes” of the particles themselves due to hydrodynamics. Line 2 could be also provoked, e.g. by shear-stress-induced agglomeration (dilatancy), particle orientation, or deformation of emulsion droplets reducing the frictional coefficient. A behavior depicted schematically by line 3 may have its origin in a shear-stress-governed breakdown of agglomerates or flocs (Lerche et al., 2001). Created smaller particles settle at a slower rate compared to the initial size of the particles. The onset of consolidation of the disperse phase in the course of separation manifests itself earlier at a higher RCA (RCF) and also results in lowering the increase of the particles’ settling velocity expected by Eq. 5 (Sobisch et al., 2006), (Usher et al., 2013). The depicted non-linearity can often be described by a simple power law. The exponents are greater than one for shear thinning (line 2) and less than one in the case of shear thickening (line 3) of the continuous phase. Finally, line 4 of Fig. 22 demonstrates the effect of a continuous phase exhibiting a yield point. Particles start to move if the shear stress created by the particles due to centrifugal force destroys the polymer structure (Lerche et al., 2003). It should be underlined that a yield point does not always imply that particles do not settle (Kuentz et al., 2002). The fact that the rheological behavior of dispersion can be probed by sedimentation techniques was successfully demonstrated by (Kuentz et al., 2003), (Jacob, 2015).

Fig. 22

Schematic drawing of particle (separation) velocity in dependence of the RCA for different classes of dispersions with respect to rheological behavior.

Kynch model

We now depart from the basic approach of Stokes and focus on moving particles in a bounded quiescent liquid. The law of volume continuity demands that settling particles displace the same volume of continuous phase as they occupy. In other words, the volume flux of particles creates a flux of continuous phase directed in the opposite direction. In literature, two major physicomathematical models are employed for the formulation of the governing equations of motion. The whole particle-liquid dispersion is assumed as a single flowing continuum—the “mixture” (“diffusion”) model—or, the continuity and momentum balance are described separately for disperse phase and continuous phase—the “two-fluid (two-phase)” model (Ungarish, 1993). It was Kynch who proposed a kinematic theory of sedimentation based on the assumption that the suspension is a continuum and the sedimentation process is represented by the continuity equation (mixture model). The approach is governed by the local volume concentration of the dispersed phase as a function of space and time, and by the so-called flux density function (Kynch, 1952), (Concha et al., 2002). On the other hand, the approach assumes that the local solid-liquid relative velocity is a function of the total volume concentration of the dispersion only. For the first time, this basic framework allows the description of the complete “sedimentation” process, linking free-settling and consolidation by a unified theory. For details, industrial applications, progress and future work to be performed in this research field, refer to, e.g. (Fitch, 1983), (Buscall et al., 1987), (Schaflinger, 1990), (Ungarish, 1993), (Bustos et al., 1999), (Lerche et al., 2001), (Concha et al., 2002), (Frömer et al., 2002), (Berres et al., 2008), (Detloff et al., 2008), (Anestis et al., 2014).

3.2 Extinction-weighted size distribution

Particle velocity distributions are experimentally measured based on the primary principle with no assumptions regarding dispersions properties, models or algorithms. Experimentally determined velocities can be transformed to a particle size distribution for gravity sedimentation based on Eq. 3 and for centrifugal sedimentation based on Eq. 7, respectively (ISO 13317-1:2001), (ISO 13318-1:2001). The result is the Stokes diameter of a settling particle, more precisely an equivalent spherical Stokes diameter. If the particle shape is known, such a shape influence may be accounted for by introducing corresponding shape factors into the basic Stokes’ law. In principle, particle size determination is merely a transformation of the calculated particle velocity distribution data. There is no change in the distribution pattern or modality. For the gravitational sedimentation transformation, this is given by Eq. 12:

  

$$x(h,t)=\sqrt{\frac{h}{K·g·t}}$$(12)

and for centrifugal techniques by Eq. 13

  

$$x(r,t)=\sqrt{\frac{1}{K·{\omega }^2·t}·\text{ln\hspace{0.17em}}\left(\frac{r}{r_{\text{m}}}\right)}$$(13)

Remember that the material constant K according to Eq. 3 depends on the density difference of the disperse and continuous phase as well as the viscosity of the continuous phase.

The amount of particles for each particle size class—cumulative distribution values Q_ext(x) (y-axis)—is calculated based on the determined extinction values at time t obtained by means of:

  

$$Q_{\text{ext}}(x)=\frac{E(h,t)}{E(h,\text{end})}$$(14)

As the concentration of each particle class is physically based on an extinction measurement (Eq. 1), the obtained size distribution is extinction-weighted and cannot be directly compared with, e.g. intensity-weighted DLS or SLS or mass-weighted acoustic data, respectively.

In contrast to sedimentation under gravity, in a centrifugal field the initial concentration of dispersion decreases with increasing radial position r due to a radially outward higher acceleration acting upon the particles. The dilution is further elevated by particle motion on radial trajectories (Fig. 23). These so-called effects of radial dilution have to be taken into account calculating the distribution density data of velocity or particle size distribution (y-axis). It was shown that the concentration change of samples does not differ in the middle of the sample cell, regardless of whether rectangular or sector-shaped cells (Fig. 23) are employed (Detloff et al., 2008).

Fig. 23

Schematic illustration of centrifugal sedimentation with regard to the geometry of the sample cell shortly after the start of separation; left: sector-shaped AUC cell; right: AC-L rectangular cell.

The mass balance

  

$$\phi (r,t)·r^2(t)=m(r)=m(r_{\text{m}})=\phi (r_{\text{m}},0)·r_{\text{m}}^2$$(15)

states that the mass in any cross-sectional area is conserved when moving from the radial position r_m to r(t) and is given by:

  

$$\phi (r,t)=\phi (r_{\text{m}},0)·{\left(\frac{r_{\text{m}}}{r(t)}\right)}^2$$(16)

Here, φ(r, t) denotes the total concentration of all particles at position r at time t.

The distribution values (y-axis; q_ext(x)) of the extinction-weighted particle size distribution in the centrifugal field are obtained by solving the following integral equation Eq. 17:

  

$$E(r,t)=E(r,0)·L·\underset{x_{\text{min}}}{\overset{x(t,r)}{\int }}\text{exp}(-2·K·x^2·{\omega }^2·t)·q_{\text{ext}}(x)\text{d}x$$(17)

This can be done in an analytic way for CTA by:

  

$$Q_{\text{ext}}(x)=\frac{{\int }_{E_{\text{min}}}^{E(r)}{r}^2\text{d}E(r)}{{\int }_{E_{\text{min}}}^{E_{\text{max}}}{r}^2\text{d}E(r)}$$(18)

and for CPA in a numeric way by:

  

$$Q_{\text{ext}}(x_i)=\frac{1}{2}·(y_i+y_{i-1,i})·E_i+\sum _{j=1}^{i-1}\left(\frac{y_i+y_{i-1,i}}{y_{j+1,i}+y_{j,i}}-\frac{y_i+y_{i-1,i}}{y_{j,i}+y_{j-1,i}}\right)·Q_{\text{ext},j}$$(19)

with:

  

$$y_i={\left(\frac{r}{r_{\text{m}}}\right)}^2,y_{i,j}=y_i^{{\left(\frac{x_i}{x_j}\right)}^2},for\hspace{0.17em}i,j=1,2,3,\dots ,N,j\le i,y_{0,i}=1\hspace{0.17em}\text{and\hspace{0.17em}}{y}_{i,i}=y_i$$

It should be noted that in the case of centrifugal separation, in addition to buoyancy, the motion of the particles is driven by a second body force—the Coriolis force. Coriolis acceleration equals the particle velocity multiplied by the angular velocity of the rotor. Its direction is perpendicular to the radius (Ungarish, 1993), (Frömer et al., 2002), (Probstein, 2003). For an estimate let us assume the centrifugal separation of large and dense cement particles (x = 10 μm, ρ_p = 3150 kg/m³) dispersed in isopropanol (η = 2.1 mPas). At an RCA = 4000 for a rotor speed of 5250 min⁻¹, computation of the ratio of Coriolis acceleration to centrifugal acceleration amounts to 0.9 %. Hence the Coriolis force for analytical centrifuges operating up to the above acceleration range can be neglected even for rather large and heavy particles.

Finally, we have to consider that the local concentration of particles at position r not only depends on directed body forces, as discussed so far, but also on particle diffusion. This becomes especially relevant for nanoparticles, as diffusion coefficients D increase inversely with particle size. A gravity-governed particle transport creates concentration gradients in the direction of the acting/applied gravity ∂φ/∂r, which causes a collective particle transport in the opposite direction of the concentration gradient by diffusion (Fick’s law). Both transport terms were combined for the first time for AUC applications by (Lamm, 1929).

  

$$\frac{\partial \phi }{\partial t}=\frac{1}{r}\frac{\partial }{\partial r}\left[rD\left(\frac{\partial \phi }{\partial r}\right)-s{\omega }^2{r}^2\phi \right]$$(20)

Eq. 20 describes the change of (particle) concentration ∂φ/∂t due to the net transport by diffusion (first term) and sedimentation (second term) in the bracket. Above equation applies when both transport terms are independent of particle concentration. For details refer to (Fujita, 1962). It should be pointed out that even for nanoparticles, diffusion does not prevent particle sedimentation. It is broadening the separation created by concentration gradients. In other words, the mean particle velocity is not changed but the particle dispersity appears more polydisperse. Diffusion may be expected to be insignificant if the dimensionless Péclet number—relating the advective transport rate to the diffusive transport rate—is greater than one (Ungarish, 1993), (Probstein, 2003).

Recently, it was shown that primary sensor signals of AC-L saved by SEPView software can be imported into AUC software packages. This will allow users to tap into the existing AUC knowledge pool (Walter et al., 2015a).

3.3 Volume-weighted size distribution

Extinction-weighted particle size distribution does not account for the optical properties of the dispersion. Thus, the different particle classes contributing to the total recorded extinction intensity are not correctly weighted. To measure the correct amount of particles present in defined size distribution classes, the differences in the refractive index between particles and continuous phase, and the geometrical particle shape have to be accounted for. The volume-weighted particle size distribution is calculated by transforming the y-axis of the extinction-weighted size distribution by Eq. 21:

  

$$\mathrm{\Delta }{Q}_{3,i}=\frac{\frac{\mathrm{\Delta }{Q}_{\text{ext},i}}{A_{\text{v},i}}}{{\sum }_{j=0}^N\frac{\mathrm{\Delta }{Q}_{\text{ext},j}}{A_{\text{v},j}}}$$(21)

Here, A_v corresponds to the volume scattering cross-section which can be obtained by applying the Mie theory (Mie, 1908), (van de Hulst, 1981). Fig. 24 compares exemplarily the extinction- and volume-weighted size distribution of polydisperse submicron silica particles. Polydispersity regarding size stays identical between the extinction- and volume-based results. However, fractions of size classes differ by about 10 %. The Mie correction assumes spherical particles. Therefore, strictly speaking, the calculated sizes should be called equivalent spherical particle sizes.

Fig. 24

Particle size distribution for polydisperse silica (SiO₂, 0.1 μm–1 μm) obtained by LUMiSizer, (RCA ramp, λ = 470 nm, ϑ = 25°C, ρ_p = 2200 kg/m³ ρ_l = 997.3 kg/m³, η = 0.899 mPa s, RI_p = 1.475-0i, RI_l = 1.3377-0i); red: Extinction-weighted; blue: Volume-weighted.

Of added interest is the fact that cuvette liquid centrifugal sedimentation methods are very flexible regarding different continuous phases (e.g. pH, ionic strength, solutions of different solutes, organic solvents, ionic liquids). It allows viscosity and density contrasts to be accounted for, as shown for the velocity in Fig. 18. The median size x_50,3 of the polystyrene particles dispersed in 8 different continuous phases amounts to x_50,3 = 1.081 μm ± 0.012 μm (Eq. 13 and Eq. 21). The accuracy (trueness, reproducibility) of sedimentation-based particle sizing (separation principle) is known to be on a very high level. An example is the round robin test (Lamberty et al., 2011), as well as reports and publications issued by the European Research Project “Nanodefine” (NanoDefine, 2018).

3.4 Number-weighted size distribution

The number-weighted particle size distribution is obtained from the volume-weighted distribution by the method of moment-notation, as described in (Leschonski et al., 1974), (ISO 9276-2:2014). It is straightforward if the volume-weighted particle size distribution is correct and the particles are spherical. For non-spherical particles, the conversion situation is more difficult because different measurement principles derive an equivalent spherical size depending on the technique used.

This topic is intensively studied by the EU project “Nanodefine” (www.nanodefine.eu) inasmuch as the EU definition for the classification of products regarding “Nanomaterials” is going to be based on counting. The proposed definition states that a material (product) has to be classified a nanomaterial if for 50 % or more of particles in the number size distribution, one or more external dimensions is in the size range of 1 nm–100 nm (Bergeson et al., 2017). Recently, the performance of 12 different measurement techniques (microscopic, ensemble-and separation-based) was investigated regarding tier 1 classification, and compared to an SEM “reference.” Altogether, 15 polydisperse RMs were included in this study. Number medians ranged from 5 nm to 2016 nm and the distributions were characterized as monomodal or polymodal for size, and spherical, non-spherical or fractal for shape, respectively. The use of liquid centrifugal sedimentation techniques was deemed appropriate for the tier 1 classification of nanomaterials. The deviation of median size values of the RM compared to SEM sizes was about 33.7 % (CPS disk centrifuge, 14 RMs measured), 28.8 % (LUMiSizer, 14 RMs) and 30.2 % (AUC, 10 RMs). For comparison, the deviation for DLS (14 RMs) amounts to 80.2 % (Babick et al., 2016a).

3.5 Multi-wavelength particle sizing

Besides wavelength, the extinction coefficient of particles at visible light depends on size and optical properties (refractive index difference with respect to the continuous phase, particle shape) at a given wavelength. If the complex refractive index is known, the scatter and absorption of light by spherical particles of different sizes can be taken into account by the Mie theory (Mie, 1908), (van de Hulst, 1981). Problems arise if the particle size or optical properties are not available.

The extinction values of dispersion depend on the wavelength. The ratio obtained at different wavelengths should provide information of the particle size. The Mie calculation shows that in general, for particles of submicron size, E_NIR/E_VIS is less than one, with the ratio depending on the particle size. The new LUMiReader^® and LUMiSizer^® models are equipped with multi-wavelength functionality, enabling the capture of extinction profiles across the entire sample height by multiple wavelengths de facto simultaneously. A decrease in the extinction ratio over time indicates, e.g. destabilization due to flocculation (Detloff et al., 2011). Note that particle size does not have to be determined directly.

Another way to circumvent the necessity of quantitative optical models consists in determining extinction-based particle size distributions at different wavelengths—as already proposed by (Weichert, 1981). The basic idea behind it is as follows. The wavelength dependency of turbidity for defined particle sizes is approximately equivalent to the size dependency of turbidity at fixed wavelengths. Hence, the multi-wavelength detection itself yields the required optical model. Extinction-weighted size distributions can be transferred into a volume-weighted distribution with no assumption regarding material parameter and particle shape. It relies on the assumptions that a) the extinction coefficient of a particle is a function of the dimensionless size x/λ, and b) extinction-weighted size distributions of different wavelengths give the same volume-weighted distribution (to be published). Fig. 25 shows the results for colloidal Al₂O₃ particles obtained by the developed multi-wavelength LUMiSizer. (Babick et al., 2016b). The result of a cumulative weighted size distribution shows good conformity between the classic Mie approach and the newly developed analysis algorithm based on extinction-weighted distributions obtained by multi-wavelength experiments. Work is currently in progress to answer questions regarding detection limits for very fine and coarse particles on wavelength, and smoothing signal noise to derive high-quality optical models.

Fig. 25

Comparison of a cumulative volume-weighted size distribution of polydisperse colloidal Al₂O₃ determined by classic analysis (Mie approach, AC at 455 nm wavelength) and new Multi-Wavelength Particle Size Analysis obtained by an MW-LUMiSizer operating at 410 nm, 455 nm and 470 nm. In comparison, the results given are obtained by the DLS technique.

4. Beyond particle sizing

4.1 Hydrodynamic particle density

Density is an important material parameter. Its determination is a routine task for liquids and solids, but a challenge for nano- and microparticles such as biological cells, brushed, porous or hydrocolloid particles, core shell particles, particles stabilized by Pickering emulsification, droplets or fractals dispersed in a liquid. It is an important parameter for particle size determination based on sedimentation techniques by gravity or analytical centrifugation (ISO 13317-1:2001), (ISO 13318-1:2001), field flow fractionation (Kirkland et al., 1983) or acoustic techniques (Horváth-Szabó et al., 1996), (ISO 20998-1:2006). In addition, there are also many other application fields where density matters. Stabilization of dispersion is achieved by density matching (e.g. liquid food products (McClements, 2004)), optimization of solid-liquid separation, e.g. recycling, mining (Callahan, 1987), or to characterize the purity or composition of structured particles (e.g. internal phase of double emulsions, homogeneity of “hollow” particles or core/shell particles (Kirsch et al., 1999). Nowadays, there is an increasing interest in the particle density to enable estimation of the mass transfer of nanoparticles atop cell layers by sedimentation to calculate doses for in-vitro nanotoxicity assessment (Kato et al., 2009), (Hinderliter et al., 2010), (DeLoid et al., 2013).

Sedimentation techniques allow determining the density of dispersed particles in situ. The particle density can be calculated from experimentally determined velocities based on Stokes’ law (Eq. 3), if the shape and particle size are known. But in most cases, especially for nanoparticles and surface-decorated particles, both material parameters are not available. In such cases, two different sedimentation approaches may be applied (Lerche, 2011), (Woehlecke et al., 2012). The first one—“isopycnic velocity interpolation approach” principally dates back to Archimedes of Syracuse (3rd century BC). The Archimedean principle or buoyancy principle states that a body or a particle stays suspended in a liquid if the densities of liquid and particle match. It sinks if the particle density is higher and it floats/creams, if the particle density is lower. There are different liquids and solutions available that allow for the density measurements of particles having a density from about 800 kg/m³ to above 3000 kg/m³. The measuring task simply consists of detecting the direction of particle migration in a series of liquids with appropriate densities—lower and higher as the expected particle density—and to interpolate the liquid density for zero migration velocity (ISO 18747-1:2018). This technique determines the buoyant density defined as the ratio of particle mass to particle volume including filled or closed pores as well as adjacent layers of liquid or other coating materials. Sedimentation methods are very attractive for detecting the direction of particle migration (sedimentation or creaming/flotation).

Fig. 26 displays as an example of the experimental data for RMs (PMMA) of different size supplied by the company Dr. Lerche KG, Berlin, Germany. The obtained particle velocities were multiplied by the corresponding continuous phase viscosity to normalize viscosity effects (Lerche et al., to be published). The density of the particles then equals the liquid density corresponding to the zero number of the y-axis (dashed lines in Fig. 26) and the mean value of particle density amounts to 1201.5 kg/m³. The density of the four particle size classes differs by only less than 0.1 %. This is to be expected, as size and shape do not enter into the equations of gravity and buoyancy forces. Therefore, reasonable agglomeration or flocculation during the experiment does not influence the density determination. The method is also robust for the chosen particle volume concentration over a wide range. On the other hand, accuracy depends on trueness of velocity determination, distance of interpolation points and precision of liquid density and viscosity data. Because of this, temperature changes during the experiment have to be avoided. Furthermore, the particles should not swell or shrink in the test solutions.

Fig. 26

Gravity sedimentation velocity of monodisperse PMMA particles of different size in dependence of the density of the continuous phase (polytungstate solutions of different concentration, particle mass fraction 0.2 % to 0.8 %, ϑ = 30°C, LUMiReader PSA, λ = 870 nm, 10-mm PC cells).

The second density determination approach—“multi-velocity approach”—is based on measurement of the mean sedimentation velocity or sedimentation velocity distribution of the particles in gravitational or centrifugal fields. It can be applied both for particles exhibiting sedimentation or flotation/creaming. This approach is already known from AUC (McCormick, 1964), (Mächtle, 1984). A draft ISO standard is currently available (ISO/DIS 18747-2, 2018). According to Stokes’ law (Eq. 3, Eq. 5), the velocity v for a given particle size depends on liquid ρ_l and particle density ρ_p, respectively, and on liquid viscosity η. The multi-velocity approach is based on the velocity determination of particles dispersed in two different liquids having a different density. Based on Stokes’ law applied to sedimentation in both dispersions, we get:

  

$${\rho }_{\text{p}}=\frac{v_1·{\eta }_1·{\rho }_{1,2}-v_2·{\eta }_2·{\rho }_{1,1}}{v_1·{\eta }_1-v_2·{\eta }_2}$$(22)

where indices 1 and 2 correspond to the two different liquids.

Fig. 27 displays the cumulative velocity distribution of 1.1-μm spherical polystyrene particles. Particles from the same batch were dispersed at (mass fraction 0.03 % m/m) in continuous phases of different densities (water and solutions of different sucrose concentrations). At higher sucrose concentration, the particles float (negative velocity) as the density of the corresponding sucrose solution is higher than the particle density. For each possible combination of different continuous phases, the particle density was calculated according to Eq. 22 (Woehlecke et al., 2012). The mean density value amounts to 1053 kg/m³ (standard deviation 0.4 %) based on all 9 liquid combinations. This value corresponds well to the known density of polystyrene (1055 kg/m³). Often, normal water H₂O and D₂O are chosen as the continuous phase.

Fig. 27

Velocity distributions of monodisperse polystyrene particles dispersed in water and 5 different concentrations of sucrose solutions (sucrose/water mass fraction: 4.1 % to 28.9 %) measured by AC-L (RCA = 480, ϑ = 4°C, 2-mm PC cells, ROI 0.5 mm width at 20 mm (sedimentation), and 2 mm and 5 mm (creaming) above bottom.

Fig. 28 displays exemplarily a cumulative velocity distribution of oil droplets dispersed in the two kinds of water. As the density difference is higher in the case of D₂O, the velocity distribution is shifted to the right. Based on the harmonic mean velocities of 26.72 μm/s and 53.64 μm/s, respectively, the density of rapeseed oil was calculated to amount to 923.6 kg/m³. It differs only by 0.5 % from 918.5 kg/m³ of the oil/emulsifier phase used to make the O/W emulsion. It should be mentioned that Eq. 22 can be applied to each percentile of the velocity distribution. Neglecting experimental uncertainties, the measured densities will be the same as long as all particles are of the same material or composition. On the other hand, the density differences for percentiles indicate that fast and slow particles have different densities and therefore differ in their composition. It has been shown that a destabilization by Ostwald ripening of density-matched emulsions leads to a density distribution of emulsion droplets (Sobisch et al., 2009). The density increase of slow (small) particles is due to the increase of the weighting agent concentration and vice versa for the faster (larger) particles. The described approach can also be employed for quality control of the homogeneity of “hollow” particles or core/shell particles, as well as for determining the internal phase of double emulsions. The multi-velocity approach relies metrologically on reliable determination of the particle velocity or velocity distribution. The smaller the difference in densities between the two continuous phases, the more accurately the particle velocity has to be determined. The chemical composition/density distribution of the particles themselves does not matter. Here too, the state of particles/dispersion should not change, e.g. the particles should neither swell nor shrink in the continuous phases, nor agglomerate or flocculate. Determination of the velocity distribution is preferably made at low volume concentrations.

Fig. 28

Cumulative creaming droplet velocity of a rapeseed oil emulsion prepared in H₂O (ρ = 997 kg/m³, left red line) and in D₂O (ρ = 1103 kg/m³, right blue line). (AC-L, RCA = 2325, ϑ = 25°C, 2-mm PC cells).

A similar approach for line start sedimentation was published by (Kamiti et al., 2012), (Neumann et al., 2013). In contrast to the homogeneous sedimentation approach, it does not operate at constant continuous phase viscosity due to the built-up density gradient, and the density evaluation is more complex. Furthermore, the decane layer on top of the spinning liquid has to be taken into account.

4.2 Characterization of magnetic particles

Magnetic micro- and nanoparticles have increasingly wide applications in information technology, magnetic fluids, selective separation, and nanomedicine (Holschuh et al., 2014), (Anker et al., 2017). There are differently designed particles. Magnetic particles may be prepared by coating, i.e. uniform-sized core particles covered by a layer consisting of dispersed magnetite in a non-magnetic material (e.g. polystyrene). The magnetite content of these objects represents 10 % to 20 % and they are paramagnetic. Nanoparticles can also be built starting with a magnetite core of a few nanometers with a surrounding “isolating” non-magnetic layer of, e.g. silica carrying mostly polymers or macromolecules for functionalization. These particles are often supermagnetic. The magnetic behavior of these particles is very complex. Their susceptibility (responsiveness) and magnetization has to be analyzed during particle design, optimization of application and quality control of the production process (Wilhelm et al., 2002), (Mykhaylyk et al., 2008), (Eichholz et al., 2014).

As previously described, STEP-Technology^® provides particle velocity as a primary experimental result. If an inhomogeneous magnetic field acts on magnetic particles, they start to move in the direction of the higher magnetic field strength in dependence of its magnetic properties (similar to manipulating particles in an inhomogeneous electrical field, dielectrophoresis (Pethig, 2017). In addition to gravity or centrifugal forces, the particles are additionally accelerated by the magnetic field and the terminal velocity depends on the directions of the corresponding force vectors (Wilhelm et al., 2002). As proposed by (Lerche et al., 2008), analytical centrifugation (e.g. of LUMiFuge / LUMiSizer-type) or earth-gravity-based separation analysis (e.g. LUMiReader) can be adapted with permanent magnets of field directions parallel or orthogonal regarding the gravity field (Fig. 29a and Fig. 29b). The strength of the magnetic field can be varied by the type of magnet and distance to the sample cell.

Fig. 29

Schematic drawing of vertical/radial (a) and orthogonal (b) magnet (M) arrangements (black arrow) with regard to earth gravity/centrifugal field (green arrow) and sample cell. Photo (c) shows the adapter to build an orthogonally directed magnet field and the sample cell with a formed bridge of magnet particles between magnets before centrifugation.

The first example of a “sedimentation-based technique” to analyze magnetic particles (Mykhaylyk et al., 2015) revolves around the analysis of magnetic lipoplexes formulated by different MNPs combined with liposomal enhancers for gene delivery to human corneal endothelium (Czugala et al., 2016). Complexes were assembled at a pDNA concentration of 10 μg/mL by 3 different MNPs but different Fe-to-pDNA w/w ratios (Tab. at Fig. 30b).

Fig. 30

Principle of the determination of magnetophoretic velocity of different MNPs (Tab. (b)) by STEP-Technology^® (customized LUMiReader, LUM GmbH, Berlin) (a, left). The magnetic field is applied from the bottom (a, right) parallel to the gravity field. The cumulative distribution of magnetophoretic velocity v_mag (b, bottom right) is determined based on the kinetics of the integral extinction profiles (b, bottom left) calculated between 2.3 mm and 7.3 mm (ROI) as indicated not greyed out in (a, left) (adopted from (Czugala et al., 2016)).

The size of magnetic objects was in the range of 0.83 μm to 1.1 μm. To confirm proper complexing and stability of these magnetic gene carriers, the magnetophoretic velocity was determined based on changes of space-and time-resolved extinction profiles both under gravity and superposed magnetic fields. The extinction profiles were registered along the vertical axis of the sample cell with a customized LUMiReader^® device equipped with 2 disk Neodymium-iron-boron magnets positioned underneath an optical cell (Fig. 30a, right). This resulted in a magnetic flux density and gradient averaged over the vertical sample height of 0.16 T and 33.5 T/m, respectively. The magnetophoretic velocity ν (Fig. 30b, right) was calculated based on an integral normalized extinction at 410 nm wavelength, E/E₀, versus time t (Fig. 30b, left). No sedimentation of the magnetic carriers was detected within 1500 s with no magnetic field application (horizontal lines at about 1.0 in Fig. 30b, left). With the magnetic field, a significant decline of the integral extinction (clarification) was observed indicating a decrease of magnetic particle concentration (magnetophoresis) depending on the type of lipoplexes (different MNPs and the amount per complex, Tab. of Fig. 30b). The calculated cumulative distribution of effective magnetophoretic velocity is shown in Fig. 30b, right). For details refer to (Mykhaylyk et al., 2015) and (Czugala et al., 2016).

Determination of magnetophoretic velocity by STEP-Technology^® as a tool to characterize assembled objects and determination of magnetic responsiveness can be applied to very different magnetic objects as depicted in Fig. 31. Magnetic fields cause magnetophoresis and change particle velocity compared to a gravity field. Mean magnetophoretic velocities amount to some tens of μm/s up to over 100 μm/s.

Fig. 31

Magnetophoretic velocities (orthogonal magnetic field with regard to earth gravity, Fig. 29b) of different MNPs and magnetic objects. (a) assemblies of MNPs; (b) micro particles; (c) siRNA-MNP; (d) pDNA-MNP; (e) virus-MNP; (f) micro bubbles; (g) labeled cells (Plank C. et al., 2011).

For the sake of completeness, besides magnetophoretic velocity distributions under gravity, analytical centrifugation can be employed to quantify the magnetization and magnetic forces between magnetic particles using magnetic fields directed orthogonally to the centrifugal field (Fig. 29b). The magnetic flux density (field strength) of different holders may vary up to 0.5 T in the middle of the sample cell and increases in the direction of the magnet surfaces. Holders are placed on the horizontal centrifuge rotor (Fig. 2). Due to the applied magnetic field, magnetic particles migrate in the direction of increasing magnetic strength and finally form a “bridge” between the magnets (Fig. 29c). If the “bridge” is exposed to an increasing centrifugal force, it slides down and starts to deform under the applied load like a two-end fixed rubber strap. Finally, at a critical centrifugal force, the magnetic interaction forces between particles are overcome and the “bridge” breaks (Lerche et al., 2008), (Eichholz et al., 2014).

5. Concluding remarks

In this review we discussed the in-situ visualization and quantification of particle movement in suspensions and emulsions using an advanced sedimentation measuring technique. STEP-Technology^® can be applied for settling or creaming both at earth gravity or elevated centrifugal gravity, covering particle sizes from a few nm up to 100 μm. Although when talking about sedimentation measurement, first of all particle size comes to mind, it was demonstrated that the particle velocity distribution itself is a very powerful particle characteristic obtained based on a first-principle method. The influence of particle shape, particle concentration, rheological behavior of continuous phase on sedimentation or separation processes can be studied. Velocity can be analyzed in terms of particle size distribution in a straightforward manner and stands out with regard to sensitivity and accuracy. Advanced sedimentation measuring techniques can also be employed for additional particle characteristics such as the in-situ determination of density of dispersed particles in suspensions and emulsions or the magnetophoretic velocity (responsiveness) of MNPs and other magnetic objects.

Due to restricted space, we had to omit particle surface characterization which is gaining increasing attention in nanoparticle technology with respect to dispersion, polar-polar and hydrogen bond interaction (Hansen Dispersibility Parameters). This information is closely related to the dispersibility of particles in different liquids (Hansen et al., 2008), (Lerche et al., 2015), (Süß et al., 2018). Classification of stable and flocculated/agglomerated dispersed phases is another interesting application of the in-situ visualization of separation phenomena by analytical photocentrifugation (Lerche et al., 2007), (Paciejewska, 2011), (Bharti et al., 2011), (Lerche et al., 2014). The latter is applicable, in contrast to the Zeta potential, also for electrokinetically soft particles (Oshima, 2012) and does not need any dilution.

What could be the next steps in cuvette-based analytical photocentrifugation? It would be beneficial to increase the rotor speed to reduce the measurement time and to increase the pressure load for consolidation and filtration experiments (Loginov et al., 2017b). Higher flexibility with respect to the multi-wavelength approach would broaden application fields and allow 2-dimensional particle characterization. The determination of the volume-based particle size with no reference to particle shape and complex refractive index is also of great interest. From a theoretical point of view, a unified sedimentation theory based on the Kynch and Lamm approach is a very challenging goal.

Finally, I would like to underline that this review would not have been possible without the permanent communication, discussion and exchange of opinions with the academic and industrial communities, which has lasted now for about 25 years. Joint work in standardization organizations, especially WG 2 “Sedimentation, classification” of ISO TC24/SC4 has sharpened the understanding of the discussed matter.

Acknowledgements

Scientific work and instrument development was partly financed by grants from the European Union, Federal Ministry of Economics and Energy of Germany (Program ZIM) and Federal State of Berlin. The author gratefully acknowledges the contributions and support of all LUM team members, especially Dipl.-Ing. Torsten Detloff, Dean Dinair (PhD), Dr. Titus Sobisch, and Dr. Holger Woehlecke (company Dr. Lerche KG). The author acknowledges the support of Dr. O. Mykhaylyk regarding characterization and data of magnetic particles, and Dipl.-Ing. Torsten Detloff for providing unpublished results and great assistance in preparing the manuscript.

Nomenclature

AC-L

analytical photocentrifuge LUMiSizer^®

Al₂O₃

aluminum oxide

AUC

analytical ultracentrifuge

CA

centrifugal acceleration

CPA

constant position approach

CTA

constant time approach

F

Froude number

G

separation number

GS-L

gravity sedimentometer LUMiReader^®

D₂O

heavy water

DLS

dynamic light scattering

H₂O

water

ISO

International Organization for Standardization

MNP

magnetic nanoparticle

NIR

near infrared

O/W

oil/water

PA

polyamide

PC

polycarbonate

pDNA

plasmid-deoxyribonucleic acid

PMMA

poly(methyl methacrylate)

PTA

particle tracking analysis

RCA

relative centrifugal acceleration

Re

Reynolds number

RM

reference material

ROI

region of interest

SC

subcommittee

SEM

scanning electron microscope

SiO₂

silica, silicon dioxide

SLS

static light scattering

TC

technical committee

VIS

visible light

WG

working group

a

acceleration (m/s²)

A_V

volumetric scattering cross-section (1/m)

B

magnetic flux density (T)

C_D

drag coefficient

D

diffusion coefficient (m²/s)

E

extinction

f

translational frictional coefficient

g

acceleration due to gravity (m/s²)

h

sedimentation distance (mm)

H

hindrance function

i

index

j

index

k

coverage factor of uncertainty equation

K

factor for material data in Stokes equation

L

optical path length (m)

m

mass (kg)

n

rotational speed (RPM)

N

end of index

q₃(x)

volume-weighted particle size distribution density

Q₃(x)

cumulative volume-weighted particle size distribution

Q_ext(x)

cumulative extinction-weighted particle size distribution

Q(v)

cumulative velocity distribution (%)

ΔQ

increment of cumulative distribution

r

position, distance from rotation center (mm)

r_m

meniscus position, filling height (mm)

r_o

outer position, position of cell bottom (mm)

r_p

position of the particle, distance of particle from rotation center (mm)

Re

Reynolds number

RI_l

fluid complex refractive index

RI_p

particle complex refractive index

p

particle position (mm)

s

sedimentation coefficient (s = 10⁻¹³ s)

t

time (s)

Δt

time interval between transmission profiles (s)

T

transmission (%)

ΔT

change in transmission (%)

v

velocity (μm/s)

x

particle size (nm)

y

coefficient

η

viscosity (mPa s)

η_rel

relative dispersions viscosity

ϑ

temperature (°C)

λ

wavelength (nm)

κ

kinematic viscosity (m²/s)

ρ_l

liquid density (kg m⁻³)

ρ_p

particle density (kg m⁻³)

φ

concentration (cm³/cm³, %)

Δφ

change in concentration (cm³/cm³, %)

ω

angular speed (rad/s)

Author’s Short Biography

Dietmar Lerche

Professor Dr. D. Lerche studied biophysics at Lomonossow University in Moscow and received his PhD and Dr. of Sci. in biophysics from the Math and Natural Science Faculty at Humboldt University Berlin. From 1989 to 1994, he served as the director of the Institute of Medical Physics and Biophysics at the Medical School Charité at HUB and received a full professorship (medical physics and biophysics). He worked in fields such as dispersion analysis, colloid chemistry, blood rheology and fluid dynamics (> 250 papers). In 1994, he founded the independent LUM GmbH with the goal of providing innovative and unique instrumentation to advance the fields of suspension and emulsion sciences. He also chairs and directs subsidiary companies in the USA, China and Japan as well as the biotech company Dr. Lerche KG, and is an active member of the ISO-TC 24 / SC4—Committee “Particle Characterization” (convener of its WG2 (sedimentation methods) and WG16 (stability)). His achievements were rewarded with different prizes, among them the Innovation Prize 2012 of the states Berlin-Brandenburg (Germany).

References

•  Allen  T., Particle Size Measurement, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
•  Anestis  G.,  Schneider  W., Kinematic-wave analysis of particle settling in tube centrifuges, PAMM, Proceedings in Applied Mathematics and Mechanics, 14 (2014) 709–710.
•  Anker  J.N.,  Mefford  T., Biomedical Applications of Magnetic Particles, CRC Press, Boca Raton, 2017.
•  Arabi  A.S.,  Sanders  R.S., Particle terminal settling velocities in non-Newtonian viscoplastic fluids, The Canadian Journal of Chemical Engineering, 94 (2016) 1092–1101.
•  Arakawa  M.,  Yokoyama  T.,  Yamaguchi  T.,  Minami  T., Comparative data of particle size distribution on flake like particles by various methods, KONA Powder and Particle Journal, 3 (1985) 10–16.
•  Babick  F., Suspensions of Colloidal Particles and Aggregates, Springer, Heidelberg New York Dordrecht London, 2016.
•  Babick  F.,  Mielke  J.,  Hodoroaba  D.,  Weigel  S.,  Wohlleben  W., Critical review manuscript with real-world performance data for counting, ensemble and separating methods including in-build mathematical conversion to number distributions submitted for publication, NanoDefine Technical Report D3.3, NanoDefine Consortium, Wageningen, 2016a.
•  Babick  F.,  Schmidt  A.,  Stintz  M.,  Detloff  T.,  Lerche  D.,  Obata  K.,  Mori  Y., Multi-wavelength photo-centrifugation for particle sizing without reference to models for the optical behavior, Intern Workshop Disp Anal & Mat Test 26th–27th September 2016, Berlin, Germany, (2016b).
•  Badolato  G.G.,  Aguilar  F.,  Schuchmann  H.P.,  Sobisch  T.,  Lerche  D., Evaluation of long term stability of model emulsions by multisample analytical centrifugation, Progress in Colloid and Polymer Science, 134 (2008) 66–73.
•  Bergeson  L.L.,  Hutton  C.N., EC Begins Consultation on Revising Recommendation on Definition of Nanomaterial, 2017, Nano and Other Emerging Chemical Technologies Blog, <https://nanotech.lawbc.com/2017/09/ec-begins-consultation-on-revising-recommendation-on-definition-of-nanomaterial/> accessed 26.03.2018.
•  Berres  S.,  Bürger  R., On the settling of a bidisperse suspension with particles having different sizes and densities, Acta Mechanica, 201 (2008) 47–62.
•  Bharti  B.,  Meissner  G.H.,  Findenegg  G.H., Aggregation of silica nanoparticles directed by adsorption of lysozyme, Langmuir, 27 (2011) 9823–9833.
•  Bickert  G., Sedimentation feinster suspendierter Partikeln im Zentrifugalfeld, Institut für Mechanische Verfahrenstechnik und Mechanik (MVM), Fakultät für Chemieingenieurwesen (Fak. f. Chemieing.), Karlsruher Institut für Technologie, Karlsruhe, 1997.
•  Boldt  S., Personal Communication, Beuth University of Applied Sciences, Applied Physics—Medical Engineering, Berlin, 2017.
•  Bonzel  H.P.,  Bradshaw  A.M.,  Ertl  G., Physics and Chemistry of Alkali Metal Adsorption, Amsterdam New York: Elsevier, 1989.
•  Brouwers  H.J.H., Viscosity of a concentrated suspension of rigid monosized particles, Physical Review E, 81 (2010) 1–11.
•  Bullard  J.W.,  Pauli  A.T.,  Garboczi  E.J.,  Matys  N.S., A comparison of viscosity–concentration relationships for emulsions, Journal of Colloid and Interface Science, 330 (2009) 186–193.
•  Buscall  R.,  White  L.R., The consolidation of concentrated suspensions. Part 1.—The theory of sedimentation, Journal of the Chemical Society, Faraday Transactions 1: Physical Chemistry in Condensed Phases, 1 (1987) 873–891.
•  Buske  N.,  Gedan  H.,  Lichtenfeld  H.,  Katz  W.,  Sonntag  H., A method for measuring particle and aggregate size distribution in colloidal dispersions, Colloid and Polymer Science, (1980) 1303–1394.
•  Bustos  M.C.,  Concha  F.,  Bürger  R.,  Tory  E.M., Sedimentation and Thickening: Phenomenological Foundation and Mathematical Theory, Kluwer Academic Publishers, Dordrecht, 1999.
•  Cai  Z.,  Wu  J.,  Du  B.,  Zhang  H., Impact of distribution of carboxymethyl substituents in the stabilizer of carboxymethyl cellulose on the stability of acidified milk drinks, Food Hydrocolloids, 76 (2018) 150–157.
•  Callahan  J., A non-toxic heavy liquid and inexpensive filters for separation of mineral grains, Journal of Sedimentary Research, 57 (1987) 765–766.
•  Cao  D.Q.,  Iritani  E.,  Katagiri  N., Flotation and sedimentation properties in centrifugal separation of emulsion–slurry, Journal of Chemical Engineering of Japan, 47 (2014) 392–398.
•  Chenouard  N.,  Smal  I.,  de Chaumont  F.,  Maska  M.,  Sbalzarini  I.F.,  Gong  Y.,  Cardinale  J.,  Carthel  C.,  Coraluppi  S.,  Winter  M.,  Cohen  A.R.,  Godinez  W.J.,  Rohr  K.,  Kalaidzidis  Y.,  Liang  L. et al., Objective comparison of particle tracking methods, Nature Methods, 11 (2014) 281–289.
•  Concha  F.,  Bürger  R., A century of research in sedimentation and thickening, KONA Powder and Particle Journal, 20 (2002) 38–70.
•  Coulter  W.H., Means for counting particles suspended in a fluid, USA Patent, (1953) US2656508A.
•  Czugala  M.,  Mykhaylyk  O.,  Böhler  P.,  Onderka  J.,  Stork  B.,  Wesselborg  S.,  Kruse  F.E.,  Plank  C.,  Singer  B.B.,  Fuchsluger  T.A., Efficient and safe gene delivery to human corneal endothelium using magnetic nanoparticles, Nanomedicine (Lond), 11 (2016) 1787–1800.
•  DeLoid  G.,  Cohen  J.M.,  Darrah  T.,  Derk  R.,  Rojanasakul  L.,  Pyrgiotakis  G.,  Wohlleben  W.,  Demokritou  P., Estimating the effective density of engineered nanomaterials for in vitro dosimetry, Nature Communications, 5 (2014) 3514.
•  Deshpande  A.P.,  Krishnan  J.M.,  Kumar  S., Rheology of Complex Fluids, Springer, New York, 2010.
•  Detloff  T., Personal Communication, LUM GmbH, Berlin, Germany, 2004.
•  Detloff  T.,  Küchler  S.,  Sobisch  T.,  Lerche  D., Is it possible to reduce cost for pilot centrifuge testing by lab scale tests?, 11th World Filtration Congress, 16th–20th April 2012, Graz, Austria, Filtech Exhibitions Germany GmbH & Co. KG, 2012.
•  Detloff  T.,  Lerche  D., Centrifugal separation in tube and disc geometries: Experiments and theoretical models, Acta Mechanica, 201 (2008) 83–94.
•  Detloff  T.,  Lerche  D.,  Sobisch  T., Particle size distribution by space or time dependent extinction profiles obtained by analytical centrifugation, Particle & Particle Systems Characterization, 23 (2006) 184–187.
•  Detloff  T.,  Sobisch  T.,  Lerche  D., Particle size distribution by space or time dependent extinction profiles obtained by analytical centrifugation (concentrated systems), Powder Technology, 174 (2007) 50–55.
•  Detloff  T.,  Sobisch  T.,  Lerche  D., Characterisation of separating dispersions by multi wavelength extinction profiles, Dispersion Letters, 2 (2011) 32–35.
•  Detloff  T.,  Sobisch  T.,  Lerche  D., Instability index, Dispersion Letters Technical, T4 (2014) 1–4.
•  Dintenfass  L., Internal Viscosity (Rigidity) of the Red Cell and Blood Viscosity Equation; Counteraction of Errors Due to Flow Instability of Plasma, Biofluid Mechanics, Plenum Publishing Corp., Boston, MA, 1980, pp. 401–416.
•  Dukhin  A.S.,  Goetz  P.J., Characterization of Liquids, Nanoand Microparticulates, and Porous Bodies Using Ultrasound, Elsevier, Amsterdam, 2010.
•  Eichholz  C.,  Knoll  J.,  Lerche  D.,  Nirschl  H., Investigations on the magnetization behavior of magnetic composite particles, Journal of Magnetism and Magnetic Materials, 368 (2014) 139–148.
•  Erne  B., Sedimentation equilibria of colloidal dispersions in ultrathin glass capillaries, Dispersion Letters, 3 (2012) 16–17.
• European Union, Commission recommendation of 18 October 2011 on the definition of nanomaterial, Official Journal of the European Union, 54 (2011) 38–40.
•  Filipe  V.,  Hawe  A.,  Jiskoot  W., Critical evaluation of nanoparticle tracking analysis (NTA) by NanoSight for the measurement of nanoparticles and protein aggregates, Pharmaceutical Research, 27 (2010) 796–810.
•  Fitch  B., Kynch theory and compression zones, AIChE Journal, 29 (1983) 940–947.
•  Frömer  D.,  Lerche  D., An experimental approach to the study of the sedimentation of dispersed particles in a centrifugal field, Archive of Applied Mechanics, 72 (2002) 85–95.
•  Fujita  H., Mathematical Theory of Sedimentation Analysis, Academic Press Inc., New York, London, 1962.
•  Guyon  E.,  Hulin  J.P.,  Petit  L.,  Mitescu  C.D., Physical Hydrodynamics, Oxford University Press, Oxford, 2001.
•  Hansen  C.M.,  Abbott  S.,  Yamamoto  H., Hansen Solubility Parameters: A User’s Handbook, fourth ed., www.Hansen-Solubility.com, 2008.
•  Happel  J.,  Brenner  H., Low Reynolds Number Hydrodynamics, Martinus Nijhoff Publishers, The Hague / Boston / Lancaster, 1983.
•  He  P.,  Mejia  A.F.,  Cheng  Z.,  Sun  D.,  Sue  H.-J.,  Dinair  D.S.,  Marquez  M., Hindrance function for sedimentation and creaming of colloidal disks, Physical Review E, 81 (2010) 026310.
•  Hinderliter  P.M.,  Minard  K.R.,  Orr  G.,  Chrisler  W.B.,  Thrall  B.D.,  Pounds  J.G.,  Teeguarden  J.G., ISDD: A computational model of particle sedimentation, diffusion and target cell dosimetry for in vitro toxicity studies, Particle and Fibre Toxicology, 7 (2010) 36.
•  Hogg  R., A spheroid model for the role of shape in particle analysis, KONA Powder and Particle Journal, 32 (2015) 227–235.
•  Holschuh  K.,  Bauer  J., Industrial production, surface modification, and application of magnetic particles, in:  Keller  K. (Ed.) Upscaling of Bio-Nano-Processes, Springer, Berlin Heidelberg, 2014, pp. 117–128.
•  Hölzer  A., Bestimmung des Widerstandes, Auftriebs und Dreh-moments und Simulation der Bewegung nichtsphärischer Partikel in Laminaren und Turbulenten Strömungen Mit Dem Lattice-Boltzmann-Verfahren, Dissertation, 2007.
•  Horn  F.M.,  Richtering  W., Viscosity of bimodal charge-stabilized polymer dispersions, Journal of Rheology, 44 (2000) 1279–1292.
•  Horváth-Szabó  G.,  Høiland  H., Compressibility determination of silica particles by ultrasound velocity and density measurements on their suspensions, Journal of Colloid and Interface Science, 177 (1996) 568–578.
• ISO 9276-2:2014, Representation of results of particle size analysis—Part 2: Calculation of average particle sizes/diameters and moments from particle size distributions, (2014).
• ISO 13317-1:2001, Determination of particle size distribution by gravitational liquid sedimentation methods—Part 1: General principles and guidelines, (2001).
• ISO 13317-3:2001, Determination of particle size distribution by gravitational liquid sedimentation methods—Part 3: X-ray gravitational technique, (2001).
• ISO 13317-4:2014, Determination of particle size distribution by gravitational liquid sedimentation methods—Part 4: Balance method, (2014).
• ISO 13318-1:2001, Determination of particle size distribution by centrifugal liquid sedimentation methods—Part 1: General principles and guidelines, (2001).
• ISO 13318-2:2007, Determination of particle size distribution by centrifugal liquid sedimentation methods—Part 2: Photocentrifuge method, (2007).
• ISO 13318-3:2004, Determination of particle size distribution by centrifugal liquid sedimentation methods—Part 3: Centrifugal X-ray method, (2004).
• ISO 13320:2009, Particle size analysis—Laser diffraction methods, (2009).
• ISO 14887:2000, Sample preparation—Dispersing procedures for powders in liquids, (2000).
• ISO 19430:2016, Particle size analysis—Particle tracking analysis (PTA) method, (2016).
• ISO 20998-1:2006, Measurement and characterization of particles by acoustic methods—Part 1: Concepts and procedures in ultrasonic attenuation spectroscopy, (2006).
• ISO 22412:2017, Particle size analysis—Dynamic light scattering (DLS), (2017).
• ISO, International Organization for Standardization, (2017).
• ISO/DIS 18747-2, Determination of the particle density by sedimentation methods—Part 2: Multivelocity approach, (2018).
• ISO 18747-1, Determination of particle density by sedimentation methods—Part 1: Isopycnic interpolation approach, (2018).
• ISO/TR 13097:2013, Guidelines for the characterization of dispersion stability, (2013).
•  Jacob  K.H., Prüfung der Entmischungsstabilität von Cremes, in:  Miller  R.,  Schäffler  M. (Eds.), Dispersionseigenschaften: 2D-Rheologie, 3D-Rheologie, Stabilität, Self-Published, LUM GmbH, Berlin, Potsdam, 2015, pp. 191–202.
•  Jillavenkatesa  A.,  Dapkunas  S.J.,  Lum  L.S.H., NIST Recommended Practice Guide: Particle Size Characterization, NIST, Washington, 2001.
•  Kamiti  M.,  Boldridge  D.,  Ndoping  L.M.,  Remsen  E.E., Simultaneous absolute determination of particle size and effective density of submicron colloids by disc centrifuge photosedimentometry, Analytical Chemistry, 84 (2012) 10526–10530.
•  Kato  H.,  Suzuki  M.,  Fujita,  Horie  M.,  Endoh  S.,  Yoshida  Y.,  Iwahashi  H.,  Takahashi,  Nakamura  A.,  Kinugasa  S., Reliable size determination of nanoparticles using dynamic light scattering method for in vitro toxicology assessment, Toxicology in Vitro, 23 (2009) 927–934.
•  Kaur  S.,  Leal  L.G., Drop deformation and break-up in concentrated suspensions, Journal of Rheology, 54 (2010) 981–1008.
•  Kaye  B.H., UK Patent, (1962) 895 222,320.
•  Kelly  R.,  Etzler  F.M., What is wrong with laser diffraction?, 2013, Donner Technologies <www.donner-tech.com/whats_wrong_with_ld.pdf> accessed 20.06.2018.
•  Kirkland  J.J.,  Dilks Jr  C.H.,  Yau  Y.W.W., Sedimentation field flow fractionation at high force fields, Journal of Chromatography A, 255 (1983) 255–271.
•  Kirsch  S.,  Doerk  A.,  Bartsch  E.,  Sillescu  H.,  Landfester  K.,  Spiess  H.W.,  Maechtle  W., Synthesis and characterization of highly cross-linked, monodisperse core–shell and inverted core–shell colloidal particles. polystyrene/poly(tert-butyl acrylate) core–shell and inverse core–shell particles, Macromolecules, 32 (1999) 4508–4518.
•  Krieger  I.M.,  Dougherty  T.J., A Mechanism for non-newtonian flow in suspensions of rigid spheres, Journal of Rheology, 3 (1959) 137–152.
•  Kuchenbecker  P.,  Gemeinert  M.,  Rabe  T., Inter-laboratory study of particle size distribution measurements by laser diffraction, Particle & Particle Systems Characterization, 29 (2012) 304–310.
•  Kuentz  M.,  Röthlisberger  D., Sedimentation analysis of aqueous microsuspensions based on near infrared transmission measurements during centrifugation. Determination of a suitable amount of gelling agent to minimise settling in the gravitational field, STP Pharma Sciences, 12 (2002) 391–396.
•  Kuentz  M.,  Röthlisberger  D., Rapid assessment of sedimentation stability in dispersions using near infrared transmission measurements during centrifugation and oscillatory rheology, European Journal of Pharmaceutics and Biopharmaceutics, 56 (2003) 355–361.
•  Kynch  G.J., A theory of sedimentation, Transactions of the Faraday Society, 48 (1952) 166–176.
•  Ladenburg  R., Über den Einfluß von Wänden auf die Bewegung einer Kugel in einer reibenden Flüssigkeit, Annalen der Physik, 328 (1907) 447–458.
•  Lamberty  A.,  Franks  K.,  Braun  A.,  Kestens  V.,  Roebben  G.,  Linsinger  T., Interlaboratory comparison for the measurement of particle size and zeta potential of silica nanoparticles in an aqueous suspension, Journal of Nanoparticle Research, 13 (2011) 7317–7329.
•  Lamm  O., Die Differentialgleichung der Ultrazentrifugierung, Arkiv för matematik, astronomi och fysik, 21B (1929) 1–4.
•  Lau  R.,  Hassan  M.S.,  Wong  W.,  Chen  T., Revisit of the wall effect on the settling of cylindrical particles in the inertial regime, Industrial & Engineering Chemistry Research, 49 (2010) 8870–8876.
•  Leith  D., Drag on nonspherical objects, Aerospace Science and Technology, 6 (1987) 153–161.
•  Lerche  D., Determination of hydrodynamic density of micro-and nano particles dispersed in liquids by migration velocities, Particulate Systems Analysis, Edinburgh UK, 2011.
•  Lerche  D.,  Bilsing  R., Separation of red blood cells at high volume concentration under low centrifugal acceleration, Biorheology, 25 (1988) 245–252.
•  Lerche  D.,  Frömer  D., Theoretical and experimental analysis of the sedimentation kinetics of concentrated red blood cell suspensions in centrifugal field: Determination of aggregation and deformation of RBC by flux density and viscosity function, Biorheology, 38 (2001) 249–262.
•  Lerche  D.,  Hennicke  G., The effect of different ionic and non-ionic X-ray contrast media on the morphological and rheological properties of human red blood cells, Clinical Hemorheology, 12 (1992) 341–355.
•  Lerche  D.,  Horvat  S.,  Sobisch  T., Efficient instrument based determination of the Hansen Solubility Parameters for talc-based pigment particles by multisample analytical centrifugation: Zero to One Scoring, Dispersion Letters, 6 (2015) 13–18.
•  Lerche  D.,  Mertens  U., Sedimentationsverhalten von Teilchen unter Einwirkung überlagerter Magnetfelder, Jahrestreffen des Fachausschusses Mechanische Flüssigkeitsabtrennung, Würzburg, 2008.
•  Lerche  D.,  Piesendel  L.,  Senge  B., Characterization of food quality and structural stability by analytical centrifugation, Proceedings of the 3rd International Symposium on Food Rheology and Structur, Zürich, Switzerland, ETH Zürich, 2003.
•  Lerche  D.,  Sobisch  T., Direct and accelerated characterization of formulation stability, Journal of Dispersion Science and Technology, 32 (2011) 1711–1811.
•  Lerche  D.,  Sobisch  T., Evaluation of particle interactions by in situ visualization of separation behaviour, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 440 (2014) 122–130.
•  Lerche  D.,  Sobisch  T.,  Küchler  S., Shelf life prediction of liquid food products based on multisample analytical centrifugation, Proc of 4th Int Symp on Food Rheology and Structure, Zürich, Switzerland, (2006).
•  Lerche  T.,  Sobisch  S., Consolidation of concentrated dispersions of nano- and microparticles determined by analytical centrifugation, Powder Technology, 174 (2007) 46–49.
•  Leschonski  K., Grundlagen und moderne Verfahren der Partikelmesstechnik, Technische Universität Clausthal, Clausthal-Zellerfeld, 1988.
•  Leschonski  K.,  Alex  W.,  Koglin  B., Teilchengrößenanalyse. 1. Darstellung und Auswertung von Teilchengrößenverteilungen, Chemie Ingenieur Technik, 46 (1974) 23–26.
•  Leung  W., Separation of dispersed suspension in rotating tube, Separation and Purification Technology, 38 (2004) 99–119.
•  Leung  W.W.F., Centrifugal Separations in Biotechnology, Oxford: Elsevier/Academic Press, 2007.
•  Lichtenfeld  H.,  Stechemesser  H.J.,  Möhwald  H., Single particle light-scattering photometry—some fields of application, Journal of Colloid and Interface Science, 276 (2004) 97–105.
•  Linsinger  T.P.J.,  Roebben  G.,  Gilliland  D.,  Calzolai  L.,  Rossi  F.,  Gibson  N.K.C., Requirements on measurements for the implementation of the European Commission definition of the term “nanomaterial”, Geel, Belgium: Joint Research Centre of the European Commission, 2012.
•  Loginov  M.,  Samper  F.,  Gésan-Guiziou  G.,  Sobisch  T.,  Lerche  D.,  Vorobiev  E., Centrifugal ultrafiltration for determination of filter cake properties of colloids, Journal of Membrane Science, 536 (2017a) 59–75.
•  Loginov  M.,  Zierau  A.,  Kavianpour  D.,  Lerche  D.,  Vorobiev  E.,  Gésan-Guiziou  G.,  Mahnic-Kalamiza  S.,  Sobisch  T., Multistep centrifugal consolidation method for characterization of filterability of aggregated concentrated suspensions, Separation and Purification Technology, 183 (2017b) 304–317.
•  Lorentz  H.A., Ein allgemeiner Satz, die Bewegung einer reibenden Flüssigkeit betreffend, nebst einigen Anwendungen derselben, Annalen der theoretischen Physik, 1 (1907) 28–42.
•  Luerkens  D.W., Theory and Application of Morphological Analysis: Fine Particles and Surfaces, CRC Press, Boca Raton Ann Arbor Boston London, 1991.
•  Luigjes  B.,  Thies-Weesie  D.M.E.,  Erné  B.H.,  Philipse  A.P., Sedimentation equilibria of ferrofluids: I. Analytical centrifugation in ultrathin glass capillaries, Journal of Physics: Condensed Matter, 24 (2012) 245103 (8pp).
•  Mächtle  W., Charakterisierung von disperisionen durch gekopplete H₂O/D₂O-ultrazentrifugenmessungen, Die Makromolekulare Chemie, 185 (1984) 1025–1039.
•  Mächtle  W.,  Börger  L., Analytical Ultracentrifugation of Polymers and Nanoparticles, Springer-Verlag GmbH, Berlin Heidelberg, 2006.
•  Macosko  C.W., Rheology: Principles, Measurements, and Applications, VCH Publishers, Inc., New York, Weinheim, Cambridge, 1994.
•  Masuda  H.,  Higashitani  K.,  Yoshida  Y., Powder Technology: Fundamentals of Particles, Powder Beds, and Particle Generation, CRC Press, Boca Raton London New York, 2006.
•  Maude  A.D.,  Whitmore  R.L., A generalized theory of sedimentation, British Journal of Applied Physics, 9 (1958) 477–482.
•  McClements  D.J., Food Emulsions: Principles, Practices, and Techniques, second ed., CRC Press, Boca Raton, 2004.
•  McCormick  H.W., Determination of latex particle size distribution by analytical ultracentrifugation, Journal of Colloid Science, 19 (1964) 173–184.
•  Mehn  D.,  Iavicoli  P.,  Cabaleiro  N.,  E.B.S.,  Caputo  F.,  Geiss  O.,  Calzolai  L.,  Rossi  F.,  Gilliland  D., Analytical ultracentrifugation for analysis of doxorubicin loaded liposomes, International Journal of Pharmaceutics, 523 (2017) 320–326.
•  Merkus  H.G., Particle Size Measurements: Fundamentals, Practice, Quality, Springer, Heidelberg New York Dordrecht London, 2009.
•  Merkus  H.G.,  Meesters  G.M.H., (Eds.) Particulate Products: Tailoring Properties for Optimal Performance, Springer, Cham, 2014.
•  Michaelis  A.S.,  Bolger  J.C., Settling rates and sediment volumes of flocculated kaolin suspensions, Industrial & Engineering Chemistry Fundamentals, 1 (1962) 24–33.
•  Mie  G., Beiträge zur Optik trüber Medien, speziell kolloidaler Goldlösungen, Annals of Physics, 25 (1908) 377–452.
•  Mykhaylyk  O.,  Lerche  D.,  Vlaskou  D.,  Schoemig  V.,  Detloff  T.,  Krause  D.,  Wolff  M.,  Joas  T.,  Plank  C., Magnetophoretic velocity determined by space- and time-resolved extinction profiles, IEEE Magnetics Letters, 6 (2015) 1–4.
•  Mykhaylyk  O.,  Zelphat  O.,  Hammerschmid  E.,  Anton  M.,  Rosenecker  J.,  Plank  C., Recent advances in magnetofection and its potential to deliver siRNAs in vitro, Methods in Molecular Biology, 487 (2008) 1–36.
•  Nadler  M.,  Mahrholz  T.,  Riedel  U.,  Schilde  C.,  Kwade  A., Preparation of colloidal carbon nanotube dispersions and their characterisation using a disc centrifuge, Carbon, 46 (2008) 1384–1392.
• NanoDefine, NanoDefine Publications, (2018) NanoDefine, 2018, NanoDefine Publications <www.nanodefine.eu/index.php/nanodefine-publications> accessed 26.03.2018.
•  Neumann  A.,  Hoyera  W.,  Wolff  M.W.,  Reichl  U.,  Pfitzner  A.,  Rotha  B., New methods for density determination of particles using CPS disc centrifuge, Colloids and Surfaces B: Biointerfaces, 104 (2013) 27–31.
•  Oshima  H., Electrical phenomena in a suspension of soft particles, Soft matter, 8 (2012) 3511–3514.
•  Paciejewska  K.M., 2011, Untersuchung des Stabilitätsverhalten von binären kolloidalen Suspensionen <http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-65050> accessed 26.03.2018.
•  Pal  R., Shear viscosity behavior of emulsions of two immiscible liquids, Journal of Colloid and Interface Science, 225 (2000) 359–366.
•  Pethig  R., Dielectrophoresis: Theory, Methodology and Biological Applications, Wiley, Hoboken, West Sussex, 2017.
•  Pettyjohn  E.S.,  Christiansen  E.B., Effect of particle shape on free-settling rates of isometric particles, Chemical engineering progress, 44 (1948) 157–172.
•  Plank  C.,  Zelphati  O.,  Mykhaylyk  O., Magnetically enhanced nucleic acid delivery. Ten years of magnetofection—Progress and prospects, Advanced Drug Delivery Reviews, 63 (2011) 1300–1331.
•  Planken  K.L.,  Cölfen  H., Analytical ultracentrifugation of colloids, Nanoscale, 2 (2010) 1849–1869.
•  Probstein  R.F., Physicochemical Hydrodynamics. An Introduction, John While & Sons Inc., Hoboken, 2003.
•  Quemada  D., Rheology of concentrated disperse systems and minimum energy dissipation principle. I. Viscosity-concentration relationship, Rheologica Acta, 16 (1977) 82–94.
•  Quemada  D., Rheology of concentrated disperse systems II. A model for non-newtonian shear viscosity in steady flows, Rheologica Acta, 17 (1978a) 632–642.
•  Quemada  D., Rheology of concentrated disperse systems III. General features of the proposed non-newtonian model. Comparison with experimental data, Rheologica Acta, 17 (1978b) 643–653.
•  Richardson  J.F.,  Zaki  W.N., Sedimentation and fluidisation. Part 1, Transactions Institution of Chemical Engineers, 32 (1954) 35–53.
•  Rodriguez  L., Personal Communication, Dr. Lerche KG, Berlin, Germany, 2017.
•  Salinas-Salas  G.,  Ruiz-Tagle-Gutiérrez  I.,  Babick  F., Análisis de la función de corrección de la velocidad de sedimentación para micro partículas / Analysis of the correction function for micro-particle sedimentation velocity, Ingeniare Revista Chilena de Ingeniería, 15 (2007) 283–290.
•  Schaflinger  U., Centrifugal separation of a mixture, Fluid Dynamics Research, 6 (1990) 213–249.
•  Schärtl  W., Light Scattering from Polymer Solutions and Nanoparticle Dispersions, Springer Verlag, Berlin Heidelberg, 2007.
•  Schuck  P.,  Perugini  M.A.,  Gonzales  N.R.,  Howlett  G.J.,  Schubert  D., Size-distribution analysis of proteins by analytical ultracentrifugation: strategies and application to model systems, Biophysical Journal, 82 (2002) 1096–1111.
•  Schuldt  U.,  Woehlecke  H.,  Lerche  D., Characterization of mechanical parameters of microbeads by means of analytical centrifugation, Food Hydrocolloids, (2018) in press. DOI: 10.1016/j.foodhyd.2018.07.005
•  Schulz  D.N.,  Glass  J.E., Polymers as Rheology Modifiers, ACS Symposium Series, American Chemical Society, Washington, 1991.
•  Schuster  B.S.,  Ensign  L.M.,  Allan  D.B.,  Suk  J.S.,  Hanes  J., Particle tracking in drug and gene delivery research: State-of-the-art applications and methods, Advanced Drug Delivery Reviews, 91 (2015) 70–91.
•  Shuler  M.L.,  Aris  R.,  Tsuchiya  H.M., Hydrodynamic focusing and electronic cell-sizing techniques, Applied microbiology, 24 (1972) 384–388.
•  Sobisch  T.,  Detloff  T.,  Lerche  D., Speeding up the evaluation of the physical stability of beverage dispersions, ISFRS 2009, Proceedings of the 5th International Symposium on Food Rheology and Structure, ed.  P.  Fischer,  M.  Pollard,  E.J.  Windhab, 2009, pp. 574–575.
•  Sobisch  T.,  Küchler  S.,  Uhl  A., Comprehensive characterization of medical nutrition—accelerated stability analysis and creaming velocity distribution, Dispersion Letters, 1 (2010) 10–13.
•  Sobisch  T.,  Lerche  D., Separation behaviour of particles in biopolymer solutions in dependence on centrifugal acceleration: Investigation of slow structuring processes in formulations, Colloid Surface A: Physicochemical and Engineering Aspects, 536 (2018) 74–81.
•  Sobisch  T.,  Lerche  D.,  Detloff  T.,  Beiser  M.,  Erk  A., Tracing the centrifugal separation of fine-particle slurries. Effect of centrifugal acceleration, particle interaction and concentration, Filtration, 6 (2006) 313–321.
•  Sommer  K., Sampling of Powders and Bulk Materials, Springer-Verlag, Berlin Heidelberg, 1986.
•  Staudinger  U.,  Krause  B.,  Steinbach  C.,  Pötschke  P.,  Voigt  B., Dispersability of multiwalled carbon nanotubes in polycarbonate-chloroform solutions, Polymer, 55 (2014) 6335–6344.
•  Steinour  H.H., Rate of sedimentation. Nonflocculated suspensions of uniform spheres, Industrial & Engineering Chemistry, 36 (1944) 618–624.
•  Stokes  G.G., On the effect of the internal friction of fluids on the motion of pendulums, Transactions of the Cambridge Philosophical Society, 9 (1851) 1–83.
•  Süß  S.,  Sobisch  T.,  Peukert  W.,  Lerche  D.,  Segets  D., Determination of Hansen parameters for particles: A standardized routine based on analytical centrifugation, Advanced Powder Technology, 7 (2018) 1550–1561.
•  Svedberg  T.,  Rinde  H., The ultra-centrifuge, a new instrument for the determination of size and distribution of size of particle in amicroscopic colloids, Journal of the American Chemical Society, 46 (1924) 2677–2693.
•  Takeda  S., Evaluation of stability of slurries by centrifugal sedimentation methods (in Japanese), Journal of the Society of Powder Technology, Japan, 49 (2012) 489–494.
•  Taylor  G.I., The viscosity of a fluid containing small drops of another fluid, Proceedings of the Royal Society of London. Series A, 138 (1932) 41–48.
•  Uchiyama  S.,  Arisaka  F.,  Stafford  W.F.,  Laue  T., Analytical Ultracentrifugation: Instrumentation, Software, and Applications, Springer Japan, Tokyo, 2016.
•  Uhl  A., Partikelgrößenverteilung von Kontrollpartikeln—Teil 1 SediTest Mikropartikel 1000 nm, Dispersion Letters Technical, T7 (2015) 1–5.
•  Ungarish, Hydrodynamics of Suspensions, Springer, Berlin, 1993.
•  Usher  S.P.,  Studer  L.J.,  Wall  R.C.,  Scales  P.J., Characterisation of dewaterability from equilibrium and transient centrifugation test data, Chemical Engineering Science, 93 (2013) 277–291.
• USP-729, Injections and implanted drug products (parenterals)—Product quality tests, United States Pharmacopeial Convention, (2016).
• USP-788, Particulate matters in injections, United States Pharmacopeial Convention, (2012).
•  van de Hulst  H.C., Light Scattering by Small Particles, Dover Publications Inc., New York, 1981.
•  Walter  J.,  Sherwood  P.J.,  Lin  W.,  Segets  D.,  Stafford  W.F.,  Peukert  W., Simultaneous analysis of hydrodynamic and optical properties using analytical ultracentrifugation equipped with multiwavelength detection, Analytical Chemistry, 87 (2015a) 3396–3403.
•  Walter  J.,  Thajudeen  T.,  Süß  S.,  Segets  D.,  Peukert  W., New possibilities of accurate particle characterisation by applying direct boundary models to analytical centrifugation, Nanoscale, 7 (2015b) 6574–6587.
•  Ward-Smith  S.,  Rawle  A., Validating laser diffraction particle sizing methods to maintain integrity, Scientist live, (2013).
•  Weichert  R., Determination of extinction efficiency and particle size distribution by photosedimentation using light of different wavelengths, Poceed 4th Particle Size Analysis Conf Sept 21st–24th, 1981, Loughborough, Wiley, (1981).
•  Werner  E., Zuckertechnik-Taschenbuch, Vol. 7. vollst. Neubearb. Aufl., Bartens, Berlin-Nikolassee, 1966.
•  Whitby  K.T., A rapid general purpose centrifuge sedimentation method for measurement of size distribution of small particles, part I.—apparatus and method, Journal of the Air Pollution Control Association, 5 (1955) 120–132.
•  White  D.J., The measurement of particle size distribution using the Single Particle Optical Sizing (SPOS) method, (2002).
•  Wiese  B., Accelerated Sedimentation for the Prediction of Long-term Storage Stability, 3 Anwenderseminar 2D- und 3D-Rheologie, Potsdam, (2010).
•  Wilhelm  C.,  Gazeau  F.,  Bacri  J.C., Magnetophoresis and ferromagnetic resonance of magnetically labeled cells, European Biophysics Journal, 32 (2002) 118–125.
•  Woehlecke  H.,  Detloff  T.,  Lerche  D., Determination of hydrodynamic density of micro- and nano particles dispersed in liquids, Dispersion Letters, 3 (2012) 12–15.
•  Zhou  W.,  Su  M.,  Cai  X., Advances in nanoparticle sizing in suspensions: dynamic light scattering and ultrasonic attenuation spectroscopy, KONA Powder and Particle Journal, 34 (2017) 168–182.