Supplementary to:
The additive effect of harmonics on conservative and dissipative interactions
Sergio Santos1,ψ, Karim R. Gadelrab1,ψ, Victor Barcons2, Josep Font2, Marco Stefancich1,
Matteo Chiesa1
1
Laboratory of Energy and Nanosciences, Masdar Institute of Science and Technology, Abu
Dhabi, UAE2 Departament de Disseny i Programació de Sistemes Electrònics, UPC Universitat Politècnica de Catalunya Av. Bases, 61, 08242 Manresa, Spain
ψ
Equally contributing authors
The equation of motion to be solved via numerical integration is
m
d 2 z m dz

 kz  Fts  F0 cos t
dt 2
Q dt
(S1)
where k is the spring constant, Q is the Q factor due to dissipation with the medium, ω=ω0 are
the drive and natural angular frequency of oscillation respectively,
m=k/ω2, Fts is the
instantaneous (net) tip-sample interaction, z is the instantaneous position of the tip relative to the
cantilever and F0 is the drive force. For Fts the tip-surface interaction (Fig. 2) we write[1, 2]
Fa  
H wR
6d w2
FAD (d ) 
d>doff (tip retraction) and
FDoff  FDon
d off  d on
(d  d on )  FDon
d>don (tip approach)
doff>d>don (tip retraction)
(S2)
(S3)
where
FDon  

Rtip  H w  H
(d on  a0 )  H 
2 
6a0  d off  a0

(S4)
and
FDoff  Coff
FAD
RH H 2O
6a02
H *R
 2
6a0
Coff≥0
(S5)
a0<d< don
(S6)
where
H* 
Hw  H
(d  a0 )  H
d off  a0
FAD  
HR
 4R
6a02
(S7)
d≤a0
(S8)
where d is the instantaneous tip-sample's surface distance, Fa is the attractive and conservative
long range force, FAD is the adhesion force, doff is the distance at which the capillary force
ruptures, don is the distance at which the capillary force breaks, H is the Hamaker constant for
the tip and the sample, Hw is the Hamaker constant corresponding to the hydrated tip-hydrated
surface interaction, dw is an effective distance of interaction that takes into account the fact that
in the presence of adsorbed water films the tip and the surface as dw=d-2h where h is the height
of the water films on the tip and sample's surfaces [1-3], a0 is an intermolecular distance that
implies that matter interpenetration cannot occur, γ is the surface energy and Coff which is
defined next. These forces have been recently shown to lead to close reproduction of phenomena
in ambient dynamic AFM[2].
The interpretation of (S2-S8) is that that when the water layers overlap as a consequence of
capillary neck formation, the force is relatively constant for a distance a0<d<don but significantly
decays with separation at the larger distances don<d<doff[1-3]. The decay in the adhesion force is
controlled by Coff where Coff=0.3 has been used here throughout. The decay in force in the long
range and on tip retraction can be clearly seen in Fig. 2. The dissipation due to this phenomenon
of hysteretic behavior coincides with the area shaded in light grey (right side) in Fig. 2. When
the capillary neck is formed the capillary force is also added[2, 4]
FCAP (d )  
2 w R tip
 d3
1
Vmen
provided capillary on and d>a0
where γw is the surface energy of water. Also, Hw≡24π(a0)2γw and H≡24π(a0)2γ[5].
(S9)
The distance don has been taken to be don=3h[2]. Additionally doff can be calculated numerically
by solving the Laplace-Young equation to give[6, 7]
1/ 3
d off  Vmen

1 2/3
Vmen
5R
(S10)
where Vmen is the volume of the meniscus or the volume of water forming the water bridge. In
(S6) Vmen is calculated using geometrical considerations[7]
4
Vmen  4Rh 2  h3
3
(S11)
From the above expressions, it is clear that for don<d<doff there is hysteresis due to capillarity. As
stated, energy is dissipated (gray colored area (right side) in Fig. 2) due to meta-stability in this
region. The conservative short range repulsive force is written as[8, 9]
FDMT  a
3/ 2
d≤a0
(S12)
where
a
4 *
E R
3
(S13)
and where E* is the effective elastic modulus of the contacting bodies (tip Et and sample Es). The
dissipation during sample deformation, i.e. d<a0, is modeled as follows.
Garcia et al.[10] identified two main dissipative processes during sample deformation, namely,
surface energy γ hysteresis and viscoelasticity. The authors found that these processes could be
well modeled by considering 1) the variation in the adhesion force FAD(A)=-4πRγa and FAD(R)=4πRγr, where the respective surface energies are γa and γr (γr ≥γa) and 2) the Voigt model for the
viscoelastic force Fη
.
F   (d )   ( R )1 / 2 
(S14)
.
where δ is the instantaneous deformation,  is the tip velocity during deformation and η is the
viscosity of the materials in contact. In the article we distinguish between the viscosity η and the
proportionality, i.e. function, of the viscous forces η(d). The dissipative term of the adhesion
force Fα can also be written as
F  4R  FAD
(S15)
where α=γr/γa-1 accounts for differences in γa and γr and where γ=γa and FAD=-4πRγ. More
recent studies also support the validity of these expressions for a variety of samples where large
values of α have been interpreted as the capacity to restructure the atomic bonds of the surfaces
in contact[11]. Together (S14) and (S15) account for the dissipation in the region of mechanical
contact (light grey area in Fig. 2 (left)). It is clear that only the viscosity (S14) depends on
velocity while the hysteresis in force (S15) does not. Both have been recovered in Fig. 4b.
In the numerical integration in Figs. 2-4, k=2N/m, Q=120, ω=2πf0 (f0=70 kHz), Et=170 GPa
(Young modulus of the tip), Es=1 GPa, a0=0.165 nm, R=7 nm, γ=7 mJ, γw= 10mJ, α=1 and
η=500 Pa∙s. The equation of motion has been solved numerically with the use of a Runge Kutta
algorithm and implemented in both C and Matlab[12]. Both implementations were equivalent
while C was more than an order of magnitude faster.
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MATLAB R2008a and SIMULINK T M, Inc., Natick, Massachusetts, US.