NIST Special Publication 260-185

Method for measuring the diameter of 

polystyrene latex reference spheres by 

atomic force microscopy

John A. Dagata

Natalia Farkas

Prem Kavuri

Andras E. Vladàr

Chung-Lin Wu

Hiroshi Itoh 

Kensei Ehara

This publication is available free of charge from:

http://dx.doi.org/10.6028/NIST.SP.260-185

NIST Special Publication 260-185

Method for measuring the diameter of 

polystyrene latex reference spheres by 

atomic force microscopy

John A. Dagata

Natalia Farkas

Prem Kavuri

Andras E. Vladàr

Engineering Physics Division 

Physical Measurement Laboratory

Chung-Lin Wu

Center for Measurement Standards, ITRI

Hsinchu, Taiwan 

Hiroshi Itoh 

Kensei Ehara

National Metrology Institute of Japan 

Tsukuba, Japan

This publication is available free of charge from:

http://dx.doi.org/10.6028/NIST.SP.260-185

July 2016

U.S. Department of Commerce 

Penny Pritzker, Secretary

National Institute of Standards and Technology 

Willie May, Under Secretary of Commerce for Standards and Technology and Director 
ii

Certain commercial entities, equipment, or materials may be identified in this

document in order to describe an experimental procedure or concept adequately.

Such identification is not intended to imply recommendation or endorsement by the

National Institute of Standards and Technology, nor is it intended to imply that the

entities, materials, or equipment are necessarily the best available for the purpose. 

National Institute of Standards and Technology Special Publication SP 260-185

Natl. Inst. Stand. Technol. Spec. Publ. 260-185, 26 pages (July 2016) 

CODEN: NSPUE2

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iii

ABSTRACT 

This report presents a correlated height and width measurement model for particle size analysis of 

spherical particles by atomic force microscopy (AFM). It is complementary to more familiar methods

based on a single value of the particle height or on a line average obtained from a close-packed particle 

array. Significant influence quantities affecting the determination of average particle size and its 

uncertainty are considered for the important case of polystyrene latex (PSL) reference materials. Particlesubstrate deformation, resulting from adhesive contact between particle and substrate during sample 

preparation, is estimated as a function of particle size. Post-processing of AFM datasets is explored as a 

means of eliminating bias due to non-steady state measurement conditions. These biases arise from 

variable particle-tip interaction caused by drift of instrumental parameters from their optimal settings 

during long acquisition times and inevitable wear of the AFM probe. Changes of the initial probe shape

are established using a Si/SiO2 multilayer tip characterizer and are updated periodically during the 

analysis of sequential data sets for combinations of several particles sizes and different probes. Finally, 

the capability of this procedure to serve as a statistical error-correction scheme in AFM particle-size 

metrology is assessed.

KEY WORDS

Atomic force microscopy

Nanoparticle diameter

Polystyrene

Standard Reference Material

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TABLE OF CONTENTS

Abstract iii

Keywords iii

Table of Contents iv

List of Tables v

List of Figures v

1 Introduction 1

2 Measurand and Measurement model 1

3 Calibration 3

3.1 AFM calibration

3.2 Determination of AFM probe shape

3.3 SEM calibration 

4 Measurement 5

4.1 Particle-substrate deformation

4.2 Particle-tip interaction

4.3 Probe geometry

4.4 Cross-sectional analysis

4.5 Dynamic error correction

4.6 Validation of the measurement model

5 Uncertainty Analysis 17

6 Summary 20

Acknowledgments 20

References 20

Appendix 23

Further consideration of particle-tip interactions

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LIST OF TABLES

Table 1: Guide to PSL particle data appearing in Fig. 9 13

Table 2: Average AFM height for PSL particle data shown in Fig. 9 15

Table 3: Estimated uncertainty budget for the measurement model 18

Table 4: Average diameter and expanded uncertainty for PSL particle data 20

LIST OF FIGURES

Figure 1: Measurement process for an AFM probe tip and a spherical particle 2

Figure 2: Schematic of the Si/SiO2 multilayer tip characterizer 4

Figure 3: AFM and SEM images of PSL reference particles used in this study 6

Figure 4: SEM images of PSL dimers observed in PSL particle solutions 7

Figure 5: Average deformation of PSL particles as a function of reduced radius 8

Figure 6: Measurement principle used to investigate particle-tip deformation 9

Figure 7: Reconstructed AFM probe tip profiles 11

Figure 8: Cross-sectional AFM measurement and fit to the model shown in Fig. 1 12

Figure 9: Correlated AFM height and width data for four PSL particle sizes 14

Figure 10: Correlated AFM height and width for 50 nm diameter PSL particles 16

Figure A1: Apparent height as a function of set point voltage 24

Figure A2: Average deformation vs set point voltage and applied force 26

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1. Introduction

Polystyrene latex (PSL) spheres represent an important historical and still very relevant class of 

reference materials (RMs) for particle sizing. A chief feature is that they are relatively inexpensive to 

manufacture, available in a wide variety of diameters, approach nearly ideal properties of sphericity, 

monodispersity, material homogeneity, and can be readily functionalized for specialized applications or 

intended use in metrological, environmental, and biomedical investigations [1-3]. Atomic force 

microscope (AFM) is one of several instruments capable of providing nanometer scale measurements and 

is under active development as a primary calibration instrument in a number of national measurement 

laboratories [4] as well as being an accessible user’s tool for routine size characterization. 

The possibility that PSL spheres might be compressible has been explored recently by Garnaes [5] 

and it was suggested that particle deformation encountered in AFM height and raft measurements may 

arise from several possible sources: Adhesive contact forces between the particle and substrate during 

sample preparation, δPS, and imaging-induced deformation of the particles by the AFM probe tip, δPT, may 

affect the height measurement. While it is possible to measure an overall height deformation, Δh = δPS + 

δPT, it is not possible to estimate these terms and their uncertainties separately without additional 

information or assumptions. Width measurements made on rafts may be subject to deformation due to 

particle-particle interactions, δPP, during lattice formation in the sample preparation stage. An additional 

concern is lattice imperfections or air gaps, δAG. Again, raft measurements yield an overall deformation, 

Δw = δPP + δAG, and do not permit estimates of these potential contributions to be determined separately.

In order to estimate uncertainties for these parameters, various assumptions about the nature of the 

interactions have been made, leading to sometimes inconsistent conclusions. For instance, Misumi et al. 

[6] were able to obtain an effective Young’s modulus from an elastic contact mechanics model for 30 nm 

and 100 nm PSL particles by assuming that δPT is insignificant compared with δPS. On the other hand, 

work by He et al. [7], based on extrapolation of high loading force data and other contact mechanics 

models, suggested that δPT is inevitable and plastic deformation may be significant after all. 

In this report we propose an alternative to height and raft measurements based on the analysis of 

correlated AFM height and width data of individual PSL spheres. Immediately, the problem simplifies 

because Δw = 0 for individual particles so we need only focus on the deconvolution of Δh in the

uncertainty analysis. The trade-off is that AFM probe geometry now enters the measurement process

explicitly, thus requiring separate evaluation. In the present article, we demonstrate that sufficient

information about the AFM probe geometry is available by the appropriate use of a tip characterizer to

make this extra effort worthwhile.

2. Measurand and measurement model

Odin et al. [8] discussed how to estimate particle size on the basis of the simple geometric 

measurement model. Its elements consist of a sphere resting on a flat substrate and a conical AFM probe 

tip. All interactions are treated as purely repulsive. Simulations for individual particles and particle rafts 

for a variety of probe types were presented in that work. The calculation may be summarized briefly as 

follows: An AFM tip of radius, Rtip, and full cone angle, α, moves along the x axis, and it encounters a 

spherical particle of radius R. The cross section generated by the Odin interaction model is marked by 

closed circles in Fig. 1a. The problem is rendered dimensionless by use of the parameter λ = Rtip/R. Then, 

the apparent width of the particle is given by 

ΔL = 2·ΔL1/2 (1)

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where

ΔL1/2 = 

(1+λ)+(1−λ)·sin(α/2)

cos(α/2)

 (2)

and the apparent height is given by 

zo = 

(1+λ)+(1−λ)·sin(α/2)

sin(α/2)

+

xo

tan(α/2)

 (3)

evaluated at the point at which it reaches its maximum value, i.e., xo = 0.

Figure 1. (a) Geometrical interaction between an AFM probe tip and a spherical particle. (b) 

Measurement process for correlated AFM height and width determination of an individual nanoparticle 

based on Ref. [9]. 

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The measurand is therefore defined as the spherical diameter, d = 2·R, obtained from self-consistent 

fitting of an individual AFM image cross section. Fitting experimental apparent height and width data in 

terms of (xo, zo) yields a single value for the corrected particle d (h) = d (w), parameterized by (Rtip, α). It 

is evident that Rtip, and α must be known in order to solve for R and we implement tip characterization as 

an essential modification of the model below. We consider a further modification of the Odin model by 

introducing corrections for δPS and δPT along the lines of the work of Garnaes as well.

The measurement process is summarized in Fig. 1b: Y(d) refers to the measurand, the diameter of a 

spherical particle, d. The instrument (transducer) output X1 is given as a function of (xo, zo) and influence 

quantities, Xi, arising from sample interaction, instrument calibration, and model assumptions. Estimates 

of the measurand, F-1(Xi), and combined uncertainty, Uc (Xi), are obtained self-consistently. Elements of 

the model are labeled with the section numbers in which they are discussed: Calibration is described in 

Sec. 3.1. The interaction model of Odin now includes explicit contributions for Xi = δPS, δPT, and the AFM 

probe-tip parameters, Rtip and α and these quantities are evaluated in Sec. 4.1, Sec. 4.2, and Sec. 4.3, 

respectively. These parameters are combined into the model as discussed in Sec. 4.4. The correlation 

coefficient, Rp, between d (h) and d (w) is shown to be a useful metric for evaluating parameter values in 

Sec. 4.5.

Sommer & Siebert emphasized the importance of recognizing whether or not a measurement process 

may adequately be described as under steady state control [9]. Nanoscale fracture and wear of an AFM 

probe tip is inevitable during imaging. The stability of the electro-mechanical control loop of the 

instrument is also subject to subtle physical and chemical changes in the operating environment, 

instrument, and sample. We follow their suggestions in the present work by classifying influence 

parameters as either static or dynamic on the time scale of the measurement.

3. Calibration

3.1 AFM calibration

A Veeco MultiMode AFM with Nanoscope IV controller, series J scanner, and v.6 software was used for 

imaging and particle size analysis. Imaging was performed in TappingMode using Bruker OTESPA (k = 

42 N/m, fo = 300 kHz) silicon probes. The microscope was shielded from air currents in a hood and the 

room was maintained under constant temperature and relative humidity, (23 + 0.5)

oC and (28 + 8) % RH, 

respectively. Control settings for AFM imaging were drive voltage = 2 V; an initial set point voltage is 

established at 85 % or approximately 1.7 V; Sensitivity of the OTESPA cantilever is typically 90 nm/V. 

[The amplitude (in nanometers) is related to the set point voltage (in volts) by a sensitivity factor (nm/V) 

which is obtained by calibration.] Free-oscillation amplitude is then approximately 180 nm and imaging 

amplitude is 150 nm. Optimized tapping-mode AFM operating conditions are established by monitoring 

topography, phase angle, and amplitude channels during imaging. For a specific AFM probe tip, the set 

point voltage value may need to be adjusted slightly to account for properties of the resonant cantilever 

and probe geometry. AFM measurement data presented here were obtained in the side-to-side scan

direction and a strategy of employing a minimum number of tip-sample approaches during the acquisition 

of the image dataset was adopted to reduce tip fracture. 

AFM scanner calibration of the lateral fast-scan, or X axis direction, was performed using a 70 nm 

pitch standard grating maintained at NIST [10, 11]. The calibrated value is stated as (70.055 + 0.027) nm. 

The calibration was further verified with a line average measurement of nominally 1 μm PSL spheres,

NIST standard reference material (SRM 1690) certified to be (895 + 8) nm in diameter. Calibration of the 

height, or Z axis, was performed using a set of NANO2 step height reference standards over the range of 

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7 nm to 700 nm [11]. The resulting X and Z values were fit to a polynomial correction, linear in the case 

of X and quadratic in the case of the Z. Calibration correction values were checked periodically during the 

measurement cycle and were found to vary by < 1 %.

3.2 Determination of AFM probe shape

Figure 2 presents an AFM image of the Si/SiO2 multilayer tip characterizer used in this study [10, 

11]. Briefly, the structure is formed by magnetron sputtering of the alternating multilayers, wafer bonding 

to form a support, cross-sectioning and polishing the specimen, and finally selective etching. This process 

produces trenches with well-defined width and depth. The depth of the etched trenches is approximately 

80 nm, feature width varies from 15 nm to 60 nm, and the between-feature spacing varies from 10 nm to 

70 nm. A cross section through the image, superimposed with the nominal values for this characterizer

structure, is shown in Fig. 2b. The AFM probe is able to access only some portions of the narrow trench; 

this information will be used to create a 2D reconstruction of the AFM probe shank. An estimate of the tip 

radius is obtained from information along the top of the knife-edge feature. 

Figure 2. Schematic of the Si/SiO2 multilayer tip characterizer. (a) AFM image and (b) cross section.

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The characterizer provides two approaches to tip estimation: In the first, cross-sectional data from 

trench structures of known dimension allow determination of tip width based on the distance from the tip 

apex. The width of a specific trench, W, and length that the tip penetrates into it, L, represent contact 

points between the tip and characterizer encoded by (L, W) in the cross sectional AFM data. In the 

second, the knife-edge indicated in Fig. 2b may be used. After subtracting the line width of the knife 

edge, cross-sectional segments are inverted in both x and y directions. This allows for a consistency check 

of the final estimate of the 2D tip profile. 

3.3 SEM calibration

An FEI scanning electron microscope (SEM) was used to image rafts and individual particles of PSL 

reference particles. A working distance of 3.7 mm, accelerating voltage of 5 keV, and beam current of 

0.34 nA were used to obtain all images. The magnification of the instrument was calibrated using a 100

nm VLSI grating sample traceable to NIST. The calibrated value is stated as (99.936 + 0.062) nm. The 

pixels-per-particle count was maintained between 40 and 48 in order that SEM image resolution would 

match that of corresponding AFM images. Threshold intensity was determined using the ImageJ 

maximum entropy function. The effect of beam exposure on particle size was determined to be 0.5 % or 

less for 100 nm PSL particles. All imaging was carried out on uncoated samples.

4. Measurement

NIST PSL SRMs 1964, 1963a, and 1691, with nominal diameters of (60, 100, and 300) nm were 

obtained from inventory. JSR PSL reference materials with nominal diameters of (30, 50, 70, and 100)

nm, were obtained from JSR Life Sciences (Tokyo). PSL nanoparticles were attached to a poly-L-lysine 

coated mica substrate (Pella) or conductive single-crystal silicon chip for imaging. The sample was 

prepared by incubating a freshly cleaved mica or KOH-etched silicon substrate with 0.01 % poly-L-lysine 

(Sigma) for 10 minutes. The substrate was then blown dry with high-purity compressed air. A 10 μL drop 

of diluted nanoparticle solution was incubated on the substrate for 5 minutes, rinsed with DI water, and 

any remaining solvent removed by gently blowing with nitrogen gas. 

AFM images of individual particles were analyzed using routines provided with the Veeco v.5 

imaging software. A first-order image flattening routine was applied to obtain a global background 

suitable for obtaining particle height measurements. Particles in each image were identified by visual 

inspection and excluded before the background was calculated. To collect a statistically meaningful 

estimate of the particle size distribution, roughly 300 < Np < 500 nanoparticles were measured for each 

size. To obtain sufficiently flat background and enough pixels/particle, typically this requires 30 images 

containing 10 particles each to be analyzed. SEM images of individual particles were analyzed using 

ImageJ [15] by first applying a minimum threshold to eliminate background noise to each image and 

collecting a distribution of area-equivalent diameters for at least 500 particles of each particle size in 

accordance with ISO 13322-1 [16]. 

Representative images of the four JSR PSL reference materials are presented in Fig. 3. There are 

inherent advantages and disadvantages to raft and individual particle measurement approaches in terms of 

sample preparation, information content, and measurement uncertainty. Either method may be used for 

raft or individual particle analysis, although the measurands will differ. 

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Figure 3. Images of nominally (30, 50, 70, and 100) nm diameter JSR PSL reference materials used in 

this study. AFM images (top) and SEM images (bottom) correspond to raft and isolated particles, 

respectively. A common scale bar appears below each column. 

4.1 Particle-substrate deformation 

PSL reference materials in solution are typically highly negatively charged colloidal systems, 

typically –70 mV < ζ < - 40 mV, depending on the solution environment. Attachment of PSL spheres onto 

a freshly cleaved mica substrate, for which ζ = - 80 mV, or a poly-L-lysine (PLL) coated substrate, for 

which ζ = + 80 mV, involves considerable rearrangement of interfacial charge. The compressibility of the 

particle assists the equilibrium electrostatic balance by allowing an increase in the contact area between 

the particle and substrate. 

We are concerned that an elastic deformation model which neglects such interactions may not provide 

a reasonable quantitative estimate for the contact area that would develop in this case. In particular, it may 

substantially underestimate the deformation that a PSL particle experiences during sample preparation. 

The elastic deformation model was adopted by Garnaes following work of Grobelny et al. [17] in order to 

obtain an estimated uncertainty for δPS. Here we will approach the problem in terms of the more general 

Maugis-Pollock (MP) model which interpolates between the limiting cases of elastic and plastic models

[18]. A theoretical discussion of this topic for AFM applications has been presented by Butt et al. [19].

We will reference our measurement results to recent experimental work by Wang et al. [20] which clearly 

reveals that deformation of PSL dimers are fully plastically deformed. 

To establish a reference point for our particle-substrate deformation measurements, we consider the 

general MP approach to contact mechanics as outlined by Butt and somewhat modify the SEM technique 

of Wang. In this case we obtain SEM images of uncoated PSL dimers for nominally (60, 100, and 300)

nm diameter NIST PSLs (SRMs 1964, 1963a, and 1691, respectively), shown in Fig. 4a. Dimers occur 

naturally in these solutions due to aging and can be located simply by surveying the sample.

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Figure 4. (a) SEM images of PSL dimers found in (60, 100, and 300) nm diameter NIST SRM sample 

solutions. Scale bars: (50, 100, 300) nm, respectively. (b) Geometrical relationship to extract deformation 

and contact radius, Rc, from AFM topography. (c) Geometrical relationship used to obtain Rc from SEM 

dimer images. 

To avoid well-known difficulties associated with determining the contact radius (or contact area) 

from electron microscopy images, we extract contact radius and deformation from SEM dimer images 

using the geometrical relationships presented in Fig. 4b and 4c. For each SEM dimer image, we obtain 

undistorted particle radii, R1, R2, measured perpendicular to the dimer axis and the length measured along 

the dimer axis, L, as shown in Fig. 4b. This approach avoids direct measurement at or near the contact 

region where SEM image intensity is significantly attenuated. All distances are then defined between 

locations on the dimer with equal secondary-electron intensity. The contact radius, Rc, can be obtained 

from R1, R2, and L by simple trigonometry. Extraction of a contact radius and particle-substrate 

deformation from AFM height measurements follows in a similar fashion, Fig. 4c, given that particle-tip 

deformation, δPT, may be neglected. 

Particle-substrate deformation from combined SEM and AFM height measurements, δPS, vs. Rc is 

plotted in Fig. 5 for NIST PSLs (filled circles). Each data point consists of an average of several

individual particle measurements, Np > 15. Averaged SEM dimer measurements, i.e., particle-particle 

deformation, δPP, vs. the reduced radius are also plotted (open circles). Elastic and plastic limits according 

to the MP model are shown in the figure as well. These limits are found in Fig. 4 of Garnaes and Fig. 3 of 

Wang; theoretical background can be found in Section 4 of Butt. 

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Figure 5. Average deformation of NIST PSL SRMs as a function of reduced radius. Theoretical behavior 

for the elastic and plastic limits is indicated by a solid curve. The dotted line represents an approximation 

for the plastic result appropriate for large reduced radius; the limiting MP result is indicated by the dashed 

horizontal line. 

Figure 5 demonstrates unambiguously that PSL attachment during sample preparation can be 

described in the limit of plastic deformation. Deformation is defined within the MP model as (2 wA/3Y)1/2

= Rc

2

/Rred, where wA is the work of adhesion, Y is the yield point, Rc is the contact radius and Rred is the 

reduced radius. The limiting value of 3.4 nm appears as the dashed horizontal line in the figure. This limit 

is consistent with the behavior of particle-tip deformation past the yield point. Taken together with wet 

and dry attachment comparisons mentioned later, we conclude that the amount of deformation is 

determined by PSL material properties and not the work of adhesion. Therefore, alternative nanoparticle 

surface attachment methods should lead to comparable particle-substrate deformation. Only the 100 nm 

NIST SRM 1963a shows a somewhat lower value than the plastic model prediction for both AFM height 

and SEM dimer results.

4.2 Particle-tip interaction

In tapping-mode AFM, variation of the set point changes the oscillation amplitude of the integrated 

tip and cantilever. Fig. 6a illustrates that AFM operation is then largely determined by the set point 

voltage which (indirectly) controls the applied imaging force. Deformation is just the measured difference 

between apparent particle heights as the set point is changed. The set point voltage also determines how 

faithfully the AFM feedback loop probe tracks topographical features; consequently, it may be affected 

by the stability of the electromechanical system and unintentional modification of the AFM probe tip. 

To establish an operational definition for a zero-deformation condition, δPT = 0, there are different 

ways to proceed: We might use the fact that the AFM can sense a minimally detectible interaction

between the particle and tip, such as with a force-distance curve, and then determine whether or not a 

measurable deformation of the particle has occurred at that point. On the other hand, such a small force 

may be insufficient for the feedback loop to maintain tracking of the particle shape during imaging, so we 

might want to consider the minimal observable deformation under imaging conditions. It is not 

immediately clear how close or far away these conditions may be. 

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Operationally, we seek an optimal value of the set point which is the maximum overlap of these two 

criteria: the minimally detectible interaction and minimal observable deformation under imaging 

conditions. This is indicated by the vertical line in Fig. 6a. In the force-distance mode (otherwise referred 

to as force-volume imaging in this work), the AFM probe approaches a particle from a position of zero 

force until a preset interaction force, the trigger point, is detected. At that point, the probe is retracted and 

the apparent height at which the interaction was detected is recorded. Decreasing the trigger point from 

the free-oscillation voltage to the imaging set point voltage provides an estimate of the first criterion. 

Imaging at the highest set point for which good real-time tracking can be maintained (arguably operator 

defined) and its observed deformation, if any, provide an estimate of the second. 

Figure 6. (a) Measurement principle used to investigate particle-tip deformation, δPT, of compressible 

particles. (b) Experimental height vs. set point voltage data. Each data point is an average of the same15 

PSL particles as the set point voltage is reduced by 0.025 V/step. 

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But what happens if the set point drifts away from, or was not properly set to, this optimal value? The 

principles of tapping-mode AFM operation and contact mechanics lead us to expect that distinctly 

different compressibility behavior will emerge as the set point is decreased (or the force is increased). The 

transition from elastic to plastic behavior, the yield point, is a familiar concept; it is indicated by a circle 

in Fig. 6a. However, the possibility that the system wanders into the regime of elastic-plastic interactions 

in which subsurface material damage may occur must be considered. Regimes of PSL compressibility that 

may be encountered during AFM imaging, elastic, elastic-plastic, and pure plastic deformations, must be 

identified precisely. 

We can remove the size dependence by normalizing and averaging individual PSL height data for a 

number of particles from a single image. Data for 15 particles from the same image stack were averaged 

and the aggregate behavior is presented in Fig. 6b. The box appearing on the upper right side of the graph 

indicates the narrow window for optimal AFM imaging of deformable particles, approximately 0.05 V. 

The region of elastic deformation extends from 1.54 V < A < 1.67 V in the figure. The region of plastic 

deformation for which the height never recovers is given by A < 1.54 V.

Additional discussion of particle-tip interaction forces is taken up in the Appendix. 

4.3 Probe geometry 

The fabrication process used to manufacture silicon AFM probes is based on selective etching of 

single-crystal silicon. In the case of OTESPA probes, this process yields an asymmetric tip geometry with 

a nominal 35° tip angle, and a 10° tip holder tilt. The term “front-to-back” is used in this article to identify 

the alignment of the long cantilever axis to the fast-scan, or microscope X axis, direction. Conversely, the 

term “side-to-side” refers to alignment of the narrow cantilever axis to the fast-scan, or microscope X 

axis, direction. The final 500 nm or so of the shank do not follow the etching geometry since oxide 

sharpening of the tip apex significantly narrows the width of the tip angle and yields a nonzero orientation 

with respect to the surface plane. In contrast the final probe geometry in the side-to-side direction is 

symmetric. In this case, the nominal tip angle of α = 35°, is reduced to 12o < α < 13° and the angular 

orientation of the probe axis with respect to the surface plane is zero degrees. 

The occurrence of tip wear during imaging is inevitable. On the one hand, catastrophic damage is 

easy to spot; more gradual processes, on the other hand, introduce uncertainty in particle width 

measurements which are more subtle. So, determination of the sensitivity of AFM particle size 

measurements to both fracture and continuous wearing of contact surfaces of the probe tip and shank must 

be considered. Here again, it is important to recognize that tip wear depends upon the scan direction of the 

tip: Tip wear is symmetric in the side-to-side direction and its angles and orientation remain unchanged 

even after extensive tip wear has occurred. By contrast, the wear pattern is highly asymmetric in the frontto-back direction and orientation of the tip with respect to the surface plane then changes rapidly. Inherent 

asymmetry of the front-to-back angle produces a rapid increase of the width of the tip as a result of even 

moderate wear. 

Wear testing of two different probes is presented for side-to-side imaging and back-to-front imaging, 

Fig. 7a and 7b, respectively, for accelerated wear conditions, i.e., set points were varied from 1.2 V to 1.5

V, compared to typical set points for OTESPA probes, 1.6 V to 1.7 V. In the case of front-to-back 

imaging, the front tip edge shows very little wear, whereas the back side suffers almost all the damage 

resulting from fracture and wear. Silicon probe tips are most likely to fracture during the engagement of 

feedback control and this operation effectively determines the initial tip apex [21]. For side-to-side 

imaging the fracture and wear patterns tend to be more symmetrical. Alignment of a reconstructed tip 

shape to an SEM image indicates that actual tips retain the characteristic probe geometry rather well 

despite undergoing some degree of arbitrary fracture and wear. This greatly simplifies the entire problem 

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of tip characterization because it allows us to adopt the tip model shown in Fig. 1a. Furthermore, the 

conical tip function, parametrized by (Rtip α), is largely independent of wear, i.e., the probe-tip angle, α, 

remains essentially constant. 

Measurement sensitivity then depends mainly on how Rtip is affected by typical fracture and wear 

patterns. Monitoring changes in the tip radius with the characterizer provides a way to estimate the rate of 

change of the effective radius with wear. No change due to wear was observed for set point values 

between 1.6 V and 1.7 V when using the characterizer. Wear behavior of the tip in the front-to-back scan 

direction is such that the overall shape of the tip deviates increasingly from the conical tip model as it 

wears and this has a large effect on the tip radius. Conversely, the average tip angle persists in the side-toside scan direction. Because of the symmetrical wear and retention of steep angles, tip wear can be 

monitored by simply correcting for the tip radius when side-to-side imaging is employed.

Figure 7. Reconstructed AFM probe tip profiles in the (a) side-to-side and (b) front-to-back tip 

directions. Profiles for a previously unused and a severely worn tip (Tip_1 and Tip_2) are shown aligned 

to a common orientation of the probe axis (dotted lines).

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4.4 Cross-sectional analysis

Figure 8 illustrates how it is possible to obtain a self-consistent estimate of the measurand, Y(d). The 

output function of the measurement is the AFM cross sectional data (xo, zo) for an individual PSL particle 

fit according to the Odin model (dashed line). The modeling result is given by the solid line. This figure 

provides a comparison of measurement data and model fits for 50 nm, 70 nm, and 100 nm diameter JSR 

PSL particles. Good correspondence between the AFM cross sectional data and the modified Odin model 

can be achieved over a wide range of particle sizes. For this example, a constant value of δPS = 3.4 nm 

from Fig. 5 was found to be a sufficiently precise value for these PSL samples. A fixed tip angle of α = 

12.5° is a good assumption for imaging in the side-to-side scan direction. Nominal values of Rtip = 7 nm 

and δPT = 0 nm are assumed initially. It is important to recognize that more information is available from

AFM cross-sectional data which allows an improved estimate of Rtip. Two-dimensional information is 

shown in the areas shaded in gray in the figure. Here the cross section encodes the particle-tip contact 

interaction without induced δPT deformation because the major component of the total force of the tip on 

the particle in these regions is non-normal and, as a consequence, δPT is insignificant. This is certainly not 

the case as the AFM probe scans near the particle apex and some evidence of particle-tip deformation is 

evident in the cross-sectional data for all three particles. 

With a good estimate for Rtip, a nonzero value of δPT near the particle apex region may be obtained by an 

iterative fitting process. Of course, if there were no prior knowledge of δPS, there would be no unique 

solution, Y (d) = 2·R, where R is the particle radius. For example, equally acceptable fits of the AFM 

measurement data can be obtained for the set of parameters {R: δPS, δPT, Rtip, α} and a value of R from 

48.1 nm to 51.5 nm can be obtained for different combinations of δPS, δPT, Rtip and α. Evidently, an 

understanding of the deformation and tip submodels is needed to avoid errors of 7 % or more. In 

particular, a value of δPS ≤ 1 nm for elastic particle-substrate deformation would result in an error of 

several percent in the determination of the particle radius. 

Figure 8. Cross-sectional fitting using the modified Odin model. (a) AFM measurement data (dashed

line) is plotted along with a fit to the model (solid line) for three particle sizes. A constant value of δPS = 

3.4 nm provides a satisfactory fit for all 3 particles. 

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13

4.5. Dynamic error correction

Analysis of particle size measurement data in metrology is based on statistical principles. For this 

reason, it is not practical, or necessary, to fit cross sections for each of the 1 700 JSR particles 

measurements described below. So, let’s change focus from an individual particle measurement as shown 

in the previous figure to a more distributional point of view. A complete fitting of (xo, zo) data is not 

necessary because a 2-parameter representation of the full AFM height, zo = zmax, and width, xo = 2· ΔL1/2, 

in the Odin model can be calculated directly if δPS, δPT, Rtip, and α are estimated reasonably well. These 

parameters are known or can be calculated from the model using a representative sampling of individual 

particles from a sequence of image files. 

An example of this error correction procedure is illustrated for the set of data files listed in Table 1 and 

plotted in Fig. 9. Figure 9a, the upper curve, consists of initial height and width values corrected for Z and 

X calibration errors only. The swarm of data points is clearly nonlinear, so calculating average size does 

not properly account for dynamically changing influence quantities. Table 1 contains a color-coded listing 

of six AFM OTESPA probes used to collect height and width data for the four JSR particles along with 

the associated number of image files. A given AFM tip may acquire imaging data for more than one 

particle-size sample. Particles within each image file are recorded sequentially and for each subset of 

particles corresponding to a given AFM tip and image file, one particle is fit to the full model and updated 

values for δPT and Rtip are then applied to calculate a corrected set of height and width values for the 

remaining image file. 

Table 1. Guide to the data appearing in Fig. 9.

Tip 

number

(#)

Color 

code

JSR PSL 

particle 

size

(nm)

Number of 

image files 

averaged

(#)

Range of set 

point voltage

(V)

Average 

tip radius

Rtip

(nm)

Average particletip deformation

δPT

(nm)

1 Red 30

50

50

100

4

3

3

2

1.687

1.687-1.637

1.662-1.612

1.662

11.0

11.0

11.0

13.0

0.0

4.9

2.4

1.1

2 Green 30

50

3

2

1.687

1.687

7.0

14.3

0.0

0.0

3 Blue 50

30

100

5

2

14

1.687-1.512

1.637

1.637-1.612

12.8

13.0

22.0

1.0

0.0

3.6

4 Gold 50

50

3

3

1.687-1.662

1.587-1.562

14.2

15.7

0.0

0.0

5 Magenta 70

70

4

4

1.637-1.587

1.612-1.462

14.5

14.8

2.1

2.4

6 Black 70 3 1.687-1.637 12.5 0.0

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14

Figure 9. Correlated AFM height and width data for the four JSR particle sizes and six different AFM 

probes listed in Table 1. (a) Apparent height and width corrected for Z and X calibration only. (b) Same 

data set after correction for δPS, δPT, and Rtip. 

Figure 9b represents the swarm of data points after δPS, δPT and Rtip correction. It reflects an alignment 

of the data along the 1:1 reference line, i.e., a (Pearson’s) correlation coefficient of Rp = +1 relation 

between corrected height and width values. This relation holds independent of particle size and other 

factors such as AFM tip condition and potential set point drift. The measurement approach proposed here 

is based on statistically relevant datasets of correlated AFM height and width measurements. Given the 

inevitability of tip wear, we seek to minimize its influence in the course of collecting Np > 500 individual 

measurements and by correcting for tip wear during the analysis stage. For a given set of image files, a 

monotonic increase in Rtip is observed due to tip wear. On the other hand, the set point voltage which 

largely determines δPT is subject to operator intervention and electro-mechanical drift of the instrument. 

The magnitude of particle-tip deformation can be seen to increase or decrease at times as a result of these 

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15

adjustments. Also note that the height and width values plotted in Fig. 9 appear discrete. This reflects the 

limited number of pixels used to acquire the image. In particular, this effect is more pronounced for larger 

particle sizes. It underscores the need to optimize a number of quality control factors during sample 

preparation and data acquisition such as the average particle density per image and the number of images 

needed for a distributional analysis. These factors influence the range of scan size, pixel size, and number 

of pixels that overlay a single particle and evidently set a lower limit on the measurement error that may 

be claimed.

It is worth emphasizing once again that the significantly better agreement between corrected height 

and width values for the 1700 particles presented in Fig. 9 could be achieved with relatively little 

additional cost: Only one cross section for a single particle contained within each of the 55 image files 

listed in Table 1 was analyzed according to Fig. 8. The set of updated δPS, δPT and Rtip parameters were 

then used with Eqns. (1) to (3) to analyze the remaining number of particles for each image file. This 

extra effort seems worthwhile to obtain such a dramatic improvement in measurement quality.

4.6 Validation of the measurement model

Particle height data from Fig. 9 is grouped according to sample size in Table 2. The average height 

corresponding to data shown in Fig. 9a, which includes a calibration correction only, is seen to be 

significantly smaller than the average height corresponding to data shown in Fig. 9b, which also includes 

the deformation corrections, δPS and δPT, determined from the dynamic error correction scheme described 

above. The height difference represents a substantial reporting error if not adequately taken into account. 

Figure 5 indicates that δPS is nearly constant across the nanometer-scale particle size range of interest 

here. Uncertainty related to the dynamic influence of δPT, however, is more difficult to specify although it 

may also make a large contribution to the overall deformation. For example, a particle height difference 

of 4 nm will occur for a 60 nm particle if a set point voltage of 1.50 V is used rather than the optimal 

value near 1.65 V, a difference of almost 9 % resulting solely from δPT. 

We believe such a contribution may account for the discrepancy between AFM height measurements 

reported by Garnaes and those obtained in this work for the NIST SRM 1963a. Mean uncorrected AFM 

height and width (h, w) estimates based on our measurement model are 96.5 nm and 98.8 nm, 

respectively. Values reported by Garnaes are 92.3 nm and 98.9 nm. Width estimates are identical but 

height estimates are considerably different. The difference, Δ = w – h, is 2.3 nm for our values and 6.6 nm

for his. This latter difference is too large to be accounted for by the fixed uncertainty estimates given by 

Garnaes, δPS = 2 nm and δPT = 0.5 nm. (This inconsistency was recognized, but not accounted for, in the 

article.)

Table 2. Average AFM height for the JSR PSL particle data shown in Fig. 9. Np is the number of 

particles of each size analyzed in Table 1. 

JSR Number of 

particles,

Np

Fig. 9a

Height and standard deviation

(nm)

Fig. 9b

Height and standard deviation

(nm)

Difference

Δ(h)

(nm)

30 545 (20.4 ±4.0) (23.8±4.0) 3.4

50 517 (39.3±7.1) (44.0±7.2) 4.7

70 328 (60.1±9.4) (64.8±9.2) 4.7

100 318 (93.8±3.8) (99.2±4.0) 5.4

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To investigate the degree to which the corrected AFM height and width data represent a selfconsistent solution, i.e., Δ = w – h = 0, we plot a subset of the 50 nm JSR data from Fig. 9 in Fig. 10. The 

height and width data are also shown as separate size distributions in blue and red, respectively. A glance 

at Table 1 indicates that these data for Np = 517 particles were collected from19 image files using 4 

different AFM probe tips from measurements made on 4 separate days.

The peaks of these distributions coincide at a diameter of (47.0 + 4.0) nm, which is larger than the 

average diameter value of (44.0 + 7.2) nm given in Table 2. This is because the particle size distribution 

for the JSR 50 nm is somewhat negatively skewed. However, this difference lies within one standard 

deviation of the average value. An (asymmetric Gaussian) fitting function from SEM analysis of the JSR 

50 nm particles, where Np = 1 596 particles, is represented by a solid black curve superimposed on the 

AFM distributions. The peak value of the SEM distribution occurs at 46.5 + 3.8 nm, equivalent to the 

AFM peak. This close correspondence is significant because the SEM measurand, defined in terms of an 

area-equivalent diameter, is not required to be the same as that from an AFM height or width crosssectional measurement. That all three values coincide is strong evidence that the abstract spherical model 

is self-consistent within the AFM measurement model and valid across imaging methods as well.

Similar results are obtained for the other JSR and NIST PSL particles although lower particle counts, 

Np, for both AFM and SEM data sets make the comparison slightly less compelling.

Figure 10. Correlated AFM height and width for the 50 nm diameter JSR particles (filled circles). This is 

a subset of the data presented in Fig. 9b. Size distributions for the height and width appear in blue and 

red, respectively. 

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5. Uncertainty analysis

Influence quantities considered to be significant for the measurement model are described briefly 

below. An example of estimated uncertainties for the 100 nm JSR particle is given in Table 3. Normal 

distributions are assumed for Type A and rectangular distributions are assumed for Type B quantities. 

Following the table, combined and expanded relative standard uncertainties, uc and Uc, respectively, are 

given for the nominally 100 nm diameter particle with estimates derived from height and width

corrections calculated separately. An asterisk indicates that the non-steady state parameter value must be

determined as part of the error-correction scheme presented in Sec. 4.5.

Influence quantities for the height measurement:

Type B: Particle-substrate deformation

We adopt the Maugis-Pollock result for pure plastic deformation of a polystyrene sphere, 3.4 nm, as a 

systematic correction of the measured apparent height. There is some further uncertainty of this value for 

a specific particle sample because of size dependence and compositional variations. From the results 

shown in Fig. 5 and measurements of other PSL samples, we include an additional random component of 

+ 0.7 nm. 

Type B: Particle-tip deformation

This quantity is sampled periodically and updated for each image file as part of the error-correction 

scheme. We can then assume image-to-image variation to be zero nanometers. Sampling of individual 

particles within a single image yields an estimate of about + 0.5 nm about the current value.

Type B: Z calibration and hysteresis

NANO2 step heights are reported to have an expanded uncertainty of Uc = + 1.1 nm or less for the 

range of artifacts used in this study. Measurements of these artifacts before and after a day’s worth of 

imaging indicated a change in calibration of no more than + 0.6 nm. Repeated sequential measurements 

of the same artifact yielded a drift of + 0.3 nm from image to image.

Type B: Background flatness

Measurement of the topographical roughness of substrates used in this work, freshly cleaved mica

with a coating of 0.01% poly-L-lysine, indicates a peak-to-valley roughness of + 0.5 nm or less for the 

range of scan sizes used in this work, 500 nm to 2 μm. 

Type A: Instrument noise and measurement repeatability

Analysis of incompressible gold particles attached to poly-L-lysine coated mica substrates indicates 

the topographic repeatability of the faceting of individual gold particles is + 0.3 nm. This reflects the 

capacity of the AFM to maintain stable feedback operation over multiple image scans. For PSL particles 

which are compressible, the repeatability increases to + 0.7 nm. For the measurements reported here, we 

assume that the former represents an instrumental noise uncertainty while the latter is relevant for the 

sample type of interest here. 

Influence quantities for the width measurement:

Type B: AFM probe radius, angle and wear rate

Nominal values provided by the manufacturer for the tip radius and probe angle of a fresh tip are 11 

nm and 12.5o. Experience with these probes indicates that a typical tip radius varies by no more than + 4 

nm from the average and the angle by less than + 0.5ofrom the average side-to-side orientation. Since the 

tip radius is sampled periodically and updated for each image file as part of the error-correction scheme, 

we assume image-to-image variation to be zero nanometers due to the measurement. A geometrical 

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18

argument suggests that the angle uncertainty contributes + 0.4 % Accumulated tip wear over a single 

image scan is below 0.8 nm/image. This estimate refers specifically to imaging in the side-to-side scan 

direction and translating without mechanically retracting the probe tip. 

Type B: X calibration and hysteresis

The ASM sample is reported to have an expanded uncertainty of Uc = + 0.027 nm. Measurement of 

this artifact before and after a day’s worth of imaging indicated a change in calibration of no more than +

0.3 nm. 

Type B; Validity of the tip model

The uniqueness of the probe geometry obtained by fitting the modified Odin model to test data 

indicates that the tip radius can be determined to within + 3 % of the particle diameter. Note that the 

particle-tip and particle-substrate deformation were allowed to vary for this estimate.

Type B; Pixel resolution

An important criterion for good particle size measurements is the rule to select the image scan size so 

that a minimum of 40 pixels spans the average particle diameter. However, as the pixel size is decreased 

to accommodate small particle sizes, the probe tip radius remains essentially constant. This mismatch is 

therefore inversely proportional to particle diameter for particles less than 100 nm in diameter, on the 

order of 1 % for 30 nm particles.

Type A: Repeatability

Repeatability of the instrument in the fast-scan direction is estimated to be about + 1.5 nm for imageto-image comparison of a well-characterized 2D grid sample. The grid has 2 μm and 3 μm periodic 

structures and this estimate is an average over the range of scan sizes used in this study.

Table 3. Influence quantities, x, and estimated uncertainties, u(x), for the measurement model. Entries 

are completed for the 100 nm JSR particle size for which the corrected average diameter estimates are

d(h) = 99.2 nm and d(w) = 98.4 nm.

a %u(x) ≡ 100 ∙ u(x)/x

Height Width

Component x

(nm)

u(x)

(nm)

%u(x)

a Component x

(nm)

u(x)

(nm)

%u(x)

a

δPS -3.4 0.7 0.7 Rtip (11+ 4) nm 0* 0 nm 0

δPT 0* 0.5 0.5 Tip wear rate

(nm/image)

0 * 0.8 nm/ image 0.8

α (12.5+ 0.5)o 0 0.4

Z calibration 0 0.6 0.6 X calibration 0 0.03nm 0.03

Z hysteresis 0 0.3 0.3 X hysteresis 0 0.3 nm 0.3

Background flatness 0 0.5 0.5 Validity of the tip 

model

0 3

Pixel resolution 0 2800·d

-2

Instrument noise 0 0.3 0.3

Repeatability 0 0.7 0.7 Repeatability 0 1.5 nm 1.5

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19

Combined Relative Standard Uncertainty:

The combined relative standard uncertainty of the Type A components for particle diameter is:

%uA(d) = √∑%u

2(xi

)

8

i=7

where u(xi) are normal-distribution standard uncertainties for components 7 and 8 listed in Table 3. As 

determined from the height calculation, %uA(d(h)) = 0.76 % and %uA(d(w)) = 1.5 % from the width 

calculation. 

The combined relative standard uncertainty of the Type B components is:

%uB(d) = √

1

3

√∑u

2(xi

)

6

i=1

where u(xi) are uniformly distributed relative standard uncertainties for components 1 to 6 listed in Table 

3. As determined from the height calculation, %uB(d(h)) = 0.69 % and %uB(d(w)) = 1.8 % from the 

width calculation. 

The complete combined relative standard uncertainty of the diameter measurements is then:

%u(d) = √uA

2

(d) + uB

2

(d)

As determined from the height calculation, %u(d(h))= 1.0 % and %u(d(w))= 2.4 % from the width 

calculation. 

Assuming a coverage factor of k = 2, an expanded relative uncertainty for the droplet 

diameter estimated in this manner is

%Uk=2

(d) = 2 ∙ %u(d)

As determined from the height calculation, %Uk=2

(d(h)) = 2.0% and %Uk=2

(d(w)) = 4.8% from the 

width calculation. Expanded uncertainties for all four JSR particle sizes are given in Table 4 of the next 

section. 

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20

6. Summary 

The measurement method proposed here provides an effective strategy for analyzing a statistically 

significant number of PSL particle size measurements obtained from AFM imaging data. A selfconsistent estimate for the average spherical diameter for particle size distributions of the four JSR PSL 

reference materials and their combined uncertainties are presented in Table 4. Height- and width-derived 

diameters agree to within 1 nm to 2 nm. Estimated uncertainty for the width-derived diameter is a factor 

of 2.2 greater than for the height-derived diameter. 

This is consistent with the generally accepted view that AFM height measurements offer higher 

precision and accuracy than width measurements. However, the key point of this demonstration is to 

show how the modified Odin model and error correction scheme efficiently makes use of data already 

available within a standard image file. It is a promising method for characterization of future generation 

PSL reference materials.

Table 4. Average diameter and expanded uncertainty derived from adjusted height and width data for 

PSL particle data presented in Fig. 9b.

Acknowledgments

The authors wish to thank Vipin Tondare and Gheorghe Stan (NIST) for providing constructive 

comments during manuscript preparation. PSL reference materials with nominal diameters of (30, 50, 70, 

and 100) nm were provided as a gift by JSR Life Sciences (Tokyo).

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Appendix: Further consideration of particle-tip interaction

The purpose of Sec. 4.2 is to understand and control particle-tip deformation as much as possible

during AFM measurement of PSL particle sizes. This is necessary in order to decouple particle-tip 

interaction from particle-substrate deformation. Our position is that separate estimates of these 

uncertainty contributions is not possible without additional information or assumptions. We present some 

relevant measurement data and novel analysis below. Relying on the latter approach, Garnaes obtained 

estimates for the PSL compressibility terms δPT and δPS by assuming elastic deformation, relying on 

earlier work in Ref. [17]. Plastic deformation was not considered because “...a possible plastic 

deformation on a nanoscale is not assessed as it has not been thoroughly analyzed in the literature.” 

Misumi employed the elastic model and, by making the additional assumption of zero particle-tip 

deformation, δPT = 0, noted this required the use of a two-part thin film - bulk Young’s modulus to fit the 

data. This assumption was motivated by previous studies on the role of the glass transition temperature,

Tg, on surface mechanical properties of polymer thin films [22, 23]. According to these ideas and the fact 

that Tg for polystyrene is about 100 oC, this assumption suggests that a thin-film layer thickness of 2.3 nm 

would form on the PSL particle surface. 

This layer would be sufficient for viscoelastic effects to appear. If we look more closely at Ref. [23], 

we find that this article connects plastic deformation of polystyrene to a glass-to-rubber transition under 

AFM tapping mode conditions. In fact, we observe a similar material flow in the front-to-back scanning 

direction of the asymmetric AFM probe in the case of PSL particles. Such observations are simply 

inconsistent with a purely elastic deformation model for AFM tapping-mode imaging of compressible 

materials.

Tamayo & Garcia [24] have discussed the complexity of AFM probe interaction with deformable 

polymer samples in tapping mode. Deformable in this case is defined by a Young’s modulus of E < 10 

GPa. An interesting point of this work is that it establishes an inverse relationship between contact time 

and average force for an oscillating tip in close proximity to a surface. Viscous effects were shown to 

substantially decrease the hardness of the tapping force below a relaxation frequency wV = E/η, where E 

is the Young’s modulus and η is the viscosity. Note that the effective range of this parameter depends on 

both sample and cantilever properties. For polystyrene with E = 3.3 GPa and η = 103 – 104 Pa·s, a tapping 

mode probe with an average cantilever resonance frequency of fo = 300 kHz is within this relaxation 

frequency range. In this case, interaction stiffness increases. On the other hand, the use of an AFM 

cantilever with a much lower fo would be subject to viscoelastic forces as well as exciting higher-order 

cantilever modes. This compromises stability of the electromechanical system making optimal 

measurements via the set point voltage difficult to set up and maintain during long periods of data 

acquisition. In addition to the contact time -- or the applied force during tapping mode operation -- the 

reduced radius, Rred

-1 = (1/Rtip + 1/Rpart)-1

, must be considered in the total load on the nanoparticle. Then, a 

sharp tip with a radius of 7 nm spends only a quarter of the time tapping at each location than one of twice 

the radius. Counterintuitively, a blunt tip produces relatively greater deformation than a sharp one under 

tapping mode imaging conditions.

Rather different elastic and plastic responses of the PSL particles to the tapping load may then appear 

depending on experimental conditions and the choice of contact mechanical model. We make this clear by 

acquiring PSL deformation data with two different set-point lowering sequences which we refer to as 

large step and small step sequences: A graph of apparent height vs. set point voltage for a sequence Aset

 A’  A”... is presented in Fig. A1a. In this case, the set point voltage is decreased by large steps, i.e., 

ΔA = 0.2 V. Data for six individual PSLs are plotted with their initial height, ho, indicated on the right 

hand side of the graph. Deformation increases linearly with decreasing set point voltage. Smaller particles 

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This publication is available free of charge from: http://dx.doi.org/10.6028/NIST.SP.260-185
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of the same particle material deform to a greater extent than larger particles. This behavior, attributable to 

the reduced radius effect noted above, is observed consistently for measurements of this kind. 

Fig. A1. Apparent height vs set point voltage for (a) large, ΔV = 0.2 V, and (b) small, ΔV = 0.025 V, 

decreases of the set point voltage. Measurement sequences for six individual PSLs are shown in each 

panel. The initially measured particle height is indicated on the right hand side of the figure.

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This publication is available free of charge from: http://dx.doi.org/10.6028/NIST.SP.260-185
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Figure A1b presents apparent height vs. set point voltage data, but for a sequence Aset  A’ Aset 

A”... in which the set-point voltage decreases by small steps, i.e., ΔA = 0.025 V. The appearance of 

these small step curves resembles the conceptual and empirical results presented earlier in Fig. 6. The 

essential point is that plastic material response to large decreases in set point voltage, Fig. A1a, should not 

be used to infer anything about relatively small elastic-plastic interactions, Fig. A1b. It is precisely such 

small excursions, caused by electromechanical drift or changes in probe shape, which are responsible for 

the time-dependent uncertainty associated with δPT.

In contrast, Ref. [7] analyzed peak-force AFM data for PSL nanoparticles using an elastic-plastic 

model due to Jackson & Green [25]. Their data exhibits a linear response which is evidently very similar 

to that shown in Fig. A1a. They concluded that AFM in general operates in a regime where large plastic 

deformation of PSLs is inevitable. Indeed, they suggested that, δPT may be considerably larger than δPS.

This conclusion is at variance with the measurements and analysis we have presented in this work. To 

attempt a resolution of the issue, we recall that, according to the conclusions of Ref. [24] cited above, 

viscoelastic effects may be more prominent at lower frequencies than higher ones. Since peak-force is a 

nonresonant, low-frequency mode compared with tapping mode, viscoelastic effects ought to play a more 

prominent role in peak-force deformation. In particular, it is inappropriate to extrapolate compressibility 

results obtained from those presented in Fig. A1a -- or, for that matter, peak-force imaging -- to tapping 

mode imaging conditions well above the yield point indicated in Fig. 6. This becomes more apparent by

averaging the data sets in Fig. A1and replotting them in Fig. A2 in terms of an applied force appropriate

for each regime. 

The large-step data in Fig. A1a, for example, can be readily interpreted on the basis of the AbbottFirestone [26] approximation for the strongly plastic regime. This model is used to transform AFM 

parameters, Fig. A2a, into a standard deformation vs force representation, shown in Fig. A2b. (FAF in the 

figure refers to the applied force calculated from the Abbott-Firestone model). We can then interpret the 

linear deformation regime as follows: Stiffness increases under tapping-mode conditions and tapping 

harder leads to deeper penetration of the plastic region into the polymer material according to Ref. [24]. 

Since the tapping frequency in our measurements is close to, or exceeds, the relaxation frequency of PSL 

[28, 29], the compounding of strong plasticity is consistent with theories from both contact mechanics and 

AFM tapping mode. 

On the other hand, calculations based on the work of Kogut & Etsion [27] predict the modification 

depth for the elastic-plastic situation, Fig. A1b: As before, AFM parameters in Fig. A2a are transformed 

into a standard deformation vs force representation in Fig. A2b. (FKE in the figure refers to the applied 

force calculated from the Kogut & Etsion model). Elastic-plastic behavior for small applied loads is 

evidently sublinear and is given by a deformation vs force relationship, d ∝ F

0.71

. 

A comparison of these results leads to the conclusion that the interaction of the probe tip with a PSL 

particle induces different non-additive material responses into the polymer material depending on the 

regime. A given applied force of 2 nN, say, in Fig. A2b, delivered in a single, direct step produces 

substantially greater deformation, almost a factor of 5 greater, than from the load deposited by a 

succession of much smaller ones, even if this sequence eventually reaches 2 nN. This seems to be a 

consequence of the fact, emphasized in Ref. [25], that hardness, H = 2.8 Sy, and therefore yield strength 

are not constant. Thus the ratio of the yield strength to elastic modulus, Sy/E, evolves through the course 

of elastic-plastic deformation through an increase in material stiffness.

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Fig. A2. Conversion of apparent height vs. set point data into a normalized deformation vs. force plot for 

the large- and small-step sequences in Fig. A2. The linear increase under strong plasticity follows the 

same behavior obtained for peak-force data in Ref. [7].

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This publication is available free of charge from: http://dx.doi.org/10.6028/NIST.SP.260-185