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Journal of Adhesion Science and Technology

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Effects of Electrostatic and Capillary Forces andSurface Deformation on Particle Detachment inTurbulent Flows

Xinyu Zhang & Goodarz Ahmadi

To cite this article: Xinyu Zhang & Goodarz Ahmadi (2011) Effects of Electrostatic and CapillaryForces and Surface Deformation on Particle Detachment in Turbulent Flows, Journal of AdhesionScience and Technology, 25:11, 1175-1210, DOI: 10.1163/016942410X549906

To link to this article: http://dx.doi.org/10.1163/016942410X549906

Published online: 02 Apr 2012.

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Journal of Adhesion Science and Technology 25 (2011) 1175–1210brill.nl/jast

Effects of Electrostatic and Capillary Forces and SurfaceDeformation on Particle Detachment in Turbulent Flows

Xinyu Zhang and Goodarz Ahmadi ∗

Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam,NY 13699-5725

Received in final form 27 August 2009; revised 1 November 2010; accepted 7 November 2010

AbstractIn this work the rolling detachment of particles from surfaces in the presence of electrostatic and capillaryforces based on the maximum adhesion resistance was studied. The effective thermodynamic work of ad-hesion, including the effects of electrostatic and capillary forces, was used in the analysis. The Johnson,Kendall and Roberts (JKR) and Derjaguin, Muller and Toporov (DMT) models for elastic interface de-formations and the Maugis–Pollock model for plastic deformation were extended to include the effect ofelectrostatic and capillary forces. Under turbulent flow conditions, the criteria for incipient rolling detach-ment were evaluated. The turbulence burst model was used to evaluate the airflow velocity near the substrate.The critical shear velocities for removal of particles of different sizes were evaluated, and the results werecompared with those without electrostatic and capillary forces. The model predictions were compared withthe available experimental data and good agreement was observed.© Koninklijke Brill NV, Leiden, 2011

KeywordsParticle adhesion, particle removal, electrostatic force, capillary force, surface tension, resuspension, elasticdeformation, plastic deformation

Nomenclature

A Hamaker constant

a contact radius, m

ae effective contact radius, m

Cc Cunningham factor, nondimensional

c̄i mean thermal speed of the ions, cm/s

* To whom correspondence should be addressed. Tel.: (315)-268-2322; Fax: (315)-268-4494; e-mail:[email protected]

© Koninklijke Brill NV, Leiden, 2011 DOI:10.1163/016942410X549906

1176 X. Zhang, G. Ahmadi / J. Adhesion Sci. Technol. 25 (2011) 1175–1210

d particle diameter, m

dc particle diameter in cgs units, cm

d+ nondimensional particle diameter

E electric field strength in mks units, V/m

Ec electric field strength in cgs units, V/cm

Ei Young’s modulus of material i, N/m2

ec electronic unit charge in cgs units, stC

e electronic unit charge in mks units, C

Fc capillary force, N

Fe electrostatic force, N

FL lift force, N

Fpo pull-off force, N

Ft drag force, N

H hardness of material, Pa

K composite Young’s modulus, N/m2

k Boltzmann constant in cgs units, ergs/K

Kn Knudsen number

Mmax maximum adhesion resistance moment due to the applied normal load,N · m

Mt hydrodynamic moment, N · m

MDMTmax maximum resistance moment evaluated by DMT model, N · m

MJKRmax maximum resistance moment evaluated by JKR model, N · m

MMPmax maximum resistance moment evaluated by Maugis–Pollock model,

N · m

n number of units of charge

Ni ion concentration, 1/cm3

P applied normal load, N

(P ·a)max maximum adhesion resistance moment, N · m

q electrical charges

X. Zhang, G. Ahmadi / J. Adhesion Sci. Technol. 25 (2011) 1175–1210 1177

t time, s

T temperature, K

u∗c minimum shear velocity needed for detaching a particle from the sub-

strate, m/s

u+M nondimensional maximum gas velocity at the mass center of the particle

uM maximum gas velocity at the mass center of the particle

u∗ shear velocity, m/s

WA thermodynamic work of adhesion, J/m2

W eA effective thermodynamic work of adhesion, J/m2

W eJKRA effective thermodynamic work of adhesion for JKR model, J/m2

W eDMTA effective thermodynamic work of adhesion for DMT model, J/m2

W eMPA effective thermodynamic work of adhesion for Maugis–Pollock model,

J/m2

y+ nondimensional distance from the wall

y distance from the wall, m

zo minimum separation distance, m

Greek Letters

α half particle–liquid contact angle, rad

αo relative approach between the particle and surface, m

ε dielectric constant of the particle, dimensionaless

ε0 permittivity, amp · s/V · m

θ wetting angle, rad

λ mean free path of air, m

ν kinematic viscosity of air, m2/s

νi Poisson’s ratio of material i

ρ density of air, kg/m3

ρp density of particle and substrate material, kg/m3

σ surface tension of water, N/m


1. Introduction

Fine particle adhesion and removal from surfaces have attracted considerable at-tention due to their numerous applications in semiconductor, pharmaceutical andxerographic industries. Despite numerous studies, the effects of electrostatic andcapillary forces on particle adhesion and removal are not fully understood.

Particle adhesion and removal have been reviewed by Corn [1], Krupp [2], Visser[3], Tabor [4], Bowling [5] and Berkeley [6]. For particles under dry conditions andin the absence of electric forces, the van der Waals force is the major contributingforce to the particle adhesion to a surface. Bradley [7] and Derjaguin [8] indepen-dently postulated the effect of contact deformation on particle adhesion. Johnson,Kendall and Roberts [9] developed a particle adhesion theory, now referred to as theJKR adhesion model, which includes the effects of the surface energy and surfacedeformation. Kendall [10] investigated the rolling friction and adhesion betweensmooth solids. Based on the Hertzian profile assumption and including the effects ofsurface energy, Derjaguin, Muller and Toporov [11] developed the so-called DMTmodel. Maugis and Pollock [12] developed a particle adhesion model to considerplastic deformation, which is referred to as the Maugis–Pollock model. Other adhe-sion models include the works by Tsai, Pui and Liu [13] and Maugis [14]. Reviewsof these earlier works were reported by Ranade [15] and Mittal [16]. More recentstudies on particle adhesion were performed by Quesnel et al. [17] and Kendall[18]. Because particle adhesion force per unit mass increases sharply as the particlesize decreases, it is very difficult to remove particles with diameters smaller than1 µm.

There has been some debate on the validity of the JKR and DMT models. Mulleret al. [19, 20] tested the regime of the JKR and DMT models using a Leonard–Jonespotential and suggested that the DMT model applies for a system with high Young’smodulus, low surface energy, and small-diameter particles, while the JKR model ismore suitable for a system with low Young’s modulus, high surface energy, andlarge-diameter particles.

For charged particles in the presence of an electric field, the electrostatic forcesstrongly affect the particle transport, as well as adhesion and detachment. Donald[21, 22], Donald and Watson [23], and Lee and Ayala [24] showed that there was astrong dependence of the adhesion force on charges carried by toner particles evenin the absence of an imposed external electric field. Donald and Watson [25] de-veloped a model to include the effect of an electric field on particle adhesion inxerographic processes. Goel and Spencer [26] analyzed the effect of the electro-static and van der Waals forces on the adhesion of toner particles. They concludedthat for large particles (about 15 µm), the electrostatic image force dominates theparticle adhesion, while for small particles (about 5 µm), the van der Waals forcebecomes the dominant force.

Hays [27, 28] studied the detachment of charged toner particles under an elec-tric field and suggested that the large adhesion force was due to the electrostaticimage force of ‘patchy’ charges that were concentrated on particle asperities. Lee


and Jaffe [29] also reported the effect of electrostatic forces on the adhesion oftoner particles in xerographic printers. Mizes [30] quantified the relative contribu-tions of non-electrostatic and electrostatic forces to the net particle adhesion force.He concluded that under large electric fields (2–8 V/µm), the electrostatic contri-butions dominated the non-electrostatic effects. Schaefer et al. [31] evaluated thedeformation of a polystyrene particle by investigating the force curve. Schaefer etal. [32] also studied the adhesion force by measuring the force needed to removedifferent particles from a variety of substrates. Tombs [33] investigated the effectof surface moisture on the electrostatic force by calculating the image force usingcomplex linear multipoles. Gady [34] and Gady et al. [35, 36] measured the inter-action forces between micro-particles and flat substrates and provided informationon the relative contributions to the net interaction due to van der Walls and electro-static forces. Rimai et al. [37] discussed the nature of the deformations which tendto be caused by van der Waals and electrostatic interactions. Rimai et al. [38] mea-sured the force needed to remove toner particles using ultracentrifugation and foundthat adhesion is dominated by van der Waals interactions for toner particles with anumber-weighted diameter of 6.5 µm and a charge of approximately 20–25 µC/g,while electrostatic interactions can also significantly contribute to the adhesionforces for highly charged particles. Soltani and Ahmadi [39] studied rough parti-cle detachment with electrostatic forces in turbulent flows. Their predictions agreedwell with the experimental results by Hays [27, 28] and Mizes [30]. Feng and Hays[40] performed a finite-element analysis of the electrostatic force on a uniformlycharged dielectric sphere resting on a dielectric-coated electrode in a detachingelectric field. Masashi and Manabu [41] measured particle adhesion force usingthe electric field detachment method. They found that the adhesion force increaseswith time elapsed after the particles are placed on a substrate. Hays and Sheflin[42] performed electric field detachment measurements on an ion-charged toner fordifferent charge levels and found that the adhesion of ion-charged toner is less thanthat of triboelectrically charged toner. They suggested that the enhanced electrosta-tic adhesion of triboelectrically charged toner is attributed to a non-uniform surfacecharge distribution.

Hinds [43] and Nicholson [44] reviewed the work on particle resuspensionprocesses. More recent studies on models of particle resuspension processes wereprovided by Wang [45], Masironi and Fish [46], Soltani and Ahmadi [47–50] andIbrahim et al. [51]. For particles in humid air, the capillary force significantly af-fects the detachment of the particles. Wei and Zhao [52, 53] found that the loadingrate and the relative humidity have a major influence on adhesion. The effect of rel-ative humidity on adhesion was also studied by Podczeck et al. [54], Busnaina andElsawy [55] and Tang and Busnaina [56]. Zimon [57] and Taheri and Bragg [58]performed experiments on particle resuspension in dry and humid air conditions.The experiments of Zimon [57] were conducted with humidity of less than 10%.Taheri and Bragg [58] carried out their experiments under the condition of normalroom temperature and humidity; their results agree well with the simulation results


found by Soltani and Ahmadi [49] for moist particle resuspension. Ibrahim et al.[53, 59] measured particle resuspension in both dry and humid air conditions. Theirexperiments were performed at a humidity of 25 ± 3% for the dry condition and67% for the humid condition. A direct atomic force microscopy (AFM) measure-ment of adhesion force was reported by Gotzinger and Peukert [60, 61]. Recently,Ahmadi et al. [62] studied particle adhesion and detachment in turbulent flows, in-cluding the effect of capillary forces. More recently, Zhang and Ahmadi [63] useda maximum adhesion resistance moment model to study particle detachment withcapillary force. They proposed an effective thermodynamic work of adhesion the-ory that includes capillary forces for particle adhesion and detachment in turbulentflows.

In this study, the rolling detachment of spherical particles in the presence ofcapillary and electrostatic forces for both elastic and plastic surface deformationswas studied. An effective thermodynamic work of adhesion model was used toaccount for the effects of capillary and electrostatic forces for hydrophilic materials.The maximum adhesion resistance moments were evaluated using the JKR and theDMT models for elastic surface deformation, while the Maugis–Pollock model wasused for the plastic surface deformation. The turbulence burst/inrush model wasused for evaluating the near-wall velocity field. The rolling detachment of sphericalparticles was studied, and the critical shear velocities for detaching particles ofvarious sizes were evaluated. The results were compared with those obtained inthe absence of the electrostatic force and the capillary force. It was shown that thecapillary and electrostatic forces significantly affected the particle adhesion and theopportunity for resuspension. The model predictions for the resuspension of glassparticles from a glass substrate with an average Boltzmann charge distribution werecompared with the available experimental data.

2. Adhesion Models

The adhesion models used in this study are similar to those described in detail andused in Zhang and Ahmadi [56].

3. Capillary Force, Electrostatic Force and Effective Thermodynamic Workof Adhesion

3.1. Capillary Force

In this section, the introduction on capillary force is briefly presented. Detailedinformation about capillary force can be found in Zhang and Ahmadi [63].

As shown in Fig. 1, for hydrophilic materials in humid air, the capillary force isdetermined by the surface tension of water σ (=0.0735 N/m, at room temperature),the particle diameter d , the wetting angle θ and the angle α as shown in Fig. 1. Thatis:

Fc = 2πσd[sinα sin(θ + α) + cos θ)]. (1)


Figure 1. Geometric features of a spherical particle attached to a surface with capillary effect.

The angle α is normally very small; therefore for small values of wetting angle θ ,the expression for the capillary force becomes [15, 55]:

Fc = 2πσd. (2)

3.2. Charge Distribution

An aerosol particle rarely has zero charge due to the high ion concentration in theatmosphere. Particles smaller than 0.1 µm in diameter do not naturally contain acharge [64]. The number of natural charges increases with particle size. Particlescan be charged through different mechanisms based on corresponding ionic at-mosphere conditions. Three different charge distributions are outlined below.

3.2.1. Boltzmann Charge DistributionBoltzmann charging occurs for small particles in a bipolar ionic atmosphere. Forparticles larger than 0.1 µm at normal condition, the average number of absolutecharges per particle can be approximated by [39]:

|n| = 2.37√

d, (3)

where d is given in µm. Therefore the average number of positive or negativecharges per particle is |n|/2 [39].

3.2.2. Diffusion Charge DistributionDiffusion charging occurs when uncharged particles obtain charges by the diffu-sion of charged unipolar gaseous ions onto their surfaces through random collisions


between ions and particles. For a particle of diameter dc during diffusion chargingtime t , the approximate number of obtained charges n is given as [43]:

n = dckt

2e2c

ln

[1 + πdc̄ie

2cNit

2kT

], (4)

where c̄i = 2.4 × 104 cm/s is the mean thermal speed of the ions, dc is particle di-ameter in cm, ec = 4.8 × 10−10 stC (electrostatic units) is the electronic unit chargein cgs units, Ni is the ion concentration, k = 1.38 × 1016 ergs/K is the Boltzmannconstant, and T is the temperature of the gas. In the subsequent analysis, a typicalvalue of Nit ≈ 108 s/cm3 is used.

3.2.3. Field Charge DistributionParticles in an electric field acquire charges due to collisions with ions which aremoving along the lines of force that intersect the particle surfaces. This process isknown as field charging. After a sufficient time for a given charging condition, thesaturation number of charges n acquired by the particle of diameter d is given as[43]:

n =[

3ε

ε + 2

][Ecd

2c

4ec

], (5)

where ε is the dielectric constant of the particle and Ec is the electric field strengthin cgs units. Equations (4) and (5) are expressed in cgs units. Field charging is thedominant mechanism for particles larger than 1 µm, and diffusion charging is thedominant mechanism for particles less than 1 µm.

3.3. Electrostatic Force

For a charged particle resting on a conducting substrate in the presence of an appliedelectric field, the electrostatic force acting on the particle is given as [39]:

Fe = qE + q2

16πε0y2+ qEd3

16y3+ 3πε0d

6E2

128y4, (6)

where ε0 = 8.859 × 10−12 amp · s/V · m is the permittivity (dielectric constant offree space). d is the particle diameter, E is the electric field strength and q is thetotal electrical charge on the particle, which is given as [39]:

q = ne, (7)

where e = 1.6 × 10−19 C is the electronic unit charge in mks units and n is thenumber of units of charge. In equation (6), y is the distance of the particle from thesurface, which is approximately half d; therefore, equation (6) becomes:

Fe = qE + q2

4πε0d2+ qE

2+ 3πε0d

2E2

8. (8)

The first term on the right-hand side of equation (8) is the Coulomb force. Thesecond term is the image force. The third term is the dielectrophoretic force on


the induced dipole due to the gradient of the field from the image charge. Thefourth term is the polarization force due to the interaction of the induced dipoleand its image. The Coulomb and dielectrophoretic forces can be either toward oraway from the substrate, depending on the charges carried by the particles and thedirection of the electric field. The image and polarization forces are always towardthe substrate.

In most cases of this study, the particles and substrates are made of the samematerial, so the electrical double layer force is neglected. This force is caused bythe contact potential and becomes important when the particle and the substrate aremade of different materials with a high level of contact potential or when particlesare smaller than 10 µm.

In order to include the effects of capillary and electrostatic forces in the adhe-sion models, we assume that the particles without a charge are deposited on thesubstrate under dry conditions. When an electric field is applied and particles getcharged, a liquid meniscus forms at the same time due to vapor condensation aroundthe particle–substrate contact. Therefore, a superposition of van der Waals, capil-lary and electrostatic forces may be assumed. The total force needed to pull-off theparticle then is Fpo + Fc + Fe, where Fpo is the pull-off force for overcoming thevan der Waals adhesion, which is a function of the thermodynamic work of adhe-sion WA. The presence of the capillary force enhances the surface energy of thematerials, while the electrostatic forces can increase or decrease the surface energy,depending on the charges carried by the particles and the direction of the electricfield; therefore, it is reasonable to account for the combined effect of van der Waalsadhesion, capillary and electrostatic forces with an effective thermodynamic workof adhesion W e

A. This effective thermodynamic work of adhesion varies, dependingon the adhesion model used.

3.4. Effective Thermodynamic Work of Adhesion

In the presence of the capillary and electrostatic forces, the effective pull-off forceis equal to the sum of the van der Waals pull-off force, the capillary force and theelectrostatic force. Then the effective thermodynamic work of adhesion may bedefined by using an equivalent JKR model for evaluating the pull-off force. That is:

3

4πW eJKR

A d = 3

4πWAd + Fc + Fe, (9)

where W eJKRA is the effective work of adhesion for the JKR model. Fc and Fe are

given by equations (2) and (8). It then follows that:

W eJKRA = WA + 8σ

3+ 2qE

πd+ q2

3π2ε0d3+ ε0dE2

2. (10)


The corresponding approximate expression for the effective contact radius is thengiven by replacing WA with W eJKR

A in the contact radius expression:

a3e = d

2K

[P + 3

2W eJKR

A πd +√

3πW eJKRA dP +

(3πW eJKR

A d

2

)2 ], (11)

where K is the composite Young’s modulus and P is the applied normal load.The effective thermodynamic work of adhesion for the DMT and the Maugis–

Pollock models can be obtained in a similar way. That is:

W eDMTA = W eMP

A = WA + 2σ + 1.5qE

πd+ q2

4π2ε0d3+ 3ε0dE2

8. (12)

The corresponding effective contact radius as estimated from the DMT model isgiven as:

a3e ≈ d

2k

(P + πW eDMT

A d). (13)

For the Maugis–Pollock model, the corresponding effective contact radius is givenas:

ae =√

P + πW eMPA d

πH, (14)

where H is the hardness of the material.

4. Detachment Models

Particle detachment from a surface can happen through three mechanisms: rolling,sliding and lifting. Wang [45] and Soltani and Ahmadi [47, 48] pointed out thatthe detachment of smooth spherical particles is more easily achieved by the rollingmotion, rather than by sliding and lifting. Therefore, only rolling detachment isdiscussed in this study.

4.1. Rolling Detachment Model

Ziskind et al. [65] developed a model for the rolling detachment of a sphere from asurface, where the detachment occurs when the hydrodynamic moment exceeds themaximum adhesion resistance moment evaluated for the JKR and DMT models forelastic surface and particle deformations. Here the approach is extended to includethe effect of capillary and electrostatic forces as well as the effect of plastic surfacedeformation.

Figure 1 shows a spherical particle attached to a planar surface in a fluid flow.The lift and gravity forces, which are very small, are neglected in this study. Inhumid air, a meniscus is formed at the particle–substrate contact. Here it is assumedthat the particle without any charge is deposited on the substrate under a dry aircondition. Then an electric field is applied, and the particle gets charged while at


the same time a liquid meniscus forms around the particle–substrate contact. Theparticle will be detached when the moment of the hydrodynamic force about thepoint ‘O’ (which is located at the rear perimeter of the contact circle) overcomesthe maximum adhesion resistance moment due to the combined adhesion, capillaryand electrostatic forces. That is:

Mt + Ft

(d

2− αo

)� (P · a)max, (15)

where Ft is the fluid drag force, αo is the relative approach between the particleand surface, Mt is the hydrodynamic moment about the center of the particle, and(P · a)max is the maximum adhesion resistance moment due to the combined adhe-sion, capillary and electrostatic forces. In most practical cases, αo can be neglectedand equation (15) becomes:

Mt + Ftd

2� (P · a)max. (16)

5. Maximum Adhesion Resistance

To develop a particle rolling detachment model, the corresponding maximum adhe-sion resistance moment needs to be evaluated. In this study, the JKR and DMTadhesion models are used for elastic surface deformations, while the Maugis–Pollock adhesion model is used for plastic surface deformation. Using an approachsimilar to that developed by Zhang and Ahmadi [63], one can obtain the maximumadhesion resistance moment for different adhesion models.

For the JKR model without capillary and electrostatic forces:

MJKRmax = 2.707

W4/3A d5/3

K1/3. (17)

For the JKR model with capillary and electrostatic forces:

MJKRmax = 2.707

W eJKRA

4/3d5/3

K1/3. (18)

For the DMT model without capillary and electrostatic forces:

MDMTmax = 1.725

W4/3A d5/3

K1/3. (19)

For the DMT model with capillary and electrostatic forces:

MDMTmax = 1.725

W eDMTA

4/3d5/3

K1/3. (20)

For the Maugis–Pollock model without capillary and electrostatic forces:

MMPmax = 2π(WAd)3/2

3√

3H. (21)


For the Maugis–Pollock model with capillary and electrostatic forces:

MMPmax = 2π(W eMP

A d)3/2

3√

3H. (22)

6. Hydrodynamic Forces and Torques

Soltani and Ahmadi [47, 48] pointed out that the particle detachment process isstrongly affected by the near-wall turbulent flow structure. This flow structure in-cludes burst and inrush processes. In this section, the peak near-wall velocity duringturbulent burst/inrush and the corresponding hydrodynamic forces and torques arebriefly outlined.

6.1. Burst/Inrush Model

Soltani and Ahmadi [47, 48] reported that the maximum instantaneous streamwisevelocity experienced locally near the wall during the turbulent burst/inrush processgiven in wall units is:

u+M = 1.72y+, (23)

where:

u+M = uM

u∗ and y+ = yu∗

ν. (24)

Here, u∗ is the shear velocity, ν is the kinematic viscosity of air and uM is themaximum velocity at a distance of y from the wall. For a particle attached to a wall,the fluid velocity at the center of the particle is obtained from equation (23), i.e.:

u+M = 0.86d+, (25)

where d+ is the nondimensional particle diameter given as:

d+ = du∗

ν. (26)

In this case, the drag force acting on the particle is given as:

Ft = 4.38πρd2u∗2

Cc, (27)

where ρ is the density of the air and Cc is the Cunningham factor given as [66, 67]:

Cc = 1 + Kn[1.257 + 0.4 exp(−1.1/Kn)]. (28)

Here the Kn is the Knudsen number defined as:

Kn = 2λ

d, (29)

where λ is the mean free path of air.


The corresponding moment of the hydrodynamic force acting on the particle isgiven as:

Mt = 1.62πρu∗2d3

Cc. (30)

The lift force acting on the particle is very small compared to the adhesion; capil-lary and electrostatic forces are therefore is neglected in this work. For a polystyreneparticle with a diameter of 10 µm in a shear velocity of 1 m/s, the pull off forces is2550 times larger than the lift force.

7. Particle Detachment

Soltani and Ahmadi [39] evaluated the minimum critical shear velocity for roughparticle detachment with electrostatic forces in turbulent flows. Soltani and Ahmadi[47] also studied the minimum critical shear velocity for removing different sizeparticles in the absence of the capillary force. Recently, Ahmadi et al. [62] reportedthe effect of the presence of capillary force on particle adhesion and detachmentin turbulent flows. They used a simplified version of moment detachment with theuse of the JKR model and rolling and sliding detachment models. Very recently,Zhang and Ahmadi [63] used a maximum adhesion resistance moment model andthe effective thermodynamic work of adhesion theory to study the minimum criticalshear velocity for particle detachment with capillary force. In this section, the criti-cal shear velocities for particle removal in humid air with capillary and electrostaticforces are evaluated with the use of the JKR, the DMT and the Maugis–Pollockmodels.

Substituting the expression for the hydrodynamic drag and torque into equa-tion (16), the critical shear velocity for rolling detachment of spherical particlesincluding the capillary and electrostatic force is obtained as:

u∗c

2 = Mmax

3.81πρd3/Cc, (31)

where Mmax is the maximum adhesion resistance moment. The critical shear veloc-ity for particle detachment according to the JKR, DMT and Maugis–Pollock modelscan be evaluated by substituting, respectively, the expressions for the maximum re-sistance moment from equations (18), (20) and (22) into equation (31) for Mmax.

8. Results

8.1. Force and Charge Analysis

In this section the results for the average Boltzmann, saturation and fixed 20 µC/gcharge distributions are given. The results for the electrostatic, capillary and pull-off forces are also presented and discussed. The material properties for polystyrene,glass, nickel and steel that were used in this work are listed in Table 1 [50, 63].


As for the dielectric constant, for polystyrene, a value of ε = 4 was used; for glass,a value of ε = 3.1 was used. Two electric fields are presented in the study, 5000 and10 000 kV/m.

8.1.1. Boltzmann, Saturation and Fixed 20 µC/g Charge DistributionsFigure 2 shows the variation of the average Boltzmann charge carried by thepolystyrene particles versus the particle diameter. It is seen that the average Boltz-mann charge increase with the increase of the particle diameter. Figure 3 shows thevariation of the saturation charge distribution and a fixed 20 µC/g charge with thediameter of polystyrene particles. Figure 3 also shows similar increasing trend as isseen in Fig. 2. The saturation and fixed 20 µC/g charge carried by the particles in-crease with the increase of the particle diameter. The saturation charge in an electricfield of 5000 kV/m is half of that in 10 000 kV/m. For particles smaller than 25 µm,the 20 µC/g charge distribution leads to smaller amount of charge as compared withthe saturation charge. However, for particles larger than 50 µm, the 20 µC/g charge

Table 1.Material properties for different combinations

Material combination Ei A WA ρp νi H

(1010 Pa) (10−20 J) (10−3 J/m2) (103 kg/m3) (107 Pa)

Polystyrene–polystyrene 0.28 6.37 10.56 1.05 0.33 6.59Glass–glass 6.9 8.5 14.1 2.18 0.2 490–665.4Glass–steel _ _ 150 _ _ 646.8Polystyrene–nickel _ _ 23.65 _ _ _

Ei : Young’s modulus of material i; A: Hamaker constant; WA: thermodynamic work of adhesion;ρp: density of material; νi : Poisson’s ratio of material i; H : hardness of material.

Figure 2. Variation of the average Boltzmann charge distribution with the particle diameter.


distribution is larger than the saturation charge distributions. Compared to Fig. 2,Fig. 3 shows that the saturation and fixed 20 µC/g charge distributions are muchhigher than the average Boltzmann charge distribution.

8.1.2. Coulomb and Dielectrophoretic ForcesFigure 4 shows the variation of the combined Coulomb forces (Coulomb force anddielectrophoretic force) with the particle diameter for polystyrene particles carryingdifferent charges in an electric field of 10 000 kV/m. It is seen that the combinedCoulomb forces increase with the increase of the particle diameter. For particles

Figure 3. Variation of the saturation and fixed 20 µC/g charge distribution with the particle diameterfor polystyrene particles.

Figure 4. Variation of the combined Coulomb forces with the particle diameter for polystyrene parti-cles carrying different charges in an electric field of 10 000 kV/m.


(a) (b)

Figure 5. Variation of the image forces with the particle diameter for polystyrene particles carryingdifferent charges.

carrying an average Boltzmann charge, the combined Coulomb forces are muchsmaller than those of particles carrying saturation and fixed 20 µC/g charges. Thereason is that the average Boltzmann charge distribution is much smaller than thesaturation and fixed 20 µC/g charge distributions, as mentioned above. Figure 4also shows that corresponding to the amount of charge carried by the particles, forparticles smaller than 50 µm, the combined Coulomb forces for particles carryingfixed 20 µC/g charge distribution are smaller than those of particles carrying satu-ration charge distributions. However, for particles larger than 50 µm carrying fixed20 µC/g charge distribution, the combined Coulomb forces are larger than for par-ticles larger than 50 µm carrying saturation charge distributions.

8.1.3. Image ForceFigure 5(a) and 5(b) shows the variation of the image forces with the particle di-ameter for polystyrene particles carrying different charges. Similar to the trend inFig. 4, Fig. 5(a) and 5(b) shows that the image forces increase with the increaseof the particle diameter. For particles carrying an average Boltzmann charge, theimage forces are much smaller than those of particles carrying saturation and fixed20 µC/g charges. The reason is that, in general, the average Boltzmann charge dis-tribution is much smaller than the saturation and fixed 20 µC/g charge distributions.Figure 5(a) and 5(b) also shows that for particles smaller than 25 µm, the imageforces for particles carrying a fixed 20 µC/g charge distribution are smaller thanthose of particles carrying saturation charge distributions; but the image forces forparticles larger than 50 µm carrying a fixed 20 µC/g charge distribution are largerthan for such particles carrying saturation charge distributions. It is also seen fromFig. 5(a) and 5(b) that the image forces for particles carrying the saturation chargedistribution in an electric field of 5000 kV/m are smaller than those for particles in


Figure 6. Variation of the polarization forces with the particle diameter for polystyrene particles indifferent electric fields.

10 000 kV/m. Compared to Fig. 4, Fig. 5(a) and 5(b) shows that the image forcesare smaller than the combined Coulomb forces.

8.1.4. Polarization ForcesFigure 6 shows the variation of the polarization forces with the particle diameterfor polystyrene particles in different electric fields. Similar to Fig. 5, Fig. 6 showsthat the polarization forces increase with the increase of the particle diameter; thepolarization forces for particles in an electric field of 5000 kV/m are smaller thanthose for particles in 10 000 kV/m. Compared to Figs 4 and 5, Fig. 6 shows thatthe polarization forces are smaller than the image forces and combined Coulombforces for particles carrying the saturation and fixed 20 µC/g charge distributions butlarger than the image forces and combined Coulomb forces for particles carryingthe average Boltzmann charge distribution.

8.1.5. Capillary and Pull-Off ForcesFigure 7 shows the variation of the capillary and pull-off forces with the particlediameter for polystyrene particles on a polystyrene substrate. It can be seen fromFig. 7 that the capillary and pull-off forces increase with the increase of the particlediameter, and capillary forces are much larger than the pull-off forces. Comparedto Figs 4, 5 and 6, Fig. 7 shows that the capillary forces can be smaller than thecombined Coulomb forces for particles carrying the saturation and fixed 20 µC/gcharge distributions. Capillary forces can also be smaller than the image forces forparticles carrying the fixed 20 µC/g charge distribution, but can be larger than thepolarization forces. In addition, capillary forces can be larger than the image forcesfor particles carrying the saturation and Boltzmann charge distributions.


Figure 7. Variation of the capillary and pull-off forces with the particle diameter for polystyreneparticles on a polystyrene substrate.

8.2. Critical Shear Velocity for Particle Removal

In this section the results for the detachment of particles of different sizes and vari-ous materials from substrates of various materials are presented and discussed. Allresults are presented in terms of critical shear velocity, u∗

c , which is the minimumshear velocity needed to remove a particle from the substrate. For the particles inthe turbulent flow, the near-wall velocity during the burst/inrush of turbulent flowwas used in the analysis.

8.2.1. Critical Shear Velocities in the Presence of Electrostatic ForcesFigures 8, 9 and 10 show different adhesion model predictions for the variationsof u∗

c with particle diameter in different electric fields for the rolling detach-ment of polystyrene particles with Boltzmann charge, saturation charge and fixed20 µC/g charge respectively from a polystyrene substrate for dry conditions. Herethe Coulomb force and dielectrophoretic force are directed towards the substrate.Figure 8 shows that the critical shear velocity decreases with the increase of the par-ticle diameter. That is, as expected, small particles are more difficult to remove thanthe larger ones. Figure 8 also shows that the critical shear velocity as predicted bythe JKR adhesion model is the largest. The predicted critical shear velocity by theDMT model is less than that by the JKR model. The predicted value by the Maugis–Pollock model, which accounts for plastic deformation, is lower than those by theJKR and the DMT models for elastic deformation. The differences are small forsmaller particles, but become relatively large for larger particles. As suggested byZhang and Ahmadi [63], these differences are due to the variations of the maxi-mum adhesion resistance moments for the JKR, the DMT and the Maugis–Pollockmodels. Figure 8 also shows that the critical shear velocities in an electric field of5000 kV/m are lower than those in 10 000 kV/m. The reason is that the electrostatic


Figure 8. Variation of the critical shear velocities with the particle diameter as predicted by differentadhesion models for resuspension of polystyrene particles with an average Boltzmann charge distrib-ution from a polystyrene substrate in the presence of different electric fields without capillary effects.Coulomb force and dielectrophoretic force are directed towards the substrate.

Figure 9. Variation of the critical shear velocities with the particle diameter as predicted by differentadhesion models for resuspension of polystyrene particles with saturation charge distribution from apolystyrene substrate in the presence of different electric fields without capillary effects. Coulombforce and dielectrophoretic force are directed towards the substrate.

forces increase with the increase of the electric field strength. The differences aresmall for smaller particles, but become large for larger particles. The reason is thatlarger particles carry more charges, as seen from equations (3) and (5). Althoughthe images force for Boltzmann charge decrease with the increase of the particle di-


Figure 10. Variation of the critical shear velocities with the particle diameter as predicted by differentadhesion models for resuspension of polystyrene particles with 20 µC/g charge distribution from apolystyrene substrate in the presence of different electric fields without capillary effects. Coulombforce and dielectrophoretic force are directed towards the substrate.

ameter, the electrostatic forces still increase overall with the increase of the particlediameter, as shown by equation (8).

Figures 9 and 10 show a similar trend as that in Fig. 8. The critical shear veloc-ity predicted by the JKR adhesion model is the largest, and the predicted value bythe Maugis–Pollock model is lower than those by the JKR and the DMT models;the differences are small for smaller particles, but become relatively large for largerparticles. The critical shear velocities in an electric field of 5000 kV/m are lowerthan those in 10 000 kV/m; for Fig. 9, the differences are small for smaller particles,but become large for larger particles. While for Fig. 10, although the differences aresmall for smaller particles and become large for larger particles, the differences dobecome smaller again for particles larger than 30 µm. As noted before, for a fixedcharge of 20 µC/g, and particles larger than 30 µm, the image force becomes com-parable to or larger than the combined Coulomb and dielectrophoretic forces. Thisimplies that the electric field effect is smaller. Figure 9 shows that the critical shearvelocity decreases with the increase of the particle diameter. Compared to Fig. 8,Fig. 9 shows higher shear velocities. The reason is the electrostatic forces actingon particles with a saturation charge are much higher than those on particles withan average Boltzmann charge. Figure 10 shows that for small particles, the criticalshear velocity decreases with the increase of the particle diameter; for large parti-cles, the critical shear velocity increases with the increase of the particle diameterdue to the higher charges that large particles obtained for a fixed charge of 20 µC/g.Compared to Fig. 8, Fig. 10 shows higher shear velocities for large particles. Forsmall particles, shear velocities in Figs 8 and 10 are almost same. Compared toFig. 9, Fig. 10 shows higher shear velocities for large particles and lower shear


Figure 11. Variation of the critical shear velocities with the particle diameter as predicted by differentadhesion models for resuspension of polystyrene particles with an average Boltzmann charge distri-bution from a polystyrene substrate in the presence of capillary effects and different electric fields.Coulomb force and dielectrophoretic force are directed towards the substrate.

velocities for small particles. This is because, for a fixed charge of 20 µC/g, theamount of charge on a particle is proportional to its mass; small particles have asmall charge, similar to those with Boltzmann charge distribution, but less thanthose with saturation charge distribution. Large particles have a much higher largercharge compared to those with Boltzmann charge and saturation charge distribu-tions. Thus, shear velocities for small particles in Figs 8 and 10 are almost same,while the small particles shown in Fig. 10 have lower shear velocities than those inFig. 9. For large particles, Fig. 10 shows higher shear velocities than those of bothFigs 8 and 9.

8.2.2. Critical Shear Velocities in the Presence of Capillary and ElectrostaticForces for Polystyrene ParticlesFigures 11, 12 and 13 show the variations of u∗

c with particle diameter as pre-dicted by different adhesion models in the presence of capillary effects and differentelectric fields for the rolling detachment of polystyrene particles with Boltzmanncharge, saturation charge and fixed 20 µC/g charge, respectively, from a polystyrenesubstrate. Here Coulomb force and dielectrophoretic force are directed towards thesubstrate. The trends are similar to those observed in Figs 8, 9 and 10. Figure 11shows that the critical shear velocity decreases with the increase of the particle di-ameter. The JKR model predictions are slightly higher than those from the DMTmodel. The predictions from the Maugis–Pollock model for u∗

c are the lowest forlarge particles, but slightly higher than those from the DMT model for small par-ticles. Compared to results presented in Fig. 8, Fig. 11 shows higher critical shearvelocities. This implies that the presence of capillary force significantly increasesthe critical shear velocity for particle rolling removal. Figure 11 also shows that the


Figure 12. Variation of the critical shear velocities with the particle diameter as predicted by differentadhesion models for resuspension of polystyrene particles with saturation charge distribution from apolystyrene substrate in the presence of capillary effects and different electric fields. Coulomb forceand dielectrophoretic force are directed towards the substrate.

Figure 13. Variation of the critical shear velocities with the particle diameter as predicted by differentadhesion models for resuspension of polystyrene particles with 20 µC/g charge distribution from apolystyrene substrate in the presence of capillary effects and different electric fields. Coulomb forceand dielectrophoretic force are directed towards the substrate.

critical shear velocities in an electric field of 5000 kV/m are lower than those in10 000 kV/m, but the differences are relatively small compared to results presentedin Fig. 8. Because the figures are in logarithmic coordinates, this does not meanthat the absolute values of the differences are smaller. This does mean that when


Coulomb and dielectrophoretic forces are directed towards the substrate, the rela-tive effects of the electrostatic forces decrease in the presence of capillary effects.

Figure 12 shows that the critical shear velocity decreases with the increase ofthe particle diameter. Compared to Fig. 11, Fig. 12 shows higher shear velocities.Figure 13 shows that, for small particles, the critical shear velocity also decreaseswith the increase of the particle diameter; however, for large particles, the criticalshear velocity increases with the increase of the particle diameter due to the highercharge that large particles obtained for a fixed charge of 20 µC/g. In contrast toFig. 11, Fig. 13 shows higher shear velocities for large particles. For small parti-cles, shear velocities in Figs 11, 12 and 13 are almost same. Compared to Fig. 12,Fig. 13 shows higher shear velocities for large particles and lower shear velocitiesfor smaller particles.

Similar to Fig. 11, Figs 12 and 13 also show that the model predictions fromthe JKR model are slightly higher than those from the DMT model. The predic-tions from the Maugis–Pollock model for u∗

c are the lowest for large particles, butslightly higher than those from the DMT model for small particles. Compared toresults presented in Figs 9 and 10, Figs 12 and 13 show higher critical shear ve-locities, which means that the presence of capillary force significantly increases thecritical shear velocity for particle rolling removal. Figures 12 and 13 also show thatthe critical shear velocities in an electric field of 5000 kV/m are lower than thosein 10 000 kV/m; the differences are small for smaller particles, but become largefor larger particles. But the differences are relatively smaller compared to resultspresented in Figs 9 and 10, which means that the relative effects of the electrostaticforces decrease in the presence of capillary effects. However, in contrast to Fig. 10,for particles larger than 30 µm, the differences in Fig. 13 do not become smalleragain. The reason is same: in the presence of capillary effects, the relative effectsof the electrostatic forces are decreased. For that reason, the small particles (Fig. 9)have higher shear velocities than those in Figs 8 and 10, while shear velocities inFigs 11, 12 and 13 are almost same.

8.2.3. Critical Shear Velocities in the Presence of Capillary and ElectrostaticForces for Glass ParticlesFigures 14, 15 and 16 show the variations of u∗

c with particle diameter as predictedby different adhesion models in the presence of capillary effects and different elec-tric fields for the rolling detachment from a glass substrate of glass particles withBoltzmann charge, saturation charge and fixed 20 µC/g charge, respectively. Herethe Coulomb and dielectrophoretic forces are directed towards the substrate. Thetrends are similar to those observed in Figs 11, 12 and 13. Figure 14 shows thatthe critical shear velocity decreases with the increase of the particle diameter. Themodel predictions from the JKR model are higher than those from the DMT model.The predictions from the Maugis–Pollock model are the lowest. Figure 14 alsoshows that the critical shear velocities in an electric field of 5000 kV/m are lowerthan those in 10 000 kV/m. Compared to Fig. 11, Fig. 14 shows lower critical shearvelocities. This implies that for particles with Boltzmann charge distribution, de-


Figure 14. Variation of the critical shear velocities with the particle diameter as predicted by differentadhesion models for resuspension of glass particles with an average Boltzmann charge distributionfrom a glass substrate in the presence of capillary effects and different electric fields. Coulomb forceand dielectrophoretic force are directed towards the substrate.

Figure 15. Variation of the critical shear velocities with the particle diameter as predicted by dif-ferent adhesion models for resuspension of glass particles with saturation charge distribution froma glass substrate in the presence of capillary effects and different electric fields. Coulomb force anddielectrophoretic force are directed towards the substrate.

taching glass particles from a glass substrate is easier than detaching polystyreneparticles from a polystyrene substrate. Figure 15 shows that the critical shear ve-locity decreases with the increase of the particle diameter. Compared to Fig. 14,Fig. 15 shows higher shear velocities for large particles. Figure 16 shows that forsmall particles, the critical shear velocity decreases with the increase of the particle


Figure 16. Variation of the critical shear velocities with the particle diameter as predicted by dif-ferent adhesion models for resuspension of glass particles with 20 µC/g charge distribution from aglass substrate in the presence of capillary effects and different electric fields. Coulomb force anddielectrophoretic force are directed towards the substrate.

diameter; for large particles, the critical shear velocity increases with the increaseof the particle diameter due to the higher number of charges that large particlesobtained for a fixed charge of 20 µC/g. Compared to Fig. 14, Fig. 16 shows highershear velocities for large particles. Compared to Fig. 15, Fig. 16 shows higher shearvelocities for large particles and slightly lower shear velocities for smaller parti-cles. For much smaller particles, shear velocities in Figs 14, 15 and 16 are almostsame.

Similar to Fig. 14, Figs 15 and 16 also show that the model predictions from theJKR model are higher than those from the DMT model. The predictions from theMaugis–Pollock model are the lowest. Compared to Fig. 12, Fig. 15 shows lowercritical shear velocities. This implies that for particles with saturation charge dis-tribution, detaching glass particles from a glass substrate is easier than detachingpolystyrene particles from a polystyrene substrate. Compared to Fig. 13, for theMaugis–Pollock model, Fig. 16 shows lower critical shear velocities. While forJKR and DMT models, Fig. 16 shows lower critical shear velocities for small parti-cles, but slightly higher critical shear velocities for large particles. Because glass isbasically an elastic material, the JKR and DMT models may be more suitable. Thisimplies that for small particles with fixed 20 µC/g charge distribution, detachingglass particles from a glass substrate is easier than detaching polystyrene particlesfrom a polystyrene substrate; for big particles, detaching glass particles from a glasssubstrate can be a little more difficult than detaching polystyrene particles from apolystyrene substrate, but there is no significant difference.


Figure 17. Variation of the critical shear velocities with the particle diameter as predicted by differentadhesion models for resuspension of polystyrene particles with saturation charge distribution from apolystyrene substrate in the presence of capillary effects and different electric fields. Coulomb forceand dielectrophoretic force are directed away from the substrate.

Figure 18. Variation of the critical shear velocities with the particle diameter as predicted by differentadhesion models for resuspension of polystyrene particles with 20 µC/g charge distribution from apolystyrene substrate in the presence of capillary effects and different electric fields. Coulomb forceand dielectrophoretic force are directed away from the substrate.

8.2.4. Critical Shear Velocities in the Presence of Capillary and UpwardElectrostatic Forces for Polystyrene ParticlesFigures 17 and 18 show the variations of u∗

c with particle diameter as predictedby the different adhesion models in the presence of capillary effects and differentelectric fields for the rolling detachment of polystyrene particles with saturation


charge and fixed 20 µC/g charge, respectively, from a polystyrene substrate. Herethe Coulomb and dielectrophoretic forces are directed away from the substrate.Figure 17 shows that the critical shear velocity decreases with the increase of theparticle diameter. The model predictions from the JKR model are slightly higherthan those from the DMT model. The predictions from the Maugis–Pollock modelfor u∗

c are the lowest for large particles, but slightly higher than those from theDMT model for small particles. Compared to results presented in Fig. 12, Fig. 17shows lower critical shear velocities for large particles, especially in an electric fieldof 10 000 kV/m. This implies that in the presence of capillary force, when directedaway from the substrate and under a strong electric field, Coulomb force and dielec-trophoretic force significantly decrease the critical shear velocity for large particlerolling removal. Figure 17 also shows that the critical shear velocities in an elec-tric field of 5000 kV/m are higher than those in 10 000 kV/m, and the differencesare relatively larger compared to results presented in Fig. 12, where Coulomb forceand dielectrophoretic force are directed towards the substrate. This means that therelative effects of the electrostatic forces increase when Coulomb force and dielec-trophoretic force are directed away from the substrate.

Figure 18 shows that the critical shear velocity in an electric field of 5000 kV/mdecreases with the increase of the particle diameter for small particles, but increasesslightly for large particles. In an electric field of 10 000 kV/m, however, the criticalshear velocity decreases with an increase of the particle diameter. The model predic-tions from the JKR model are slightly higher than those from the DMT model. Thepredictions from the Maugis–Pollock model for u∗

c are the lowest for large particles,but slightly higher than those from the DMT model for small particles. Comparedto results presented in Fig. 13, Fig. 18 shows lower critical shear velocities for largeparticles, especially in an electric field of 10 000 kV/m. This implies that in thepresence of capillary force, when directed away from the substrate especially un-der a strong electric field, Coulomb force and dielectrophoretic force significantlydecrease the critical shear velocity for large particle rolling removal. Figure 18 alsoshows that the critical shear velocities in an electric field of 5000 kV/m are higherthan those in 10 000 kV/m, and the differences are quite larger compared to resultspresented in Fig. 13, where Coulomb force and dielectrophoretic force are directedtowards the substrate. This means that the relative effects of the electrostatic forcesincrease when Coulomb force and dielectrophoretic force are directed away fromthe substrate.

8.2.5. Critical Shear Velocities in the Presence of Capillary and UpwardElectrostatic Forces for Glass ParticlesFigures 19 and 20 show the variations of u∗

c with particle diameter as predictedby the different adhesion models in the presence of capillary effects and differentelectric fields for the rolling detachment of glass particles with saturation chargeand fixed 20 µC/g charge, respectively, from a glass substrate. Here the Coulombforce and the dielectrophoretic force are directed away from the substrate. Figure 19shows that the critical shear velocity decreases with an increase of the particle di-


Figure 19. Variation of the critical shear velocities with the particle diameter as predicted by dif-ferent adhesion models for resuspension of glass particles with saturation charge distribution froma glass substrate in the presence of capillary effects and different electric fields. Coulomb force anddielectrophoretic force are directed away from the substrate.

Figure 20. Variation of the critical shear velocities with the particle diameter as predicted by dif-ferent adhesion models for resuspension of glass particles with 20 µC/g charge distribution from aglass substrate in the presence of capillary effects and different electric fields. Coulomb force anddielectrophoretic force are directed away from the substrate.

ameter. The model predictions from the JKR model are higher than those fromthe DMT and Maugis–Pollock models. The predictions from the Maugis–Pollockmodel for u∗

c are the lowest. Compared to results presented in Fig. 15, Fig. 19shows lower critical shear velocities for large particles, especially in an electric fieldof 10 000 kV/m. This implies that in the presence of capillary force, when directed


away from the substrate and under a strong electric field, Coulomb force and dielec-trophoretic force significantly decrease the critical shear velocity for large particlerolling removal. Figure 19 also shows that the critical shear velocities in an elec-tric field of 5000 kV/m are higher than those in 10 000 kV/m, and the differencesare relatively larger compared to results presented in Fig. 15, where Coulomb forceand dielectrophoretic force are directed towards the substrate. This means againthat the relative effects of the electrostatic forces increase when Coulomb force anddielectrophoretic force are directed away from the substrate.

Figure 20 shows that the critical shear velocity decreases with the increase of theparticle diameter for small particles, but increases for large particles. The modelpredictions from the JKR model are slightly higher than those from the DMTand Maugis–Pollock models. The predictions from the Maugis–Pollock model foru∗

c are the lowest. Compared to the results presented in Fig. 16, Fig. 20 showslower critical shear velocities for large particles, especially in an electric field of10 000 kV/m. This implies that in the presence of a capillary force, Coulomb forceand dielectrophoretic force, when directed away from the substrate especially un-der a strong electric field, significantly decrease the critical shear velocity for largeparticle rolling removal. Figure 20 also shows that the critical shear velocities in anelectric field of 5000 kV/m are higher than those in 10 000 kV/m, and the differ-ences are relatively larger compared to results presented in Fig. 16, where Coulombforce and dielectrophoretic force are directed towards the substrate. This againmeans that the relative effects of the electrostatic forces increase when Coulombforce and dielectrophoretic force are directed away from the substrate.

8.2.6. Critical Shear Velocities in the Presence of Capillary and ElectrostaticForces for Glass Particles on a Steel SubstrateFigure 21 shows the variations of u∗

c with particle diameter as predicted by differentadhesion models in the presence of capillary effects and different electric fieldsfor the rolling detachment of glass particles with saturation charge from a steelsubstrate. Here the Coulomb and dielectrophoretic forces are directed away fromthe substrate. Figure 21 shows that the critical shear velocity decreases with anincrease of the particle diameter. The model predictions from the JKR model arehigher than those from the DMT and Maugis–Pollock models. The predictions fromthe Maugis–Pollock model for u∗

c are the lowest. Compared to results presented inFig. 19, Fig. 21 shows higher critical shear velocities.

Figure 21 also shows that the critical shear velocities in an electric field of5000 kV/m are slightly higher than those in 10 000 kV/m, but the differences arerelatively smaller compared to the results presented in Fig. 19. This means that inthe presence of capillary force, when Coulomb force and dielectrophoretic force aredirected away from the substrate and under an electric field, the relative effects ofthe electrostatic forces are material dependent. The effect for rolling detachment ofglass particles with saturation charge from a steel substrate is smaller than that forrolling detachment of glass particles with saturation charge from a glass substrate.


Figure 21. Variation of the critical shear velocities with the particle diameter as predicted by dif-ferent adhesion models for resuspension of glass particles with saturation charge distribution froma steel substrate in the presence of capillary effects and different electric fields. Coulomb force anddielectrophoretic force are directed away from the substrate.

Figure 22. Variation of the critical shear velocities with the particle diameter as predicted by differ-ent adhesion models for resuspension of polystyrene particles from a polystyrene substrate withoutelectrostatic effects.

8.2.7. Critical Shear Velocities in the Presence of Capillary Force in the Absenceof Electrical EffectsFigure 22 shows the variations of critical shear velocities with particle diameter aspredicted by different adhesion models for the rolling detachment of polystyreneparticles from a polystyrene substrate with or without capillary effects in the con-dition without electrostatic forces. It can be seen from Fig. 22 that the critical shear


velocity decreases with the increase of the particle diameter. For both cases, withor without capillary effects, the model predictions from the JKR model are higherthan those from the DMT and Maugis–Pollock models. For the case without capil-lary effects, the predictions from the Maugis–Pollock model for u∗

c are the lowest;while for the case with capillary effects, the predictions from the Maugis–Pollockmodel for u∗

c are the lowest for large particles, but slightly higher than those fromthe DMT model for small particles. It can be also seen from Fig. 22 that capillaryeffects significantly increase the critical shear velocity. Comparing the three lineswithout the capillary effects in Fig. 22 with those in Figs 8, 9 and 10, shows that theelectrostatic forces have a major effect on the increase of the critical shear velocityfor large particle detachment.

8.3. Comparison with Experimental Data

This section compares the model predictions with the experimental data of Taheriand Bragg [58] and Hays [27]. The experiment of Taheri and Bragg was concernedwith the resuspension of glass particles from a smooth glass surface under normalroom temperature and humidity for a range of air velocities between 2 and 130 m/s.Their results agree well with the predictions by Soltani and Ahmadi [49] for parti-cle resuspension in moist air, which implies the existence of a capillary force. Theexperiments of Hays [27] studied the electrical detachment of charged 13 µm tonerparticles under an electric field from a nickel carrier bead with an average chargeof 3 × 10−14 C, where the Coulomb force and dielectrophoretic force are directedaway from the substrate. They found, for toner particles carrying different charges,corresponding critical electric detachment fields under which the particles’ elec-trostatic forces exactly balance the adhesion forces, thus allowing particles to bedetached.

Figure 23 shows the comparison of the critical shear velocities as predicted byeach of the three adhesion models with the experimental data of Taheri and Bragg[58] for resuspension of glass particles with an average Boltzmann charge distribu-tion from a glass substrate in the presence of capillary effects. It can be seen fromFig. 23 that the critical shear velocity decreases with the increase of the particle di-ameter, and the model predictions for the JKR theory are slightly higher than thosefor the DMT theory, while the predictions from the Maugis–Pollock model leadto the lowest critical shear velocities. It also can be seen that the predicted criticalshear velocities from the JKR and DMT models with capillary force agree verywell with the experimental data of Taheri and Bragg [58] under a humid air con-dition, while the predicted critical shear velocity from the Maugis–Pollock modelwith capillary force is slightly lower than the experimental data. The reason is theMaugis–Pollock model accounts for the plastic deformation at the particle–surfacecontact, but glass–glass contact may be better modeled with an elastic deformationmodel such as the JKR or DMT theory.

Figure 24 shows the comparison of the critical electric detachment fields for13 µm particles by the JKR model with the experimental data of Hays [27] for


Figure 23. Comparison of the critical shear velocities as predicted by different adhesion models withthe experimental data of Taheri and Bragg [58] for resuspension of glass particles with an averageBoltzmann charge distribution from a glass substrate in the presence of capillary effects.

Figure 24. Comparison of the critical electric detachment fields for 13 µm particles predicted by JKRwith the experimental data of Hays [27] for toner (PSL) particles on a nickel carrier bead without flowand capillary effects. Coulomb force and dielectrophoretic force are directed away from the substrate.

toner (PSL) particles on a nickel carrier bead without flow or capillary effects. Itcan be seen from Fig. 24 that the critical electric detachment fields increase withthe increase of the charges carried by the particles. The reason is that Coulomband dielectrophoretic forces are related to the amount of charge and the electricfield, while image force is related to the square of the amount of charge. When thecharge increases, the image force toward the substrate increases much faster thanthe Coulomb and dielectrophoretic forces increase; therefore, an increased electricfield is needed to increase the Coulomb and dielectrophoretic forces further to bal-


ance the image force. Although the polarization force is related to the square of theelectric field, polarization force is too small to play an important role. Figure 24shows that the predicted electric detachment fields from the JKR model are higherthan the experimental data. The reason is that the toner particles are not smooth par-ticles. The coarse surface roughness of the toner particles will decrease the particleadhesion force and therefore decrease the electric detachment fields.

The analysis performed here was focused on micron size particles. However,the model can also be applied to sub-100 nm particles. The fluid drag and hydro-dynamic moment then need to be evaluated using appropriate corrections for slipvelocity and/or with the use of the molecular dynamics method.

It should be noted that the presence of electric field may vary the liquid contactangle due to the electrowetting effect. According to [68, 69] the electrowetting ef-fects could be quite important. In this study, however, we assumed that the particlesare deposited on the substrate under dry condition. Then the liquid meniscus formsdue to vapor condensation around the particle–substrate contact area. Therefore, theparticles and the substrate are not separated by a liquid film, and the van der Waalsforces between the particles and the substrate are not affected by the presence ofthe liquid meniscus. When the particles are charged also the electrowetting effectsare neglected in the present analysis.

9. Conclusions

Particle re-suspension including the effects of capillary and electrostatic forcesbased on the maximum adhesion resistance in turbulent flows was studied. Theeffective thermodynamic work of adhesion including the effects of electrostatic andcapillary forces was used in the analysis. The JKR, DMT and Maugis–Pollock mod-els were extended to include the effect of electrostatic and capillary forces. Thecritical shear velocities for removal of particles of different sizes were evaluated.The model predictions were compared with the available experimental data. Basedon the results presented in this work, the following conclusions are drawn:

• The capillary forces significantly increase the critical shear velocity for parti-cle detachment; while the electrostatic forces only have major effects on theincreases of the critical shear velocity for large particle detachment.

• The critical shear velocity predicted by the JKR adhesion model is somewhathigher than that predicted by the DMT and the Maugis–Pollock models.

• In general, the Maugis–Pollock model leads to the lowest critical shear velocity.For the rolling detachment of polystyrene particles from a polystyrene substratein the presence of capillary effects, the predictions from the Maugis–Pollockmodel for u∗

c are the lowest for large particles, but slightly higher than thosefrom the DMT model for small particles.


• For particles in an electric field of 5000 or 10 000 kV/m with Boltzmann chargeor saturation charge distribution, the critical shear velocity decreases with theincrease of the particle diameter; for particles with fixed 20 µC/g charge distri-bution, the critical shear velocity also decreases with the increase of the particlediameter for small particles, but increases with the increase of the particle di-ameter for large particles.

• When Coulomb force and dielectrophoretic force are directed towards the sub-strate, the relative effects of the electrostatic forces decrease in the presence ofcapillary effects.

• When Coulomb force and dielectrophoretic force are directed towards the sub-strate, the critical shear velocities for large particles with fixed 20 µC/g chargesare higher than those with Boltzmann and saturation charges; while the criticalshear velocities for smaller particles with fixed 20 µC/g charges are higher thanthose with Boltzmann charge but lower than those with saturation charge; forvery small particles, the critical shear velocities for particles with Boltzmann,saturation and fixed 20 µC/g charges are almost same.

• In the presence of capillary force, when Coulomb force and dielectrophoreticforce are directed away from the substrate, the relative effects of the electrosta-tic forces increase. These relative effects are material dependent. Under a strongelectric field, Coulomb force and dielectrophoretic force significantly decreasethe critical shear velocity for large particle rolling removal.

Acknowledgements

The financial support of the Environmental Protection Agency (EPA) and the NYS-TAR Center of Excellence at Syracuse University is gratefully acknowledged.

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