Theory of adhesion: role of surface roughness.

TLDR: In this article, surface roughness influences the adhesion between elastic solids and a Tabor number which depends on the length scale or magnification is introduced to give information about the nature of adhesion at different length scales.

Abstract: We discuss how surface roughness influences the adhesion between elastic solids. We introduce a Tabor number which depends on the length scale or magnification, and which gives information about the nature of the adhesion at different length scales. We consider two limiting cases relevant for (a) elastically hard solids with weak (or long ranged) adhesive interaction (DMT-limit) and (b) elastically soft solids with strong (or short ranged) adhesive interaction (JKR-limit). For the former cases we study the nature of the adhesion using different adhesive force laws (F ∼ u−n, n = 1.5–4, where u is the wall-wall separation). In general, adhesion may switch from DMT-like at short length scales to JKR-like at large (macroscopic) length scale. We compare the theory predictions to results of exact numerical simulations and find good agreement between theory and simulation results.

Figures

FIG. 5. The Tabor length parameter dT as a function of the wavevector (log10− log10 scale). For the surface with the power spectrum given in Fig. 3 and with the elastic modulus E = 1012 Pa (ν = 0.5) and work of adhesion γ = 0.3 J/m2.

FIG. 4. Surface topography of one realization of a surface with the surface roughness power spectrum shown in Fig. 3. The difference between the lowest and highest point is about 5 nm, i.e., about 10 times higher than the rms roughness 0.6 nm (see Appendix A in Ref. 6).

FIG. 3. The surface roughness power spectrum C(q) as a function of the wavevector q (log10 − log10 scale), used in the present calculations. The power spectrum corresponds to a surface with the rms roughness amplitude 0.6 nm, the rms slope 0.0035 and the Hurst exponent H = 0.8.

FIG. 21. Normalized (projected) contact area A/A0 = (Ar + Aa) as a function of the applied pressure pN. For the same parameters as in Fig. 19.

FIG. 22. Applied pressure pN as a function of the average interfacial separation ū. For the same parameters as in Fig. 19.

FIG. 19. Normalized (projected) area of repulsive contact Ar/A0 as a function of the applied pressure pN, and for γ = 0.2 J/m2. For an elastic solid with Er = 1.33 × 1012 GPa and for the surface roughness of Fig. 18 (with q0 = 2.5 105 m−1, qr = 4q0, H = 0.8, resulting in C0 = 5.24 × 10−32 m4), at different truncation frequencies q1 = 64, 128, 256, 512 q0.

Full text

Theory of adhesion: Role of surface roughness
B. N. J. Persson and M. Scaraggi
Citation: The Journal of Chemical Physics 141, 124701 (2014); doi: 10.1063/1.4895789
View online: http://dx.doi.org/10.1063/1.4895789
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THE JOURNAL OF CHEMICAL PHYSICS 141, 124701 (2014)
Theory of adhesion: Role of surface roughness
B. N. J. Persson
1
and M. Scaraggi
1,2
1
PGI, FZ-Jülich, 52425 Jülich, Germany
2
DII, Universitá del Salento, 73100 Monteroni-Lecce, Italy
(Received 13 May 2014; accepted 4 September 2014; published online 23 September 2014)
We discuss how surface roughness influences the adhesion between elastic solids. We introduce a
Tabor number which depends on the length scale or magnification, and which gives information
about the nature of the adhesion at different length scales. We consider two limiting cases rele-
vant for (a) elastically hard solids with weak (or long ranged) adhesive interaction (DMT-limit) and
(b) elastically soft solids with strong (or short ranged) adhesive interaction (JKR-limit). For the
former cases we study the nature of the adhesion using different adhesive force laws (F ∼ u
−n
,
n = 1.5–4, where u is the wall-wall separation). In general, adhesion may switch from DMT-like at
short length scales to JKR-like at l arge (macroscopic) length scale. We compare the theory predic-
tions to results of exact numerical simulations and find good agreement between theory and simula-
tion results. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4895789]
I. INTRODUCTION
Surface roughness has a huge influence on the adhesion
and friction between macroscopic solid objects.
1–6
Most in-
teractions are short ranged and become unimportant when the
separation between solid surfaces exceeds a few atomic dis-
tances, i.e., at separations of order nm. This is trivially true
for chemical bonds (covalent or metallic bonds) but also holds
for the more long-ranged van der Waals interaction. One im-
portant exception is charged bodies. For uncharged solids, if
the surface roughness amplitude is much larger than the de-
cay length of the wall-wall interaction potential and if the
solids are elastically stiff enough, no macroscopic adhesion
will prevail, as is the case in most practical cases. Only for
very smooth surfaces, or elastically very soft solids (which
can deform and make almost perfect contact at the contacting
interface without storing up a large elastic energy) adhesion
can be observed for macroscopic solids.
7
In this paper we will discuss how surface roughness influ-
ences adhesion between macroscopic solids. We consider two
limiting cases, which are valid for elastically hard and weakly
interacting solids (Deryagin, Muller, and Toporov, DMT-
limit)
8
and for elastically soft or strongly interacting solids
(Johnson, Kendall, and Roberts, JKR-limit).
9
This problem
has been studied before but usually using the Greenwood-
Williamson (GW)
10, 11
type of asperity models (see, e.g.,
Refs. 7 and 12), whereas our treatment is based on the Pers-
son contact mechanics theory. The latter theory is (approx-
imately) valid even close to complete contact (which often
prevails when adhesion is important).
13, 14
For surfaces with roughness on many length scales (as
is always the case in reality) the GW-type of asperity con-
tact models, even when one includes the long-range elastic
lateral coupling between the asperities, are not valid. First,
one cannot approximate all the asperities as spherical (or el-
lipsoidal) cups with the same radius of curvature as done in
the GW theory. Also, in many adhesion problems the con-
tact is almost complete, and for this limit the GW theory fails
qualitatively.
Asperity models can only be used as long as the contact
area is small compared to the nominal contact area, and even
in this limit these models have severe problems for surfaces
with roughness on many length scales.
15–17
Recently, several numerical simulation studies of ad-
hesion between randomly rough surfaces have been pub-
lished. Pastewka and Robbins
18
studied the adhesion between
rough surfaces in the DMT-limit, and presented a criterion for
macroscopic adhesion. They emphasized the role of the range
of the adhesive interaction, which we also find is important in
the DMT-limit and when the surface roughness amplitude is
small (see below). Medina and Dini
19
studied the adhesion
between an elastic sphere with smooth surface and a rigid
randomly rough substrate surface. They observed strong con-
tact hysteresis in the JKR-limit (relative smooth surfaces) and
very small contact hysteresis in the DMT-limit which prevails
for small roughness. Analytical theories of contact mechan-
ics have been compared to Molecular Dynamics calculations
for an elastic ball in contact with two-dimensional (2D) ran-
domly rough surfaces in Ref. 20. Numerically exact studies
for adhesion between 1D surface roughness was presented in
Ref. 21. Experimental adhesion data for rough surfaces
have been compared to analytical theory predictions in
Refs. 22–24.
In this paper we present a more general study for the ad-
hesive contact between solids with nominal flat but randomly
rough surfaces, f or a large range of system parameters. Ex-
act simulation results are compared to theory predictions in
both the JKR and DMT limits, but with the main focus on the
DMT limit. We also introduce a scale-dependent Tabor num-
ber which gives information about the condition for the valid-
ity of the DMT and JKR limits. For macroscopic solids one
expect in many cases that the JKR limit is accurate for the
contact mechanics at long length scales, but the DMT-limit
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124701-2 B. N. J. Persson and M. Scaraggi J. Chem. Phys. 141, 124701 (2014)
may still prevail at short length scales (for small asperity-
contact regions).
Many practical or natural adhesive systems involve ef-
fects which usually are not considered in adhesion models,
and which we will not address in this paper. In particu-
lar, biological applications typically involve complex struc-
tured surfaces (e.g., hierarchical fiber-and-plate structures)
with anisotropic elastic properties, which are elastically soft
on all relevant length scales.
25–28
Instead of directly relying
on molecular bonding over atomic dimension, many biologi-
cal systems adhere mainly via capillary bridges.
29–31
We will
also not discuss either the adhesion between charged objects,
which must be treated by special methods which takes into ac-
count the long-range nature of the Coulomb interaction.
32–35
In this paper we first briefly review (Sec. II) two limiting
models of adhesion for smooth surfaces. In Sec. III we show
how the same limiting cases can be studied analytically for
randomly rough surfaces using the Persson contact mechanics
model. Numerical results obtained using the analytical theory
are presented in Sec. IV, and compared to simulation results
in Sec. V. Section VI contains a discussion and Sec. VII con-
tains the summary and conclusion.
II. ADHESION OF BALL ON FLAT (REVIEW)
Analytical studies of adhesion have been presented for
smooth surfaces for bodies of simple geometrical shape, the
most important case being the contact between spherical bod-
ies. For a sphere in contact with a flat surface two limit-
ing cases are of particular importance, usually referred to as
the DMT theory
8
and the JKR theory,
9
see Fig. 1. Analyti-
cal results for intermediate-range adhesion were presented by
Maugis
36, 37
and the ball-flat adhesion problem has also been
studied in detail using numerical methods.
38, 39
A particularly
detailed simulation study was recently published by Müser
who also included negative work of adhesion (repulsive wall-
wall interaction).
40
(a)
elastic ball
(b)
d
T
DMT-limit JKR-limit
F
N
F
N
FIG. 1. (a) In the DMT theory the elastic deformation field is calculated with
the adhesion included only as an additional load F
ad
acting on the sphere.
Thus the contact area is determined by Hertz theory with the external load
F
N
+ F
ad
. The adhesive load F
ad
is obtained by integrating the adhesive
stress over the ball non-contact area. (b) In the JKR theory the adhesion force
is assumed to have infinitesimal spatial extent, and is included only in the
contact area as an interfacial binding energy E
ad
= γ A. The shape of the
elastic body is obtained by minimizing the total energy −E
ad
+ U
el
,where
U
el
is the elastic deformation energy.
Consider an elastic ball (e.g., a rubber ball) with the ra-
dius R, Young’s elastic modulus E (and Poisson ratio ν), in
adhesive contact with a flat rigid substrate. Let γ = γ
1
+ γ
2
− γ
12
be the Dupré’s work of adhesion and let d
c
be the
spatial extent of the wall-wall interaction potential (typically
of order atomic distance). The DMT theory is valid when ad-
hesive stress σ
ad
≈ γ /d
c
is much smaller than the stress in
the contact region, which is of order
σ
c
≈

γ E
2
R

1/3
.
In the opposite limit the JKR theory is valid. In the DMT
theory the elastic deformation field is calculated with the ad-
hesion included only as an additional load F
ad
acting on the
sphere. Thus the contact area is determined by Hertz theory
with the external load F
0
= F
N
+ F
ad
, where F
N
is the ac-
tual load on the ball (see Fig. 1). The adhesion load F
ad
is
obtained by integrating the adhesion stress over the ball non-
contact area.
The JKR theory neglects the extent of the interaction po-
tential and assumes interaction between the solids only in the
contact area. The deformation field in the JKR theory is ob-
tained by minimizing the total energy given by the sum of
the (repulsive) elastic deformation energy and the (attractive)
binding energy E
ad
= γ A, where A is the contact area. In
this theory the contribution to binding energy from the non-
contact region is neglected.
Since σ
ad
≈ γ /d
c
we can define the Tabor number:
μ
T
=
σ
ad
σ
c
=

Rγ
2
E
2
r
d
3
c

1/3
=
d
T
d
c
,
where
d
T
=

Rγ
2
E
2
r

1/3
,
where E
r
= E/(1 − ν
2
) is the effective elastic modulus. The
DMT and JKR limits correspond to μ
T
 1 and μ
T
 1, or,
equivalently, d
T
 d
c
and d
T
 d
c
, respectively. In the JKR-
limit the Tabor length d
T
can be considered as the height of the
neck which is formed at the contact line (see Fig. 1(b)). This
neck height must be much larger than the length d
c
, which
characterizes the spatial extent of the wall-wall interaction, in
order for the JKR-limit to prevail.
At vanishing external load, F
N
= 0, the JKR theory pre-
dicts the contact area:
A
JKR
= π

9πR
2
γ
2E
r

2/3
.
This contact area is a factor 3
2/3
≈ 2.1 larger than obtained
from the DMT theory. In the JKR theory the force necessary
to remove the ball f rom the flat (the pull-off force) is given by
F
c
=
3π
2
γ R (1)
which is a factor of 3/4 times smaller than predicted by
the DMT theory. Also the pull-off processes differ: in the
JKR theory an elastic instability occurs where the contact
area abruptly decreases, while in the DMT theory the con-
tact area decreases continuously, until the ball just touches
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124701-3 B. N. J. Persson and M. Scaraggi J. Chem. Phys. 141, 124701 (2014)
the substrate in a single point, at which point the pull-force is
maximal.
For the sphere-flat case the pull-off force in the DMT-
limit is independent of the range of the wall-wall interaction
potential. However, this is not the case for other geometries
where in fact the contact mechanics depends remarkably sen-
sitively on the interaction range. As a result the interaction
between rough surfaces in the DMT-limit will depend on the
force law as we will demonstrate below for power law inter-
action p
ad
∼ u
−n
.
In an exact treatment, as a function of the external load
F
N
, the total energy E
tot
=−E
ad
+ U
el
must have a minimum
at F
N
= 0. This is the case in the JKR theory but in gen-
eral not for the DMT theory. Instead, the DMT theory is only
valid for very stiff solids and in this limiting case the total en-
ergy minimum condition is almost satisfied. Nevertheless, one
cannot expect dE
tot
/dF
N
(F
N
= 0) = 0 to be exactly obeyed in
any (approximate) theory which does not minimize the total
energy.
The results above assume perfectly smooth surfaces. The
JKR (and DMT) theory results can, however, be applied also
to surfaces with roughness assuming that the wavelength λ of
the most longest (relevant) surface roughness component is
much smaller than the diameter of the contact region. In that
case one only needs to replace the work of adhesion γ for
flat surfaces with an effective work of adhesion γ
eff
obtained
for the rough surfaces. We will now describe how one can
calculate γ
eff
.
III. THEORY: BASIC EQUATIONS
We now show how surface roughness can be taken into
account in adhesive contact mechanics. We consider two lim-
iting cases similar to the JKR and DMT theories for adhe-
sion of a ball on a flat. The theory presented below is not
based on the standard Greenwood-Williamson
10, 11
picture in-
volving contact between asperities, but on the Persson contact
mechanics theory.
A. JKR-limit (review)
In the JKR-limit the spatial extent of the wall-wall inter-
action potential is neglected so the interaction is fully char-
acterized by the work of adhesion γ . The contact between
randomly rough surfaces in this limit was studied in Ref. 41
and here we only review the most important results.
In order for two elastic solids with rough surfaces to make
adhesive contact it is necessary to deform the surfaces elasti-
cally, otherwise they would only make contact in three points
and the adhesion would vanish, at least if the spatial extend
of the adhesion force is neglected. Deforming the surfaces to
increase the contact area A results in some interfacial bonding
−γ A (where γ = γ
1
+ γ
2
− γ
12
is the change in the in-
terfacial energy per unit area upon contact), but it costs elastic
deformation energy U
el
, which reduces the effective binding.
That is, during the removal of the block from the substrate
the elastic compression energy stored at the interface is given
back and helps to break the adhesive bonds in the area of real
contact. Most macroscopic solids do not adhere with any mea-
surable force, which implies that the total interfacial energy
−γ A + U
el
vanishes, or nearly vanishes, in most cases.
The contact mechanics theory of Persson
6, 41–47
can be
used to calculate (approximately) the stress distribution at the
interface, the area of real contact, and the interfacial sepa-
ration between the solid walls.
42, 43
In this theory the inter-
face is studied at different magnifications ζ = L/λ, where
L is the linear size of the system and λ the resolution. We
define the wavevectors q = 2π/λ and q
0
= 2π/L so that
ζ = q/q
0
. The theory focuses on the probability distribution
P(σ , ζ )ofstressesσ acting at the interface when the system is
studied at the magnification ζ .InRef.42 an approximate dif-
fusion equation of motion was derived for P(σ , ζ ).Tosolve
this equation one needs boundary conditions. If we assume
that, when studying the system at the lowest magnification ζ
= 1 (where no surface roughness can be observed, i.e., the
surfaces appear perfectly smooth), the stress at the interface
is constant and equal to p
N
= F
N
/A
0
, where F
N
is the load and
A
0
the nominal contact area, then P(σ ,1)= δ(σ − p
N
). In ad-
dition to this “initial condition” we need two boundary con-
ditions along the σ -axis. Since there can be no infinitely large
stress at the interface we require P(σ , ζ ) → 0asσ →∞.For
adhesive contact, which interests us here, tensile stress occurs
at the interface close to the boundary lines of the contact re-
gions. In this case we have the boundary condition P[−σ
a
(ζ ),
ζ ] = 0, where σ
a
> 0 is the largest (locally averaged at mag-
nification ζ ) tensile stress possible. Hence, the detachment
stress σ
a
(ζ ) depends on the magnification and can be related
to the effective interfacial energy (per unit area) γ
eff
(ζ )using
the theory of cracks.
6
The effective interfacial binding energy
γ
eff
(ζ )A(ζ ) = γ A(ζ
1
)η − U
el
(ζ ),
where A(ζ ) denotes the (projected) contact area at the magni-
fication ζ , and A(ζ
1
)η is the real contact area, which is larger
than the projected contact area A(ζ
1
), i.e., η ≥ 1 (e.g., if the
rigid solid is rough and the elastic solid has a flat surface η
> 1, see Ref. 41 for an expression for η). U
el
(ζ ) is the elastic
energy stored at the interface due to the elastic deformation of
the solids on length scale shorter than λ = L/ζ , necessary in
order to bring the solids into adhesive contact.
The area of apparent contact (projected on the xy-plane)
at the magnification ζ , A(ζ ), normalized by the nominal con-
tact area A
0
, can be obtained from
A(ζ )
A
0
=

∞
−σ
a
(ζ )
dσ P(σ, ζ )
Finally, we note that the effective interfacial energy to be
used in the JKR expression for the pull-off force (1) is the
macroscopic effective interfacial energy corresponding to the
magnification ζ = 1 (here we assume that the reference length
L is of order the diameter of the JKR contact region). Thus in
the numerical results presented in Sec. IV we only study the
area of contact A(ζ
1
) and the macroscopic interfacial energy
γ
eff
= γ
eff
(1), which satisfies
γ
eff
A
0
= γ A(ζ
1
)η − U
el
(1).
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124701-4 B. N. J. Persson and M. Scaraggi J. Chem. Phys. 141, 124701 (2014)
B. DMT-limit
The DMT-limit depends on the interaction potential be-
tween the solid walls. In this section we show how this limit-
ing case can be studied using the Persson contact mechanics
theory.
Let p
N
= F
N
/A
0
be the applied pressure (which can be
both positive and negative). In the DMT-limit one assumes
that the elastic deformation of the solids is the same as in the
absence of an adhesive interaction, except that the external
load F
N
is replaced with an effective load. The latter contains
the contribution to the normal force from the adhesive force
acting in the non-contact interfacial surface area: F
0
= F
N
+ F
ad
. If we divide this equation by the nominal contact area
A
0
we get
p
0
= p
N
+ p
ad
.
The adhesive pressure
p
ad
=
1
A
0

n.c.
d
2
xp
a
[u(x)], (2)
where p
a
(u) is the interaction force per unit area when two flat
surfaces are separated by the distance u.In(2) the integral is
over the non-contact (n.c.) area. In this study we assume that
there is a force per unit area (pressure) between the surfaces
given by (u ≥ 0 , see, e.g., Fig. 2):
p
a
= B

d
c
u + d
c

n
− α

d
c
u + d
c

m

, (3)
where the cut-off d
c
is a typical bond length and α a number,
which we take to be either 0 or 1. The interaction potential
between solid surfaces can have different form depending on
the nature of the interaction (e.g., Coulomb interaction be-
tween charged walls, van der Waals interaction or chemical
interaction), so we consider a rather general case which al-
low us to vary the extent of the attractive and repulsive part
of the potential. van der Waals i nteraction correspond to n
= 3 and is used in most of the calculations. Coulomb interac-
tion between uniformly charged surfaces would correspond to
n = 0. Chemical bonding is short-ranged and would qualita-
tively correspond to a large n (n > 3) but in this case an expo-
nential interaction, ∼exp(−z/a), between the walls would be
separation u (nm)
p
a
(GPa)
Δγ = 0.2 J/m
2
d
c
= 1 nm
0 0.5 1 1.5 2 2.5 3
-0.4
-0.3
-0.2
-0.1
0
0.1
α = 0
α = 1
FIG. 2. The adhesive pressure for n = 3andm = 9andα = 1 (red line)
and α = 0 (blue line). In the calculation we assumed γ = 0.2J/m
2
and
d
c
= 1nm.
more realistic. The parameter B is determined by the work of
adhesion (per unit surface area):

∞
0
du p
ad
(u) = Bd
c
(m − 1) − (n − 1)α
(m − 1)(n − 1)
= γ ,
so that
B =
γ
d
c
(m − 1)(n − 1)
(m − 1) − α(n − 1)
.
If P(u) denotes the distribution of interfacial separations then
we can also write (2) as
p
ad
=

∞
0
+
du p
a
(u)P (u).
In Ref. 45 we have derived an expression for P(u)usingthe
Persson contact mechanics theory. In the numerical results
presented below we have used the expression for P(u)given
by Eq. (17) in Ref. 45 (seealsointhefollowing).
The effective interfacial energy can in the DMT-limit be
calculated using
γ
eff
=

∞
0
+
du φ(u)P (u) + γ
A
A
0
−
U
el
A
0
,
where A = A
r
is the (repulsive) contact area and where φ(u)is
the interaction potential per unit surface area for flat surfaces
separated by the distance u and given by
φ( u) =

∞
u
du p
a
(u).
Thus in the present case
φ( u) =
Bd
c
n − 1

d
c
u + d
c

n−1
−
Bd
c
α
m − 1

d
c
u + d
c

m−1
.
For u = 0 an infinite hard wall occurs and we define the (re-
pulsive) contact area A
r
when the surface separation u = 0. In
the calculations below we use n = 3 and m = 9 and α = 0
(Sec. IV) and α = 1 (Sec. V). The interaction pressure for
these two cases are shown in Fig. 2.
The probability distribution of interfacial separations
P(u) can be calculated as follows: We define u
1
(ζ )tobe
the (average) height separating the surfaces which appear to
come into contact when the magnification decreases from ζ to
ζ − ζ , where ζ is a small (infinitesimal) change in the
magnification. u
1
(ζ ) is a monotonically decreasing function
of ζ , and can be calculated from the average interfacial sepa-
ration
¯
u(ζ ) and the contact area A(ζ ) using (see Ref. 44)
u
1
(ζ ) =
¯
u(ζ ) +
¯
u

(ζ )A(ζ )/A

(ζ ).
The equation for the average interfacial separation
¯
u(ζ )is
given in Ref. 44. The (apparent) relative contact area A(ζ )/A
0
at the magnification ζ is given by
A(ζ )
A
0
= erf

p
0
2G(ζ )
1/2

,
where
G(ζ ) =
π
4

E
1 − ν
2

2

ζq
0
q
0
dqq
3
C(q),
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134.94.122.242 On: Mon, 29 Jun 2015 08:17:33

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Theory of adhesion: role of surface roughness" ?

In this paper, the surface roughness influences the adhesion between elastic solids, and a Tabor number is introduced to give information about the nature of the contact mechanics at different length scales. 

Q2. What is the effect of the wall-wall interaction potential on the surface of a solid?

For charged bodies, due to the long range of the coulomb interaction, the wall-wall interaction potential is important for any wall-wall separation. 

Q3. What is the effect of deforming the surfaces to increase the contact area?

Deforming the surfaces to increase the contact area A results in some interfacial bonding − γ A (where γ = γ 1 + γ 2 − γ 12 is the change in the interfacial energy per unit area upon contact), but it costs elastic deformation energy Uel, which reduces the effective binding. 

Q4. What is the elastic energy stored at the interface due to the elastic deformation of the solids?

Uel(ζ ) is the elastic energy stored at the interface due to the elastic deformation of the solids on length scale shorter than λ = L/ζ , necessary in order to bring the solids into adhesive contact. 

Q5. What is the work of adhesion for smooth surfaces?

The effective work of adhesion to be used in macroscopic adhesion applications, i.e., the pull-off of a ball from a flat (Sec. II) is γ eff for the applied pressure pN = 0, and in all cases in Fig. 6 γ eff(pN = 0) is less than half of the work of adhesion γ for smooth surfaces. 

Q6. What is the effect of the adhesive on the contact area?

Fig. 6(a) shows that in the JKR-limit the contact area as a function of pN increases much faster with increasing pN than in the absence of adhesion (green line), i.e., even if no adhesion manifests itself during pull-off, the contact area and hence other properties such as the friction force, may be strongly enhanced by the adhesive interaction. 

Q7. What is the theory of adhesion hysteresis?

The authors note that adhesion problems which are JKR-like for large length scales and DMT-like for short length scales can be approximately treated using the theory presented above: 

Q8. What is the slope of the eff(pN) curve?

For other surfaces which have dT(q) dc or dT(q) dc for all q, the JKR-like and DMT-like theories may be more accurate and the slope of the γ eff(pN) curve negative. 

Q9. What is the reason why the slope of the eff(pN) curves is?

This is easy to understand since when the wall-wall separation is larger than the highest asperity the solid walls will only interact with the long-ranged attractive wall-wall potential and increasing the separation to infinity will always require a finite amount of work making γ eff(pN = 0) always non-zero in the DMT-limit. 

Q10. What is the simplest way to study the adhesion problem?

1. Analytical results for intermediate-range adhesion were presented by Maugis36, 37 and the ball-flat adhesion problem has also been studied in detail using numerical methods. 

Q11. How many mesh points are used to extend the roughness spectral components of Fig.?

To do so, the authors gradually extend the numerically calculated roughness spectral components of Fig. 12, as shown in Fig. 18, up to a system size of 224 mesh points. 

Q12. What is the effect of radiating the elastic energy into the solids?

not all the stored elastic energy Uel may be used to break adhesive bonds during pull-off but some fraction of it may be radiated as elastic waves (phonons) into the solids.