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Particle Adhesion and Detachment Models

Figure 1 shows the schematic of a particle of diameter d attached to a flat surface.Here P is the external force exerted on the particle, a is the contact radius and adF is the

adhesion force. The classical Hertz contact theory provides for the elastic deformation ofbodies in contact, but neglects the adhesion force. Several models for particle adhesionto flat surfaces were developed in the past that improves the Hertz model by including theeffect of adhesion (van der Waals) force.

JKR Model

Johnson-Kandall-Roberts (1971) developed a model (The JKR Model) thatincluded the effect of adhesion force on the deformation of an elastic sphere in contact toan elastic half space. Accordingly, the contact radius is given as

˙˙

˚

˘

ÍÍ

Î

Ș¯

ˆÁË

Ê p+p+p+=

2

AAA

3

2

dW3dPW3dW

23

PK2d

a (1)

Here AW is the thermodynamic work of adhesion, and K is the composite Young'smodulus given as

adF

P

O

a

d

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1

2

22

1

21

E

1

E

1

34

K-

˙˚

˘ÍÎ

È u-+

u-= (2)

In Equation (2), E is the elastic modulus, u is the Poisson ratio, and subscript 1 and 2refer to the materials of the sphere and substrate.

In the absence of surface forces, ,0WA = and Equation (1) reduced to theclassical Hertz model. That is

K2dP

a3 = (3)

Pull-Off Force

The JKR model predicts that the force needed to remove the particle (the pull-offforce) is given as

dW43

F AJKRpo p= (4)

Contact Radius at Zero Force

The contact radius at zero external force may be obtained by setting P = 0 inEquation (1). That is,

3

12

A0 K2

dW3a ˜̃

¯

ˆÁÁË

Ê p= (5)

Contact Radius at Separation

The contact radius at the separation is obtained by setting JKRpoFP -= in Equation

(1). The corresponding contact radius is given by

3/10

3

12

A

4

a

K8

dW3a =˜̃

¯

ˆÁÁË

Ê p= (6)

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DMT Model

Derjaguin-Muller-Toporov (1975) assumed that the Hertz deformation anddeveloped another model that included the effect of adhesion force. According to theDMT model, the poll-off force is given as

dWF ADMTPo p= , ( JKR

PoDMTPo F

34

F = ) (7)

Contact Radius at Zero Force

The contact radius at zero external force is given as

3

12

A0 K2

dWa ˜̃

¯

ˆÁÁË

Ê p= (Hertz contact radius under adhesion force) (8)

Contact Radius at Separation

The DMT model predicts that the contact radius at the separation is zero. That is

0a = (at separation) (9)

Maugis-Pollock

While the JKR and the DMT models assume elastic deformation, there are experimentaldata that suggests, in many cases, plastic deformation occurs. Maugis-Pollock developedA model that included the plastic deformation effects. Accordingly, the relationshipbetween the contact radius and external force is given as

HadWP 2A p=p+ (10)

where H is hardness and

Y3H = , (11)

with Y being the yield strength.

Note that variations of contact radius with particle diameter at equilibrium, that isin the absence of external force, for elastic and plastic deformation are different. That is

3

2

0 d~a (elastic), 2

1

0 d~a (plastic) (12)

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Thermodynamic Work of Adhesion

The thermodynamic work of adhesion (surface energy per unit area) is given as

20

A z12

AW

p= (13)

where A is the Hamaker constant and oz is the minimum separation distance.

Non-Dimensional Forms

Nondimensional form of the relationship between contact radius and the externalforce and the corresponding moment are described in the section.

JKR Model

Equation (1) in nondimensional form may be restated as

**3* P21P1a -+-= (14)

where the nondimensional external force and contact radius are defined as

dW23

PP

A

*

p-= ,

3

12

A

*

K4

dW3

aa

˜̃¯

ˆÁÁË

Ê p

= (15)

Variation of the nondimensional contact radius with the nondimensional force is shownin Figure 2. Note that for 0P* = , Equation (14) and Figure 2 shows that 26.1a*

0 = .

The corresponding resistance moment about point O in Figure 1 as a function ofnondimensional force is given as

3/1*****JKR* )P21P1(PaPM -+-== (16)

Figure 3 shows the variation of the resistance moment as predicted by the JKR model.The corresponding maximum resistance moment then is given by

42.0M JKR*max = (17)

and

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5.0dW

23

FFP

A

JKRJKR*

po*max =

p== (18)

The resistance moment at *P is 397.0M JKR* = . Also

63.0aP *0

*max = (19)

DMT

For DMT Model, the approximate expression for the contact radius is given as

( )dWPK2

da A

3 p+ª (20)

or

3

2P)

K4/dW3

a(a *3

2A

3* +-=p

= (21)

Variation of the nondimensional contact radius with the nondimensional force aspredicted by the DMT model is shown in Figure 2 and is compared with the JKR model.Note that for 0P* = , Equation (21) and Figure 2 shows that 874.0a*

0 = .

The corresponding resistance moment as a function of nondimensional force aspredicted by the DMT model is given as

3/1**DMT* )P3/2(PM -= (22)

The variation of the resistance moment as predicted by the DMT model is also shown inFigure 3. The corresponding maximum resistance moment is

28.0M DMT*max = (23)

Note also that the maximum force (the poll-off force) is given by

32

dW2

3F

FP

A

DMTDMT*

po*max =

p== (24)

and

58.0aP *0

*max = (25)

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Comparing Equations (17) and (23) shows that the JKR model predicts a largerresistance moment. That is

DMT*max

JKR*max M5.142.0M == , ( 28.0MDMT

max = ) (26)

The resistance moment predicted by the JKR and the DMT models in dimensional formare given as

3

1

3

5

3

4

max 63.2

K

dWM AJKR = ,

3

1

3

5

3

4

max 83.1

K

dWM ADMT = (27)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

a*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 P*

JKR

DMT

Figure 2. Variations of contact radius with the exerted force.

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JKR

DMT

0.0

0.1

0.2

0.3

0.4

0.5

M*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 P*

Figure 3. Variations of resistance moment with the exerted force.