Settling Velocities of Small Microplastic Fragments and Fibers

There is only sparse empirical data on the settling velocity of small, nonbuoyant microplastics thus far, although it is an important parameter governing their vertical transport within aquatic environments. This study reports the settling velocities of 4031 exemplary microplastic particles. Focusing on the environmentally most prevalent particle shapes, irregular microplastic fragments of four different polymer types (9–289 μm) and five discrete length fractions (50–600 μm) of common nylon and polyester fibers are investigated, respectively. All settling experiments are carried out in quiescent water by using a specialized optical imaging setup. The method has been previously validated in order to minimize disruptive factors, e.g., thermal convection or particle interactions, and thus enable the precise measurements of the velocities of individual microplastic particles (0.003–9.094 mm/s). Based on the obtained data, ten existing models for predicting a particle’s terminal settling velocity are assessed. It is concluded that models, which were specifically deduced from empirical data on larger microplastics, fail to provide accurate predictions for small microplastics. Instead, a different approach is highlighted as a viable option for computing settling velocities across the microplastics continuum in terms of size, density, and shape.


S1 Scanning electron microscope images
Scanning electron microscope (SEM) images of all investigated particles were acquired on a Gemini DSM 982 (Zeiss, Germany) equipped with DISS6 (point electronic, Germany) and are partly shown in the original publication.The following tableaus comprise the images of microplastic (MP) fragments (Figure S1), PA 6.6 fibers (Figure S2) and PET fibers (Figure S3), respectively.Each sample is depicted at three different magnifications as indicated by scale bars.

S2.1 Particle size cutoffs
Settling experiments with MP fragments were conducted in several sets with gradually reduced runtime in order to measure an increased number of larger MP particles.The runtime of the first set of experiments was fixed based on a longer, preliminary run with the respective samples.It was chosen to be long enough to capture even the sample's particles with the lowest settling velocity.Usually this corresponds to the smallest size fraction of the sample, while always acknowledging the lower size limit for particle detection of 10 µm, which was previously described for this setup by Dittmar et al. 1 .
Reducing the duration of an experimental run thus leads to a carryover of slow-settling, mostly smaller particles that were dosed during previous runs.As sufficient numbers of these particles were already measured during sets of experiments with increased runtimes, they were generally excluded from further evaluation by introducing respective size cutoffs.More so, as their settling velocities are probably affected by interactions with larger, recently dosed particles.This is also apparent from Figure S8 and Figure S9, which depict measured particles below the size cutoffs as wella significant proportion of which does not comply with criteria set for the maximum modeled velocity deviation (see section S2.3).Some measured velocities even notably exceed Stokes' law and the Kaskas equation 2 , which are plotted for the considered particle size range and density, respectively.Both equations describe the theoretical settling velocities of a sphere, with Stokes' law being only valid within the laminar flow regime.
The size cutoffs (cf.Table 1 of the main publication) refer to the equivalent circular diameter (ECD) of a particle and were based on the maximum size of all particles from sets of longer experimental runs, which were detected after the respectively considered runtime elapsed.

S2.2 Visual inspection
All particle tracking results were assessed by a human observer to identify and exclude potential agglomerates, contaminations or incomplete detections.Examples for the described categories are depicted in Figure S4.They were taken from data acquired during experiments with fibers.In each case, the region of interest (ROI) of the raw image as well as the detected contour is shown.Some of the investigated polymer materials are translucent.Opposing particle surfaces can thus appear transparent on raw images if they are even and aligned in parallel to the image plane.Such transparencies occasionally prevent the image processing algorithm from capturing a particle contour in its entirety.Yet, these incomplete detections occurred only rarely, e.g. in some cases in which fibers are aligned orthogonal to the image plane as it is shown in Figure S4.Regarding MP fragments, less than 1% of all particles were not detected completelyoften even at just one of several frames.Only for PET fragments the number of incomplete detections is slightly increased (1.5%).It is evident from SEM images (cf. Figure S1) that the investigated PET fragments often exhibit a rather flat shape with even surfaces, which probably promotes such transparencies.In compliance with partly imposed size cutoffs (cf.Table 2 of the main publication), 3603 particles have been successfully tracked during experiments with MP fragments.31 of these particles (0.9%) were not further considered after visual inspection due to incomplete detection (28 particles) or due to being agglomerates (3 particles).No particles were identified as contaminations.Although the irregular shapes of fragments would only allow to spot prominent contaminations, e.g.large fibrous particles, a negligible contamination rate is very likely and in line with previous results: Validation experiments with spherical and thus distinctive particles already revealed a mean contamination rate of only 0.5% using the identical measuring setup 1 .
With respect to results from experiments with MP fibers, the number of particles excluded after visual inspection is notably higher.In total, 947 particle trajectories were obtained, 187 of which (19.7%) had to be rejectedeither as contaminations (114 particles, 12.0%), agglomerates (58 particles, 6.1%) or due to incomplete contour detection (15 particles, 1.6%).Classification results of all investigated length fractions are depicted in detail in Figure S5 for PA 6.6 fibers and in Figure S6 for PET fibers, respectively.
A classification as agglomerate must not only be caused by insufficient stabilization of particles, but could instead originate from fiber cutting artifacts that connected two filaments.It might as well be the observed joint settling of fibers without actual adhering forces, which probably occurs more regularly with increased fiber length and would thus be consistent with the trend visible in Figure S5 and Figure S6.Only 1.6% of all fibers were incompletely detected in at least one frame and were thus excluded from further evaluation, too.It could be argued, that incomplete detections have a minor impact in the case of fibers, because uniform length is assumed for every investigated fraction anyway.Still, the exclusion was done analogously to the handling of results from experiments with MP fragments.Compared to validation experiments with spherical particles (0.5%) 1 which can be visually well distinguished from other particles, tooa substantially larger share of particles was classified as contaminations during experiments with fibers (12.0%).Yet, there are various reasons for this increase: On the one hand, the applied particle doses were very low (18.5 µg) in order to account for monodispersity and minimize interactions between the settling fibers 1 .Thus, the proportion of particle contaminations versus targeted fibers potentially increases.Yet more importantly, the protocol for embedding and cutting fibers is laborious (cf.instructions available from the Zenodo repository 3 or original protocol by Cole 4 ) and consequently involves several potential sources of particle contaminations.Despite minimizing airborne contamination by covering samples as much as possible during preparation and fiber embedding, contamination cannot be fully preventedespecially as the protocol requires intensive handling of samples and many cutting operations.The obtained cryosections have to be suspended to fully dissolve the embedding matrix and thus not only recover target particles but potential contaminations, too.
The experiment with 100 µm long PET fibers was additionally contaminated with several PMMA spheres (cf.example given in Figure S4 and respective fraction in Figure S6) from an immediately preceding settling experiment.Although the settling column and dosing funnel was always cleaned intensively between experiments as detailed in a previous publication 1 , the previously applied doses of PMMA spheres up to 500 mg were extremely high and apparently caused the carryover of some particles.Yet, the situation was manageable, since the fibers are distinct and thus enable impeccable identification of contaminations.The problem was fixed with the subsequent cleansing and was exceptional in the sense that the concerned doses of PMMA spheres were between 2 and 5 orders of magnitude above the doses used in every other conducted experiment.

S2.3 Applying empirical model for particle-particle interactions
The conducted experiments aim at measuring the settling velocities of single particles.Yet, in order to handle particles of the targeted size, they have to be dosed in suspensionthus allowing interactions between settling particles, which potentially affect measured velocities.This was already demonstrated for spherical particles of varying density (1.05-2.46g/cm³) using the identical experimental setup.An empirical model was based on this data, which describes the increase of a particle's settling velocity due to interactions with other particles. 1This model was now applied to all data from experiments with MP fragments and fibers.It does not only consider particles, which are in-focus, but all particles that are initially tracked via image processing.Details are given in the original publication 1 and its Supporting Information.For each target particle, the deviation from its nominal single particle settling velocity was computed as model output.Although this predicted deviation is not accurate enough to directly correct measurements (e.g. because the model was only calibrated with spheres), it indicates the quality of the respective measurementespecially if strict criteria are applied.Therefore, any measurement of a particle, which exceeds a predicted absolute velocity deviation of 0.01 mm/s as well as a relative deviation of 5% with respect to the actually measured velocity, was not considered.In total, this excluded 94 MP fibers (12.4%) and 207 MP fragments (5.8%) from further evaluation.Details are depicted in Figures S7-S12.Overall, the proportion of excluded measurements is mostly low, indicating a good choice of particle doses.For Set 3 of PMMA fragments, a reduced dose might have been beneficial.

S3 Particle size and shape descriptors
Implementing theoretical formulas for predicting a particle's terminal settling requires to parameterize its size and shape.The tested models (see section S4) presuppose a number of different descriptors for size and shape.6][7] Instead, the MP fragments are dosed in suspension.For each tracked MP fragment, size and shape descriptors have thus to be estimated solely based on the respective particle contours obtained from the raw images.
For a given contour C, its area   , the circularity  , as well as length  and width  of the fitted rotated bounding rectangle with minimum area are extracted. , is computed following an areabased approach previously described by Dittmar et al. 1 (see section S6 of the respective Supporting Information).This circumvents discretization artifacts that occur when determining the perimeter of a pixel contour. 8A correction of  , to approximate a common perimeter-based circularity measure with contour perimeter   ) is adopted from Dittmar et al. 1 as well: ,   and   of all contours serve as a starting point for estimating the necessary size and shape descriptors.Initially, the average circularity   * is calculated from all  , * . Moreover, a mean diameter  2 ̅̅̅̅̅ is computed from the   of three characteristic contours selected from all contours of the tracked particleconsidering the contours with the smallest (  ), the biggest (  ) and the intermediate (  ) contour area.  is determined as the contour area, which is closest to the average of   and   . 2 ̅̅̅̅̅ is then given as the average of the respective equivalent circular diameters deduced from the contour areas: Consequently,  ,  is computed following Bagheri et al. 9 (as already described in the main publication, cf.Equation 2): Furthermore, the longest (  ), intermediate (  ) and shortest (  ) axis lengths of a MP fragment are estimated as the minimum, mean and maximum of all   and   , respectively.The equivalent diameter of a sphere with identical surface area  , , that is required for computing the sphericity  of MP fragments (cf.Table S1), is approximated from   ,   and   .For this, the approach proposed by Dellino et al. 10 is employed, which was used for the model of Dioguardia et al. 11 as well: With respect to MP fibers, discrete length fractions were investigated in separate experiments.
Consequently, diameter  and mean lengths  were determined microscopically (cf.section 'Particle samples' of the main publication) and could be uniformly assumed for the measured fibers of each fraction.This approach is superior to pure image analysis: Due to the high aspect ratio of fibrous particles, successfully extracting their length  from 2D raw images is strongly dependent on their orientation to the image plane.
Assuming cylindrical shape with diameter  and mean length , all size and shape parameters for MP fibers can now be calculated according to their respective original definitions (with length of intermediate axis   = ).For instance, the equivalent diameter   is given as: Table S1 comprises the computation of all remaining descriptors for MP fragments and MP fibers, respectively.All models requiring a certain descriptor are indicated with reference to the respective subsection of section S4 that details their implementation.for MP fragments and, most notably, the diameter  as   =  for MP fibers.
Inserting   and   into Equation 1 of the main publication enables the implicit calculation of   .

S4.2 Goral et al. (2023)
Goral et al. 7 present separate expressions for the drag coefficient of MP fragments and fibers: The authors specify the Reynolds number  to be > 0.4 or ≥ 1 to ensure the validity of Equation S8 or S9, respectively.The sphericity measure , which is required in Equation S8, is approximated here as: For MP fragments,  is computed from the diameter  = √    instead of   .Inserting the respective   into Equation 1 of the main publication enables the implicit calculation of   .

S4.3 Yu et al. (2022)
Yu et al. 12 propose a terminal settling velocity model based on the correction of the drag coefficient for a sphere of equivalent volume.First, a dimensionless diameter  * is computed from   : The drag coefficient of a sphere of equivalent volume ( , ) is calculated by: To account for particle shape,   is derived from  , using Corey shape factor  and sphericity  (cf.Table S1): With the calibrated parameters  1 = −0.25, 2 = 0.03,  3 = 0.33 and  4 = 0.25.The terminal settling velocity   is now obtained as follows:

S4.4 Zhang & Choi (2022)
Zhang & Choi 13 present a general drag coefficient for MP fragments, fibers and films based on the Aschenbrenner shape factor  (cf.Table S1): The authors decided to compute the equivalent diameter   from the maximum projection area as: Inserting   into Equation 1 of the main publication allows the implicit calculation of   .

S4.5 Kaiser et al. (2019)
Kaiser et al. 14 proposed an empirical equation for predicting settling velocities of small MP fragments: One coefficient was originally given as 55.283 mm -1 , but was converted to SI units here for conformity reasons.

S4.7 Bagheri & Bonadonna (2016)
Bagheri & Bonadonna 16 derived an expression for the drag coefficient of non-spherical particles: with   and   calculated as follows: and parameters  and  given as: Elongation  and flatness  are used as main shape descriptors for calculating   and   : Regarding MP fibers,       is consequently given as  2  (also see section S3).  is calculated as described in the main publication, which refers to an approximation method that was originally proposed by Bagheri et al. 9 in a previous paper.Finally,   can be calculated implicitly after inserting   into Equation 1 of the main publication.

S4.8 Dioguardi et al. (2018)
Dioguardi et al. 11 proposed a general expression for the drag coefficient of irregularly-shaped particles, that was deduced from data over a wide range of Reynolds numbers (0.03 ≤  ≤ 10,000): Particle shape is characterized by the shape factor Ψ, which is defined as the ratio between the sphericity  and the circularity Χ (cf.Table S1).  is inserted into Equation 1 of the main publication to implicitly calculate   .

S4.9 Komar (1980)
Komar 17 studied particles settling at low Reynolds numbers and derived separate formulas for the drag coefficient of ellipsoidal particles and cylinders, respectively.
The expression for ellipsoids is valid for  < 1 and is basically a correction of Stokes law' based on the Janke shape factor  (see Table S1).Here, it was implemented to describe the settling of MP fragments: ,  = 24  ⁄ •  −0.380 (S28) Inserting Equation S28 into Equation 1 of the main publication, gives an explicit formula for   : With respect to MP fibers, Komar's equation for the drag coefficient of cylinders was applied, which is stated to be valid for  < 2: ,  = 16.88  ⁄ • (/)

S4.10 Su et al. (2022)
Su et al. 18 conceptualize a variety of particle shapes as super-ellipsoidal particles, which are determined by their shape parameter , elongation  and flatness . can take values between 0 (cubic) and 2 (octahedral) and is set to 1.0 or 0.6 in order to represent MP fragments or MP fibers, respectively (also see Figure S21 taken from the original publication of Su et al. 18 ).
Further considered are the dimensionless diameter  * (computed from   , see Equation S11) and the dimensionless velocity  * , which is defined by: Su et al. 18 proposed an expression for explicitly calculating  * from  * , ,  and : with  1 ,  2 ,  3 and  4 defined as functions of  * again: 1 = 0.11  * + 0.17

S5 Inclination angle of fibers
The inclination of fibersconceived in the following as the angle () between its central axis and the orthogonal plane in direction of gravityis computed as exemplified in Figure S13.As shown, the height () was derived from the detected particle contour.Diameter () and length () were uniformly assumed for each polymer type or length fraction, respectively, and were determined from microscope images beforehand (detailed in the main publication).Thus, the computation is only an estimate.Moreover, it presupposes a non-curved fiber.This assumption appears to be valid for the detected particles.Only few 600 µm PA 6.6 fibers exhibit a very slight curvature, which is still tolerable.Figure S14 depicts the measured settling velocity and the average computed inclination angle for each detected fiber.

S6 Measured drag coefficients
The drag coefficient   can be computed individually for each observed particle from its measured settling velocity   and equivalent diameter   by rearranging Equation 1given in the main text as:

S7 Comparison of terminal settling velocity models
Ten different models for computing a particle's terminal velocity were tested based on the empirical data acquired for small MP fragments and fibers.The computation of necessary size and shape descriptors is detailed in section S3, whereas the implementation of each model is described in section S4.In the following, the models are additionally numbered according to the subsections of S4 (Model 1-10).As detailed in the main publication (cf.section 'Terminal settling velocity models'), the performance of each model is assessed by means of the logarithmic coefficient of determination (  2 ), the coefficient of determination ( 2 ) and the average absolute relative error (|AE|).
Four different definitions of the equivalent diameter were tested for each model (excluding for the computation of   and  itself, if a respective diameter was decidedly specified by the authors).This was done as the characterization of particle size and shape from images is prone to error for irregular particles (see respective discussions in the main manuscript).Thus, the optimal definition of the equivalent diameter might deviate from the one, which was originally specified for the respective model.The tested diameters were collected from all models and are annotated and computed as follows for MP fragments: and  3 = √4     ⁄ Consequently, the respective diameters for MP fibers, described as cylinders with diameter  and length , are defined as: ,  2 =  1 and  3 = √4  ⁄ The original specification of  for each model is indicated by the gray background in Table S2-Table S4 (with   as default setting).Waldschläger & Schüttrumpf 5 (Model 1, see S4.1), Khatmullia & Isachenko 15 (Model 6, see S4.6) and Komar 17 (Model 9, see S4.9) proposed specific models for settling fibers based on their actual diameters .When considering MP fibers and these models,  was thus not varied but always set as  following the respective authors (also indicated in Table S2-Table S4).
Figure S17 (Model 1-5) and Figure S18 (Model 6-10) compare the predictions of each model with the respectively measured settling velocities.As Goral et al. 7 , Dioguardi et al. 11 and Komar 17 specified Reynolds number limits for their models (cf.S4.2, S4.8 and S4.9), Figure S19 similarly shows predictions versus measurements for those three models, but excludes all particles that do not meet the respective criteria for .
The figures only display   2 ,  2 and |AE| as calculated for the entire data, that is respectively shown.
The subsequent tables comprise the performance measures as computed for all data, MP fragments and MP fibers (Table S2) and for each of the investigated particle samples, respectively: PS fragments, PMMA fragments and PET fragments (Table S3) as well as PVC fragments, PA 6.6 fibers and PET fibers (Table S4).S8 Testing the model of Su et al. 18

for large microplastics
The terminal settling velocity model of Su et al. 18 appears to be suitable for predicting the settling velocities of small MP fragments and fibers tested in this study (see section S7).Yet, in contrast to most of the other tested models, it has not been tested for other data sets on larger MP particles thus far. Figure S20 depicts settling velocities of ten different types of MP particles as measured by Goral et al. 7 versus corresponding predictions according to Su et al. 18 .In order to apply the model, a shape parameter  had to be set to characterize the shape of a particle within the proposed framework of super-ellipsoidsapart from determining the descriptors elongation  and flatness .
The shape parameter  can take values between 0.0 (cube) and 2.0 (octahedron)with 1.0 describing spherical shape.This is illustrated by Figure S21, which was extracted from the original publication of Su et al. 18 .Although the choice of  is not always straight forward with respect to each possibly considered particle shape (e.g.tetrahedrons), it appears to be a very sensitive parameter with respect to the results for large MP. Figure S20a shows the predictions according to the values for  that were initially chosen by the authors (ranging from 0.2 to 1.4 for the considered particle shapes) and includes inaccurate predictions for certain particle typesas is indicated by an overall negative coefficient of determination.This changes, if  is only varied between 0.8 and 1.0 as is depicted in Figure S20b yet, these choices are not coherent with the original definition of .

Figure S3 .
Figure S3.SEM images of length fractions of PET fibers.Note magnifications (columns).

Figure S4 .
Figure S4.Exemplary visual classifications of particles from settling experiments with fibers (columns).The relevant section of the raw image (upper row) and the corresponding detected contour (lower row) are depicted, respectively.Note scale bar as well.

Figure S5 .
Figure S5.Results of visual inspection of particles detected in settling experiments with PA 6.6 fibers.Relative fractions of different classification options are plotted as bars for each investigated length fraction.Absolute numbers of classified particles are indicated above the bars, respectively.

Figure S6 .
Figure S6.Results of visual inspection of particles detected in settling experiments with PET fibers.Relative fractions of different classification options are plotted as bars for each investigated length fraction.Absolute numbers of classified particles are indicated above the bars, respectively.

Figure S7 .
Figure S7.PS fragments with criteria for modeled deviation from single particle settling velocity.

Figure S8 .
Figure S8.PMMA fragments with criteria for modeled deviation from single particle settling velocity.

Figure S9 .
Figure S9.PET fragments with criteria for modeled deviation from single particle settling velocity.

Figure S10 .
Figure S10.PVC fragments with criteria for modeled deviation from single particle settling velocity.

Figure S11 .
Figure S11.Detected PA 6.6 fibers with applied criteria for modeled deviation from single particle settling velocity.All investigated length fractions (50, 100, 200, 400 or 600 µm, respectively) shown as indicated on the right hand-side.

Figure S12 .
Figure S12.Detected PET fibers with applied criteria for modeled deviation from single particle settling velocity.All investigated length fractions (50, 100, 200, 400 or 600 µm, respectively) shown as indicated on the right hand-side.

Figure S13 .
Figure S13.Computation of the inclination angle  of an observed fiber of known length .

Figure S14 .
Figure S14.Measured settling velocities versus inclination angles of all investigated fibers.Theoretical velocities calculated for cylinders according to Komar17 are included for both polymers and all length fractions, respectively.
Figure S15 depicts the distribution of all measured drag coefficients, while Figure S16 compares the measured drag coefficients with respective Reynolds numbers (cf.Equation 2 of the main text) with Stokes' law providing a reasonable lower estimate as was discussed in the main text.A table of all measured drag coefficients and Reynolds numbers is provided in the associated Zenodo repository 3 as a file named 'all_drag_coefficients.csv'.

Figure S15 .
Figure S15.Distribution of drag coefficients as measured for MP fragments (left-hand side) and fibers (right-hand side).Density plots created via log-transformed kernel density estimation.

Figure S16 .
Figure S16.Drag coefficient versus Reynolds number as computed for all measured MP fragments and fibers.

Figure S17 .
Figure S17.Measured versus predicted settling velocities of different models (rows, Models 1-5)and for various diameter approximations as input (columns).

Figure S18 .
Figure S18.Measured versus predicted settling velocities of different models (rows, Models 6-10) and for various diameter approximations as input (columns).

Figure S19 .
Figure S19.Measured versus predicted settling velocities for all models with specified Reynolds number limits (rows, Model 2, 8 and 9) and for various diameter estimates as input (columns).All data, that does not meet respective  criteria is depicted pale and excluded from the calculation of the given performance measures.

Figure S20 .Figure S21 .
Figure S20.Measured velocities of different types of large MPs (data by Goral et al.7 ) and corresponding predicted velocities according to Su et al.18 .Initial and alternative choices of shape parameter  for the considered particle shapes are depicted in panel a and b, respectively, and specified in the figure's legend.

Table S1 .
Computation of size and shape descriptors for MP fragments and MP fibers.Not Waldschläger & Schüttrumpf 5 give separate expressions for the drag coefficients of MP fragments and MP fibers, respectively.Both require the Corey shape factor  (see Table S1): Opposed to the definitions for   from the main publication, Waldschläger & Schüttrumpf 5 propose to use   = √      3 mentioned before were: Particle surface (  ), surface of a sphere of identical volume ( ℎ ), perimeter of maximum projection (  ) and perimeter of a circle of equivalent area (  ).
is defined as cylinder length  in this contextboth for computing  as well as for explicitly calculating   via inserting Equation S30 into Equation 1 of the main publication:

Table S3 .
Model performances for all samples, PS, PMMA and PET fragments, respectively. limits for M2, M8 and M9 were both applied (annotated, see footnote) as well as not implemented.

Table S4 .
Model performances for PVC fragments, PA 6.6 fibers and PET fibers, respectively. limits for M2, M8 and M9 were both applied (annotated, see footnote) as well as not implemented.