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Relating friction and adhesion for single smooth contact by means of 

models and experiments

M.A. Yaqoob a,b,n, M.B. De Rooij b, D.J. Schipper b

a Materials innovation institute, Mekelweg 2, 2628 CD, Delft, The Netherlandsb Laboratory of surface technology and tribology, University of Twente, Drienerlolaan 5, 7522 NB, Enschede, The Netherlands

a r t i c l e i n f o

 Article history:

Received 9 July 2012Received in revised form

3 January 2013

Accepted 8 January 2013Available online 23 January 2013

Keywords:

Contact mechanics

Maugis–Dugdale model

Adhesion force measurements

Friction force measurements

a b s t r a c t

The relation between adhesion and the friction force measurements for a single smooth contact has

been studied by means of sliding and pull-off experiments in ambient as well as under high vacuum(HV) conditions. The experimental results have been analysed in several ways using contact models.

First, it is shown how the appropriate adhesion regime can be determined based on the analysis of 

Maugis–Dugdale (M–D). Further, a modified M–D model incorporating meniscus forces are discussed.

This model is able to calculate the contact radius for cases where meniscus forces due to the presence

of water are dominating. The approach will be illustrated by means of adhesion and friction force

measurements between a silicon ball and a glass flat surface (Si–glass interface) performed on a novel

designed vacuum based adhesion and friction tester. Under HV conditions, meniscus forces are not

dominating and only van der Waals interactions are present. Combining the models with the measured

adhesion and friction force data, the work of adhesion and the shear stress present between the Si–glass

interface can be determined. According to the theory, the extent of the adhesive zone increases with an

increase in the relative humidity (RH). When comparing HV conditions with ambient conditions, it has

been found that the adhesion as well as friction force is significantly lower under high vacuum

conditions for the Si–glass interface. When comparing modelling results with experiments, it can be

concluded that the trends from theoretical predictions are in good agreement with the measurements,

both for the HV and ambient regime.&   2013 Elsevier Ltd. All rights reserved.

1. Introduction

In order to understand adhesive and frictional behaviour

of rough surfaces, fundamental understanding at single asperity

level is important. A ball on a flat configuration can be used to

perform adhesion and friction force measurements for a single

asperity contact by pull-off and sliding experiments, respectively.

An important aspect for the interpretation of these experiments is

to identify the adhesive regime in which the single asperity

contact is operating.Depending on the magnitude of the adhesive forces compared

to the applied load and to the elastic properties of the materials in

contact, the contact situation can be described with several

contact models, like JKR, DMT, Maugis Dugdale and Hertz. The

Hertzian model does not consider adhesion in or outside the

contact area   [1,2] and assumes that there are no surface forces

acting in the contact. Typically at high normal loads the behaviour

according to Hertz has been shown to fit experiments because the

adhesion forces are low compared to the applied load under these

conditions. However, the Hertz model fails to predict the area of 

contact at very low or zero normal load due to the significance of 

surface forces. Adhesive interactions between the surfaces were

brought into consideration when Johnson, Kendall and Roberts

(JKR) proposed their model   [3]. The Johnson–Kendall–Roberts

(JKR) model describes the effect of strong short-range interactions

between materials with relatively low elastic modulus and largeradius of curvature [1,4]. This model shows that there is a finite

contact area between the surfaces under zero normal load and

also predicts that there is an external force required to separate

two bodies of given surface energies and geometry [3,5]. The JKR 

model assumes that the adhesive interaction is inside the contact

zone. Another model including adhesive interactions was pro-

posed by Derjaguin, Muller and Toporov (DMT). In this model, it

was assumed that adhesive interactions are present outside the

contact area.

A dimensionless parameter   mT , called the Tabor parameter

representing the ratio between the gap outside the contact zone

and the equilibrium distance between atoms has been presented

Contents lists available at ScienceDirect

journal homepage:   www.elsevier.com/locate/triboint

Tribology International

0301-679X/$- see front matter &  2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.triboint.2013.01.005

n Corresponding author at: Materials innovation institute,, Mekelweg 2, 2628

CD, Delft, The Netherlands. Tel.:  þ 31 53 489 4325; fax:  þ 31 53 486 3471.

E-mail addresses:   adeelyaqoob@yahoo.com,

m.a.yaqoob@ctw.utwente.nl (M.A. Yaqoob),

m.b.derooij@utwente.nl (M.B. De Rooij), d.j.schipper@utwente.nl (D.J. Schipper).

Tribology International 65 (2013) 228–234

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in [6]. In mechanical terms, this parameter represents the ratio of 

the magnitude of the elastic deformation to the range of adhesive

forces. This parameter established the range of applicability of the

two models and suggested that the JKR and DMT models are at

the limiting cases of  mT . Maugis   [7]   provided a solution to the

contact problem with interactions inside and outside the contact

zone using the Dugdale approximation (constant adhesive stress

outside the contact zone) and is known as Maugis–Dugdale

(M–D) model. The M–D model showed the transition of a contactproblem from DMT to JKR as two opposite ends of a continuous

spectrum based on a parameter  l  (the Maugis parameter), which

is equivalent to the Tabor parameter (l¼1.16mT ). An adhesion

map has been reported by Johnson and Greenwood [4,8] based on

the Maugis model. If  l45, the JKR analysis becomes appropriate

and when lo0.1, the DMT model is applicable. In the intermedi-

ate range 0.14l45 the Maugis model has to be applied.

Further, a modified M–D model incorporating the meniscus

forces has been developed in [9]. This modification of the model is

required since the JKR, DMT and M–D models are assuming solid–

solid adhesive contacts where van der Waals forces are dominant.

However, if two hydrophilic surfaces are brought in contact with

each other under humid environment the meniscus forces will be

dominating the adhesive interaction  [2,10]. The M–D model has

been modified using the Kelvin and Young–Laplace equation and

has been used to calculate the contact areas for dry as well as

humid contact conditions. However, the validity of the Kelvin

equation is questionable at low RH   [11,12]. Therefore, the

modified M–D model cannot be used in relatively dry situations,

(e.g. in HV conditions) as it has been discussed in  [9,13].

To formulate the relation between the normal load dependent

friction force and the contact area for JKR and DMT cases is

straight forward since they provide simple expressions to calcu-

late the contact area as a function of applied normal load.

However, the intermediate case of M–D model is more complex.

Efforts have been made to generate a simplified equation to fit the

contact area and the normal load   [14,15]. In   [14]   a simplified

equation to calculate the normal load dependent contact area has

been reported. The equation also uses the JKR and DMT models as

the limiting cases of the equation and the intermediate values can

be used to calculate the contact area for M–D model.

Here, an approach will be presented to completely analyse

a single smooth adhesive contact operating in ambient and HV

conditions by combining pull-off experiments, friction experi-

ments and a theoretical analysis. Using this approach, the adhe-

sive contact can be analysed to a fuller extent. For example, the

relevant adhesive regime cannot only be predicted using the

Tabor parameter, but also be validated using pull-off experiments

in combination with sliding experiments conducted at different

loads. If the relevant adhesive regime is determined and the

relevant contact model selected, more detailed aspects of the

adhesive contact, like the extent of the adhesive zone, can be

determined. Further, relations can be found between the adhesivebehaviour and the frictional behaviour.

2. Theoretical background

The adhesion force or pull-off force can be calculated using

 JKR, DMT or M–D model using the following general relation for

the pull off force for a sphere-on-flat contact [9,13]:

F a ¼ npW 12R   ð1Þ

in this equation, n  has a value between 1.5 and 2,  W 12 is the work

of adhesion between two surfaces and   R   is the radius of the

spherical surface. If the JKR theory is applicable, then n ¼1.5, if the

DMT theory is applicable, then  n ¼2. In case that the M–D theory

is applicable an intermediate value of  n  should be used.

For a single smooth contact (a smooth sphere in contact with

a smooth flat surface as shown in   Fig. 1) the contact area can

be calculated using the JKR, DMT or M–D models depending on

which one describes the actual adhesive conditions. Using Eq.  (2)

one can calculate the contact area using JKR model and using

Eq. (3)   one can calculate the contact area using DMT model.

 Að JKR Þ ¼ p  3R

4K 

2=3

F N  þ3pW 12Rþ

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6pW 12RF N  þ ð3pW 12RÞ2

q  2=3

ð2Þ

 AðDMT Þ ¼ p  3R

4K 

2=3

F N  þ2pW 12Rð Þ2=3

ð3Þ

The M–D equations are more difficult to utilise, since Maugis’

formulation lacks a single expression relating only  a  and  F N . Eqs.

(4)–(7) are needed to solve the M–D model [7]:

la2

2  m22

s1mþ ffiffiffiffiffiffiffiffiffiffiffiffiffim21

p n oþ

 4l2a

3

 ffiffiffiffiffiffiffiffiffiffiffiffiffim21

p   s1mmþ1

n o¼ 1

ð4Þ

P  ¼ a3la2 ffiffiffiffiffiffiffiffiffiffiffiffiffim21

p   þm2s1m

n o  ð5Þ

d ¼ a2

4

3la

 ffiffiffiffiffiffiffiffiffiffiffiffiffim21

p n o  ð6Þ

where

a a  4K 

3pW 12R2

1=3

;   c   c   4K 

3pW 12R2

1=3

;   m¼ c =a;

P    F N pW 12R

;   d d  16K 2

9p2W 122R

!1=3

l so9R

2pW 12K 2

1=3

ð7Þ

in these equations,  a  and  c  are the radii of contact and adhesivezones, respectively, as shown in Fig. 1.  K   is the reduced modulus

of elasticity of two materials and   R   is the radius of the sphere.

F N  is the total load and  W 12  and  d  are the work of adhesion and

deformation, respectively. Also,  l  and  so  are the Maugis (elasti-

city) parameter and constant adhesive stress outside the contact,

respectively. If  so¼1.03W 12/ z 0   is assumed then   l¼1.16mT . One

needs to solve two equations simultaneously to solve the M–D

model, therefore, by knowing the values of  l  and  F N   and solving

Eqs. (4) and (5) one can find the solutions for  a  and  c .

In the analysis, Eq.   (7)   can be used to calculate the Maugis’

parameter for a solid–solid (dry) contact, but Eq.  (7)  needs to be

modified if the adhesive interaction is due to meniscus forces.

Fig. 1.  A sphere in contact with a flat surface under a certain applied normal load

F N   in humid environment. The solid–solid contact radius   a   and the meniscus

radius c  are also shown.

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A modification in this respect is given in   [9], using the Kelvin

equation:

r m ¼  gLV V m

R g T log   p ps

  ð8Þ

where,   V m   is the molar volume,   R g   is the gas constant,   T   is the

absolute temperature, gLV  is the surface tension of liquid (water)

and   p/ ps   is the relative humidity (RH). As explained above, the

M–D model used the Dugdale approximation of constant adhesive

stress outside the contact. Therefore, the adhesive stress due to

capillary formation inside the meniscus is also considered con-

stant. This adhesive stress is the capillary pressure or Laplace

pressure which is given by the Young–Laplace equation  [9]:

so ¼ pcap ¼ gLV r m

ð9Þ

where, pcap  is the capillary pressure and  r m is the mean radius of 

the meniscus. Similarly, since two similar surfaces are considered

the work of adhesion  W 12 can be approximated by:

W 12 ¼ 2gLV    ð10Þ

Putting Eqs. (8)–(10) in  (7)  we get;

lcap ¼  9Rg2

LV 

4pK 2r 3m

!1=3

ð11Þ

lcap ¼ 9RR3

 g T 3

½logð p= psÞ3

4pK 2V 3mgLV 

!1=3

ð12Þ

The results of the calculated dimensionless  a  and dimension-

less c  as a function of the dimensionless load for different RH are

shown in Fig. 2  (a) and (b), respectively. These values have been

calculated for a 5 mm silicon ball in contact with a flat glass

surface. The corresponding   l   values for different RH are also

shown. It can be seen from Fig. 2(a) that as the RH increases the

value of the solid–solid contact radius  a   is decreasing. However,

the value of the adhesive radius increases with an increase in RH

as shown in   Fig. 2(b). This trend in   a   and   c   is because of the

increase in the amount of water with an increase in the RH

around the contact. As the amount of water increases it will tend

to condensate around the contact forming a meniscus and the

meniscus will grow with the increase in the RH. It can also be

seen that the value of solid–solid contact radius  a and the value of 

adhesive radius c  are approaching each other when the RH is 5%.

This indeed, indicates that there under these conditions there is

no adhesive zone around the contact due to meniscus and all the

adhesion force is contributed by the solid–solid contact.

The value of  l also decreases with the increase in the RH. Using

the adhesion map reported in   [4,8] and the values of   l   the

appropriate contact mechanics model can be selected. In   Fig. 3

the adhesion map has been shown with the marked span of thel   values for 1–90% RH for a 5 mm silicon ball and a flat glass

surface. This shows that for a dry contact, such as in HV

conditions, the JKR model can be used to calculate the contact

area. However, as the RH increases, the JKR model cannot be used

and the appropriate choice to calculate the adhesion force and

contact radius will be M–D model. It is important to mention here

that the  l   values for different material combinations change the

operational regime depicted in the adhesion map. Similarly, the  l

value is also dependent on the radius of the ball in contact with a

flat and will also affect the choice of the contact model. Therefore,

basic analysis for predicting the contact model remains the same

and the material parameters and the geometry can influence the

choice of the appropriate contact model. It can also be seen from

Fig. 3 that the JKR, DMT and M–D models are applicable for low

values of the applied normal load and for higher loads the Hertz

model is applicable.

3. Experiments

Adhesion and static friction experiments have been performed

in ambient as well as in HV conditions to study the relationship

between adhesion and friction force. Furthermore, the appropri-

ate adhesion regime and the contact model determined by

M–D analysis can be verified using these experiments. Finally,

the work of adhesion and the shear strength at the interface can

be determined.

The experiments have been performed on a novel designed

vacuum based adhesion and friction tester (VAFT). The detailed

design and the working of the VAFT has been reported elsewhere

[16,17]. A brief description of the experimental setup, materials

Fig. 2.  (a) Dimensionless radius of the solid–solid contact and (b) the dimension-

less radius of the adhesive zone as a function of dimensionless load for different

values of RH. The corresponding  l  values for different RH are also shown.

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preparation and the experimental methods are discussed in the

following sections.

 3.1. Experimental setup

The experiments have been performed with a 5 mm silicon

ball against a flat glass surface in a ball on flat configuration as

shown in Fig. 1. The setup is capable of working in high vacuum

(HV) conditions (20   1C and 10

6 mbar) as well as in ambientconditions (20   1C, 1 bar and 50% Relative Humidity (RH)). The

VAFT comprises of three positioning stages and two capacitive

sensors along with a force measuring mechanism as shown in

Fig. 4(c). The setup has a ball on flat configuration and represents

a single smooth contact. The ball is mounted on the indenter and

the indenter along with the force measuring mechanism (FMM) is

mounted on one of the positioning stages which can move in   Z 

direction as shown in  Fig. 4(c). This positioning stage is used to

make contact with the flat surface and to apply a normal load. The

flat surface is placed on an XY stage. The  X  positioning stage is

used to apply a tangential load for friction measurements. The

accuracy of both  X  and  Z  stage is 20 nm with a stroke of 20 mm.The Y  stage is used to perform multiple parallel measurements on

the flat surface and has a stroke of 20 mm as well.

A cantilever along with the indenter has been attached to a

force measuring mechanism (FMM) that can measure the normal

load and the tangential load with the help of capacitive sensors as

shown in Fig. 4(c). The assembly has been placed inside a vacuum

chamber as shown in   Fig. 4(b) and the vacuum chamber has

been placed on a table with air mounts as shown in  Fig. 4(a). The

vacuum pumps have been placed on two separate platforms

which are different than that of the vacuum chamber to reduce

the transfer of vibrations due to vacuum pumps. Further, anti-

vibration bellows have also been attached between the turbo

pump and the vacuum chamber to reduce the transfer of 

vibrations.

 3.2. Sample preparation

Since this study focuses on the single smooth contacts, it is

required to reduce the effect of mechanical interlocking due to

surface roughness as much as possible. As discussed before a

smooth silicon ball of 5 mm diameter with the RMS surface

roughness of 2–3 nm and a flat float glass surface with the RMS

roughness of 0.7 nm has been used as a sample for performing the

adhesion and friction experiments. A flat glass surface has been

cleaned in an ultrasonic bath of acetone for 15 min and then dried

and placed on an XY stage. The silicon ball, after cleaning in an

ultrasonic bath of acetone for 15 min, has been glued to an

indenter that has been mounted on a cantilever. The ball has

been checked for a clean and smooth spot under KeyenceFig. 3.   An adhesion map from   [4,8] showing the span of   l   values for a 5 mm

silicon ball against a flat glass surface from 1–90% of RH.

Fig. 4.  (a) Complete design showing the vacuum chamber and the vacuum pumps. (b) Adhesion and friction tester mounted inside the vacuum chamber and (c) Internal

view of the VAFT showing all the components [16,17].

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Confocal Microscope before applying the glue. Once a good spot is

found the ball has been glued very carefully.

 3.3. Experimental procedure

The pull-off measurements have been performed in the ambi-

ent conditions. From these experiments, the adhesion force

between the silicon–glass interface is obtained. The average value

of five pull-off measurements has been used to calculate the workof adhesion. The static friction measurements have been per-

formed when the ball is in contact with the flat surface. The

normal force is increased by a small force step, that is a fractional

part of the final desired normal load, which is set to 25 mN. After

every force step a friction experiment has been performed and the

static friction force has been measured. Each friction experiment

consists of one forward and backward scan cycle. The scan speed

has been kept constant to 100 nm/s and the scan length is 10 mm.

These measurements have been performed until the final desired

load (25 mN) has been applied and the static friction force for all

the force steps have been measured.

Similar experiments have been performed when the setup

is operating in HV conditions. This is required because at HV the

RH inside the vacuum chamber reduces to a desired low level of 1–2%, which has been measured with a humidity sensor. The

change in the RH influences the adhesion and well as friction

force measurements.

4. Results and discussion

A typical force–displacement curve in normal direction in

ambient as well as in vacuum is shown in  Fig. 5. The measure-

ment shows the normal displacement of the Z–axis stage and the

measured applied normal load. In both measurements the applied

normal load is 10 mN and the resulting pull-off force is 1.05 mN

in ambient and 0.39 mN in high vacuum conditions as shown in

the insets of  Fig. 5. Therefore, it is evident from these measure-

ments that the pull-off force changes significantly if the environ-

mental conditions are changed. As mentioned before, a sequence

of five such measurements were performed each in ambient and

vacuum to get the average value of pull-off force. The average

value of the pull-off force has been used to calculate the work of 

adhesion and the contact radius. First, Eq. (1)  is used to calculate

the work of adhesion  W 12 of the surfaces in contact in ambient as

well as in HV conditions.

Once the W 12 is known the contact areas according to the JKR 

and DMT theories have been calculated using Eqs.   (2) and (3),

respectively. When comparing HV with ambient, the calculated

contact areas based on the measured adhesion force as a function

of applied normal load is shown in Fig. 6. It can be seen that the

contact area is decreased when the system operates in HV

conditions as compared with the area in ambient conditions. Thiscan also be seen in Fig. 2(b) where the radius of the adhesive zone

c  is decreasing as the RH is decreased. As discussed before, the JKR 

and DMT models are the limiting cases of the M–D model, and the

actual contact area based on the M–D model lies between the area

of solid (JKR) and dashed (DMT) lines in  Fig. 6.

After calculating the contact area using both limiting adhesive

models, the interfacial shear strength of the surfaces in contact

has been calculated. The shear stress has been calculated using

the measured static friction force  F  f  for a single asperity contact

using the following relation:

F  f  ¼ t A   ð13Þ

where,   A   is the contact area calculated for the JKR and DMT

models obtained from pull-off measurements and   t   is the

interfacial shear strength to be determined. The average value

of the shear stress has been calculated for the JKR and DMT

models for ambient as well as in HV condition and will be called

t JKR and  tDMT , respectively.

Carpick, Ogletree and Salmeron (COS)   [14]   provided anapproximate general equation for easily describing the contact

area. The general equation can be used for curve fitting and

provides a rapid method of determining the value of the transi-

tion parameter describing the range of surface forces. They

showed that the Maugis’ formulation could be approximated

using the following formula to determine the contact radius:

a ¼ a0ðaÞ

aþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þðF N =F aðaÞÞp 

  1 þa

!2=3

_a0ðaÞ ¼  3pW 12R

2 þ6pW 12aR2

2K 

" #1=3

ð14Þ

where, a  is the transition parameter, a0 is the contact area at zero

load and   F a   is the pull-off force. Note that   a¼1 corresponds

exactly to the JKR case, and a¼0 corresponds exactly to the DMT

Fig. 5.   A typical force–displacement curve measured with VAFT in (a) ambient

and (b) high vacuum conditions. The insets shows the snap-in and pull-off points

in both measurements.

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case. Included in Eq. (14) is the fact that  F a and  a0 depend on a  as

well. Eq. (14) is referred to as the generalised transition equation

or COS equation. For intermediate cases (0oao1), the general-

ised transition equation corresponds very closely to the Maugis’

solution for the transition regime (0.1olo5) which is indicated

in Fig. 3. The actual contact radius has been calculated using Eq.

(14)  by putting the values of measured average  F a,  W 12  and the

free fitting transition parameter  a.

A typical friction force measurement has been shown in  Fig. 7

where the force–displacement curve has been shown both mea-

sured in ambient as well as in HV. The applied normal load for the

measurements shown in Fig. 7 is 15 mN and the resulting friction

force is 7.6 mN and 1.55 mN in ambient and HV, respectively.

Similar friction force measurements were performed by stepwise

increasing the normal load and the measurement results along

with the data fit using Eq.  (14) are shown in  Fig. 8.

In Fig. 8 the measured static friction force is plotted against the

applied normal load when operating in ambient as well as in HV

conditions. The data fit on these measured points is plotted using

a procedure explained above. Once the value of contact radius is

known for all the values of applied normal load, the value of the

average interfacial shear strength calculated for the JKR and DMT

models has been used as a boundary for the shear stress in

Eq. (13). The calculated values of work of adhesion and average

shear stress for ambient as well as for HV conditions are shown in

Table 1. The values for the JKR and DMT models are shown in

Table 1 along with the values used to generate the data fit on the

measurement data. By changing the values of  a  (0oao1) and t(t JKRototDMT ) the friction force can be calculated as a function

of applied normal load for JKR, DMT and M–D models. It can be

seen from   Table 1   that the values of shear stress satisfy the

condition of   t JKRototDMT . It is clear from   Fig. 8   that for HV

conditions the measured data followed the JKR trend, as the value

of   a   used is approximately 1. However, for measurements

performed in ambient conditions the data fit follows the M–D

model as the value of  a  is approximately 0.35. A diagram showing

each step in plotting these data fits on the measured data has

been shown in Fig. 9.

The value of work of adhesion calculated to plot the data fit on

the measured data shown in Table 1 for ambient conditions is equal

to the surface tension of water (73 mJ/m2). Therefore, it is clear that

the meniscus forces are dominant in ambient conditions and the

pull-off force required to break the contact is contributed by the

surface tension of water. However, in HV the low values of work of 

adhesion have been calculated. The possible reason for these low

values is that even at HV conditions there are molecular thick

adsorbed layers present and it is known that the surface energy

reduces if there are adsorbed layers present on the surfaces [2]. The

presence of the interfacial layers of hydrocarbons or contaminantscan reduce the work of adhesion. Since the friction measurements

shown are the first loading/unloading cycle, therefore, the value of 

work of adhesion of the interface due to the presence of interfacial

layers is important to calculate. This is required to mimic the real life

situations in many contact mechanics applications where surfaces

are not always perfectly clean.

It can also be seen that the predicted model explained in

Section 2 corresponds to the trend of the measured static friction

force. Solving the modified   lcap   value for ambient conditions

(RHE45%) and solving the  l  value for HV conditions (RHo5%)

in the M–D model the appropriate contact mechanics model can

be selected using Fig. 3. The predicted model can be verified using

the adhesion and friction measurements and plotting the data fit

for the predicted contact mechanics model on the measured data

Fig. 6.   The calculated contact area based on adhesion force measurements data for

the Si–glass interface. The JKR and DMT areas for both ambient and HV conditions

are shown.

Fig. 7.   A typical force–displacement curve showing the horizontal displacement

and the friction force for Si–glass interface in (a) ambient and (b) HV conditions.

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using the procedure mentioned above. It can also be seen that

the measured static friction force in ambient is much higher than

the static friction force measured in HV. As shown in Fig. 5  the

adhesion force in ambient conditions is also much higher than

the adhesion force in HV conditions. Therefore, it can be stated

that due to the higher value of adhesion force in ambient

conditions a higher value of friction force has been experienced.

5. Conclusion

A model has been discussed to select the contact mechanics

model appropriate to interpret the adhesion and friction force

measurements performed in different environmental conditions

based on the M–D and modified M–D model. Combining adhesion

and friction force measurements the work of adhesion and the

shear stress at the interface can be calculated. The model shows

that the solid–solid contact radius  a   decreases with the increasein the RH whereas, the adhesive radius or total contact radius  c 

increases with the increase in RH. This can be verified from the

measurement results that the contact radius increases when the

system is operating in ambient as compared to when operating in

HV condition. It has been shown that the predicted contact

mechanics model fits very well to the measured values of the

normal load dependent static friction force. Adhesion and static

friction force are much higher when measured in ambient than in

HV conditions. This is because the contact area and the shear

stress in ambient conditions is larger than the contact area and

shear stress in HV conditions.

 Acknowledgements

This research was carried out under project number

MC7.06284 in the framework of the Research Programme of the

Materials innovation institute M2i (www.m2i.nl). Financial sup-

port for carrying out this research from the M2i is gratefully

acknowledged.

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Fig. 8.   Measured static friction force versus normal load data, with a 5 mm

diameter silicon ball and a float glass flat surface. Fit of data to the JKR and M–Dmodel using Eqs. (13) and (14). The two free fitting parameters are the interfacial

shear strength t  and the transition parameter  a.

 Table 1

Calculated work of adhesion and the shear stress from the measured adhesion and

friction force.

Model Work of adhesion W 12 (mJ/m2) Shear stress t  (MPa)

Ambient High vacuum Ambient High vacuum

 JKR 88.5 20 23.6 8.2

DMT 66.3 15 32 9.6

Fit 73.7 19.3 27 8.5

Fig. 9.  A step by step process to calculate the data fit on the measured friction

force data. During the process the work of adhesion and shear stress in the

interface has also been calculated.

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