analysis of particle size distribution by particle tracking

Analysis of Particle Size Distribution by Particle Tracking

Christiane Finder, Michael Wohlgemuth, Christian Mayer*(Received: 4 March 2004; accepted: 7 September 2004)

Abstract

Particle tracking is performed using a combination ofdark field or fluorescence video microscopy withautomatic image analysis. The optical detection togeth-er with the image analysis software allows for the timeresolved localization of individual particles with diam-eters between 100 and 1000 nm. Observation of theirBrownian motion over a set of time intervals leads tothe determination of their mean square displacements

under the given room temperature and viscosity. Here-by, the radii of a set of particles visible within a givenoptical frame are derived simultaneously. Rapid dataanalysis leads to reliable particle size histograms. Theapplicability of this method is demonstrated on poly-styrene latices and PMMA nanospheres with radiibetween 51 nm and 202 nm.

Keywords: image analysis, microscopy, particle size, particle tracking, size distribution

1 Introduction

Reliable and fast determination of particle size distribu-tions in the sub-micrometer range still poses a certainchallenge for common analytical equipment. Compara-tive studies [1 – 3] revealed remarkable differencesbetween size distribution functions derived from differ-ent techniques. Generally, the most fruitful approachesalso tend to be the most expensive and time consuming[4]. While dynamic light scattering offers the advantageof being fast and convenient, it lacks precision whendealing with multi-modal distributions. On the otherhand, many reliable techniques such as transmissionelectron microscopy suffer from the disadvantage oflengthy and sophisticated sample preparation procedures.A variety of methods for particle size distributionanalysis make use of the dependence of Brownianmotion on the particle size. The most basic observationtechnique is particle tracking, leading to the determi-nation of a characteristic mean square displacement withrespect to a given time interval Dt. A more sophisticatedapproach is represented by dynamic light scattering,where Brownian motion of the particles is sensed by thefrequency shift of scattered light. In this case, a completeparticle ensemble is analyzed collectively, leading to an

autocorrelation function that is linked to the particle sizedistribution. This procedure in general has proven to befast and convenient. However, it often fails in case ofcomplex or multi-modal size distributions. Under suchcircumstances, an analysis of the size distribution func-tion from the characteristics of Brownian motion requiresthe independent detection of a number of individualparticles [5 –8], leading to the assignment of individualmean square displacements and individual particle sizes.In the following, a method is presented which is based onsimultaneous motion tracking of several individualparticles in a particle ensemble. It is suitable for particlesin the sub-micrometer range, requires no specific samplepreparation, and in some cases even allows for a selectiveanalysis of components in particle mixtures. The instru-mentation includes an optical microscope equipped withtransmitted dark field or incident fluorescence illumina-tion together with a CCD camera and an on-line imageanalysis. The applicability of the method is demonstratedon four different polystyrene latices with radii between51 nm and 202 nm as well as on PPMA nanosphereslabeled with Cumarin 6 with a radius of 165 nm.

2 Materials and Methods

2.1 Polystyrene Latices

Three different polystyrene latices (Nanosphere� SizeStandards, Duke Scientific Corporation, Palo Alto,

� 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/ppsc.200400948

* C. Finder, Dr. M. Wohlgemuth, Prof. Dr. C. Mayer (corre-sponding author), University of Duisburg-Essen, Fakult�t 4,Institute of Chemistry, 47057 Duisburg (Germany).E-mail: [email protected]

372 Part. Part. Syst. Charact. 21 (2004) 372 – 378

California, USA) were used as reference samples fordark field microscopy:

i) type 3100A: nominal radius 50 nm, certified radius51 nm� 1.5 nm with a standard deviation of 1.5 nm(TEM) and hydrodynamic radii (PCS) between50 nm and 52.5 nm.

ii) type 3240A: nominal radius 120 nm, certified radius120 nm� 3 nm with a standard deviation of 1.65 nm(TEM) and hydrodynamic radii (PCS) between119.5 nm and 122.5 nm.

iii) type 3400A: nominal radius 200 nm, certified radius202 nm� 2 nm with a standard deviation of 2.95 nm(TEM) and hydrodynamic radii (PCS) between207.5 nm and 217.5 nm.

The aqueous suspensions have a density of 1.05 g/cm3

and contain 1% solid material by weight.As reference samples for fluorescence microscopy,fluorescent latices are used:

i) Fluorescent polystyrene latex (Nanosphere� SizeStandards, Duke Scientific Corporation, Palo Alto,California, USA) type G200: nominal radius 100 nm,standard deviation 1 nm, excitation maximum at468 nm, emission maximum at 508 nm, density ofsuspension 1.05 g/cm3, 1% solids in suspension.

ii) PMMA nanospheres labeled with Cumarin 6 wereprepared by Zentel et al. [9].

They exhibit a mean radius of 165 nm as determined bySEM images, an excitation maximum at 458 nm and anemission maximum at 505 nm.

2.2 Microscopy

Time resolved optical detection of nanoparticles isperformed using two different commercial microscopes.For observation under dark field illumination, a BIOL-AR (PZO, Poland) is equipped with a 40�waterimmersion objective with a numerical aperture of 0.75and an oil immersion dark field cardioid condenser with anumerical aperture of 1.2 – 1.4. The illuminating light ispassed through a water-filled trough as an infrared filterto reduce heat uptake by aqueous samples. In addition,the sample illumination period is restricted to the actualduration of the measurement in order to minimizeundesired sample heating. Fluorescing particles areanalyzed with a Leitz Orthoplan microscope using a50�objective with a numerical aperture of 0.85. Theinstrument is equipped with a supplementary UV lightsource (HBO 103W/2, Osram, Germany) and a com-mercial filter system (I2/3, Leica, Germany). The filter

system consists of an excitation filter in the range of 450 –490 nm, a dichroic mirror at 510 nm and a long pass filtertransmitting wavelengths above 515 nm. A high-resolu-tion CCD camera (KP-F1, Hitachi, Japan) with aresolution of 782(H)� 582(V) of effective pixels isconnected to either one of the microscopes. Real timedata from the camera are fed into a frame grabber card(Oculus F/64-DSP, Coreco, Canada) installed in a regularpersonal computer. For frame grabbing and first imagerestoration, the software PicColor� (FIBUS, D�sseldorf,Germany) is applied.

2.3 Sample Preparation

The original particle dispersions are diluted to allow forthe observation of individual particles over an extendedperiod of time. Generally, 0.03 mL of the originaldispersion was added to 100 ml of water to obtainparticle concentrations in the range of 1011 to 1013

particles/L for dark field observation and 109 to 1011

particles/L for detection under fluorescence conditions.Based on the original dispersions containing 1 w% solidparticles with a density of 1.05 g/cm3, samples of thefollowing concentrations are prepared for dark fieldobservation:

a) C51nm ¼ 3.06 · 1012 particles/Lb) C100nm¼ 2.17 · 1011 particles/Lc) C120nm¼ 4.78 · 1011 particles/Ld) C202nm¼ 1.04 · 1011 particles/L

A particle dispersion with bimodal size distribution hasbeen prepared by creating a mixture between twoparticle standards (120 nm and 202 nm), yielding theconcentrations:

e) C120nm¼ 2.41 · 1011 particles/L andC202nm¼ 3.91 · 1011 particles/L.

Observation under fluorescence conditions is possible atsignificantly lower particle concentrations. The followingsamples have been used:

f) C100nm¼ 7.24 · 109 particles/Lg) C165nmffi 1010 particles/L

In each case, a sample droplet is given onto a microscopicslide and covered with a cover glass, as for commonmicroscopic analysis. In order to avoid interactionsbetween particles and the glass surface, all cover glassesand microscopic slides that come into contact with thesample have been cleaned with sulfuric acid and coatedwith octadecyltrichlorosilane (Aldrich Chemical Com-

373Part. Part. Syst. Charact. 21 (2004) 372 – 378

pany) under dry nitrogen in a glove box as described bySchaertl et al. [5]. To assure stable conditions in long-term observations, the edges of the cover glass are sealedwith paraffin to prevent excessive evaporation of thesample solvent.

2.4 Particle Tracking

The procedure for individual particle tracking followsmethods that have been applied in the past [5 – 8]. Ofcourse, the size of the nanoparticles is far below theresolution of the microscope. However, dark field as wellas fluorescence microscopy allows for the localization ofthe particles with respect to two spatial dimensions x andy. With the given magnification factor and particleconcentration, 10 to 50 particles, depending on theirsize, are observed simultaneously. The actual particletracking is performed by taking 218 images at a rate of12.5 pictures per second, corresponding to a time intervalDt of 0.08 s between successive images. The resultingvideo data set is temporarily stored on an internal bufferof the frame grabber card. In the following, the originalimages are automatically analyzed by the image proc-essing software according to the following steps:

a) All mobile, illuminated objects n are identified. Thecenters of gravity of their two-dimensional projec-tions are calculated and their positions xn(t) and yn(t)listed in particle track lists Ln as a function of time.

b) An individual particle track list Ln is automaticallyfinished at a point of time tn, end when the correspond-ing particle n cannot be identified at t¼ tn, endþDt orif its track crosses the path of another particle m suchthat the assignment of the two following particletracks starting with t¼ tn, endþDt is unclear. Underthese circumstances, all data xn(t) and yn(t) of theindividual list Ln (as well as Lm in the second case) areignored for t> tn, end.

c) An additional track list starts if a new mobile object isdetected at any point of time during the course of themeasurement.

For the analysis of nanoparticles which appear at a verylow contrast (such as particles with a radius smaller100 nm or in case of observation in the fluorescencemode) an image restoration is required. This process isdivided into two steps:

i) To eliminate static objects, the first image of asequence B(x, y)0 is subtracted from all followingimages B(x, y)z and a medium grey value of 128 isadded to the resulting data sets in order to avoidnegative grey values:

Bðx; yÞz0 ¼ Bðx; yÞz � Bðx; yÞ0� �

þ 128: ð1Þ

ii) The new images B(x, y)z’ are flattened with a 5� 5Gaussian filter.

The resulting image set lacks any immobile structuresand shows improved contrast for mobile objects. It isanalyzed by the steps a) to c) as described above.

In the following computation, the one-dimensional meansquare displacement hS2in for each individual particle n iscalculated from the track lists Ln according to:

S2h in¼1

2kmax

Xkmax

k¼1

n½xnðkDtÞ � xnðkDt � DtÞ�2

þ½ynðkDtÞ � ynðkDt � DtÞ�2o ð2Þ

where kmax denotes the number of valid time steps in thegiven track list Ln (with kmax� 217 in the given case). Themean square displacement of a given particle n within thetime period Dt is directly related to its hydrodynamicradius an by Einstein�s equation:

an ¼RTDt

3phaNA S2h inð3Þ

with the gas constant R, the temperature T, the apparentviscosity of the continuous medium ha and the Avogadroconstant NA. This given, the calculation of the particleradius from the mean square displacement requiresknowledge of the factor (T/ha). The apparent viscosity ofthe medium within the flat sample volume is related tothe viscosity h1 of the bulk medium and depends on thehydrodynamic radius of the particles as well as on thedistance d between the glass surfaces [10]:

ha ¼h1c

ð4aÞ

leading to

an ¼cRTD

3ph1NA S2h inð4bÞ

with

c ¼ 1� 9anð16dþ 9anÞ16ðd� 2anÞð8dþ 9anÞ

ln16d� 7an

25an: ð4cÞ

With the correction factor c, the apparent viscositybecomes dependent on the particle radius an. As Eq. (4c)includes a transcendent function of an, it is not possible togive a simple expression for the particle radius bycombining Eqs. (4b) and (4c). In practice, the hydro-dynamic radius is most easily accessible by iteration: a


starting value c1 for the correction factor is calculatedusing an estimated particle radius an1 based on Eq. (4c).The new particle radius an2 resulting from Eq. (4b) is thenused to determine the second approximation c2 for thecorrection factor, which leads to a third approximationfor the particle radius an3, and so on. The resulting seriesan1, an2, an3 ... usually converges rapidly; the deviationbetween an4 and an3 is generally smaller than 0.1%.If the experimental parameters T and h1 in Eq. (4b) arenot known, the factor T/h1 may be determined bycalibration with particles of a given hydrodynamicradius: A particle standard with a narrow size distribu-tion and a mean particle size as is dispersed in the samesolvent and observed under conditions identical to theactual sample, which mainly refers to the illuminationprocedure. Alternatively, fluorescent latices may beadded directly to the observed sample and measuredindependently under application of UV light. Using Eqs.(4b) and (4c), the factor T/h1 is then derived from as, d,hS2is, and Dt in a single step and can be used for thecalculation of unknown particle radii as described above.In principle, the apparent viscosity of the liquid mediumis also influenced by the presence of the particlesthemselves. However, with the low particle concentra-tions given for the observed samples, this effect isextremely small and may be neglected for the followingprocedures.The resulting set of hydrodynamic radii an finally servesas a basis to generate a particle size histogram and acorresponding number average. Evidently, the quality ofthese data largely depends on the overall number ofindividual radii an. As the number of particles within asingle microscopic image field is limited, it is generallynecessary to repeat the observation step several times toobtain a sufficient amount of particle track lists Ln. Ourexperience has shown that at least 10 observation stepsare necessary to yield a reliable particle size distribution.A total number of 30 observations is recommended foroptimal results.

3 Results

3.1 Observation under Dark Field Conditions

Measurements on the various particle size standardshave been run at sample temperatures between T¼ 293.4and T¼ 295.9 K, during the experiment each value waskept stable within a tolerance of�0.2 K. The distances dbetween the microscopic slide and the cover glass variedbetween 2 and 5 mm.Figure 1a – d shows particle size histograms obtained byparticle tracking under dark field observation on fourparticle size standards. Each bar denotes the percentageof particles found within a given radius interval with atotal width of 4 nm. On average, each individual particlesize which enters the histogram relies on 50 steps in time.This corresponds to an ensemble of 100 independentvalues (Dx)2 and (Dy)2 for one-dimensional squaredisplacements, with typical standard deviations of 8.5%to 13.8% for each (Dx)2 and (Dy)2. Based on this data set,the corresponding individual particle radius can beassigned to a given radius interval of 4 nm with an errorprobability of less than 1%. The resulting numberaverages of the particle sizes are in good accordancewith the nominal hydrodynamic radii of the particlestandards (Table 1). The particle size distribution func-tions come close to Gaussian distributions as shown inFigure 1a – d. The polydispersities are characterized bythe half height widths of the fitted Gaussian distributionfunctions (Table 1). As expected, the values continu-ously increase with increasing particle size.Corresponding results on a particle sample showing abimodal size distribution are shown in Figure 2 and listedin Table 1. The sample represents a mixture of twonanoparticle dispersions with particle radii of 120 nmand 202 nm, respectively. The number averages of theparticles size contributions obtained by two Gaussian fits(129.8 nm and 203.7 nm) are in good accordance with thedata previously determined on the isolated samples

Table 1: Experimentally obtained particle size data of particle standards in comparison with nominal data.

particle standard(type)

expected concentration(particles/L)

hydrodynamic radius (nm)according to product inf.

radius (number av.) byparticle tracking (nm)

Gaussian distributionwidth (nm)

observed by dark field illumination:

51 nm 3.06 · 1012 50 – 52.5 59.7 16.4100 nm 2.17 · 1011 99 – 101 103.0 17.7120 nm 4.78 · 1011 119.5 – 122.5 127.5 29.7202 nm 1.04 · 1011 207.5 – 217.5 208.5 42.7120 nmþ 202 nm 2.41 · 1011 119.5 – 122.5 129.8 22.8

3.91 · 1011 207.5 – 217.5 203.7 40.8

observed under fluorescence conditions:

100 nm 7.24 · 109 99 – 101 100.8 21.4165 nm � 1010 no data available 169.9 32.9


(127.5 nm and 208.5 nm, Table 1). The relative amountsof both contributions could be determined from thearea of the two Gaussian fit functions. The ratio ofC120/C202¼ 1/1.67 as obtained by the particle trackingmethod corresponds well to the ratio of 1/1.62 which isexpected from the preparation procedure (see section 2.3).

3.2 Observation under Fluorescence Conditions

Two kinds of nanoparticles with different radii arestudied by particle tracking using fluorescence micros-copy. The resulting size histograms are shown in Figure 3,numerical data listed in Table 1.

The particle size distribution for the particle size stand-ard with a radius of 100 nm is represented in Figure 3a.The average radius is 100.8 nm, which is almost identicalto the radius given in the product information sheet(100 nm) and to the one obtained from particle trackingusing dark field microscopy (103.0 nm). Figure 3b showsthe corresponding result for PMMA nanospheres la-beled with Cumarin 6. The resulting hydrodynamicradius of 169.9 nm is in good accordance with the valueof 165 nm as determined by SEM (Table 1). The widthsof both particle size distributions are similar to thoseobserved for similar particle sizes obtained under darkfield observation.

Fig. 1: Particle size distribution of polystyrene latices as determined using dark field microscopy: a) 51 nm, b) 100 nm, c) 120 nm,d) 202 nm. The thin lines represent Gaussian functions fitted to the outlines of the histograms.


4 Discussion

Optical particle tracking is a relatively old-fashionedapproach for the determination of particle radii. How-ever, together with modern image analysis, it mayrepresent a new and promising option for a rapid particlesize analysis of small amounts of particle dispersions. Itsspecial advantage in comparison with dynamic lightscattering is its ability to observe a large number ofindividual particles (rather than a collective autocorre-lation function) which leads to a very high reliability ofthe resulting size distribution pattern.The particle size distribution functions for unimodalsamples (Figures 1 and 3) demonstrate the accuratenessof the presented method independent from the illumi-nation technique. The obtained number averages gen-erally coincide with the nominal radii (Table 1). Devia-tions vary between 1% (for the 100 nm standard) and15% (for the 51 nm standard) with typical values of 5%.With individual particles of a given radius a beingobserved over approximately 50 time intervals Dt, thestatistical basis is good enough to allow for an unambig-uous assignment to a given size interval n defined by(an� 2 nm)< a< (anþ 2 nm). As expected, all particlesize histograms in Figures 1 and 3 resemble Gaussiandistribution functions. Their width obviously depends onthe average particle size: it continuously grows withincreasing particle radius (Table 1) which is in accord-ance with the expected variation pattern connected tothe given particle preparation process.

A critical test of the reliability of any method for particlesize determination is a measurement on a bimodalsample of known composition. The result of an analysison a mixture of two particle standards with 120 nm and202 nm radii is shown in Figure 2. Obviously, the methodallows for the separation of the two size contributions.The overlap between the two distribution functions issmall and the two maxima are easily determined. Inaddition, the characteristics of the individual particle sizehistograms for both particle standards are clearly repro-duced (Table 1): the deviation between the peak maxima(127.5 nm and 208.5 nm for the isolated standards,

Fig. 2: Particle size distribution of a mixture of two polystyrenelatices as determined using dark field microscopy. The samplecontains particles with a radius of 120 nm and particles with aradius of 202 nm in a 1/1.62 ratio. The thin lines represent twoGaussian functions fitted to the outlines of the two separateparticle contributions.

Fig. 3: Particle size distribution of fluorescent particles as deter-mined using fluorescence observation conditions: a) 100 nm,b) 165 nm. The thin lines represent Gaussian functions fitted tothe outlines of the histograms.


129.8 nm and 203.7 nm for the mixture) is smaller than3%. In addition, the widths of the fitted Gaussianfunctions even have slightly decreased from 29.7 nmand 42.7 nm (isolated standards) to 22.8 and 40.8 nm(mixture). This demonstrates the ability of the particletracking approach for an independent observation ofmultiple components of different size in a homogene-ously mixed system. The presence of smaller or largermobile objects has no significant influence on thedetection of an individual particle. The ratio of the areasof the two Gaussian contributions in Figure 2 (1/1.67) isstrikingly similar to the value expected from the particleconcentrations (1/1.62), so the method yields reliablequantitative data of different size contributions.The capability of the particle tracking technique hasbeen demonstrated under dark field and fluorescenceconditions. In general, any illumination technique maybe applied that is capable of following the particlemotion. In case of fluorescent particles, the method canbe used to determine their particle size distribution evenin presence of other particles.The free choice of the illumination technique allows forthe examination of a wide range of suspended particles.The described method is optimized for particles with adiameter of 100 to 1000 nm, which represents the mostinteresting range for carrier particles used in pharma-ceutical applications. Larger particles require a largertime interval Dt between the pictures, because thedisplacement has to be in the range of several pixels toguarantee a reliable determination of the particlemotion. Alternatively, the size range may be extendedto both sides by an appropriate shift of the observationtemperature or an adequate choice of the fluid medium.

5 Conclusion

Particle tracking analysis by optical microscopy com-bined with image analysis allows for a fast determinationof particle size distribution functions in the sub-micro-meter range. The method can be used under variousillumination conditions as long as particles can be locatedand their motion can be followed. It yields detailedparticle size histograms of high quantitative reliabilityeven for multimodal size distributions.

6 Acknowledgements

We thank Prof. Dr. Clivia Sotomayor Torres for thegenerous donation of the fluorescent 165 nm nano-

spheres, Manfred Z�hres for helpful assistance on ouroptical equipment, and Dr. Reinert H. G. M�ller for hisfriendly support in dealing with different problems ofdigital image analysis.

7 Nomenclature

an m hydrodynamic radius of a particle nB(x, y)z – grey value for the pixel x/y of an

image zC particles/L particle concentration per volume

of dispersionha kg m�1s�1 apparent viscosity in a given finite

sample volumeh1 kg m�1s�1 viscosity of the medium in an infin-

ite volumeNA mol�1 Avogadro constantSh in m mean square displacement of a

particle n

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analysis of particle size distribution by particle tracking

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