Critical rolling angle of microparticlesBahman Farzi, Chaitanya K. P. Vallabh, James D. Stephens, and Cetin Cetinkaya Citation: Applied Physics Letters 108, 111602 (2016); doi: 10.1063/1.4944043 View online: http://dx.doi.org/10.1063/1.4944043 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/108/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The Critical Angle Can Override the Brewster Angle Phys. Teach. 47, 34 (2009); 10.1119/1.3049877 Critical angle laser refractometer Rev. Sci. Instrum. 77, 035101 (2006); 10.1063/1.2173790 Ultrasonic holography at the critical angle J. Acoust. Soc. Am. 56, 459 (1974); 10.1121/1.1903278 Critical‐Angle Reflectivity J. Acoust. Soc. Am. 45, 793 (1969); 10.1121/1.1911481 On Small‐Angle Critical Scattering J. Chem. Phys. 37, 1514 (1962); 10.1063/1.1733317

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Critical rolling angle of microparticles

Bahman Farzi, Chaitanya K. P. Vallabh, James D. Stephens, and Cetin Cetinkayaa)

Photo-Acoustics Research Laboratory, Department of Mechanical and Aeronautical Engineering,Center for Advanced Materials Processing, Clarkson University, Potsdam, New York 13699-5725, USA

(Received 23 December 2015; accepted 2 March 2016; published online 15 March 2016)

At the micrometer-scale and below, particle adhesion becomes particularly relevant as van der

Waals force often dominates volume and surface proportional forces. The rolling resistance of

microparticles and their critical rolling angles prior to the initiation of free-rolling and/or complete

detachment are critical in numerous industrial processes and natural phenomenon involving particle

adhesion and granular dynamics. The current work describes a non-contact measurement approach

for determining the critical rolling angle of a single microparticle under the influence of a contact-

point base-excitation generated by a transient displacement field of a prescribed surface acoustic

wave pulse and reports the critical rolling angle data for a set of polystyrene latex microparticles.VC 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4944043]

In microparticle detachment and removal, rolling requires

substantially less effort than an out-of-plane lift-off removal

mechanism. The moment balance of forces acting on a parti-

cle in obtaining a detachment criterion has been traditionally

used for predicting the onset of rolling-based macro-/ nano--

particle detachment. This basic rolling detachment criterion

assumes neither the build-up of a resisting moment nor a criti-

cal leaning angle of the particle to rolling at the adhesion

bond prior to free rolling. However, it has been demonstrated

both analytically1 and experimentally2,3 that the adhesion

bond between a microparticle and a surface indeed creates

elastic resistance against rolling initiation. Similarly, the

stability and configuration of a network of adhered micropar-

ticles both in free space and on a substrate depend on the roll-

ing moment resistance of the bonds of its individual particles.

In addition, development of accurate modeling of critical lean-

ing angle of the particles enable numerical simulation of a

wide range of such granular dynamics problems with the dis-

crete element method (DEM) or other types of particle-scale

simulation approaches.

An elastic particle in contact with a dry flat elastic sub-

strate induces short-range (i.e., van der Waals) and elastic

forces between the particle and the substrate, leading to

(elastic) restitutive deformation of the particle and the sub-

strate at the point of contact. In out-of-plane deformation,

this force-displacement relationship is now referred to as the

JKR (Johnson, Kendall, and Roberts) theory.4 Later in 1997,

it was generalized for material pairs with broader mechanical

properties.5

In addition to the out-of-plane (one-dimensional) adhe-

sion force, when an external lateral force (or rolling moment)

is applied on the particle, a restitutive moment at the contact

point is induced, leading to a two-degree of freedom (planar)

elastic system. Above a critical value of leaning, this moment

results in free-rolling or sliding of the particle on the substrate.

Rolling motion involves the change of contact area at the

leading edge and at the trailing edge of the particle-substrate

interface (contact zone), resulting in an asymmetry in the pres-

sure distribution on the contact zone. That is, the leading edge

of the contact area establishes new contact as the peeling of

the trailing edge takes place. This asymmetric pressure distri-

bution causes an elastic restitutive moment, and consequently,

the particle undergoes free rotational oscillations with respect

to its equilibrium point if the bond is intact. Based on a two-

dimensional elasto-adhesion analysis, an expression for the

rolling resistance moment as a function of the leaning angle

when subjected to a rolling moment is reported by Dominik

and Tielens1 in 1995 for the first time. Later in 2005, the

effect was observed experimentally by interferometrically

detecting resonance frequencies of rocking microparticles2

and in 2007 by lateral static pushing in an AFM-like set-up.3

Currently, there exists no analytical expression for predicting

critical angles for microparticle rolling, while the rolling stiff-

ness constant of this initial elastic deformation/leaning is pre-

dicted analytically. The elastic rolling resistance is now well

established for microparticles, yet, to-date, no non-contact/

non-invasive method for determining the critical rolling angle

prior to free-rolling has been reported.

Here we detail an experimental approach and report crit-

ical rolling angle data for a set of 30 microparticles. In the

reported experiments, Rayleigh Surface Acoustic Waves

(SAWs) are utilized as an excitation mechanism, and inter-

ferometry and image processing are used as detection and

monitoring techniques for capturing the micro-/nano-scale

motions and vibrational responses of the microparticles base-

excited by a SAW field (Fig. 1). In order to determine the crit-

ical rolling angles in a systematic manner, an applied SAW

field with a prescribed amplitude, frequency, and duty cycle

is the well controlled excitation mechanism to transfer rota-

tional momentum to the microparticle deposited on a dielec-

tric substrate with a high level of precision.

The instrumentation diagram of the experimental set-up

for acquiring the SAW-induced rocking, rolling, and drifting

motions of a single microparticle was previously reported in

Ref. 7. A similar non-contact ultrasonic experimental set-up is

used here to observe the particle vibrational dynamics and

drifting motion simultaneously under prescribed Rayleigh

a)Author to whom correspondence should be addressed. Electronic mail:

cetin@clarkson.edu. Telephone: (315) 268-6514. Fax: (315) 268-6695.

0003-6951/2016/108(11)/111602/5/$30.00 VC 2016 AIP Publishing LLC108, 111602-1

APPLIED PHYSICS LETTERS 108, 111602 (2016)

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SAW fields. The adhesion property of each microparticle is

extracted from its transient rocking motion response, obtained

by the Laser Doppler Vibrometer (LDV).2 The same approach

was also adopted for characterizing the SAW field by meas-

uring the out-of-plane displacement of the substrate surface.

The drifting micro-scale motions of the particles are captured

by a camera integrated with the microscope and are image-

processed for determining their trajectories. It is observed that,

as expected from the surface displacement field under the elas-

tic SAWs, most particles roll/move towards the source of the

SAW field, and the critical angle depends on the history of

motion and surface properties.

As previously reported,2,6 the natural frequencies for the

out-of-plane (fo) and rocking (fR) of a particle with a radius

of R, a mass density of q, and a work-of-adhesion value of

WA are related by

fo ¼1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27

20

1

qR3

3WAK2R2

4p2

� �1=3s

and fR ¼1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi45

14

WA

qR3

s;

(1)

where K ¼ 43ðð1� �2

pÞ=Ep þ ð1� �2s Þ=EsÞ�1

is the stiffness

coefficient of the particle (p)-substrate (s) adhesion bond, Ep

and Es are the Young’s moduli, and �p and �s are the Poisson’s

ratios of the particle and the substrate materials, respectively.

The leaning angle of a particle increases by increasing the

external excitation rolling force/moment. When the leaning

angle of the particle reaches a critical point that the resisting

moment of the adhesion bond is unable to restore the particle

to its equilibrium position, the substrate-particle adhesion bond

breaks, the particle dislodges from its contact zone, and starts

to drift with slip.

For determining the critical leaning angle of a particle, the

values of the forces and the resulting resisting rolling moment

acting on the particle at the onset of initiation of the particle

motion are required. The restitution moment is generated by

the lateral force acting on the particle arising from the lateral

(in-plane) displacement of the substrate due to the SAW field.

Since the amplitudes of the in-plane displacement (ux) and the

out-of-plane displacements (uy) of the substrate due to the

SAW field influence are close, the magnitudes of their second

derivatives (acceleration) are expected to be comparable. The

lateral force (Fx) and the out-of-plane force (Fy) acting on the

particle are thus taken to be approximately equal. To calculate

the out-of-plane force acting on the particle resulting from the

contact base SAW excitation, the out-of-plane acceleration of

the particle needs to be estimated. Assuming that the particle

undergoes a harmonic motion under the influence of the sub-

strate displacement field, the amplitude of the out-of-plane

acceleration is approximated as ay ¼ �2uyppðpfcÞ2, where uypp

is the peak-to-peak amplitude of the out-of-plane displacement

of the substrate and fc is the cyclical base-excitation frequency.

The out-of-plane force acting on the particle is approximated

as Fy ¼ mpay. Taking the particle mass as mp¼ 4/3pqR3 in

this expression, the out-of-plane force acting on the particle

becomes: Fy ¼ 163

p3uyppq f 2c R3. The maximum rolling

moment resulting from the lateral force with respect to the

particle-substrate contact point (Point C in Fig. 1(a)) becomes

Mcz ¼ FxR ffi FyR ¼ 16

3p3uyppq f 2

c R4: (2)

The particle leaning angle (hl) and the rolling moment of the

particle are related by the following linear approximate rela-

tion: Mcz ¼ kbhl, where kb is the linear equivalent bending

stiffness of the elastic bond (Fig. 1(a)). The bending stiffness

of the bond (kb) is approximated as:6 kb ¼ 645

p3qf 2R R5. The

rolling moment and leaning angle of the particle reach to

their critical leaning levels at the onset of the breakage of the

particle-substrate bond and the initiation of the particle drift-

ing are called critical rolling moment (Mcrcz) and critical lean-

ing angle (hcrl ), respectively. Using these equations and

assuming the linearity of the bond stiffness, the critical lean-

ing angle of the particle at the onset of its free-drifting is

expressed in terms of measured (uypp and WA) at a particular

set value of fc and known quantities (R and q) as

hcrl ¼

Mcrcz

kb¼ 14

27p2

uyppqf 2c R2

WA: (3)

In the experimental approach detailed below, the term uypp is

linearly increased by increasing the amplitude of the SAW

FIG. 1. (a) The leaning of a microparticle under the influence of a SAW field.

(b) Schematics of the experimental set-up depicting a spherical microparticle

on a soda-lime glass substrate subjected to a Rayleigh SAW pulse. Not to scale.

111602-2 Farzi et al. Appl. Phys. Lett. 108, 111602 (2016)

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17:15:27

field until the particle detachment and rolling/translation/

drift are observed. The corresponding angle hcrl calculated

with Eq. (3) is reported as the critical rolling angle. The

micrometer-scale drift motion of each particle is observed

and captured using the camera integrated into the optical

microscope.

In the reported experiments, a set of 30 commercially avail-

able NIST (National Institute of Standards and Technology)

traceable polystyrene latex (PSL) (DC-15 DryCal, Thermo

Fisher Scientific, Inc., Waltham, Massachusetts, USA) spher-

ical microparticles with an average diameter of 14.9 lm

(14.9 6 0.6 lm) are dry deposited on a soda-lime glass pho-

tomask substrate (6� 6� 0.25 in.). The mass density of

the particle material is taken as q¼ 1050 kg/m3, leading to

the mass of an average particle of m¼ 1.86 ng. Adopting the

approach detailed in Ref. 7, the average rocking resonance

frequency of PSL particles on the soda lime glass substrate is

determined as fR ffi 97.58 kHz for ten microparticles from

the sample set to form a baseline. Using Eq. (1), the corre-

sponding average work-of-adhesion between the PSL parti-

cle and the soda lime glass substrate materials is calculated

as WA¼ 51.80 mJ/m2. Prior to the experiments, the soda-

lime glass substrate is cleaned in a four-step process.

Initially, the substrate is washed and rinsed with de-ionized

water. Then acetone is used for rinsing the substrate to

remove any bulk organic residues from its surface. Next, the

substrate is washed and rinsed with 2% Helmenex II solution

to complete the stripping of any residues from the surface.

Following the final rinsing, the substrate is dried by com-

pressed air. The acoustic wedge integrated with an ultrasonic

pressure transducer (2.25 MHz) is mounted on the soda-lime

glass substrate using double-sided sticky tape. Using a pip-

ette tip, the PSL particles are dry deposited on the substrate.

The ultrasonic pressure transducer is excited using a set of

electronic instruments.7 A function generator is employed as

the trigger source for the prescribed waveform sweep gener-

ator which produces the excitation pulses, followed by inser-

tion to an RF amplifier with a fixed gain of Af¼ 55 dB. The

generated high voltage pulse is finally delivered to the trans-

ducer to create the SAW field on the substrate surface.

Under the influence of the generated SAW field, the par-

ticle starts to rock and/or drift on the substrate depending on

the displacement amplitude. The rocking motion of the parti-

cle and the out-of-plane displacement of the substrate surface

(for characterizing the SAW field) are obtained by the LDV

(Fig. 2(a)). The adhesion properties of the particles are

extracted from their rocking motion resonance frequency

using Eq. (1) (Fig. 2(b)). Furthermore, by employing the

camera integrated with the optical microscope, the drifting

behavior of the particles is captured in a video file for further

analysis utilizing image processing techniques. The ambient

temperature and the relative humidity during the experiment

are measured at 27 �C and 23% RH, respectively.

The excitation of the ultrasonic pressure transducer for

creating the SAW field takes place in 21 sequential burst

cycles (Nmaxb ¼ 21). Each triggering burst cycle (Nb) includes a

train of triggering pulses with a duty cycle of D¼ 75%, an

active duration of ta¼ 15 s, a passive duration of tp¼ 5 s, and a

burst period of Tb¼ taþ tp¼ 20 s. The triggering pulses are

generated during the active time of the burst cycle. In the

passive duration, no triggering pulse is generated, allowing the

particles sufficient time to settle prior to the start of each burst

cycle. The trigger source for the prescribed excitation is a

second function generator that delivers square pulses with an

amplitude of Vtp¼þ4.5 V, occurring in a frequency of

ftp¼ 500 Hz and, thus, a period of Ttp¼ 2 ms; the number of

triggering pulses-per-burst cycle is then Npb¼ ta� ftp¼ 7500.

A sweep generator is triggered with the triggering pulses,

and at each triggering pulse (event), it generates a single exci-

tation pulse with a negative square pulse, resulting in a central

excitation frequency fp¼ 2.25 MHz, which is also the central

frequency (fc) of the ultrasonic transducer. During the experi-

ments, the voltage amplitude of the excitation pulses (Vp) is

increased manually by a voltage increment of (DVp ffi 30 V) at

the start of each burst cycle during the passive duration (tp).

The amplitude of the excitation pulses in each burst cycle is

kept constant throughout the active time (ta) and increased

incrementally after each successive active time up to

Vmaxepp ¼ 643.4 V occurring at Nmax

b ¼ 21. After a particle is sta-

tionary (immobile) for the duration of one or more burst

cycles, the initiation of motion is marked as a translational

(drifting) Motion Onset Incident (MOI) (see Fig. 3 for

Particles 20 and 25 (Np¼ 20 and 25)). The general direction of

the particle motion is towards the SAW source, indicating that

particles make rolling motion, rather than sliding. When it

occurs, sliding motion of a particle in a SAW field results in a

translational motion in a direction of the SAW source.

The current experimental set-up enables the acquisition

of the data at two length-scales: (i) the planar drifting motion

FIG. 2. (a) Transient responses of the substrate (uy(t)) to excitation pulses

for two burst cycles (Nb¼ 1 (solid), and 21 (dashed)) for Vepp¼ 35.6, and

643.4 V, respectively. Inset: Spectrum of the substrate response for the burst

cycle Nb¼ 21. (b). Frequency spectra of the particle (solid) and the substrate

response (dashed) to the SAW field at Veppmax¼ 211.4 V for Nb¼ 9 for

Particle Np¼ 8. Inset: Corresponding transient responses.

111602-3 Farzi et al. Appl. Phys. Lett. 108, 111602 (2016)

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of a microparticle on the substrate on the micrometer-scale

and (ii) transient out-of-plane oscillations of the apex of

each microparticle on the nanometer-scale. Using the ImageJsoftware (V1.48, National Institutes of Health, Bethesda,

Maryland, USA), image processing is performed on the

recorded video files to analyze the motion of the particles for

detecting the initiation of the substrate-particle bond breakage

and determining the trajectories of the particles drifting under

the influence of the designed SAW field. The nanometer-scale

out-of-plane response of each particle in the sample set and

the substrate is acquired as a time series (waveforms) using

the probe laser beam to accurately measure and characterize

the SAW field (Fig. 2). The adhesion value of each particle is

extracted from the corresponding waveform. The transient

out-of-plane displacement responses of the particles in the

sample set and the substrate in their temporal domain are

acquired, digitized, and averaged employing the digitizing os-

cilloscope and subsequently saved in the computer for signal

processing and further analysis.

The results indicate that most of the particles demon-

strate a rocking motion around an average rocking resonance

frequency of fRffi 100 kHz. The average particle work of ad-

hesion, calculated using Eq. (1), together with the out-of-

plane response measurements of the substrate at the initiation

time of the drifting of the particles was incorporated in Eqs.

(2) and (3) to approximate the critical leaning angle of the

microparticle.

The presented data indicate that the current approach

yields critical leaning angles of microparticles under rocking.

In addition, it is observed that, as expected, most of the par-

ticles tend to roll/drift towards the SAW field source when

the contact bond is sufficiently strong. The microparticles in

the sample set are categorized in three groups based on their

approximate critical leaning angles (Fig. 4). It is found that

the critical leaning angles of 47% of the microparticles in the

sample set fall between 0.9� and 1.2� (Group I), while 30%

of the particles have the critical leaning angles between 2.0�

and 4.2� (Group III) and 20% between 5.3� and 7.8� (Group

III). The behavior of the particles and their critical leaning

angles are not only a function of the particle surface proper-

ties but also a function of the local surface properties of the

substrate. The observed grouping is attributed to such prop-

erty variations at nano-scale and local flaws. Only 3% of the

particles under study exhibited no drift motion; hence no

critical leaning angle data for them is reported. For some of

the particles, a relaxation time of 5 s (i.e., the delay time at

the start of each burst cycle) appears to be enough to newly

form strong bonds with the substrate to immobilize a previ-

ously drifting particle from moving/drifting again under the

influence of sequential SAW bursts. An analysis of the tra-

jectories of the particles motion indicates that in general a

majority of the microparticles tend to move towards the

SAW field source. They however could change their direc-

tions, speeds, and accelerations, and stop and restart their

drifting motion at different excitation amplitudes, indicating

the inhomogeneity of the surface properties of the particles

(as discussed in Refs. 8 and 9), the substrate, and possible

electrical charge distributions on the particles affecting their

motion. The electric charge distribution on a particle could

be due to its initial electrical charge and/or a developed elec-

trical charge during the experiment due to the triboelectric

effect and their frictional drifting on the substrate surface.

The authors gratefully acknowledge financial support

through grants from the National Science Foundation (NSF)

(Award Nos. 1066877 and 1200839). Partial funding was

provided by Clarkson University and CAMP (Center for

FIG. 3. Trajectories of two drifting of particles (Particles 20 (a) and 25 (b))

subjected to 21 burst cycles of SAW field. Each circled number indicates the

location of a particle when it is subjected to a particular SAW burst cycles.

A field of view of 2380� 1520 lm.

FIG. 4. The critical leaning angles (hcrl ) versus the particle numbers (Np)

according the first MOI are sorted and categorized into three groups.

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Advanced Materials Processing). Thanks are due to Dr.

Grazyna Kmiecik-Lawrynowicz and Dr. Santokh Badesh of

Xerox Corporation, and Dr. Maura Sweeney of Stratasys,

Inc., for stimulating discussions and guidance.

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Sci. Technol. 25(4–5), 407–434 (2011).7C. K. P. Vallabh, J. D. Stephens, and C. Cetinkaya, Appl. Phys. Lett. 107,

041607 (2015).8M. W. Reeks and D. Hall, J. Aerosol Sci. 32, 1 (2001).9P. Vainhstein, G. Ziskind, M. Fichman, and C. Guttfinger, Phys. Rev. Lett.

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111602-5 Farzi et al. Appl. Phys. Lett. 108, 111602 (2016)

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