# Stickiness of randomly rough surfaces with high fractal dimension: is there
a fractal limit?

G. Violano  Department of Mechanics, Mathematics and Management, Polytechnic
University of Bari, Via E. Orabona, 4, 70125, Bari, Italy  A. Papangelo
Department of Mechanics, Mathematics and Management, Polytechnic University of
Bari, Via E. Orabona, 4, 70125, Bari, Italy  Hamburg University of Technology,
Department of Mechanical Engineering, Am Schwarzenberg-Campus 1, 21073
Hamburg, Germany  M. Ciavarella  Department of Mechanics, Mathematics and
Management, Polytechnic University of Bari, Via E. Orabona, 4, 70125, Bari,
Italy  Hamburg University of Technology, Department of Mechanical Engineering,
Am Schwarzenberg-Campus 1, 21073 Hamburg, Germany

###### Abstract

Two surfaces are ”sticky” if breaking their mutual contact requires a finite
tensile force. At low fractal dimensions $D$, there is consensus stickiness
does not depend on the upper truncation frequency of roughness spectrum (or
”magnification”). As debate is still open for the case at high $D$, we exploit
BAM theory of Ciavarella and Persson-Tosatti theory, to derive criteria for
all fractal dimensions. For high $D$, we show that stickiness is more
influenced by short wavelength roughness with respect to the low $D$ case. BAM
converges at high magnifications to a simple criterion which depends only on
$D$, in agreement with theories that includes Lennard-Jones traction-gap law,
while Persson-Tosatti disagrees because of its simplifying approximations.

stickiness criterion, adhesion, Dalhquist criterion, surface roughness

##  I Introduction

The interplay of roughness and adhesion is certainly a problem of great
interest. Various industries are interested in robust adhesion, like in tapes
made of soft polymers Creton1996 ; Creton2016 ; m2016 ; Dahlquist1969 ;
Dahlquist , sometimes imitating Nature, where very elaborate strategies have
evolved Autumn ; Gao1 ; Gao2 ; Dening2014 ; Gorb . In general, roughness
explains the ”adhesion paradox” Kendall2001 , namely that all surfaces should
be strongly adhered to each other because of the strength of van der Vaals
forces. With ”stickiness”, we denote the possibility of sustaining macroscopic
tensile pressures or else non-zero contact area without load Violano1 .

For smooth bulk solids without hierarchical structures, stickiness is
generally reached only for small elastic Young modulus (smaller than about $1$
$\mathrm{MPa}$ according to the 3M criterion of Dahlquist Dahlquist1969 ;
Dahlquist ). However, real solids are characterized by surface roughness that
could extend on several length scales: in principle, for example, we could
span kilometers to nanometers and obtain 12 orders of magnitude — which is way
beyond the computational capabilities of present numerical codes. Seven
decades of almost pure power law roughness spectrum have been observed in ref.
dalvi , limiting the measurement to nanometer amplitude of roughness. With the
development of faster computers, simulations of contact mechanics with
roughness have become possible Ciavarella2018adh ; Vakis2018 ; Aff2018 .
However, due to mentioned high computational costs, only a very limited range
of roughness wavelengths can be considered even in state-of-the-art technology
as in the ”contact challenge” recently setup by Martin Müser CMC2017 ,
spanning only about 3 orders of magnitude of roughness. In such sense, the
study of adhesion between surfaces with broad roughness spectra is still a
challenge. Fortunately, we have theoretical models that can predict the
behaviour for much broader spectra, and here we attempt for the first time in
the best of the authors’ knowledge, the discussion of the limit for infinitely
broad spectra, in the case of arbitrary fractal dimension. The case of low
fractal dimension is probably more common (Persson2014 ), but it is important
to have a general understanding of the contact problem, also for reference to
validate future computational and experimental efforts.

The very early models of rough contact mechanics used the concept of
”asperities”. Fuller and Tabor (FT) FT  involved a combination of factors
depending on the long wavelength content of surface roughness, namely the root
mean square (rms) roughness amplitude $h_{\mathrm{rms}}$, and factors
depending on the small wavelength component, namely the rms curvature
$h_{\mathrm{rms}}^{\prime\prime}$.

More recently, Pastewka & Robbins (PR) PR2014  performed Boundary Element
Method (BEM) simulations of adhesive rough contact between fractal surfaces
having roughness spectra of about 3 orders of magnitude in wavelengths (from
nano to micrometer scale). From an extrapolation of their numerical results,
they extrapolated tentatively that stickiness should depend only on small
scale features of surface roughness, i.e. rms slope
$h_{\mathrm{rms}}^{\prime}$ and rms curvature
$h_{\mathrm{rms}}^{\prime\prime}$. The quantities $h_{\mathrm{rms}}^{\prime}$
and $h_{\mathrm{rms}}^{\prime\prime}$ are strongly related to the smallest
wavelength $\lambda_{1}=2\pi/q_{1}$ of roughness spectrum, which can be
expressed in terms of its Power Spectral Density (PSD). With $q_{1}$ we have
denoted the cut-off wavevector value. The ratio
$\zeta=q_{1}/q_{\mathrm{L}}=\lambda_{\mathrm{L}}/\lambda_{1}$ is the so-called
magnification, being $q_{\mathrm{L}}=2\pi/\lambda_{\mathrm{L}}$ the wavevector
of roughness PSD that corresponds to the biggest wavelength
$\lambda_{\mathrm{L}}$.

Recently, Ciavarella ciava2020  has compared three different adhesion theories
of rough contact mechanics, namely Persson & Tosatti (PT) PT2001 , Persson &
Scaraggi (PS) PS2014  as re-elaborated by Violano et al. Violano1 , and BAM
Ciavarella2018 . In Refs. ciava2020 ; Violano1  it is shown that for self-
affine fractal surfaces with power-law PSD and low fractal dimensions
($D\lesssim 2.4$), stickiness is magnification-independent, depending mainly
on $h_{\mathrm{rms}}$ and $\lambda_{\mathrm{L}}$ , and therefore noticed that
the PR observation was affected by limitations in the computer capabilities
and the use of narrow spectra. This result is in agreement with the
theoretical findings of ref. JSB2017 , where a full Lennard-Jones traction-gap
law is exploited to study the adhesive contact between randomly rough surfaces
with broad roughness spectrum. In particular, it is shown that the probability
distribution of gaps converges with $\zeta$. The nominal mean traction at the
interface is computed by convolution of the probability distribution of gaps
with the Lennard-Jones traction-gap law, showing that the effect of smaller
roughness wavelength on interfacial tractions becomes negligible at high
$\zeta$.

Moreover, Violano et al. Violano2  studied the adhesive rough contact between
self-affine fractal surfaces with an advanced multiasperity model, where
adhesion is implemented according to Derjaguin, Muller & Toporov (DMT) force-
approach DMT . In their simulations, the pull-off force is almost independent
on the value of $h_{\mathrm{rms}}^{\prime}$, again in agreement with most of
the criteria introduced in refs. ciava2020 ; Violano1 . Specifically, adhesion
is rapidly destroyed by a little increase of $h_{\mathrm{rms}}$.

The very recent experimental investigations of Tiwari et al. tiwari  are also
consistent with the former results. In ref. tiwari , normal contact adhesion
experiments are carried out on rough samples with different $h_{\mathrm{rms}}$
and similar rms slope $h_{\mathrm{rms}}^{\prime}$. Experiments confirmed that
increase in $h_{\mathrm{rms}}$ without any change in
$h_{\mathrm{rms}}^{\prime}$ or $h_{\mathrm{rms}}^{\prime\prime}$ ”kills”
adhesion, leading to vanishing pull-off force, confirming with definitive
evidence that criteria like PT and PS as re-elaborated by Violano et al.
Violano1 , and BAM ciava2020  seem correct (at low fractal dimensions). The
former is a remarkable result, if one takes into account the intrinsic
difficulty in defining a value of $h_{\mathrm{rms}}^{\prime}$, or
$h_{\mathrm{rms}}^{\prime\prime}$ which is related to the sensitivity of
roughness measurement instrumentation dalvi ; ciavaFT ; Disc .

The debate is still open for the case at high fractal dimensions $D$. In this
work, we extend BAM and PT stickiness criteria to all fractal dimensions. We
shall use sometimes instead of the fractal dimension the Hurst exponent
$H=3-D$. We show that at low $H$, PT and BAM criteria may show a
magnification-dependence. Moreover, the two criteria predict that stickiness
could increase with $H$. This trend is in agreement with recent numerical
Violano3 ; popov2019  and theoretical findings JTB2018 , according to which
the pull-off force increases with $H$. The dependence with the Hurst exponent
is much more evident for PT solution when high magnifications are considered.

But the ”fractal limit” behaviour, i.e. the trend for $\zeta\rightarrow\infty$
is qualitatively different for PT and BAM, as we shall explore.

##  II Methods

In this paper, we use BAM theory Ciavarella2018  and the Persson and Tosatti
(PT) theory PT2001 , to derive stickiness criteria for arbitrarily wide power
law spectra, with low Hurst exponent $H$ (i.e. high fractal dimension).

###  II.1 BAM stickiness criterion

BAM Ciavarella2018  is developed loosely speaking in the framework of the DMT
theory DMT , in the sense that the repulsive adhesiveless solution and the
attractive adhesive one can be found separately. In particular, adhesion acts
in an attractive area outside of the compressive contact area and does not
deform the contact shape. BAM assumes a Maugis law of attraction Maugis ,
which permits an elegant and trivial estimate of the total force of attraction
in closed form Ciavarella2018 ; cp19 . In the simple case of spherical
contact, the model returns exactly the DMT solution by estimating the area of
attraction as the increase of the bearing area geometrical intersection, when
the indentation is increased by the Maugis range of attraction. In the case of
frictionless contact between randomly rough surfaces, BAM exploits the
repulsive adhesiveless Yang & Persson formulation YangPersson2008  in its
asymptotic version. The relation between the squeezing repulsive pressure
$p_{\mathrm{rep}}$ and the mean separation $u$ is

$p_{\mathrm{rep}}\left(u\right)=\beta E^{\ast}\exp\left(-u/u_{0}\right)$ (1)

where $u_{0}$ and $\beta$ are given by Yang & Persson formulation
YangPersson2008 ; cp19 . Fig. 1a shows that the normalized repulsive pressure
$\hat{p}=p_{\mathrm{rep}}/(E^{\ast}q_{\mathrm{L}}h_{\mathrm{rms}})$ rapidly
converges with $\zeta$ for $H=0.8$. Notice that, with
$E^{\ast}=E/\left(1-\nu^{2}\right)$ we have denoted the plane strain elastic
modulus, being $\nu$ the Poisson’s ratio. The pressure $\hat{p}(\zeta)$ still
converges when very high magnifications are reached for $H=0.3$ (fig. 1b). We
are interested in a range of normalized mean separation
$1<\bar{u}/h_{\mathrm{rms}}<3$. For greater values one could have finite
effects due to poor statistics of the surface. In the following, we will show
BAM predictions for a normalized mean separation $\bar{u}/h_{\mathrm{rms}}=2$.

Figure 1: The normalized repulsive pressure $\hat{p}$ as a function of
$\zeta$, for mean interfacial separation $\bar{u}/h_{\mathrm{rms}}=[1,2,3]$
and for $H=0.8$ (a) and $H=0.3$ (b). The quantity $\bar{u}/h_{\mathrm{rms}}$
increases in the direction indicated by the arrow.

Following ref. Ciavarella2018 ; cp19 , summing up repulsive and attractive
contributions for the random gaussian surface roughness, BAM gives to a very
elementary closed form solution

$\frac{\sigma\left(u\right)}{\sigma_{\mathrm{0}}}\simeq\beta
E^{\ast}\exp\left(-u/u_{0}\right)-\frac{1}{2}\left[Erfc\left(\frac{u-\epsilon}{\sqrt{2}h_{\mathrm{rms}}}\right)-Erfc\left(\frac{u}{\sqrt{2}h_{\mathrm{rms}}}\right)\right]$
(2)

where $Erfc$ is the complementary error function,
$\sigma\left(u\right)/\sigma_{\mathrm{0}}$ is the ratio between the actual
stress acting on the surface with respect to the theoretical stress of the
material $\sigma_{\mathrm{0}}=\Delta\gamma/\epsilon$, being $\Delta\gamma$ the
surface energy and $\epsilon$ the range of attractive forces. As pointed out
in Ref. ciava2020 , it is not possible to obtain a closed form solution for
BAM stickiness criterion. However, one can predict the decay in the pull-off
pressure $\sigma_{\mathrm{po}}$ with $h_{\mathrm{rms}}$ from eq. (2).
Stickiness can be defined when abrupt decay of pull-off pressure by 5 orders
of magnitude (for example) with respect to the theoretical stress is found
ciava2020 .

Figure 2: a) The $(h_{\mathrm{rms}}/\epsilon)_{\mathrm{thresh}}$ vs.
$\lambda_{\mathrm{L}}l_{\mathrm{a}}/\epsilon^{2}$ relation for BAM theory (red
solid line). Black dashed lines represents a power-law fit. Results are shown
for $H=[0.3,0.4,0.5,0.6,0.7,0.8]$ and $\zeta=10^{7}$. The Hurst exponent
increases in the direction indicated by the arrow. b) The
$\beta_{\mathrm{BAM}}\left(H\right)$ parameter that should be used in eq. (5)
for obtaining the power-law fit. The black dashed line denotes the linear fit
$\beta_{\mathrm{BAM}}\left(H\right)=0.9H-0.1$.

Fig. 2a shows the prediction of BAM model of the stickiness threshold
$(h_{\mathrm{rms}}/\epsilon)_{\mathrm{thresh}}$ as a function of the quantity
$l_{\mathrm{a}}\lambda_{\mathrm{L}}/\epsilon^{2}$, being
$l_{\mathrm{a}}=\Delta\gamma/E^{\ast}$ a characteristic length scale for
adhesion. Notice in particular that $l_{\mathrm{a}}/\epsilon\simeq 0.05$ for
the classical Lennard-Jones force-separation law. This results in a BAM
criterion, which generalize previous results of Ciavarella ciava2020

$\frac{h_{\mathrm{rms}}}{\epsilon}>\beta_{\mathrm{BAM}}\left(H,\zeta\right)\sqrt{\frac{l_{\mathrm{a}}}{\epsilon}\frac{\lambda_{\mathrm{L}}}{\epsilon}}\text{.}$
(3)

where $\beta_{\mathrm{BAM}}\left(H,\zeta\right)$ is a function of both $H$ and
$\zeta$. However, let us explore first the limit $\zeta\rightarrow\infty$.
This ”fractal limit” is well defined for BAM, since

$\lim_{\zeta\rightarrow\infty}\beta_{\mathrm{BAM}}\left(H,\zeta\right)=\beta_{\mathrm{BAM}}^{\infty}\left(H\right)$
(4)

exists and gives an effective stickiness boundary very close to that of the
low fractal dimensions, confirming the finding of the theories based on more
refined use of Lennard-Jones laws, as in refs. JSB2017 ; JTB2018 .

Results for $\beta_{\mathrm{BAM}}^{\infty}\left(H\right)$ are shown for
$H=[0.3,0.4,0.5,0.6,0.7,0.8]$. and $\zeta=10^{7}$ in fig. 2a. We will show in
the section dedicated to results, that BAM solution has already converged for
this value of the magnification, independently on the value of $H$. We can use
a power-law fit (black dashed lines) for approximating the red curves in fig.
2a. This leads to the following new and universal general BAM stickiness
criterion (but consider $H\in\left[0.3,0.8\right]$ as outside this range
results may show obviously some deviations)

$\frac{h_{\mathrm{rms}}}{\epsilon}>\left(0.9H-0.1\right)\sqrt{\frac{l_{\mathrm{a}}}{\epsilon}\frac{\lambda_{\mathrm{L}}}{\epsilon}}\text{.}$
(5)

where a linear fit for
$\beta_{\mathrm{BAM}}^{\infty}\left(H\right)=\beta_{\mathrm{BAM}}\left(H,\zeta>10^{7}\right)$
has been used. The effect of fractal dimension $D$ i.e. of $H$ is therefore
quite modest, contrary to the effect predicted by the PT theory, as we explore
in the next paragraph.

###  II.2 Persson and Tosatti stickiness criterion

PT theory PT2001  studies the adhesive contact between a randomly rough rigid
surface and an elastic half-space. PT argue with a energy balance between the
state of full contact and that of complete loss of contact that the effective
energy available at pull-off is

$\Delta\gamma_{\mathrm{eff}}=\frac{A}{A_{0}}\Delta\gamma-\frac{U_{\mathrm{el}}}{A_{0}}$
(6)

where $A$ is not the real contact area, but rather an area in full contact,
increased with respect to the nominal one $A_{0}$, because of an effect of
roughness-induced increase of contact area, so that $\frac{A}{A_{0}}>1$. We
shall neglect this effect in the following, as there is no consensus on the
importance of the area-term $\frac{A}{A_{0}}$, and in the interest of
simplicity. $\Delta\gamma$ is the surface energy corresponding to the smooth
case. Finally, $U_{\mathrm{el}}$ is the elastic strain energy stored in the
half-space having roughness with isotropic power spectrum $C\left(q\right)$
when this is squeezed flat

$\frac{U_{\mathrm{el}}\left(\zeta\right)}{A_{0}}=\frac{\pi
E^{\ast}}{2}\int_{q_{0}}^{q_{1}}q^{2}C\left(q\right)dq=E^{\ast}l\left(\zeta\right)\text{.}$
(7)

In eq. (7), we have integrated over wavevectors in the range $q_{0}$, $q_{1}$.
In the following we will assume $q_{0}=q_{\mathrm{L}}$. Notice that the
intrinsic assumption made in computing the elastic energy is that the
wavelengths in the integration range correspond to waves of roughness being
squeezed completely flat, when in contact against the rigid countersurface.
This is an approximation close to the JKR theory JKR , which becomes
questionable for very small scales, where we expect the actual range of
attractive forces being important. Indeed, the JKR model retains the Signorini
dichotomy between regions of contact and separation (see the review paper
Ciavarella2018adh ), but relaxes the requirement that contact tractions be
non-tensile. In the single sinusoid case, JKR model may cause deviations from
a more precise solution that includes the Lennard-Jones force-separation law,
when the amplitude of the sine wave becomes of the order of the attractive
range JTB2018 . Indeed, it is intuitive to expect that very small roughness
will induce almost no change in the pressure distribution, rather than a
strong effect as considered in eq. (7) when Hurst exponent is $H<0.5$. We have
introduced in (7) a length scale $l\left(\zeta\right)$ . In the fractal limit
$\zeta\rightarrow\infty$, the elastic energy
$U_{\mathrm{el}}\left(\zeta\right)$ is unbounded for surfaces with fractal
dimension $D\geq 2.5$ ($H\leq 0.5$), (see ref. Ciavarella2018 ). As a
consequence, PT theory would lead to vanishing adhesion even for arbitrarily
small rms height $h_{\mathrm{rms}}$. This is, again, due to the JKR-like
approximation.

For pure power law PSD $C\left(q\right)=Zq^{-2H-2}$

$l\left(\zeta\right)=\frac{\pi}{2}\int_{q_{\mathrm{L}}}^{q_{1}}q^{2}C\left(q\right)dq=\frac{\pi
Z}{2}\int_{q_{\mathrm{L}}}^{q_{1}}q^{-2H}dq=\frac{\pi}{2}\frac{h_{0}^{2}}{\lambda_{\mathrm{L}}}f\left(H,\zeta\right)$
(8)

where

$f\left(H,\zeta\right)=H\frac{\zeta^{-2H+1}-1}{-2H+1}$ (9)

is a function introduced in ref. PT2001  and
$h_{0}^{2}=2h_{\mathrm{rms}}^{2}$.

As stressed by PT, as long as $\Delta\gamma_{\mathrm{eff}}<0$, ”a finite pull-
off force will be necessary in order to separate the bodies”. In such case,
the stickiness limit is
$\Delta\gamma_{\mathrm{eff}}=\Delta\gamma-\frac{U_{\mathrm{el}}}{A_{0}}=0$,
which leads to the following stickiness criterion

$\frac{h_{\mathrm{rms}}}{\epsilon}>\beta_{\mathrm{PT}}\left(H,\zeta\right)\sqrt{\frac{l_{\mathrm{a}}}{\epsilon}\frac{\lambda_{\mathrm{L}}}{\epsilon}}$
(10)

with $\beta_{\mathrm{PT}}=\sqrt{\frac{f^{-1}\left(H,\zeta\right)}{\pi}}$. This
is quativelively very close to the BAM criterion, except of course for the
prefactor. For $H=0.8$, rapid convergence is obtained to
$\beta_{\mathrm{PT}}=0.49$, while for $H=0.3$ there is no convergence (see
fig. 3). This absence of convergence however is due to PT simple JKR-like
assumption which eventually at large $\zeta$ becomes questionable, when one
considers the actual Lennard-Jones distribution of forces at small scales, as
in refs. JSB2017 ; JTB2018 . This problem does not occur in the BAM model,
because the repulsive adhesionless contact gives converging results cp19 ,
while the adhesive force estimate is based on an entirely different idea,
purely geometrical one, which gives no dependence on the fractal dimension (or
Hurst exponent).

Figure 3: The $\beta_{\mathrm{PT}}(H,\zeta)$ parameter as a function of
$\zeta$, for $H=0.8$ (blue solid line) and $H=0.3$ (red solid line).

##  III Results

Figs. 4a-b show a comparison of the stickiness criteria (BAM, PT) for $H=0.8$
and $H=0.3$, respectively. As already shown in Ref. ciava2020 , for $H=0.8$,
BAM and PT solutions return very close results and rapidly converge with
$\zeta$. As a result, stickiness thresholds collapse into a single curve,
which is independent on the value of $\zeta$. The same trend is observed in
Ref. Violano1 , where a stickiness criterion has been derived from PS theory
PS2014 .

However, for $H=0.3$ (fig. 4b) and contrary to BAM, the PT criterion is
strongly magnification-dependent. This is due to the assumption in its
derivation, that we discussed. In particular, the stickiness threshold
decreases with increasing $\zeta$ both in PT and BAM solution, but it
converges for BAM only with increasing $\zeta$. This is in agreement with the
Lennard-Jones based theory developed in refs. JSB2017 ; JTB2018 , according to
which the probability distribution of gaps converges when smaller and smaller
roughness wavelengths are involved in the adhesive contact. We don’t report a
precise comparison with the theoretical findings of refs. JSB2017 ; JTB2018 ,
since for this theory stickiness criterion has not been derived, and we
haven’t implemented the relatively complex recursive integration procedure
involved.

Once again, to explain the convergence in the BAM model, we have shown in
figs. 2a-b the convergence of the mean repulsive pressure
$p_{\mathrm{rep}}\left(u\right)$ with $\zeta$, for both $H=0.3$ and $H=0.8$,
which corresponds to the convergence of BAM stickiness threshold in figs.
4a-b.

Figure 4: The stickiness criteria according to BAM theory (red solid line) and
PT theory (blue dashed line). The stickiness threshold divides the plot in two
regions: ”sticky” and ”not sticky”. Results are shown for $H=0.8$ (a) and
$H=0.3$ (b) with magnifications $\zeta=[10^{3},10^{4},10^{5}]$. The
magnification increases in the direction indicated by the arrow.

In order to understand the dependence with $\zeta$ at all fractal dimensions,
we show in fig. 5a the stickiness threshold as a function of the Hurst
exponent $H$ and for fixed values of the quantity
$\frac{\lambda_{\mathrm{L}}}{\epsilon}\frac{l_{\mathrm{a}}}{\epsilon}=10^{6}$
and $\zeta=[10^{2},10^{3},10^{5},10^{7},10^{9}]$. Fig. 5a confirms that PT
solution is very close to BAM at high Hurst exponent and at low
magnifications, where the JKR approximation gives no problem to the strain
energy calculation in the very simple and elegant PT criterion. In general,
the two criteria suggest that stickiness increases with increasing $H$.
However, this effect is much less noticeable for BAM criterion.

The discrepancies between BAM and PT are observed for low $H$, especially at
high magnifications, where eventually the two criteria differ quantitatively
and qualitatively, as discussed. As an example, fig. 5b shows the stickiness
threshold as a function of $\zeta$, for $H=[0.3,0.5]$ and $0.8$ with
$\frac{\lambda_{\mathrm{L}}}{\epsilon}\frac{l_{\mathrm{a}}}{\epsilon}=10^{6}$.

Figure 5: a) The $(h_{\mathrm{rms}}/\epsilon)_{\mathrm{thresh}}$ vs. $H$
relation for BAM theory (red solid line) and PT theory (blue dashed line).
Results are shown for $\zeta=[10^{2},10^{3},10^{5},10^{7},10^{9}]$ and
$\lambda_{\mathrm{L}}l_{\mathrm{a}}/\epsilon^{2}=10^{6}$. The magnification
increases in the direction indicated by the arrow. b) The
$(h_{\mathrm{rms}}/\epsilon)_{\mathrm{thresh}}$ vs. $\zeta$ relation for BAM
theory (red solid line) and PT theory (blue dashed line). Results are shown
for $H=[0.3,0.5,0.8]$ and
$\lambda_{\mathrm{L}}l_{\mathrm{a}}/\epsilon^{2}=10^{6}$.

##  IV Discussion

The results of BAM stickiness criteria we have derived are in general semi-
quantitative agreement with the Literature. In particular, in refs. JSB2017 ;
JTB2018 , a semi-analytical model for studying the adhesive contact between
fractal surfaces is developed. In such model, the details of the Lennard-Jones
force-separation law are considered, and neither DMT nor JKR type of
approximations are made, as instead done in BAM or PT, respectively. In ref.
JSB2017 , it is found that the gap distribution between adhesive surfaces
converges with magnification, in contrast with the JKR type of assumption made
by PT. Moreover, the effect of the fractal dimension is studied in ref.
JTB2018 . It is found that the pull-off force depends weakly on the fractal
dimension but, for relatively high rms roughness amplitude, it increases with
$H$.

In Ref. popov2019 , Li et al. used a BEM numerical code for predicting
adhesion between rough elastic spheres. They found that a finite pull-off
force can be detected for higher and higher surface roughness as $H$ is
increased. Fig. 6a shows the pull-off force $F_{\mathrm{po}}$ obtained in Ref.
popov2019 , normalized with respect to the JKR JKR  pull-off force
$F_{\mathrm{JKR}}=1.5\pi\Delta\gamma R$ for a smooth sphere of radius $R$.
Results are collected for $H=[0.1,0.3,0.5,0.7]$ and for increasing normalized
roughness parameter $h/R$, which is proportional to $h_{\mathrm{rms}}$. In
fig. 6a, we can select a different $(h/R)_{{}_{\mathrm{thresh}}}$ for each
value of $H$ that corresponds to vanishing pull-off force. Fig. 6b rearranges
$(h/R)_{{}_{\mathrm{thresh}}}$ as a function of $H$. As expected,
$(h/R)_{{}_{\mathrm{thresh}}}$ increases with $H$ and this is consistent with
our findings in fig. 5a.

Figure 6: a) The pull-off force $F_{\mathrm{po}}/F_{\mathrm{JKR}\text{ }}$vs.
the normalized surface roughness $h/R$. Results are shown for
$H=[0.1,0.3,0.5,0.7]$ and are taken from fig. 7A of Ref. popov2019 . The black
empty squares indicate the values of $(h/R)_{\mathrm{thresh}}$ for which
vanishing pull-off force is detected. b) The $(h/R)_{\mathrm{thresh}}$ values
as a function of $H$.

##  V Conclusions

We have compared BAM and PT stickiness criteria in the case of large fractal
dimensions, and we found that PT theory leads to a stickiness criterion which
strongly depends on the truncating large wavevector, showing less and less
stickiness if fine scale details are added. Such dependence is due to the
strong simplifying assumption in the theory which neglects the effective
Lennard-Jones distribution of the forces at the contact, and corresponds to a
JKR approximation which cannot hold at small scales. BAM doesn’t consider
Lennard-Jones forces either, but results in a converged result because it
estimates the effect of adhesion with a pure geometrical construction, while
the repulsive pressure converges with the magnification. PT and BAM criteria
return similar results at high $H$ (or low $D$) because the JKR approximation
in that case does not affect the convergence of the elastic strain energy.

The BAM solution shows a very simple universal result for stickiness, which
depends weakly on fractal dimension, and deserves of course numerical or
experimental final confirmation, which explains why we left the problem open
as in our title. At present, there are no experimental or detailed numerical
investigations on the effect of $\zeta$ on stickiness at very large
magnifications and at low $H$. However, BAM and Lennard-Jones based theories
JSB2017 ; JTB2018  seem to suggest the same theoretical limit. In other words,
our tentative conclusion is that BAM is qualitatively correct.

We showed that, for fixed magnification and rms roughness amplitude,
stickiness increases with $H$. This result is in agreement with the analytical
findings of refs. JSB2017 ; JTB2018 , according to which the pull-off force
could enhance with $H$, for relatively high rms values. The same trend has
been observed with numerical simulations in Refs. Violano3 ; popov2019 .

## Acknowledgements

AP is thankful to the DFG (German Research Foundation) for funding the project
PA 3303/1-1. AP acknowledges support from ”PON Ricerca e Innovazione 2014-2020
- Azione I.2 - D.D. n. 407, 27/02/2018, bando AIM. MC is thankful to Prof. JR
Barber and Mr. Junki Joe of University of Michigan for some discussions about
the fractal limit in their models. All authors acknowledge support from the
Italian Ministry of Education, University and Research (MIUR) under the
program ”Departments of Excellence” (L.232/2016).

## References

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