OPINION
published: 22 January 2020
doi: 10.3389/fmech.2019.00073
Frontiers in Mechanical Engineering | www.frontiersin.org 1January 2020 | Volume 5 | Article 73
Edited by:
Yu Tian,
Tsinghua University, China
Reviewed by:
Liran Ma,
Tsinghua University, China
Noshir Sheriar Pesika,
Tulane University, United States
*Correspondence:
Valentin L. Popov
Specialty section:
This article was submitted to
Tribology,
a section of the journal
Frontiers in Mechanical Engineering
Received: 05 December 2019
Accepted: 24 December 2019
Published: 22 January 2020
Citation:
Popov VL (2020) Contacts With
Negative Work of “Adhesion” and
Superlubricity. Front. Mech. Eng. 5:73.
doi: 10.3389/fmech.2019.00073
Contacts With Negative Work of
“Adhesion” and Superlubricity
Valentin L. Popov*
Department of System Dynamics and Friction Physics, Technische Universität Berlin, Berlin, Germany
Keywords: negative work of adhesion, electrohaptics, superlubricity, van der Waals forces, Hertzian contact
INTRODUCTION
Van der Waals forces between solids in vacuum are always attractive and are considered as the
main source of adhesion. However, in the presence of an intermediate medium, they can also be
repelling (Dzyaloshinskii et al., 1961) which means that the “work of adhesion” becomes negative.
Similarly to the case of adhesion, the interaction range of these forces can be either comparable (or
larger) than the minimum characteristic length scale of the contact problem or it can be negligible
compared with all characteristic length scales. We call this latter case the “JKR-approximation,” as
the JKR theory of adhesion (Johnson et al., 1971) is also valid in this limit. The repelling interaction
can also be due to the presence (and squeezing out) of a thin fluid layer between solids as considered
in Müser (2014). In the papers Popov and Hess (2018) and Heß and Popov (2019), it was shown
that the contact of two oppositely charged surfaces at a constant voltage is equivalent to the adhesive
contact with an effective van der Waals interaction. Similarly, the contact of the bodies with the
same charge would be equivalent to repelling van der Waals forces with a negative work of adhesion.
Further kinds of repelling forces may be solvation, structural, and hydration forces (Israelachvili,
2011). In the following, we speak about van der Waals forces, but they are thought as representative
for a larger class of long range repelling forces.
We argue that in the JKR approximation, the Hertz’ solution of the contact problem with a
repelling van der Waals interaction, remains practically unchanged. However, the contact area falls
apart into the area of “weak (van der Waals) interaction” and “strong (rigid wall) interaction.” It is
speculated that if the normal force is smaller than a critical value at which the core region of strong
interaction disappears, a macroscopic superlubricity state of the contact may be observed.
ATTRACTIVE AND REPELLING VAN DER WAALS FORCES
The interaction force between neutral molecules is often modeled as a superposition of the
very sharp increasing “core repulsion” ∝1/r13 and a weaker van der Waals “tail” ∝r−7(the
corresponding potential is known as Lennard-Jones-Potential), Figure 1A, left:
F(r)=12 ·w
r0r0
r13
−r0
r7for attractive van der Waals forces, (1)
where r0is the equilibrium distance, at which the core force and the van der Waals force become
equal and wis the work of adhesion (the work needed to separate the molecules starting with their
equilibrium position).
For two bodies with a plane surface at distance z, Equation (1) is modified to an equation of the
interaction stress:
σ(z)=8·1γ
3z0z0
z9
−z0
z3for attractive van der Waals forces (2)
Popov Negative Work of Adhesion and Superlubricity
FIGURE 1 | (A) Left: Model intermolecular force in the case of attractive van der Waals force (blue line) and repelling van der Waals force (red line), in relative units. The
distance r=1 corresponds to the equilibrium in the case of attractive van der Waals forces. At distances smaller than r=1 the repelling force increases steeply: At
these distances, one can qualitatively speak about a “rigid wall.” At distances r>1, there exist weak long-range van der Waals force, either positive or negative. The
work needed to separate the surfaces starting from the “rigid wall” position r=1 are shown by filled areas (blue filled area—positive work of adhesion in an adhesive
contact, red filled area—“negative work of adhesion” in the repelling case). (A) Right: The normal stress (pressure) distribution in a Hertzian contact. If the range of
action of both core “rigid wall” force and van der Waals force are negligible compared with all characteristic length scales of the contact problem, the repelling van der
Waals forces do not influence the contact problem and do not change the stress distribution. However, on the microscopic scale, in the regions where the stress is
larger then a critical stress σ0(see the text of paper for details), the surfaces are in “direct rigid wall contact” and feel strong atomic corrugation potential. In the regions
where the stress is smaller than σ0, they “levitate” due to van der Waals force and see only weak corrugation. By decreasing the normal force, one can achieve the
state in which the normal stress is smaller than σ0in the whole contact area. This state corresponds to the state of superlubricity. (B) Schematic illustration of the
contact configuration on the example of a simple model adhesive stress of “Dugdale type” (Dugdale, 1960) with a constant repelling stress σ0up to the distance hc
(scheme on the left). In the center: Macroscopic shape of a soft sphere in contact with a rigid surface (black line) and the Hertzian stress distribution (brown dashed
line). On the right: Microscopic view of the contact gap in the undercritical and overcritical cases. From the macroscopic point of view, the gap in both cases is zero,
from the micrsocopic point of view, the bodies can be either in direct rigid wall contact (where the local Hertzian pressure is larger σ0or “levitate” in the distance hc
where the Hertzian pressure is smaller σ0.
where z0is the equilibrium distance between the bodies (of the
order of r0), and 1γ is the (positive) specific work of adhesion
(work of separation of two surfaces per unit area).
While in vacuum the van der Waals forces are always
attractive, in the presence of an intermediate medium between
two bodies, they can also become repelling—if the dielectric
constant of the intermediate medium lies between the dielectric
constants of the contacting bodes (Dzyaloshinskii et al., 1961).
In this case, Equation (1) for the interaction force is modified by
changing the sign of the van der Waals force, Figure 1A, left:
F(r)=12 ·w
r0r0
r13
+r0
r7
for repelling van der Waals forces. (3)
In this equation, r0loses its meaning of the equilibrium position
(which without external force does not exist anymore), but can
still be considered as a distance characterizing the transition from
the “core potential” to the “van der Waals potential.” At smaller
distances, the repelling force increase very steeply and can be
qualitatively considered as a “rigid wall,” while at larger distances
it describes long range repelling van der Waals force. For bodies
with plane surfaces, the corresponding modification of Equation
(2) reads
σ(z)=8·1γ
3z0z0
z9
+z0
z3
for repelling van der Waals forces. (4)
Frontiers in Mechanical Engineering | www.frontiersin.org 2January 2020 | Volume 5 | Article 73
Popov Negative Work of Adhesion and Superlubricity
Therein, 1γ is not the work of adhesion anymore but has
to be considered just formally as a coefficient determining the
amplitude of the interaction. However, if we calculate the “work
needed to separate” the bodies starting with the distance z0
(the presumable position of the “rigid wall”), we get the specific
“negative work of adhesion”
1γrepelling = − (5/3)1γ . (5)
In the past, there were only a few attempts to study contact
mechanics in the presence of explicit surface interaction potential
(“soft walls”) (e.g., Hughes and White, 1979; Vinogradova and
Feuillebois, 2003; Müser, 2014). However, these works were
focused on the normal interaction while we would like to discuss
the implications of the surface interactions to the tangential
force (friction).
INFLUENCE OF REPELLING VAN DER
WAALS FORCES ON CONTACT AND
FRICTION
As Martin Müser writes in Müser (2014), “For repulsive
contacts,..., there is obviously no finite contact radius at
zero normal load ...The repelled rigid tip simply “hovers”
at (infinitely) large distance over an undeformed elastic
manifold ...”. Let us consider this absolutely correct statement
more closely. It is correct that the repelling van der Waals forces
will keep the surfaces at “infinite large distance” which physically
means at “very large distance.” However, from the macroscopic
point of view, this “very large distance” may be smaller than
any other characteristic length of the contact problem and thus
can be considered as being zero (JKR-approximation). In the
contact with repelling forces, the smallest characteristic length is
the indentation depth, so the range of van der Waals interactions
is assumed to be smaller than the indentation depth.
Whether the “very large distance” is zero or infinite—depends
on the quantities, which we are interested in. If we consider
the contact problem itself and the range of repelling van der
Waals forces can be neglected, then they have no influence
on the contact problem at all. In particular, all displacements
and stress distributions will remain the same as in the classical
“rigid wall” Hertz contact problem. However, if we consider
the tangential forces (caused by the microscopic tangential
corrugation potential), the “very large distance” can again be
considered as infinite which leads to a vanishing force of friction.
In order for the macroscopic frictional force to disappear, it is not
even necessary that the tangential corrugation potential becomes
zero; it is enough that it becomes smaller than some critical value,
as has been shown theoretically and experimentally in Socoliuc
et al. (2004).
When two surfaces approach each other, the interaction stress
(4) increases monotonously and accepts at the distance z=z0
the value σ0=(16/5)1γrepelling/z0. If the local elastic stress
in the material is larger than this critical stress, then, roughly
speaking, the bodies are in “rigid wall contact.” If the local stress
is much smaller than this critical stress, then the surfaces “hover”
at “infinitely large distance.” In a Hertzian contact, all parts of the
contact where the stress is larger than the critical one, will be in
“direct rigid wall contact,” and feel strong corrugation potential,
while the areas where the stress is essentially smaller then σ0will
be held apart by the repelling van der Waals forces and feel only a
very weak corrugation potential. This is illustrated in Figure 1A,
right for the example of a Hertzian contact. The contact area is
divided into two parts: the inner part of “rigid wall contact” and
high friction and the outer part of levitation due to repelling van
der Waals forces and a weak (or zero) friction force. Figure 1B
schematically illustrates the macroscopic and microscopic views
of the contact for the case of a simplified repelling stress of
“Dugdale type” (Dugdale, 1960).
The most interesting conclusion is that if the Hertzian
stress is smaller than the critical stress needed to bring the
surfaces into the “rigid wall contact,” then they “levitate” in
the whole contact area. This inevitably should lead to a small
or vanishing macroscopic frictional force—the macroscopic
state of superlubricity. The critical stress depends on particular
mechanism of repulsion force. A very rough estimation can
be made by assuming the value of 1γ ∼4·10−2J/m2for
the surface energy [which is “typical” for polymers and fluids
(Popov, 2017)] and 1l∼10−9m as the “levitation” distance
needed for suppressing the tangential corrugation potential. The
critical stress will then be on the order of σc∼1γ /1l≈
40 MPa. In Ge et al. (2019a), the values up to 600 MPa
were reported.
CONCLUSION
Let us briefly summarize the main points of the above-
sketched picture:
1. If one does not look at the contact so closely, the “adhesive”
repulsion looks just like a hard wall, similarly to the actual hard
wall of the Pauli principle.
2. Therefore, in this “JKR-limit” nothing changes in the solution
of the contact problem.
3. Nevertheless, if one looks more closely, the adhesive repulsion
(as opposed to the hard wall) has some reach, so this long-
range tail of the repelling force can hold the surfaces apart.
4. There is (almost) no friction in the areas that are levitating due
to “negative adhesion.”
5. If the maximum stress in the whole contact area is smaller
then the “critical stress of levitation,” the system transits
into the state of very low (or vanishing) friction—state
of superlubricity.
The key prerequisite of the described mechanism of
superlubricity is the presence of repelling long range interaction
forces which are able to hold the surfaces apart so that they do
not feel the corrugation potential. This repulsion can be achieved
in different ways:
- As repelling van der Waals force due to an intermediate
medium with dielectric constant lying between the dielectric
constancies of both contacting bodies. This mechanics can
only be active for a contact of two bodies having different
dielectric constants.
Frontiers in Mechanical Engineering | www.frontiersin.org 3January 2020 | Volume 5 | Article 73
Popov Negative Work of Adhesion and Superlubricity
- As thermodynamic repulsion due to a layer of free or grafted
macromolecules between the bodies so that the entropy of
the intermediate layer decreases at small distances due to
stronger confinement.
- As electrical repulsion due to external electrical voltage (so that
both surfaces receive the charge of the same sign).
- As repulsion due to electrical double layer (Guldbrand et al.,
1984).
- Possible is also an effective repulsion due to thermal
fluctuations (Müser et al., 2019).
The necessity of an intermediate layer for achieving negative
work of adhesion leads to the conclusion that the kinetic friction
will also be essentially dependent on the rheology or viscosity of
this intermediate medium. The importance of the local pressure
brings the problematics of the surface roughness in play. The
flattening of roughness and lowering local stresses may be one
of the reasons of the necessity of the wearing-in process for
achieving the superlubricity state (Ge et al., 2019a).
In my opinion, the above simple picture can help a lot for
both qualitative and quantitative physical understanding of the
transition into the superlubricity state—both in the case of fluid
superlubricity (Ge et al., 2019b) and solid state superlubricity
(Erdemir and Eryilmaz, 2014) as well of the tuning of friction by
electric fields (Krim, 2019;Figure 1).
AUTHOR CONTRIBUTIONS
The author confirms being the sole contributor of this work and
has approved it for publication.
FUNDING
This work was funded by the German Research Foundation
(DFG PO 810/55-1).
ACKNOWLEDGMENTS
I am grateful to Emanuel Willert and Qiang Li for discussions.
I acknowledge support from the German Research Foundation
and the Open Access Publication Funds of TU Berlin.
REFERENCES
Dugdale, D. S. (1960). Yielding of steel sheets containing slits. J. Mech. Phys. Solids
8, 100–104. doi: 10.1016/0022-5096(60)90013-2
Dzyaloshinskii, I. E., Lifshitz, E. M., and Pitaevskii, L. P. (1961).
General theory of van der Waals’ forces. Sov. Phys. Usp. 4, 153–176.
doi: 10.1070/PU1961v004n02ABEH003330
Erdemir, A., and Eryilmaz, O. (2014). Achieving superlubricity in DLC films
by controlling bulk, surf ace, and tribochemistry. Friction 2, 140–155.
doi: 10.1007/S40544-014-0055-1
Ge, X., Li, J., and Luo, J. (2019b). Macroscale superlubricity achieved
with various liquid molecules: a review. Front. Mech. Eng. 5:2.
doi: 10.3389/fmech.2019.00002
Ge, X., Li, J., Wang, H., Zhang, C., Liu, Y., and Luo, J. (2019a). Macroscale
superlubricity under extreme pressure enabled by the combination
of graphene-oxide nanosheets with ionic liquid. Carbon 151, 76–83
doi: 10.1016/j.carbon.2019.05.070
Guldbrand, L., Jönsson, B., Håkan Wennerström, H., and Linse, P. (1984).
Electrical double layer forces. A Monte Carlo study. J. Chem. Phys. 80:2221.
doi: 10.1063/1.446912
Heß, M., and Popov, V. L. (2019). Voltage-induced friction with application to
electrovibration. Lubricants 7:102. doi: 10.3390/lubricants7120102
Hughes, B. D., and White, L. R. (1979). ‘Soft’ contact problems in linear elasticity,
Q. J. Mech. Appl. Math. 32, 445–471. doi: 10.1093/qjmam/32.4.445
Israelachvili, J. N. (2011). Intermolecular and Surface Forces, 3rd Edn.
Waltham, MA; San Diego, CA; Oxford; Amsterdam: Elsevier.
doi: 10.1016/C2009-0-21560-1
Johnson, K. L., Kendall, K., and Roberts, A. D. (1971). Surface energy
and the contact of elastic solids. Proc. R. Soc. Lond. A 324, 301–313.
doi: 10.1098/rspa.1971.0141
Krim, J. (2019). Controlling friction with external electric or magnetic fields: 25
examples. Front. Mech. Eng. 5:22. doi: 10.3389/fmech.2019.00022
Müser, M. H. (2014). Single-asperity contact mechanics with positive and negative
work of adhesion: influence of finite-range interactions and a continuum
description for the squeeze-out of wetting fluids, Beilstein J. Nanotechnol. 5,
419–437. doi: 10.3762/bjnano.5.50
Müser, M. H., Zhou, Y., and Wang, A. (2019). How thermal fluctuations affect
hard-wall repulsion and thereby Hertzian contact mechanics. Front. Mech. Eng.
5:67. doi: 10.3389/fmech.2019.00067
Popov, V. L. (2017). Contact Mechanics and Friction: Physical
Principles and Applications, 2nd Edn. Berlin; Heidelberg: Springer.
doi: 10.1007/978-3-642-10803-7
Popov, V. L., and Hess, M. (2018). Voltage induced friction in a
contact of a finger and a touchscreen with a thin dielectric coating.
arXiv[Preprint].arXiv:1805.08714
Socoliuc, A., Bennewitz, R., Gnecco, E., and Meyer, E. (2004). Transition from
stick-slip to continuous sliding in atomic friction: entering a new regime of
ultralow friction. Phys. Rev. Lett. 92:134301. doi: 10.1103/PhysRevLett.92.1
34301
Vinogradova, O. I., and Feuillebois, F. J. (2003). Interaction of elastic bodies
via surface forces 2. Exponential decay. Colloid Interface Sci. 268, 464–475.
doi: 10.1016/j.jcis.2003.09.00
Conflict of Interest: The author declares that the research was conducted in the
absence of any commercial or financial relationships that could be construed as a
potential conflict of interest.
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