Particle Size Measurements for Spheres

With Diameters of 50 nm to 400 nm

Michelle K. Donnelly 

George W. Mulholland

NISTIR 6935

Particle Size Measurements for Spheres

With Diameters of 50 nm to 400 nm

Michelle K. Donnelly 

George W. Mulholland

Building and Fire Research Laboratory

Gaithersburg, Maryland 20899

August 2003

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration

Phillip J. Bond, Under Secretary for Technology

National Institute of Standards and Technology

Arden L. Bement, Jr., Director

NISTIR 6935
Particle Size Measurements for Spheres 

With Diameters of 50 nm to 400 nm 

Michelle K. Donnelly and George W. Mulholland 

Building and Fire Research Laboratory 

National Institute of Standards and Technology 

Gaithersburg, Maryland 20899 USA 

ABSTRACT 

This paper describes a calibration service NIST is offering that measures the effective 

diameter of particles suspended in liquids using electrical differential mobility analysis. 

Measurements are for particles with diameters in the range of 50 nm to 400 nm. Electrical 

differential mobility analysis separates aerosol particles based on the particle size dependence of 

their electrical mobility. This measurement method was chosen because it offers lower 

uncertainty for particles smaller than 400 nm compared to other methods. Detailed descriptions 

of the calibration system, operating procedures, and analysis method are included. Customer 

information regarding test sample requirements, measurement uncertainty analysis, and quality 

control procedures is also included in the paper. 
1. Description of Service

 NIST is offering a calibration service to measure the effective diameter of particles 

suspended in liquids. The measurement method consists of nebulizing the particle suspension 

and measuring the size distribution of the particles in aerosol form by electrical differential 

mobility analysis. The service is primarily designed for measuring polystyrene latex (PSL) 

spheres in water with a narrow size distribution. Customers targeted are those who need to 

demonstrate NIST Traceability1 or who cannot use other techniques to measure the particles. 

Particles of mean diameter size in the range of 50 nm to 400 nm are measured with an expanded 

uncertainty (95 % confidence interval) of 1.5 % of the peak size. Plans are to expand the 

measurement ability to range from 20 nm to 800 nm. The calibration facility will be able to 

provide a certified peak particle diameter and size distribution results (not certified). In the 

future we plan to provide certified information regarding the standard deviation of the size 

distribution. 

Customers are asked to provide samples in aqueous suspensions in quantities of 5 mL to 

50 mL of concentrated solutions (0.5 % volume fraction or higher). Dilute solutions will require 

at least 200 mL. The container should be insulated and shipped by next day mail to avoid 

extended exposure to high or low temperatures. 

2. Design Philosophy and Theory

 This particle size calibration method uses the physical principle of electrical differential 

mobility analysis, which separates aerosol particles based on the particle size dependence of their 

electrical mobility. The particle’s electrical mobility Z is equal to the velocity U of a charged 

particle divided by the electrical field strength E. 

 E

U Z = (1) 

The electric mobility is computed from a balance of the electrical force due to the charge on the 

particle and the drag force given by Stoke’s law: 

( )

3

p

p

C D

UD

neE πμ = (2) 

 

where n is the number of charges, e is the charge of an electron, μ is the gas viscosity, Dp is the 

particle diameter, and C(Dp) is theCunningham slip correction. The factor C(Dp) approaches 

one for a particle that is large compared to the mean free path of the gas, and increases as the 

particle size decreases. From equations (1) and (2), the following expression is obtained for the 

electrical mobility for the case of singly charged particles (n=1): 

p

p

D

eC( D ) Z

3πμ = (3) 

 

1

 NIST has developed an organizational policy on traceability. Official statements regarding this policy and further 

information can be found at the website: http://www.nist.gov/traceability 
The electrical mobility for our measurements is obtained by introducing a small flow of 

charged aerosol particles into a stream of clean air. The combination flows between two co-axial 

cylinders with a radial electrical field. A small flow through a narrow slit in the center electrode 

enables high resolution electrical mobility measurements. By scanning the voltage on the center 

electrode thereby scanning the electrical field strength, and measuring the number concentration 

of the aerosol exiting the slit, the electrical mobility distribution of the aerosol is determined. 

Special considerations were given to the measurement system in order to obtain accurate 

measurements using the differential mobility analysis method. Most importantly, particle 

measurements are made relative to NIST Standard Reference Material 1963 (SRM 1963). 

SRM 1963 is a 100.7 nm PSL sphere solution traceable to the wavelength standard. The 

system also uses a steady aerosol generator to provide a constant concentration of particles. A 

carefully constructed recirculation system keeps the sheath flow rate balanced with the excess 

flow rate, and maintains a flow temperature near room temperature. An electronic barometric 

pressure gage and thermistors provide accurate temperature and pressure measurements for 

computing the viscosity and the Cunningham slip correction. 

The differential mobility analysis method was chosen to measure the size of particles 

because it offers lower uncertainty for particles smaller than 400 nm compared to other methods. 

The relative expanded uncertainty for particles in the 50 nm to 400 nm size range is ± 1.5 % of 

the peak size. This method offers good statistics since each test measures thousands of particles. 

In addition, the transfer function for the NIST differential mobility analyzer has been determined 

and is used to help assess the measurement uncertainty. 

 Differential mobility analysis has crucial advantages compared to other measurement 

methods, such as electron microscopy and light scattering. Differential mobility analysis has 

lower uncertainties than electron microscopy for the measurement of PSL spheres. The dominant 

uncertainties for microscopy are magnification uncertainty, determination of the edge of the 

particle, and electron beam damage to the PSL sphere. Additional issues with electron 

microscopy are the relatively small number of particles sized and the collection of a 

representative sample. 

While elastic light scattering methods have been shown to provide low uncertainty 

measurements for particle sizes greater than 500 nm, their uncertainty increases with decreasing 

particle size because of the lack of structure in the scattering pattern. Quasi-elastic light 

scattering is sensitive to particle sizes as small as 10 nm to 30 nm; however, this method 

provides a particle size weighted by the scattering efficiency of the particle. While there are 

methods for determining the number average diameter, there can be large uncertainties if there is 

a small population of off-size particles. All the particles contribute to the mean size and it is 

difficult to remove the effect of the off-size particles on the results. 

While the differential mobility analysis method provides the lowest uncertainty, the other 

methods provide complementary information. For example, electron microcopy provides particle 

shape information that cannot be obtained by the differential mobility analysis method. 

 

3. Description of System

 The equipment comprising the calibration facility consists of five critical components as 

shown in the block diagram in Figure 1: the aerosol generator, the electrostatic classifier, the 

condensation particle counter, the recirculation system, and the data acquisition system. All of 

the components, excluding the recirculation system are commercially available equipment. 
 Aerosol

Generator Electrostatic

 Classifier

Recirculation

 System

Condensation

 Particle

 Counter

 Data

Acquisition

Figure 1. Equipment for particle measurements. 

The PSL aerosol is created using an Aeromaster Constant Number PSL Standard Particle 

Generator manufactured by JSR Corporation2

. The generator is a pneumatic atomizer that 

operates by using a clean air stream to nebulize the liquid solution containing the PSL particles. 

The liquid aerosol passes through a heated tube where the liquid evaporates leaving only the 

solid PSL particles as an aerosol. The flow then enters a diluter where it joins a clean air stream. 

The flow passes through a bipolar charger to reduce the droplet charge. This generator is very 

stable and maintains a steady particle concentration during testing. The generator is equipped 

with an air filtration system to ensure that the pressurized input air is free of moisture and 

particles. Inside the generator, the air passes through a mist trap, a dryer, a coarse filter, and 

finally a High Efficiency Particulate Air (HEPA) filter before reaching the atomizer or the 

diluter. 

The flow leaving the aerosol generator enters an integrating chamber. The chamber has a 

volume of approximately 14 L and serves to dampen any short-term fluctuation in the flow. The 

flow leaves the chamber and reaches the classifier via a path containing regulating vents. The 

vents are adjusted so only the desired flow rate enters the classifier and the remaining flow is 

 

2

 Certain trade names and company products are mentioned in the text or identified in an illustration in order to 

adequately specify the experimental procedure and equipment used. In no case does such identification imply 

recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the 

products are necessarily the best available for the purpose. 
sent to the exhaust. This is necessary since the aerosol generator supplies a flow of 

approximately 16 L/min (267 cm3

/s) while the typical flow into the classifier is less than 2 L/min 

(33 cm3

/s). 

The electrostatic classifier used in these experiments is a model 3071A Electrostatic 

Classifier manufactured by TSI Incorporated. The classifier separates aerosol particles based on 

their electrical mobility, which is dependent on their charge state and particle size. This allows 

for a flow of monosize particles to exit the classifier. The classifier contains a bipolar charger 

(TSI Model 3077 Aerosol Neutralizer) consisting of 2 mCi (7.4 x 107

 Bq) of Kr-85 radioactive 

gas contained in a capillary tube. Here the particles collide with bipolar ions resulting in an 

equilibrium charge distribution that is a function of the particle size. For example, 100 nm 

particles would emerge from the bipolar charger with 42.6 % of the particles uncharged, 24.1 % 

with a +1 charge, 24.1 % with a –1 charge, and smaller percentages with multiply charged 

particles. 

After passing through the bipolar charger, the aerosol flows to the differential mobility 

analyzer (DMA), also located inside the classifier. The DMA is a long cylindrical chamber with 

a radius of 1.958 cm. Inside the chamber is a 0.937 cm radius rod, concentric to the walls of the 

chamber so that an annulus is formed between the rod and the chamber walls. The rod voltage 

can be adjusted from 0 V to -10,000 V. The outer cylindrical chamber is kept at ground 

potential, allowing for an electrical field to develop within the annulus. The aerosol flow enters 

the top of the chamber and is joined by a sheath flow of clean air. Both flows travel through the 

annulus to the bottom of the chamber. Along the way, charged particles move towards the center 

rod due to the voltage potential. A small slit in the rod allows for the passage of particles with 

electrical mobility, Z, as computed by the following equation: 















 =

1

2

2 r

r ln

VL

q Z sheath

π

 (4) 

where qsheath is the sheath flow, V is the rod voltage, L is the length from the aerosol entrance to 

the slit, and r1 and r2 are the inner and outer radii of the annulus, respectively. By adjusting the 

voltage, particles of a narrow mobility range are selected. A nearly monosize particle output is 

achieved by selecting particles based on mobility results. The flow comprised of monosize 

particles passes through the rod opening and exits the classifier, then flows into the CPC where 

the number of particles is counted. The rest of the flow leaves the classifier through an excess 

flow outlet and enters the recirculation system. 

A recirculation system pumps the sheath air through the classifier, draws out the excess 

air and then conditions it before returning it as the sheath air flow. Recirculating the excess air 

into the sheath inlet ensures that the excess and sheath flow rates are equal. This significantly 

reduces uncertainties in the size calculations that would be present if these flows were not 

matched and needed to be measured independently. The recirculation system was not supplied 

with the TSI model 3071A Electrostatic Classifier. It was built separately at NIST for use with 

the electrostatic classifier. 

A schematic diagram of the recirculation system is shown in Figure 2. Excess air leaves 

the classifier, passes through an adjustable needle valve and is then filtered through an ultra high 

efficiency pleated membrane cartridge filter to remove any particles. From the filter, the flow 

enters a buffer tank. The tank is a brass cylinder 40 cm long with a volume of 6 L. The buffer 

tank serves to dampen the pulsations caused by the pump. After the buffer tank are two 
Filter

 Valve

Buffer

Tank

Electrostatic

 Classifier

 Gel

Drier

 Heat

Exchanger

Water

 Bath

Pumps

Filter

Buffer

Tank

 Valve

Figure 2. Recirculation system. 
diaphragm pumps connected in parallel. The operator can choose to have only one pump operate 

for low flow rates, or have two pumps operate for higher flow rates. After the pumps, the flow 

travels through coils submerged in a water bath. Since moving through the pumps heats the 

flow, the water bath is needed to reduce the temperature of the flow. The bath itself is cooled by 

a second set of coils carrying chilled water. After leaving the bath, the flow enters another buffer 

tank, the same size as the first one, to further dampen the effects of the pump. From there the 

flow travels through a drier packed with silica gel to remove moisture from the flow. It then 

travels through a coiled section that acts as a heat exchanger to allow the flow to reach room 

temperature. The conditioned flow reaches a final needle valve where it is split into two flows. 

Part of the flow is sent through another pleated membrane cartridge filter, to remove any residual 

particles, and then into the top of the classifier as the sheath flow. The rest of the flow is 

diverted back into the recirculation system, joining the excess air as it leaves the classifier. 

Adjusting the needle valve regulates the sheath flow by adjusting how much of the recirculated 

air enters the classifier. 

 In addition to the recirculation system, equipment was added to the electrostatic 

classifier to obtain accurate pressure and temperature measurements. The barometric pressure is 

measured using a Mensor Corporation Model 4011 digital pressure transducer. The pressure 

transducer has an accuracy of 0.010 % over the range of 70 kPa to 140 kPa. The transducer 

provides a digital output to the computer at a rate of 1 Hz. Two ultra-stable thermistors provide 

temperature measurements of the sheath flow, accurate to ± 0.01 °C. The thermistors are type 

CSP Thermoprobes manufactured by Thermometrics, Incorporated with NIST traceable 

calibrations. One thermistor is located in the upper sheath flow just before it enters the DMA 

and the other is located after the DMA exit, where the sheath flow becomes the excess flow. The 

DMA rod was calibrated by using a Spellman HUD-100-1 precision resistor ladder to step down 

the voltage, which was then measured using a Fluke Corporation 8060A digital multimeter. The 

DMA output signal was then adjusted so that it matched the actual voltage as measured by the 

multimeter. 

The particle concentration is determined using a model 302201 Condensation Particle 

Counter (CPC) manufactured by TSI Incorporated. The CPC detects particles by condensing 

supersaturated butanol vapor onto the particles to make them appear larger before they enter the 

optical sensing zone where they are counted. The CPC is capable of detecting particles of size 

7 nm and larger. 

The data acquisition system consists of a desktop computer equipped with data 

acquisition boards and software for communication with the instruments. Communication with 

the CPC and the electrostatic classifier is accomplished via an RS-232 serial communications 

port. Information from the digital pressure transducer is collected using a National Instruments 

PCI-6503 digital input/output board. Thermistors are connected to a National Instruments 

TBX-68T Isothermal Terminal Block, which relays the temperature information to a National 

Instruments 4351 PCI board. A customized data acquisition software program is used to control 

the instruments and collect the data. Another custom program performs data analysis and 

computations. Both programs are written using National Instruments LabVIEW software. 

Detailed descriptions of the programs are included in section 4. 

An important element of this measurement process is the NIST SRM 1963 used to 

calibrate the system. By using the SRM, we are able to accurately determine the flow rates and 

establish consistent diameter measurements even if environmental conditions change during 

testing. The SRM 1963 is sized at 100.7 nm, which is in the midrange of the particle sizes 
measured. The SRM 1963 has been sized relative to the NIST one micrometer standard, which 

is sized relative to the wavelength standard. Use of the SRM allows us to obtain accurate 

particle measurements and to identify possible problems or inconsistencies in the equipment. 

One potential source of concern using SRM 1963 is the agglomeration of the particles in the 

suspension so that the SRM cannot be used for calibrating the system. It is also crucial that the 

dilution of the SRM particles be made with water containing very small amounts of nonvolatile impurities, less than 1 μg impurities per g of water, to minimize the size of the residue 

layer surrounding the PSL spheres as the water in the droplets evaporates. 

4. Operating Procedures and Assessment of Uncertainty

Detailed Operating Procedure

The following is a complete description of the procedures for actual measurements to calibrate a 

customer’s particle sample. 

System start-up: 

Turn on CPC and let it warm up. Display will indicate when it is ready. 

 Turn on electrostatic classifier and make sure rod voltage is set to zero. 

 Turn on aerosol particle generator. 

 Supply inlet air to particle generator at a pressure between 240 kPa and 275 kPa. 

Set the aerosol particle generator pressure to 177 kPa (1.8 kg/cm2

) and flow rate to 

16.0 L/min (267 cm3

/s).

 Turn on recirculation pump (or pumps) and turn on cooling water for water bath. 

Set the electrostatic classifier flows to desired rates. Typically, the aerosol flow into the 

classifier is set between 0.5 L/min (8.3 cm3

/s) and 2.0 L/min (33.3 cm3/s) and the sheath 

flow is set to either 10 L/min (167 cm3

/s) or 20 L/min (333 cm3/s). The aerosol flow into 

the classifier is controlled by adjusting the vents in-line before the classifier so that the 

correct amount of flow is vented, leaving the desired flow for the classifier. The sheath 

flow is controlled by adjusting the two valves in the recirculation system. For higher 

sheath flows, both recirculation pumps will need to be used. The two flow settings will 

impact one another, so rechecking and readjusting is necessary until both are at their 

desired flow rates. 

Rinse a clean testing bottle 6 times with filtered, deionized water. 

Fill the test bottle with at least 45 mL of filtered, deionized water and place test bottle in 

the particle generator sample compartment. Lock the test bottle in place, making sure the 

connection between the test bottle and the generator is sealed by the o-ring. 

Allow system to run for at least 30 min with filtered, deionized water before the first 

calibration to warm up the equipment, prime the system, and stabilize flows and 

temperatures. 

Prepare an SRM 1963 particle suspension: 

Rinse a 250 mL polyethylene container 6 times with filtered, deionized water. 

Add 200 mL of filtered, deionized water to the container. 

Add 4 drops of SRM 1963 to container, and then shake to mix. 

 Place the container in ultrasonic bath for 2 min to further mix contents. 

 SRM concentration will be approximately 5 x 109

 particles/mL. 
First calibration run using SRM 1963 sample: 

 Place the SRM 1963 container in ultrasonic bath for 60 s. 

Rinse a clean testing bottle 6 times with filtered, deionized water. 

Pour at least 45 mL of SRM solution into test bottle and place bottle in particle 

generator sample compartment. Lock the test bottle in place, and check that the 

connection is sealed by the o-ring. 

 Check particle generator pressure and flow and adjust if necessary. 

Adjust aerosol flow regulating vents, and recirculation system flow controls so that the 

electrostatic classifier aerosol inlet flow and sheath flow are at desired settings. 

Rechecking and readjusting may be necessary until both are at their desired flow rates. 

Open the data collection program on the computer. 

Input a list of classifier voltage settings for the SRM sample into the computer. 

 Select data filename for SRM 1963 calibration file and input into the computer. 

Record the filename, aerosol flow setting, sheath flow setting, and SRM 1963 mixture 

information in the laboratory notebook. 

 Start data collection program. 

Program will record the voltage, particle concentration, barometric pressure, and 

both temperature measurements at a frequency of 1 Hz. 

All data are written to specified data file. 

Readings are collected at each voltage setting for 45 s before changing to the next 

voltage setting. 

Voltage is reset to zero at end of test. 

Sample measurement test: 

 Sample to be tested should contain at least 5 x 109

 particles/mL. 

Remove particle generator test bottle containing SRM 1963. 

Place the container of test sample in ultrasonic bath for 60 s. 

Rinse a clean testing bottle 6 times with filtered, deionized water. 

Pour at least 45 mL of test sample solution into clean, rinsed test bottle and secure bottle 

in generator sample compartment, making sure the connection is sealed by the o-ring. 

Do NOT make any adjustments to any flow or pressure settings on either the particle 

generator or the classifier. 

 Input list of desired classifier voltage settings for the sample into the computer. 

 Select filename for data collection file and input into the computer. 

Record the filename, aerosol flow setting, sheath flow setting, and test sample identifying 

information in the laboratory notebook. 

 Start data collection program, it will scan the voltages and collect data as before. 

Repeat sample measurement test for up to three samples. 

After measuring the test samples, do another SRM 1963 calibration run, but do NOT adjust any 

of the flows: 

Remove particle generator test bottle containing sample. 

Place the SRM 1963 container in ultrasonic bath for 60 s. 

Rinse a clean testing bottle 6 times with filtered, deionized water. 
Pour at least 45 mL of SRM solution into clean, rinsed test bottle and place the bottle in 

the particle generator sample compartment. Lock the bottle in place, and check that the 

connection is sealed by the o-ring. 

Do NOT adjust any flow or pressure settings on either the particle generator or the 

classifier. 

 Input a list of classifier SRM voltage settings to the computer. 

 Select data filename for SRM calibration file and input into the computer. 

Record the filename, aerosol flow setting, sheath flow setting, and SRM mixture 

information in the laboratory notebook. 

Start data collection program, it will scan the voltages and collect data as before. 

Explanation of Computer Programs for Data Collection and Analysis

A customized data acquisition software program written using National Instruments 

LabVIEW software is used for data collection. The program allows the user to input a sequence 

of voltage settings for the electrostatic classifier. The user can also select a unique data file name 

for each test. The software program sets the DMA rod to the desired voltage and then collects 

readings of the temperatures, barometric pressure, and CPC particle concentration at a frequency 

of 1 Hz. Measurements are collected for a total of 45 s at each voltage setting. A real-time 

display of the readings, including the CPC particle concentration allows inspection of the data 

during testing so the experimenter can easily identify if the system is working properly. All of 

the measurements are recorded in the specified data file in spreadsheet format for easy 

processing or plotting. 

After measurement data are collected, the files are analyzed using a program also written 

using National Instruments LabVIEW software. Calculations for the diameter size are made 

using a software program written to automatically perform the computations based on the 

theoretical equations outlined in the previous section. All of the calculations are made using the 

best available constants for values such as viscosity, mean free path, the charge of an electron, 

and the Cunningham slip correction. 

First, an accurate sheath flow rate is computed using the SRM 1963 calibration runs 

collected before and after each set of sample tests. The sheath flow is calculated from the ratio 

of the theoretical mobility and the experimental mobility. Using the measured barometric 

pressure and temperature, and setting the diameter to 100.7 nm for the SRM particles, the 

theoretical mobility is calculated by equation (3). Next, from the experimental data for the 

SRM 1963 runs, the voltage corresponding to the peak particle concentration is determined by 

fitting a third order polynomial to and solving for the voltage setting at the peak. From this 

voltage, and using the nominal value for the sheath flow rate, the experimental mobility is 

computed using equation (4). Multiplying the nominal sheath flow by the ratio of the theoretical 

mobility to the experimental mobility gives the corrected sheath flow measurement. An average 

of the corrected sheath flows for the two SRM runs is used as the sheath flow rate in the test 

sample calculations. A detailed description of the sheath flow calculations, including constants 

and intermediate values is given in the appendix.

The test sample data files are reduced to eliminate the transient data collected after a 

voltage change. The particle concentration generally reaches a stable reading 30 s after a new 

voltage is set; therefore, the final 15 s of data for each voltage setting are used in the calculations. 

Mean values are then computed for the temperature, pressure and particle concentration at each 
voltage setting. Using the temperature and pressure, the viscosity and mean free path are 

calculated. Next, the mobility is computed from equation (4) for each voltage setting, using the 

sheath flow as calculated from the SRM files. Finally, the particle diameter is calculated using 

the terms computed as given in equation (3) of the previous section. Because the slip correction, 

C, is a function of the diameter, an iterative process is used to solve for the particle diameter. A 

detailed description of the diameter calculations, including the iterative process, is given in the 

appendix. Once the particle diameter for each voltage setting is calculated, a graph of the 

particle number concentration verses the corresponding particle diameter is then plotted. The 

peak diameter size is determined by fitting a third order polynomial to the graph and calculating 

the diameter at the peak of the curve fit. In cases where samples have a broad size distribution, a 

correction factor is used that is based on the instrument convolution integral and involves the 

product of the transfer function times, the charging probability, and the size distribution. 

Uncertainty Analysis

The uncertainty analysis presented below is based on a paper by Mulholland and 

Fernandez3

. The components of uncertainty are divided into two categories: Type A are those 

evaluated by statistical methods and Type B are those evaluated by other means. These types 

roughly correspond to random and systematic effects. For the particle sizing measurements, the 

Type A uncertainty is determined from the measurement repeatability and sample variability. 

The Type B uncertainty includes the uncertainty in the SRM 1963 and the uncertainty in the 

various physical quantities appearing in equations (1-3). The Type A and Type B uncertainty are 

combined by a root-sum-of-squares to determine the combined uncertainty. Taking twice the 

combined uncertainty gives the expanded uncertainty, which is the total uncertainty statement for 

the calibration. 

Type A Uncertainty

Two components contribute to the Type A uncertainty. One is the homogeneity of the 

sample: that is, sample-to-sample variability. The second is concerned with the measurement 

repeatability. Both of these components were obtained by making three repeat measurements on 

each of three different samples. The results are shown in Figure 3. It is seen that all the results 

are within about 1 % of the average of the peak diameters with the exception of one apparent 

outlier for the nominal 90 nm particle size. It is evident that the spread in the results are 

somewhat greater for the 90 nm particles relative to the larger sizes. 

 The analysis of variance shows no significant difference among diameters from the 

three different samples from each particle size. This finding translates into a between-sample 

deviation of zero. 

 The mean of the nine values of peak diameter, Dpeak(avg), and the standard deviation of 

the mean, σ(A), are computed for each of the three particle sizes with the following 

results: 92.4 nm, 0.30 nm; 126.9 nm, 0.12 nm; 217.7 nm, 0.21 nm. The standard deviation 

σ(A) is the total type A uncertainty. 

 

3

 Mulholland, G.W. and Fernandez, M., “Accurate Size Measurement of Monosize Calibration Spheres by 

Differential Mobility Analysis,” American Institute of Physics Conference Proceedings 449, Characterization and 

Metrology for ULSI Technology: 1998 International Conference, edited by D.G. Seiler, A.C. Diebold, W.M. Bullis, 

T.J. Shaffner, R. McDonald, and E.J. Walters, pp. 819-823, Gaithersburg, Maryland, March 1998. 
0.98

0.99

1

1.01

1.02

1.03

0 1 2 3 4 5 6 7 8 9 10

90 nm 

125 nm

220 nm 

D/D

avg

Test #

Sample 1 Sample 2 Sample 3

Figure 3. The ratio of the measured peaks in the size distribution to the average for each particle 

size are plotted to show the repeatability and sample to sample variability. 

Type B Uncertainty

 For five of the uncertainty components - voltage, the calibration particle size, pressure, 

temperature, and viscosity - the uncertainty analysis is straightforward. First an estimated 

standard deviation of the variable is obtained and then the resulting effect of changing the 

variable by one standard deviation on the particle diameter is obtained using equations (1-3). 

Less direct analysis is required for slip correction, DMA resolution, and data analysis 

methodology. Here a brief account of the analysis is presented and the unique features of the 

study described. A more detailed description of the uncertainty analysis is contained in a paper 

describing the measurement of SRM 1963 by differential mobility analysis.4

Voltage Calibration 

 Because errors in the DMA voltage readings were observed in the range of 1 % to 3 % 

in a previous study, a high voltage (1000 V to 10000 V) calibration facility was set up. The 

facility consists of a high voltage divider and a digital voltmeter. The high voltage divider 

(Spellman High Voltage Electronics Corp., Model HVD-100-1) has a standard deviation equal to 

0.05 % of the nominal reading. The 10000 to 1 divider output was used resulting in output 

voltages to the digital voltmeter in the range of 0.1 V to 1.0 V. The digital voltmeter (Fluke 

Corporation, 8060A) also has a standard deviation of 0.05 % of the nominal reading over this 

range. Combining in quadrature the two standard deviations leads to a total uncertainty for the 

voltage calibration of 0.071 % of the reading. In computing the total uncertainty associated with 

the voltage measurement, the 1.0 V resolution of the DMA voltmeter is also included. 

 The effect of the change of voltage on particle size is determined via the particle 

mobility equations. As explained in more detail by Mulholland et al. (1999), the voltage 

 

4

 Mulholland, G.W., Bryner, N.P., and Croarkin, C., “Measurement of the 100 nm NIST SRM 1963 by Differential 

Mobility Analysis,” Aerosol Science and Technology, Vol. 31, pp. 39-55, 1999. 
uncertainty affects the measurement of the unknown particle size directly but also indirectly 

through the calibration measurement of the 100 nm SRM. A change in the voltage for the 

calibration measurement will affect the corrected flow, which will, in turn, affect the measured 

particle size. Table 1 contains the estimated voltage uncertainty and the resulting uncertainty in 

the particle size measurement. 

Particle Standard 

The 100 nm SRM has a combined uncertainty of 0.47 nm and this uncertainty has the 

largest effect on the overall uncertainty in the calibration particles. Changing the diameter of the 

100 nm calibration particle changes the corrected flow, which, in turn, affects all of the derived 

particle sizes. As seen in Table 1, the effect is about 0.5 % for each of the three sizes. 

Pressure, Temperature, and Viscosity 

 The uncertainty in the pressure affects the mean free path λ, which affects the slip 

correction. The 4 × 103

 Pa uncertainty in the pressure results in changes of 0.13 % to 0.17 % in 

the three particle diameters. The temperature and viscosity uncertainties, listed for 

completeness, have a negligible effect on the overall uncertainty. 

Slip Correction 

 The effect of the uncertainty in the slip correction on the particle size is subtle. As seen 

from equations (2) and (3), the value of the slip correction affects the particle size, but the 

particle size also affects the slip correction. Two separate effects of uncertainty associated with 

the slip correction have been analyzed. The first, listed as Slip Correction A, is a result of the 

uncertainty in the constants A1, A2, and A3 as determined by Allen and Raabe5

. The second, Slip 

Correction B, is the larger of the two and results from assessing the effect of using two different 

expressions for the slip correction. For the 200 nm particle, the effect of the slip correction 

uncertainty (0.5 %) is as large as the effect of the 100.7 nm SRM. 

DMA Resolution/Data Analysis

 There are two issues regarding the data analysis that are discussed here even though 

their impact on the uncertainty analysis is negligible. First, the size distribution output of the 

DMA is broadened relative to the true size distribution. Second, there is a possibility that the 

peak size would be shifted by a change in the voltages selected for the analysis. Both of these 

effects were estimated using the DMA transfer function and assuming Gaussian size distributions 

for the three calibration particles. In one set of calculations the voltages were also adjusted by 

20 V to 50 V. As shown in Table 1, the largest effect was only 0.06 % of the particle size. 

Computation of Total Uncertainty 

 The total Type B uncertainty, σ(B), is obtained as the root-sum-of-squares of the 

individual standard deviations. The total Type A and Type B uncertainty are also combined as a 

root-sum-of-squares to obtain the combined uncertainty, u(Dpeak). The expanded uncertainty 

U(Dpeak), defined such that there is an approximately 95 % level of confidence that the true 

average peak diameter is within ± U(Dpeak) of the measured average, is calculated as 2u( Dpeak). 

 

5

 Allen, M.D. and Raabe, O.G., “Slip Correction Measurements of Spherical Solid Aerosol Particles in an Improved 

Millikan Apparatus,” Aerosol Science and Technology, Vol. 4, pp. 269-286, 1985. 
Summary of Uncertainty Analysis

 The values of the average peak diameters Dpeak (avg) and the associated expanded 

uncertainty U(Dpeak) are the following: 92.4 nm ± 1.1 nm, 126.9 nm ± 1.4 nm, and 217.7 nm ± 

3.4 nm. We believe that these three sizes together with the 100.7 nm SRM are the most 

accurately characterized particles in the size range less than 250 nm. The use of these materials 

together with other commercially available size standards based on these materials is expected to 

greatly improve the accuracy of calibration and measurement equipment. 

Table 1. Uncertainties of Nominal 90 nm, 125 nm, and 220 nm Calibration Particles 

Variable y, nominal value σ(y) σ(90), nm σ(125), nm σ(220), nm 

Voltage 

 1400 V 1.4 V 0.05 

 2520 V 2.0 V 0.06 

 5900 V 4.3 V 0.11 

 1650 V 1.5 V 0.05 0.06 0.11 

Pressure 

 101 kPa 4 kPa 0.16 0.21 0.29 

Temperature 

 22.0 o

C 0.5 oC 0.02 0.01 0.04 

Viscosity 

 1.8277×10-5 Pa · s 7.3×10-9 Pa · s 0.02 0.03 0.06 

100.7 nm SRM 0.47 nm 0.43 0.63 1.20 

Slip Correction A 0.07 0.11 0.17 

Slip Correction B 0.05 0.20 1.16 

DMA Resolution 0.06 0.04 0.09 

Spline Fit 0.01 0.06 0.05 

σ(B), total class B uncertainty 0.47 0.71 1.71 

σ(A), total class A uncertainty 0.30 0.12 0.21 

u(Dpeak), combined uncertainty 0.56 0.72 1.72 

U(Dpeak),expanded uncertainty 1.12 1.44 3.44 

5. Quality Control Procedures

 In order to ensure that the calibrations are performed correctly, only qualified personnel 

will be authorized to perform the calibration measurements. They are trained to monitor the 

performance of the system and to recognize any malfunctions with the equipment that may cause 

inaccurate results. The report of calibration will also be reviewed by certified personnel to 

ensure that it is accurate in every respect. 

 Because the customer will be supplying a test sample that will be consumed during the 

measurement process, there will be no device or sample to be retuned to the customer. The 

customer will receive a report detailing the size measurement results. 

 The checking mechanism for determining if the equipment is operating properly during 

the measurement is the calibration of the equipment using the SRM 1963. Calibrations with the 

SRM are performed before, after, and between regular measurements. Inconsistencies in the 
results of SRM calibration tests performed over several months indicate system troubles 

including calibration of equipment or leaks in the flow. A recalibration of the DMA voltage 

using the resistor ladder is performed periodically to maintain accurate voltage measurements. 

Recalibration of the thermistors and digital pressure transducer are completed periodically using 

resources available at NIST. Calibration documentation will be update as necessary to include 

any changes to the equipment or procedure. 

6. Safety Considerations

 

The calibration process involves possible safety hazards, therefore, precautions must be 

taken when performing these measurements. The electrostatic classifier contains a high voltage 

DC source with a maximum voltage of 10000 V. This voltage is contained within the inner 

column with the outer column grounded so the user is not exposed to the high voltage. The 

current associated with the high voltage is relatively small (maximum of 100 μA). Caution must 

be exercised when calibrating the voltage using a voltage divider and a precision voltmeter since 

a connection will be made from the high voltage power supply and the divider circuit. 

The electrostatic classifier contains a Kr-85 radioactive source, which emits β radiation. 

The source strength is 2 mCi (7.4 x 107

 Bq). It is contained within a capillary tube and there is 

no external exposure. Still, proper handling, inventorying, and shipping procedures need to be 

followed. 

The condensing liquid for the condensation particle counter is n-butanol. The Materials 

Safety Data Sheet advises that n-butanol is a severe irritant, that can cause damage through 

inhalation, ingestion or skin absorption. Inhalation or ingestion of n-butanol can result in severe 

internal injuries and possible death. Contact should be avoided since it can cause damage to eyes 

or skin. During normal system operations, the n-butanol is fully contained within sealed lines 

and containers. The experimenter has potential contact with the n-butanol only when filling or 

draining the containers, and safety gloves, safety glasses, and a laboratory coat are worn at these 

times. Additionally, the containers will be filled or drained only under a well-ventilated hood. 

The condensation particle counter exhaust that contains n-butanol vapor is vented from the lab to 

avoid exposure by the experimenter. 

The excess flow from the aerosol generator and the electrostatic classifier is also vented from the 

laboratory, since the flows contain a large fraction of aerosol particles in the size range that may 

be respired deep into the lungs if inhaled. 

7. Education and Training Considerations

There are no plans for providing training material at this time. If there is a need for more 

detailed information by other laboratories providing size calibration, training material will be 

provided in the future. 
APPENDIX

Sheath Flow Calculations

T = Temperature from data file [K] 

P = Pressure from data file [mm Hg], (barometric pressure + pressure drop in DMA) 

e = 1.60219×10-19, charge of one electron [C] 

Theoretical Mobility

Set Diameter to 100.7 nm: 9 100.7 10− D = × [m] 

Compute Viscosity : ( )

110.4

1.4618 10 6 1.5

+

× =

−

T

T

μ [kg / m·s] 

Compute Mean Free Path: ( )( )





 



 +

× =

−

T

P

T MFP

110.4 1

2.3709 10 7

 [m] 

Compute Knudsen number: D

MFP Kn 2 = 

Compute Cunningham Slip Correction: 



 



 − = + + ) 0.999 1 1.142 0.558exp( Kn

C Kn

Compute Theoretical Mobility: ( )D

eC Ztheory 3πμ = [C·s/kg] 

Experimental Mobility

L = 0.4444 [m], length from aerosol entrance to slit in DMA 

r1 = 0.009370 [m], inner radius of DMA annulus 

r2 = 0.019580 [m], outer radius of DMA annulus 

qnominal = nominal sheath flow [m3

/s] 

Vpeak = voltage corresponding to the peak particle concentration from SRM test 

Compute Experimental Mobility: 













 =

1

min 2

exp ln

2 r

r

V L

q Z

peak

no al

erimental

π

 [C·s/kg] 

Corrected Sheath Flow

( ) no al

erimental

theory

sheath q Z

Z

q min

exp

= [m3/s] 
Diameter Calculations

qsheath = corrected sheath flow as computed from above [m3/s] 

V = Voltage from data file [V] 

T = Temperature from data file [K] 

P = Pressure from data file [mm Hg], (barometric pressure + pressure drop in DMA) 

e = 1.60219×10-19, charge of one electron [C] 

L = 0.4444 [m], length from aerosol entrance to slit in DMA 

r1 = 0.009370 [m], inner radius of DMA annulus 

r2 = 0.019580 [m], outer radius of DMA annulus 

Compute Viscosity : ( )

110.4

1.4618 10 6 1.5

+

× =

−

T

T

μ [kg / m·s] 

Compute Mean Free Path : ( )( )





 



 +

× =

−

T

P

T MFP

110.4 1

2.3709 10 7

 [m] 

Compute Mobility: ln( ) 2 1

2

r

r

VL

q Z sheath

π = [C·s/kg] 

Compute intermediate term PAR (solve for D without Slip Correction): Z

e PAR

3πμ = [m] 

Calculate Initial Guess for Diameter: ( ( ))

2

+ + 8× ×1.142 = PAR PAR PAR MFP

Dinitial [m] 

Start Iterative Process to Determine Particle Diameter

Compute Knudsen number: Dinitial

MFP Kn 2 =

Compute Cunningham Slip Correction: 



 



 − = + + ) 0.999 1 1.142 0.558exp( Kn

C Kn

Compute Diameter with Slip Correction: D C(PAR) slip = [m] 

Compare Diameter with Slip Correction to Initial Guess: 

slip

initial slip

D

D D

Comparison − =
If Comparison is not less than 1 x 10-14 [m] then Calculate Di as: 

2

initial slip

i

D D

D

+ = [m] 

Using Di Repeat the Iterative Process as follows until the Comparison between Di and Dslip is 

less than 1 x 10-14 [m]. 

Repeat Iterative Process

Compute Knudsen number: Di

MFP Kn 2 =

Compute Cunningham Slip Correction: 



 



 − = + + ) 0.999 1 1.142 0.558exp( Kn

C Kn

Compute Diameter with Slip Correction: D C(PAR) slip = [m] 

Compare Diameter with Slip Correction to Initial Guess: 

slip

i slip

D

D D

Comparison − =

Compute a new value for Di by: 2

i slip

i

D D

D

+ = [m] 

If Comparison is not less than 1 x 10-14 [m] then go back and repeat the iterative process again, 

using the new value for Di, and starting with computing a new Knudsen Number. 

 If Comparison is less than 1 x 10-14 [m] then you have solved for the particle diameter: 

Dparticle = Di.