exceptional divisors in the chain of length $s-9$ again represent classes
$h-e_{1}-e_{2},e_{2}-e_{3},e_{3}-e_{4}$ or
$h-e_{1}-e_{2},e_{2}-e_{3},e_{1}-e_{2}$. Blowing down exceptional spheres,
this configuration descends to a single conic with two lines, which has a
unique symplectic isotopy class by [GS19, Theorem 1.5]. Thus this yields a
unique symplectic filling when $9\leq s\leq 11$. The two symplectic fillings
corresponding to the two homological embeddings when $s=12$ can be
distinguished by their intersection forms, as their non-torsion homology is
generated by $e_{1}-e_{2}-e_{3}-e_{4}+2e_{5}+2e_{6}$ or $e_{4}-e_{5}-e_{6}$
respectively as in the previous case. Similarly, we obtain no possible
homological embeddings when there are more than two $-2$-spheres in the chain
so when $s\geq 13$, there are no symplectic fillings of $(Y_{s},\xi_{s})$. ∎

###  7.3. Genus 3

When we allow the algebraic genus of the rational cuspidal curve to increase
to $3$, there are more options for the types of cusps. The first option is to
have a single cusp of type $(3,4)$ (multiplicity sequence $[3]$), where we
consider self-intersection numbers $s\geq 10$. The second is to have a single
cusp of type $(2,7)$ (multiplicity sequence $[2,2,2]$). The third option is to
have two cusps, one a simple cusp (multiplicity sequence $[2]$) and the other
of type $(2,5)$ (multiplicity sequence $[2,2]$). The fourth option is to have
three simple cusps (each with multiplicity sequence $[2]$). In the second,
third, and fourth options, we can classify fillings when the self-intersection
number satisfies $s\geq 13$.

######  Proposition 7.4.

Let $C$ be a rational cuspidal curve with self-intersection $s$ and a cusp of
type $(3,4)$ being the unique singularity, and let $(Y_{s},\xi_{s})$ be the
corresponding contact manifold.

  * •

When $10\leq s\leq 16$, $(Y_{s},\xi_{s})$ has a unique minimal symplectic
filling $W$ and $b_{2}(W)=16-s$.

  * •

When $s\geq 17$, $(Y_{s},\xi_{s})$ is not symplectically fillable.

###### Proof.

That there are no fillings when $s\geq 17$ follows from Proposition 6.2.

Figure 17. A blow-up of a curve $C$ with a one cusp of type $(3,4)$ and self-
intersection $s$, so that the proper transform $\tilde{C}$ has self-
intersection $+1$.

For $10\leq s\leq 16$, we proceed as in the lower genus cases. Blow up an
embedding of $C$ into $X$ once at the cusp to get the minimal smooth
resolution so that the exceptional sphere $E$ is tangent with multiplicity $3$
to the proper transform of $C$. We blow-up $s-10$ additional times at a
different point so that $\tilde{C}$ has self-intersection $1$, yielding a
chain of $s-10$ exceptional spheres as in Figure 17. Identifying $\tilde{C}$
with $\mathbb{C}\mathbb{P}^{1}$ by Theorem 4.1 so $[\tilde{C}]=h$, by Lemma
4.3, we see that $[E]=3h-2e_{1}-e_{2}-\cdots-e_{7}$, and the chain of $s-8$
exceptional spheres represent the classes in (a truncation of) the sequence
$h-e_{1}-e_{2},e_{2}-e_{3},e_{3}-e_{4},e_{4}-e_{5},e_{5}-e_{6},e_{6}-e_{7}$.
Blowing down exceptional spheres representing $e_{1},\cdots,e_{7}$ using Lemma
4.2, $E$ descends to a cubic curve with either a node or a cusp, $\tilde{C}$
descends to an inflection line, and the first divisor in the chain descends to
a line passing through the singular point of the cubic and one other point.
Without the last line, this configuration has a unique symplectic isotopy
class by [GS19, Proposition 5.11]. We get a unique symplectic isotopy class
for the configuration which includes the last line by [GS19, Proposition 5.1].
Thus there is a unique relatively minimal symplectic embedding of this blow-up
of $C$ into a closed manifold and that manifold must be
$\mathbb{C}\mathbb{P}^{2}\\#7\overline{\mathbb{C}\mathbb{P}}\,\\!^{2}$. We
conclude that $(Y_{s},\xi_{s})$ has a unique minimal symplectic filling with
$b_{2}=16-s$. ∎

######  Proposition 7.5.

Let $C$ be a rational cuspidal curve with self-intersection $s$ and either a
unique $(2,7)$ cusp, or one $(2,3)$ cusp and one $(2,5)$ cusp, and let
$(Y_{s},\xi_{s})$ be the corresponding contact manifold.

  * •

When $13\leq s\leq 15$, $(Y_{s},\xi_{s})$ has two minimal symplectic fillings
$W$ and $W^{\prime}$. $b_{2}(W)=16-s$ and $b_{2}(W^{\prime})=17-s$.

  * •

When $s=16$, $(Y_{s},\xi_{s})$ has three minimal symplectic fillings $W$,
$W^{\prime}$ and $W^{\prime\prime}$. $b_{2}(W)=16-s$ and
$b_{2}(W^{\prime})=b_{2}(W^{\prime\prime})=17-s$.

  * •

When $s\geq 17$, $(Y_{s},\xi_{s})$ is not symplectically fillable.

###### Proof.

Figure 18. A blow-up of a curve $C$ with a one cusp of type $(2,7)$ and self-
intersection $s$, so that the proper transform $\tilde{C}$ has self-
intersection $+1$. Figure 19. A blow-up of a curve $C$ with a one cusp of type
$(2,5)$ and one of type $(2,3)$ and self-intersection $s$, so that the proper
transform $\tilde{C}$ has self-intersection $+1$.

Proceeding as in prior cases, we blow up an embedding of $C$ to its minimal
resolution, then blow up $s-13$ additional times at another point until
$\tilde{C}$ has self-intersection $1$ and can be identified with
$\mathbb{C}\mathbb{P}^{1}$. The three exceptional divisors from the minimal
resolution must have homology classes as specified by the options shown in
Figures 18 and 19 (depending on whether there are $1$ or $2$ cusps). Note that
in both cases, there are two options for the homology class of the exceptional
divisor $E^{\prime\prime}$: $e_{6}-e_{7}$ or $e_{4}-e_{5}$. Also for both
cases, the homology classes of the divisors in the extra chain must represent
either $h-e_{1}-e_{2},e_{2}-e_{3},e_{3}-e_{4}$ or
$h-e_{1}-e_{2},e_{2}-e_{3},e_{1}-e_{2}$. If $[E^{\prime\prime}]=e_{4}-e_{5}$,
only the second option is possible. When $[E^{\prime\prime}]=e_{6}-e_{7}$,
both options are possible, but note that they only differ when all three
components are included corresponding to the $s=16$ case. Thus, we get three
cases when $s=16$ and two cases when $13\leq s\leq 15$. Note, there are no
allowed homological embeddings if the chain is longer, so $(Y_{s},\xi_{s})$
has no fillings when $s\geq 17$. Each configuration descends under blow-downs
to a configuration in $\mathbb{C}\mathbb{P}^{2}$ with a unique symplectic
isotopy class by [GS19, Theorem 1.5 and Proposition 5.1], so each homological
embedding corresponds to a unique minimal filling. If
$[E^{\prime\prime}]=e_{6}-e_{7}$, we have a relatively minimal symplectic
embedding into
$\mathbb{C}\mathbb{P}^{2}\\#7\overline{\mathbb{C}\mathbb{P}}\,\\!^{2}$. If
$[E^{\prime\prime}]=e_{4}-e_{5}$ the relatively minimal embedding is into
$\mathbb{C}\mathbb{P}^{2}\\#6\overline{\mathbb{C}\mathbb{P}}\,\\!^{2}$.
Symplectic fillings complementary an embedding in
$\mathbb{C}\mathbb{P}^{2}\\#7\overline{\mathbb{C}\mathbb{P}}\,\\!^{2}$ will
have $b_{2}=17-s$, and those which are complementary to an embedding in
$\mathbb{C}\mathbb{P}^{2}\\#6\overline{\mathbb{C}\mathbb{P}}\,\\!^{2}$ will
have $b_{2}=16-s$. Thus, when $13\leq s\leq 15$, the two fillings have
different $b_{2}$ so they are clearly not diffeomorphic. When $s=16$, there
are two potential fillings with $b_{2}=17-s$. To distinguish these, we look at
the square of the generator for the homology of the complement. In one case
the generator is $e_{1}-e_{2}-e_{3}-e_{4}+2e_{5}+2e_{6}+2e_{7}$ of square
$-16$ and in the other case the generator is $e_{4}-e_{5}-e_{6}-e_{7}$ of
square $-4$. ∎

######  Remark 7.6.

When $s=16$, in the minimal symplectic filling with $b_{2}=17-s=1$ where the
homology class of $E^{\prime\prime}$ agrees with the option for the filling
with $b_{2}=16-s=0$, we see that there is a symplectic $-4$-sphere in this
filling representing the class $e_{4}-e_{5}-e_{6}-e_{7}$. Rationally blowing
down this $-4$-sphere will yield another filling which is necessarily the
unique rational homology ball filling. The other minimal symplectic filling
with $b_{2}=17-s=1$ cannot be rationally blown-down (since there is no
$-4$-sphere in a manifold with intersection form $\langle-16\rangle$).

######  Proposition 7.7.

Let $C$ be a rational cuspidal curve with self-intersection $s$ and three
simple cusps, and let $(Y_{s},\xi_{s})$ be the corresponding contact manifold.

  * •

When $13\leq s\leq 16$, $(Y_{s},\xi_{s})$ has a unique minimal symplectic
filling $W$ and $b_{2}(W)=16-s$.

  * •

When $s\geq 17$, $(Y_{s},\xi_{s})$ is not symplectically fillable.

###### Proof.

The determination of homological embeddings has a similar classification to
the previous case. What is different in this situation is that not all of
these homological embeddings correspond to actual geometric embeddings, and we
need to prove a symplectic isotopy result for the one which does. The possible
homological embeddings are shown in Figure 20, where the additional chain of
$s-13$ exceptional divisors represent classes $h-e_{1}-e_{2}$, $e_{2}-e_{3}$,
and either $e_{3}-e_{4}$ or $e_{1}-e_{2}$. Note that when
$[E^{\prime\prime}]=2h-e_{1}-e_{2}-e_{3}-e_{5}-e_{6}$, and $s=16$, the last
divisor in the chain must represent $e_{1}-e_{2}$ to be disjoint from
$E^{\prime\prime}$. Therefore we have three homological embeddings.

Figure 20. A blow-up of a curve $C$ with a three cusps of type $(2,3)$ and
self-intersection $s$, so that the proper transform $\tilde{C}$ has self-
intersection $+1$.

For the two homological embeddings where $[E^{\prime\prime}]=2h-e_{1}-\cdots-
e_{4}-e_{7}$, if we suppose this can be realized symplectically in
$\mathbb{C}\mathbb{P}^{2}\\#7\overline{\mathbb{C}\mathbb{P}}\,\\!^{2}$, then
we can blow down exceptional spheres so that the image in
$\mathbb{C}\mathbb{P}^{2}$ of the resolution becomes three symplectic conics
in a pencil (all three intersecting at 4 points transversally or all three
intersecting at one point tangentially and two points transversally) with a
symplectic line tangent to all three conics. This configuration is obstructed
by [GS19, Proposition 5.23] (note that the statement refers to a pencil with
four transverse intersections, but the same proof applies to the case where
the pencil has one base point of multiplicity two–in the birational derivation
blow up once at each base point in the pencil). Therefore these homological
embeddings cannot be symplectically realized, so there are no corresponding
symplectic fillings.

For the last remaining homological embedding where
$[E^{\prime\prime}]=2h-e_{1}-e_{2}-e_{3}-e_{5}-e_{6}$ and the chain represents
a (truncation of) $h-e_{1}-e_{2}$, $e_{2}-e_{3}$, $e_{1}-e_{2}$, for any
symplectic realization of this embedding into
$\mathbb{C}\mathbb{P}^{2}\\#6\overline{\mathbb{C}\mathbb{P}}\,\\!^{2}$, there
will exist exceptional spheres so that the result blows down to a
configuration in $\mathbb{C}\mathbb{P}^{2}$ of three symplectic conics
intersecting at one common triple tangency (the image of $e_{1}$ and $e_{2}$),
one transverse triple point (the image of $e_{3}$) and a transverse double
point for each of the three pairs (the images of $e_{4},e_{5},e_{6}$),
together with one symplectic line tangent at the common tangency of the conics
and another a symplectic line tangent to all three conics at non-intersection
points. To verify this configuration has a unique symplectic isotopy class in
$\mathbb{C}\mathbb{P}^{2}$, we add a symplectic line through the two points
which pass through all three conics–this does not change the symplectic
isotopy classification by [GS19, Proposition 5.1]. Then we find a birational
equivalence as in Figure 21 by (1) blowing up twice at the tangency of the
conics and once at the triple intersection of the conics, then (2) blowing
down the proper transforms of the two lines which passed through intersection
points of the conics (which became $-1$-spheres after step (1)), and (3)
blowing down the remaining $-1$-sphere which is the image under the blow-downs
of step (2) of an exceptional $-2$-sphere introduced in step (1). The
resulting configuration contains a single conic (the image of the line which
had three tangencies to the original conics), and five lines—four of which are
tangent to the conic (the images of the three original conics and the image of
one of the exceptional divisors), and one of which passes transversally
through the tangency point of the fourth line with the conic (and otherwise
has generic transverse double point intersections). This configuration has a
unique symplectic isotopy class because it can be built from a single conic by
iteratively applying [GS19, Proposition 5.1]. Therefore, there is a unique
minimal symplectic filling corresponding to this homological embedding. ∎

Figure 21. Birational equivalence.

##  8\. Further speculations and questions

We collect in this section some ideas and questions for further investigation.

###  8.1. Handlebodies

In Section 2 we gave explicit Stein handlebody descriptions for all rational
homology ball fillings of the cuspidal contact structures associated to the
$\mathcal{A}_{p}$ and $\mathcal{B}_{p}$ families. Similar handlebody
descriptions are known for rational homology balls bounded by lens spaces or
by connected sums of lens spaces [LM14], thus covering also the two Fibonacci
families of [FLMN07, Theorem 1.1] described in the introduction.

######  Question 8.1.

Can one find an explicit Stein handlebody description of the rational homology
ball fillings of $\mathcal{E}_{3}$ and $\mathcal{E}_{6}$?

In [AGLL20], there is a (somewhat non-explicit) handle decomposition of a
topological rational homology ball bounding the corresponding Seifert fibered
spaces. Starting from [AGLL20, Figure 7], one can recover a handlebody diagram
for a rational homology ball bounding the cuspidal manifold of type
$\mathcal{E}_{3}$, as shown in Figure 22. Does this handlebody support a Stein
structure?

Figure 22. The handlebody description of a rational homology ball whose
boundary is the cuspidal manifold of type $\mathcal{E}_{3}$. The framing of
the $2$-handle, in Gompf’s convention from [Gom98], is $-9$.

Note that it would suffice to Legendrian-realize the attaching curve of the
2-handle so that the contact framing is one less than the topological framing:
that is, we are looking to find a Legendrian realization of the link of Figure
22 where the 2-handle has Thurston–Bennequin invariant $-8$ (in Gompf’s
convention [Gom98]). By the argument of Proposition 3.1, the contact structure
obtained on the boundary would automatically be the canonical one (which is
the cuspidal one, up to conjugation), and by the classification of fillings,
such a diagram would represent the unique rational homology ball filling of
$(Y_{C},\xi_{C})$ where $C$ is a curve of type $\mathcal{E}_{3}$.

###  8.2. Lefschetz fibrations and open books

In Sections 2 and 3 we have given explicit Stein handlebody decompositions for
at least one filling for each curve of type $\mathcal{A}_{p}$,
$\mathcal{B}_{p}$, $\mathcal{E}_{3}$, and $\mathcal{E}_{6}$. For the Fibonacci
families, such descriptions were already known to Honda [Hon00] and Giroux
[Gir00] (see also [LM14]).

######  Question 8.2.

Can one construct explicit Lefschetz fibrations on these fillings?

As a byproduct, one would also be able to recover an open book decomposition
for the cuspidal contact structures in this case. In fact, we conjecture that
the diagrams in Figures 23 and 24 depict the vanishing cycles of the fillings
in the $\mathcal{A}_{p}$ and $\mathcal{B}_{p}$ cases. We verified that the
underlying topology is the expected one.

Moreover, adding the orange vanishing cycle in Figure 23 (respectively the
orange vanishing cycle on the left of Figure 24) corresponds to attaching a
Weinstein $2$-handle along a Legendrian knot in $S^{1}\times S^{2}$ which is
in the same smooth isotopy class as the Legendrian knot in Figure 1
(respectively Figure 4). We see this by observing that Lefschetz fibrations
without these orange curves destabilize to the trivial Lefschetz fibration on
the annulus. In the $\mathcal{A}_{p}$ case, the Legendrian represented by the
orange vanishing cycle is stabilized $p-1$ times, and the unstabilized version
would correspond to the rational homology ball appearing in the standard
rational blow-down corresponding to a daisy relation. Starting from a Stein
handlebody diagram of this standard rational blow-down, it is not clear
whether the Lefschetz fibration depicted represents positive or negative
stabilizations of the Legendrian in that diagram, so there is some ambiguity
remaining on whether this is the correct Lefschetz fibration. Similar
arguments can be made in the $\mathcal{B}_{p}$ case, and a similar ambuiguity
remains.

Figure 23. Lefschetz fibration for the filling corresponding to the
$\mathcal{A}_{p}$ family.

Figure 24. Lefschetz fibrations for the two fillings corresponding to the
$\mathcal{B}_{p}$ family.

If the proposed open book decompositions do indeed describe the correct
contact structures, we would also obtain that all cuspidal contact structures
of types $\mathcal{A}_{p}$, $\mathcal{B}_{p}$, or Fibonacci are supported by a
planar open book. By contrast, by [GGP20, Theorem 1.2], we know that in the
cases of $\mathcal{E}_{3}$ and $\mathcal{E}_{6}$ the cuspidal contact
structures _cannot_ be planar. On the other hand, Etnyre and Ozbagci have
proved that these contact structures are supported by an open book of genus 1
[EO06].

## References

  * [AGLL20] Paolo Aceto, Marco Golla, Kyle Larson, and Ana G. Lecuona.  Surgerires on torus knots, rational balls, and cabling.  Preprint arXiv:2008.06760, 2020.
  * [BCG16] József Bodnár, Daniele Celoria, and Marco Golla.  Cuspidal curves and Heegaard Floer homology.  Proc. Lond. Math. Soc., 112(3):512–548, 2016.
  * [BO16] Mohan Bhupal and Burak Ozbagci.  Symplectic fillings of lens spaces as Lefschetz fibrations.  J. Eur. Math. Soc. (JEMS), 18(7):1515–1535, 2016.
  * [BS11] Mohan Bhupal and András I. Stipsicz.  Weighted homogeneous singularities and rational homology disk smoothings.  Amer. J. Math., 133(5):1259–1297, 2011.
  * [EG91] Yakov Eliashberg and Mikhael Gromov.  Convex symplectic manifolds.  In Several complex variables and complex geometry, Part 2, Proceedings of Symposia in Pure Mathematics, pages 135–162, 1991.
  * [Eli90] Yakov Eliashberg.  Filling by holomorphic discs and its applications.  In Geometry of low-dimensional manifolds, 2 (Durham, 1989), volume 151 of London Math. Soc. Lecture Note Ser., pages 45–67. Cambridge Univ. Press, Cambridge, 1990.
  * [EO06] John B. Etnyre and Burak Ozbagci.  Open books and plumbings.  Int. Math. Res. Not. IMRN, 2006:72710, 2006.
  * [FLMN07] Javier Fernández de Bobadilla, Ignacio Luengo, Alejandro Melle Hernández, and Andras Némethi.  Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair.  In Real and complex singularities, Trends Math., pages 31–45. Birkhäuser, Basel, 2007.
  * [FS97] Ronald Fintushel and Ronald J. Stern.  Rational blowdowns of smooth $4$-manifolds.  J. Differential Geom., 46(2):181–235, 1997.
  * [FS06] Ronald Fintushel and Ronald J. Stern.  Double node neighborhoods and families of simply connected 4-manifolds with $b^{+}=1$.  J. Amer. Math. Soc., 19(1):171–180, 2006.
  * [GGP20] Paolo Ghiggini, Marco Golla, and Olga Plamenevskaya.  Surface singularities and planar contact structures.  Ann. Inst. Fourier (Grenoble), 70(4):1791–1823, 2020.
  * [Ghi08] Paolo Ghiggini.  On tight contact structures with negative maximal twisting number on small Seifert manifolds.  Algebr. Geom. Topol., 8(1):381–396, 2008.
  * [Gir00] Emmanuel Giroux.  Structures de contact en dimension trois et bifurcations des feuilletages de surfaces.  Invent. Math., 141(3):615–689, 2000.
  * [GK20] Marco Golla and Fabien Kütle.  Symplectic isotopy of rational cuspidal sextics and septics.  Preprint arXiv:2008.10923, 2020.
  * [Gom98] Robert E. Gompf.  Handlebody construction of Stein surfaces.  Ann. of Math. (2), 148(2):619–693, 1998.
  * [GS99] Robert E. Gompf and András I. Stipsicz.  $4$-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 1999.
  * [GS19] Marco Golla and Laura Starkston.  Symplectic isotopy of rational curves.  Preprint, arXiv:1907.06787, 2019.
  * [Har69] Robin Hartshorne.  Curves with high self-intersection on algebraic surfaces.  Publ. Math. I.H.E.S., 36:111–125, 1969.
  * [Hon00] Ko Honda.  On the classification of tight contact structures, I.  Geom. Topol., 4(1):309–368, 2000.
  * [Kap79] Steve J. Kaplan.  Constructing framed $4$-manifolds with given almost framed boundaries.  Trans. Amer. Math. Soc., 254:237–263, 1979.
  * [Küt21] Fabien Kütle.  Courbes symplectiques de haute auto-intersection dans les surfaces symplectiques.  PhD thesis, Université de Nantes, 2021.  Preprint, arXiv:2111.04836.
  * [Lis08] Paolo Lisca.  On symplectic fillings of lens spaces.  Trans. Amer. Math. Soc., 360(2):765–799, 2008.
  * [Liu14] Tiankai Liu.  On planar rational cuspidal curves.  PhD thesis, Massachussets Institute of Technology, 2014.
  * [LM14] Yankı Lekili and Maksim Maydanskiy.  The symplectic topology of some rational homology balls.  Comment. Math. Helv., 89(3):571–596, 2014.
  * [LS04] Paolo Lisca and András I. Stipsicz.  Ozsváth-Szabó invariants and tight contact three-manifolds. I.  Geom. Topol., 8:925–945, 2004.
  * [Mat18] Irena Matkovič.  Classification of tight contact structures on small Seifert fibered $L$-spaces.  Algebr. Geom. Topol., 18(1):111–152, 2018.
  * [McD90] Dusa McDuff.  The structure of rational and ruled symplectic $4$-manifolds.  J. Amer. Math. Soc., 3(3):679–712, 1990.
  * [Mos71] Louise Moser.  Elementary surgery along a torus knot.  Pacific J. Math., 38(3):737–745, 1971.
  * [OO99] Hiroshi Ohta and Kaoru Ono.  Simple singularities and topology of symplectically filling 4-manifold.  Comment. Math. Helv., 74:575–590, 1999.
  * [OO05] Hiroshi Ohta and Kaoru Ono.  Symplectic $4$-manifolds containing singular rational curves with $(2,3)$-cusp.  Sémin. Congr., 10:233–241, 2005.
  * [OS12] Brendan Owens and Sašo Strle.  Dehn surgeries and negative-definite four-manifolds.  Selecta Math. (N.S.), 18(4):839–854, 2012.
  * [OSz03] Peter S. Ozsváth and Zoltán Szabó.  Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary.  Adv. Math., 173(2):179–261, 2003.
  * [OSz04] Peter Steven Ozsváth and Zoltán Szabó.  Holomorphic disks and topological invariants for closed three-manifolds.  Ann. of Math. (2), 159(3):1027–1158, 2004.
  * [OSz05] Peter S. Ozsváth and Zoltán Szabó.  Heegaard Floer homology and contact structures.  Duke Math. J., 129(1):39–61, 2005.
  * [Par97] Jongil Park.  Seiberg-Witten invariants of generalised rational blow-downs.  Bull. Austral. Math. Soc., 56(3):363–384, 1997.
  * [Par05] Jongil Park.  Simply connected symplectic 4-manifolds with $b^{+}_{2}=1$ and $c^{2}_{1}=2$.  Invent. Math., 159(3):657–667, 2005.
  * [PSS05] Jongil Park, András I. Stipsicz, and Zoltán Szabó.  Exotic smooth structures on $\mathbb{CP}^{2}\\#5\overline{\mathbb{CP}^{2}}$.  Math. Res. Lett., 12(5-6):701–712, 2005.
  * [SS05] András I. Stipsicz and Zoltán Szabó.  An exotic smooth structure on $\mathbb{C}\mathbb{P}^{2}\\#6\overline{\mathbb{C}\mathbb{P}^{2}}$.  Geom. Topol., 9:813–832, 2005.
  * [SSW08] András I. Stipsicz, Zoltán Szabó, and Jonathan Wahl.  Rational blowdowns and smoothings of surface singularities.  J. Topol., 1(2):477–517, 2008.
  * [Sym98] Margaret Symington.  Symplectic rational blowdowns.  J. Differential Geom., 50(3):505–518, 1998.
  * [Sym01] Margaret Symington.  Generalized symplectic rational blowdowns.  Algebr. Geom. Topol., 1:503–518, 2001.
  * [Tos20] Bülent Tosun.  Tight small Seifert fibered spaces with $e_{0}=-2$.  Algebr. Geom. Topol., 20(1):1–27, 2020.
  * [Tur02] Vladimir Turaev.  Torsions of $3$-dimensional manifolds, volume 208 of Progress in Mathematics.  Birkhäuser Verlag, Basel, 2002.
  * [Wal04] C. T. C. Wall.  Singular points of plane curves, volume 63.  Cambridge University Press, 2004.
  * [Wen10] Chris Wendl.  Strongly fillable contact manifolds and $J$-holomorphic foliations.  Duke Math. J., 151(3):337–384, 2010.


# Can one predict a drop contact angle?

Marion Silvestrini∗,1,2 Antonio Tinti2 Alberto Giacomello2 Carolina Brito1
1Instituto de Física  Universidade Federal do Rio Grande do Sul  Av. Bento
Gonçalves 9500  CEP 91501-970  Porto Alegre  Brazil
2Dipartimento di Ingegneria Meccanica e Aerospaziale  Università di Roma “La
Sapienza  00184 Rome  Italy

###### Abstract

The study of wetting phenomena is of great interest due to the multifaceted
technological applications of hydrophobic and hydrophilic surfaces. The
theoretical approaches proposed by Wenzel and later by Cassie and Baxter to
describe the behaviour of a droplet of water on a rough solid were extensively
used and improved to characterize the apparent contact angle of a droplet.
However, the equilibrium hypothesis implied in these models means that they
are not always predictive of experimental contact angles due to strong
metastabilities typically occurring on heterogeneous surfaces. A predictive
scheme for contact angle is thus urgently needed both to characterise a
surface by contact angle measurements and to design superhydrophobic and
oleophobic surfaces with the desired properties, e.g., contact angle
hysteresis. In this work a combination of Monte Carlo simulation and the
string method is employed to calculate the free energy profile of a liquid
droplet deposited on a pillared surface. For the analyzed surfaces, we show
that there is only one minimum of the free energy that corresponds to the
superhydrophobic wetting state while the wet state can present multiple
minima. Furthermore, when the surface roughness decreases the amount of local
minima observed in the free energy profile increases.

∗email<EMAIL_ADDRESS>
##  1 Introduction

Contact angle is perhaps the most immediate way to characterise the
wettability of a given surface, due to the ease of depositing a drop and
estimate the angle formed by it. Indeed, the Young equation allows one to
relate the surface tensions with the contact angle on an ideally smooth
surface [1]. However, the crucial question is whether this observable is a
reliable one in the case of actual surfaces, in which heterogeneities are
inevitably present.

Surface heterogeneities can dramatically change the contact angle as compared
to the case of an ideal, homogeneous surface. In particular, topographical
heterogeneities tend to magnify the intrinsic wettability of the surface,
making them “superhydrophobic” or “superhydrophilic” [2]. Since the pioneering
work of Wenzel [3] and Cassie and Baxter [4], the concept of _apparent_
contact angle has been used, which is meant to predict the angle measured at
the actual surface accounting for the effect of both wettability and
heterogeneity. Since then, a continued effort has been spent to provide
accurate models for predicting the apparent contact angles based on the
chemical and topographical characteristics of the surface. The main ambition
of this line of research is finding a rapid and inexpensive way to
characterise surface wettabilities via contact angle measurements.

The standard approach to develop wetting models for heterogeneous surfaces
consists in assuming a wetting state of the surface and predict the contact
angle based on the homogenised surface energy of such composite interface. The
task becomes more involved when a surface allows multiple visibly different
states, e.g., for superhydrophobic surfaces in which the roughness can be
fully wet (Wenzel, W) or dry (Cassie-Baxter, CB).

Importantly, major difficulties arise in relating the apparent contact angle
and surface characteristics, most importantly due to _static_ contact angle
hysteresis (CAH), i.e., a scatter of the contact angle values which depends on
the experimental procedure or history. Already in 1964, Johnson and Dettre [5]
showed that an idealised sinusoidal roughness can be an important source of
CAH; they further showed that the W state is connected to larger CAH than the
CB state. More recent work further showed that CAH emerges from micro and
nanoscale phenomena which are beyond the scope of Cassie and Wenzel equations
[6]. A heated debate originated from the question whether “Cassie and Wenzel
were wrong” [7, 8, 9]. The main merit of such discussion was to highlight that
the experimental contact angle may be determined by features localised at the
three-phase contact line where the drop meets the surface. In fact, the
classical CB and W models assume an equilibrium hypothesis in which the
droplet is allowed to relax to the absolute free energy minimum. In other
words, due to the rough free energy landscape featuring multiple minima, the
measured contact angle may significantly depart from the value predicted based
on averaged surface characteristics and the commonly used wetting models [10,
11, 12].

Several attempts have been made to relate the characteristics of single
defects to CAH, starting from the pioneering work of Joanny and de Gennes [13,
14, 15, 16]. However, densely packed heterogeneities [17], together with the
finite size of the drop and the multiple wetting states possible on
superhydrophobic surfaces [18, 19, 20, 21, 22, 23] considerably complicates
the matter. Indeed to date, notwithstanding the theoretical and practical
importance, the analysis of the free energy landscape connected to wetting of
a realistic surface by a droplet is still lacking. Thus making a connection
between measured contact angles and available wetting models is highly needed
in the community. At the same time, the difficulties connected to the
exploration of rough free energy landscapes [24] represent a major obstacle to
develop the computational tools required for engineering textured surfaces
with the desired wetting properties. In this work, we fill such a gap,
providing a generic framework for reconstructing the free energy connected to
wetting of a pillared superhydrophobic surface based on the combination of the
cellular Potts model [25] and the (zero temperature) string method for rare
events [26, 27, 28]. This approach allows us to assess the variation of CAH
with the pillar spacing, to connect CAH with the existence of multiple free
energy minima, and to verify the energetic origin of the superhydrophobicity
of the CB state. In addition, the presented approach provides useful
guidelines for contact angle measurements on actual surfaces and for designing
surfaces with tailored wetting properties, notably superhydrophobic ones.

The paper is organised as follows. In Section 2 we present the numerical setup
used: a Monte Carlo scheme combined with the string method. Section 3 contains
the results of the free energy calculations and the physical origin of its
saddle points. In Section 4, we discuss the origin of contact angle hysteresis
and compare the CB and W models with the simulations. The paper is then
concluded in Section 5.

##  2 Numerical Model

The main aim of our simulation campaign is connecting the apparent contact
angles attainable on a given surface with its physical and chemical
characteristics. We focus on a hydrophobic surface decorated with a three-
dimensional array of pillars with the geometry shown in Figure 1-a. A three-
dimensional spherical droplet with volume $V_{0}=4/3\pi R_{0}^{3}$ is placed
over the surface. We use a mesoscale wetting model, the cellular Potts model
[25, 29, 30], which has been successfully used in studying the wetting of
structured surfaces [31, 32, 33, 34, 35]. Its low computational cost as
compared to, e.g., molecular dynamics, allows computations of relatively large
3D drops via Monte Carlo minimisation.

We combine the MC minimisation with the string method [26, 28] in order to
calculate the most probable path leading from the superhydrophobic CB to the
fully wet W state or vice versa. The string method, combined with atomistic
simulations, diffuse interface methods, or sharp interface models, has proven
instrumental to clarify the origin of metastability in wetting of structured
surfaces [23, 36, 37, 38]. We will employ this approach to overcome the free
energy barriers, to establish the mechanism of breakdown/recovery of the CB
state, and to identify intermediate minima.

Figure 1: Sketch of the surface decorated with pillars. (a) Geometrical
parameters defining the texture: $\mathrm{w}$ is the pillar width,
$\mathrm{h}$ is the pillar height, $\mathrm{a}$ is the interpillar distance
and $\mathrm{d}=\mathrm{w}+\mathrm{a}$. (b) Top view of the surface. The blue
region delimits one cavity, used in the definition of the collective
variables.

###  2.1 The cellular Potts model

The Cellular Potts model (CPM) assumes a three state system in a simple cubic
lattice, with each site representing a different state: solid, liquid, and
gas. We employ this technique to simulate a three-dimensional droplet in
contact with air and a solid surface. The Hamiltonian of the system is given
by:

$H=\frac{1}{2}\sum\limits_{\langle{\rm i,j}\rangle}E_{s_{\rm i},s_{\rm
j}}\left(1-\delta_{s_{\rm i},s_{\rm
j}}\right)+\frac{\kappa}{2}\sum\limits_{l=1}^{l=N}\left(z_{l}-z_{l}^{T}\right)^{2},$
(1)

where $s_{\rm i}\in\\{0,1,2\\}$ is the state of a given site in the lattice,
gas (G), liquid (L), or solid (S). The first term on the right hand side
describes the surface interaction between the three states, with ${E_{s_{\rm
i},s_{\rm j}}=J_{\scriptscriptstyle\rm ij}\cdot a_{\scriptscriptstyle\rm L}}$,
where $J_{\scriptscriptstyle\rm ij}$ is the site-site coupling between $i$ and
$j$ type sites, $a_{\scriptscriptstyle\rm L}=L^{2}$ is the unit area and $L$
is the lattice size. The summation ranges over the $n_{\scriptscriptstyle\rm
neigh}=26$ nearest and next-nearest lattice sites of the 3D simple cubic
lattice. Notice that the energy contribution will be non-zero only when
$s_{\rm i}\neq s_{\rm j}$. Gravitational contributions to the energy are
neglected due to small size of the droplets hereby investigated [39, 34, 33].

The second term in Equation (1) corresponds to a harmonic bias used to drive
the system across metastabilities allowing to more effectively explore phase
space [27] using general descriptors, the _collective variables_ $z$, which
are function of the lattice sites. Biasing is used because, in an unrestrained
simulation ($\kappa=0$), drops relax and remain trapped in the closest minimum
state. Simulation modeling a droplet deposited on a textured surface without
any bias were previously studied in Refs. [33, 40], where the authors indeed
showed that the final contact angle depends on the initial wetting
configuration.

The collective variables $z$ that appear in Equation (1) should be chosen
carefully [41] in order to correctly discriminate among relevant
configurations of the system. More in detail $z^{T}$ represents the target
value for the collective variable, while $\kappa$ is stiffness of the harmonic
constraint. In this work we shall identify $z_{l}$ with liquid-phase
occupation of $N$ interpillar grooves. The usage of such discrete density
indicators is customary when dealing with wettability problems [42].

The energy scale is set once $J_{\scriptscriptstyle\rm ij}$ is defined. In
this work spin-spin couplings are chosen to render the same ratios between the
interfacial surface tensions as in Ref. [43], in which a droplet of water is
studied on a surface composed by Polydimethylsiloxane (PMDS) with the
following surface tensions $\sigma_{\scriptscriptstyle\rm ij}$ between the gas
(G), liquid (L), and solid (S): $\sigma_{\scriptscriptstyle\rm GL}=70\times
10^{-3}$ N/m, $\sigma_{\scriptscriptstyle\rm SG}=25\times 10^{-3}$ N/m, and
$\sigma_{\scriptscriptstyle\rm SL}=53\times 10^{-3}$ N/m; the last value was
calculated by Young’s equation, $\sigma_{\scriptscriptstyle\rm
GL}\cos\theta_{\scriptscriptstyle\rm Y}=\sigma_{\scriptscriptstyle\rm
SG}-\sigma_{\scriptscriptstyle\rm SL}$ with $\theta_{\scriptscriptstyle\rm
Y}=114^{\circ}$. The data will be presented in dimensionless form using the
lattice size $L$ as unit length, $a_{\scriptscriptstyle\rm L}=L^{2}$ as unit
area, and $J_{\mathrm{GL}}\cdot a_{\scriptscriptstyle\rm L}$ as unit energy,
referred as e.u. In such reduced units, $J_{\mathrm{GL}}=1$,
$J_{\mathrm{SG}}=0.36$, and $J_{\mathrm{SL}}=0.76$. For the forcing parameter
we use $\kappa\simeq 0.004$, which drives the system to the target
configuration while allowing some fluctuation. More details about the choice
of $\kappa$ is reported in the _Supporting Information_ (SI). In order to
convert the dimensionless units to physical ones, one can for instance assume
$L=1\,\mu$m and the energy for the gas-liquid interface of water, which yields
$\sigma_{\scriptscriptstyle\rm GL}a_{\scriptscriptstyle\rm L}=7\times
10^{-14}$ J.

To evolve the system we use the Metropolis-Hastings algorithm, which consists
in changing the state of two random sites at the gas-liquid interface with an
acceptance rate equal to min$\\{1,{\rm exp}[-\beta\Delta H]\\}$, where
$\beta=1/T$ is the inverse of the effective temperature of the CPM, which
introduces some noise, allowing for a more effective exploration of the phase
space. We set $T=4.8$ throughout the article, which allows the system to
fluctuate with an acceptance rate of approximately 9$\%$. The attempted MC
moves consist in swaps between liquid and gas sites, which guarantees that the
volume of the droplet is constant throughout the simulation.

###  2.2 The string method

In order to attempt a simulation of the wetting of rough surfaces it is
crucial to tackle the challenge associated with the presence of
metastabilities. Previous methods to sample rough free energy landscapes
required projecting to a low dimensional representation of the free energy as
a function of a handful of parameters and reconstructing the full landscape as
a function of such collective parameters. It is easy to understand how this
task requires a computational effort which is exponential in the number of
collective variables thus limiting the applicability of this class of methods
typically to two/three variables. Such low dimensional representation of the
energetic landscape often results in a poor description of the phenomena [41].
The (zero temperature) string method in collective variables was first
introduced by Vanden-Eijnden and collaborators [27] and is a path method that
requires only the computation of the _local_ gradient of the free energy
landscape at certain points along the path; its computational cost thus only
grows linearly with the number of collective variables.

The string method allows for a fast and convenient identification of the
minimum free energy path (MEP) connecting two metastable regions of the rough
landscape, along with providing a measure of the free energy along the path,
without requiring to fully sample a high dimensional variable space. This is
achieved by iteratively refining a discretized guess path (i.e., the string).
In this work the collective variables (CV) represent the liquid occupation of
each cavity, as represented by the blue region in Figure 1-b. The available
volume in the cavity is given by $(\mathrm{d}^{2}-\mathrm{w}^{2})\mathrm{h}$.
The algorithm proceeds as follows:

  1. 1.

We run $N_{r}$ Monte Carlo (MC) replicas, each biased to explore the vicinity
of a $z^{T}_{l,i}$ point in the collective variable space. Each replica
corresponds a string point in the collective variable space.

  2. 2.

The $N_{r}$ replicas run for $10^{5}$ MC moves allowing to sample the mean
biasing forces and the local free energy gradient at each string point.

  3. 3.

The string is updated by a convenient gradient descent of the string points
and reparametrized [28] so that one has a new set of $z^{T}_{l,i}$. The
process is iterated from point 1.

Refinement of the initial string guess is repeated until convergence is
reached, as evaluated by a suitable threshold in the changes to the path/path
energy. In all runs convergence was obtained within 20 iterations. At
convergence, numerical integration of the free energy gradients allows for the
reconstruction of the free energy profile along the string:

$\Delta\mathcal{F}_{j}=\sum_{i=1}^{j}\sum_{l=1}^{l=N}-\kappa\langle(z_{l,i}-z^{T}_{l,i})\rangle\Delta
z^{T}_{l,i}\text{ ,}$ (2)

with $\Delta\mathcal{F}_{j}$ the free energy at point $z^{T}_{l,j}$, the index
$l$ running over the $N$ collective variables, $i$ up to the current replica
$j$, with $j\leq N_{r}-1$, and $\Delta z^{T}_{l,i}=z^{T}_{l,i+1}-z^{T}_{l,i}$.
The results shown in this work are averages over the distinct realizations of
the simulation and $\langle\dots\rangle$ represents the average over MC steps.

##  3 Results

We consider three substrates with same pillar height $\mathrm{h}=10L$ and
width $\mathrm{w}=5L$ and different distances between pillars: substrate
referred to as S1 presents a low interpillar distance ($\mathrm{a}=5L$), S2 an
intermediate value ($\mathrm{a}=8L$), while S3 has the largest interpillar
distance ($\mathrm{a}=11L$). The volume of the spherical droplet is defined by
imposing an initial radius of $R_{0}=50L$. We first show the rough free energy
landscape connected with wetting of the three surfaces, characterizing the
droplet configuration at the minima and maxima of the free energy. We then
introduce a possible physical explanation of the roughness in the free energy
and end this section discussing the connection between the free energy
landscape and contact angle hysteresis.

###  3.1 Rough free energy of a hydrophobic pillared surface

Figure 2-a reports, for substrates S1, S2, and S3, the free energy of the drop
as a function of the total filling of the cavities below the drop, which is
defined as the sum of the target filling of each cavity divided by the total
volume of the droplet: $f=\sum_{l}z_{l}^{T}/V_{0}$. The minima in the free
energy correspond to stable (global) and metastable (local) states of the
system. The free energy extrema are numerically identified from the data;
minima are indicated in Figure 2-a as black circles, while maxima by red
triangles. Since some minima are very shallow and may be affected by numerical
accuracy we performed numerical MC minimisation using different initial
conditions close to the putative minima, to confirm that the system can indeed
reside in such states; empty symbols denote shallower minima which will not be
further discussed. Figure 2-b shows the fraction of water wetting the bottom
area. The vertical dotted line in the figure roughly separates two regimes: on
the left, the liquid does not touch the bottom of the substrate, indicating
that the corresponding configurations of the droplet are associated to the
superhydrophobic CB state, while, on the right of the line, the liquid reaches
to the bottom of the substrate and the droplet configurations are associated
with the wet W state(s). Importantly, the free energy landscape appears rough
for all considered surfaces, with markedly different trends for the three
interpillar distances: monotonically growing for S1, almost at the CB-W
coexistence, with additional high free energy minima, for S2, and with a
significant W basin with multiple local minima for S3.

Figure 2: (a) Free energy profile as a function of the liquid filling the
cavities for the three substrates. Minima are identified with black circles
and maxima by red triangles; empty symbols denote very shallow minima (and
related maxima) in which the system does not remain after standard MC
minimisation. (b) Fraction of the bottom area of the substrate in contact with
the liquid, normalized by the total substrate area. (c) Lateral view of the
minima and maxima in the wetting of substrate S3. The inset in (a) defines the
left and right barriers, $H^{L}$ and $H^{R}$, respectively.

Each point on the free energy profiles corresponds to a droplet configuration,
whose sequence thus defines a wetting path, see Figure 2-c; in particular,
these paths represent the most probable way in which the transition from CB to
one of the various W states (or vice versa) occurs. Other paths may exist
connecting minima, especially in such complex landscape, see e.g. Ref. [44].
Figure 2-c presents a lateral view of the 3D droplet in correspondence of the
minima and maxima of substrate S3. Minimum I corresponds to the CB state, with
air trapped between the pillars – this numbering is the same for the three
substrates. As the droplet starts infiltrating the substrate, it touches the
bottom for the first time at point II, where free energy increases to a
maximum. A similar behaviour was also reported in molecular dynamics
simulations [23, 45, 46, 47]. As the filling level increases, the free energy
of the substrate presents several local minima which correspond to the
progressive infiltration of liquid within the pillars: lateral views of the
droplet in Figure 2-c clearly show full wetting of 3 lines of cavities at
minimum III, 4 lines at V, 5 lines at VII. For filling levels above 7 lines of
cavities, the free energy does not present any other minima for this droplet
size.

The maxima separating the mentioned minima are found to be associated to an
incomplete wetting of some of the cavities at the drop perimeter; we will
further analyse their origin in Section 3.2. We note that the wetting of the
substrate in 3D is a more complex problem than what can be inferred from a
lateral view, which is however convenient to picture the main features of the
process; we thus make available in the _Supporting Information_ videos in 2D
and 3D of the droplet wetting the 3 substrates.

The inset of Figure 2-a defines the barrier $H^{L}$, which is the difference
in $\Delta\mathcal{F}$ between a minimum and the consecutive maximum on its
left and $H^{R}$, being the difference in $\Delta\mathcal{F}$ between a
minimum and the first maximum on its right. Using these definitions, we found
that the barrier $H^{R}$ and $H^{L}$ are typically of the same order of
magnitude for substrate S3, while for S1 $H^{L}\ll H^{R}$. We will come back
to this point later.

These observations raise several questions, which are addressed in the next
section: why some substrates present global minima at the CB state and others
in the W configuration? What is the physical origin and significance of the
local minima and the intervening maxima?

###  3.2 Physical origin of the minima and maxima of the free energy

The wetting state corresponding to the global free energy minimum is different
for the considered substrates and can be rationalized using a continuous
global energy model [33]. This model takes into account the energy of creating
interfaces between the liquid, the gas, and the solid of a spherical droplet
with fixed volume $V_{0}$ placed on a surface. The droplet is allowed to
display two wetting states: it either stays in the superhydrophobic CB state
or it homogeneously wets the surface in the W state. We refer to these
configurations as ideal Cassie-Baxter, CBI and ideal Wenzel, WI, states. The
difference in energy of a system with and without the droplet on the surface
can be written as [33]:

$\displaystyle\Delta E^{\bf{CB^{I}}}$ $\displaystyle=$
$\displaystyle\sigma_{\scriptscriptstyle\rm
GL}~{}\left[~{}S_{\scriptscriptstyle\rm CB}-\pi B_{\scriptscriptstyle\rm
CB}^{2}\left(\phi_{\scriptscriptstyle\rm S}\cos\theta_{\scriptscriptstyle\rm
Y}-(1-\phi_{\scriptscriptstyle\rm S})\right)~{}\right],$ (3)
$\displaystyle\Delta E^{\bf{W^{I}}}$ $\displaystyle=$
$\displaystyle\sigma_{\scriptscriptstyle\rm
GL}~{}\left[~{}S_{\scriptscriptstyle\rm W}-\pi B_{\scriptscriptstyle\rm
W}^{2}\,r\cos\theta_{\scriptscriptstyle\rm Y}~{}\right],$ (4)

where $S_{\rm s}=2\pi{R^{\rm s}}^{2}\left[1-\cos(\theta_{\scriptscriptstyle\rm
C}^{\rm s})\right]$ is the surface of the spherical cap in contact with air,
$B^{\rm s}=R^{\rm s}\sin(\theta_{\scriptscriptstyle\rm C}^{\rm s})$ is the
base radius, $R^{\rm s}$ the radius of the droplet, and
$\theta_{\scriptscriptstyle\rm C}^{\rm s}$ its contact angle in the state $s$.
$\phi_{\scriptscriptstyle\rm S}=\mathrm{w}^{2}/\mathrm{d}^{2}$ is the fraction
of solid surface area wet by the liquid (or pillar density) and
$r=1+4\mathrm{w}\mathrm{h}/\mathrm{d}^{2}$ is the surface roughness ratio.

The main difference between the model defined by Equation (4) and the Wenzel
one [3] is that the former one considers a droplet of finite size, which means
that the interface of the cap and the lower interface in contact with the
substrate compete in the minimisation of the total free energy. We will return
to this point in Section 4. Concerning the CB state, it is instead found that
results are equivalent when one explicitly considers the cap as in Equation
(3) or simply homogenises the surface free energy as in the original CB theory
[4].

Figures 3-a and b show lateral views of droplets minimising the equations of
the homogenised model and $\Delta\mathcal{F}$ obtained by MC simulations for
surface S1 and S3, respectively. $\Delta\mathcal{F}$ of the surface S1
presents only one minimum for which the simulated configuration is shown in
blue together with the minimum of the solution of Equation (3), in red.
$\Delta\mathcal{F}$ of surface S3 has multiple minima and two of them are
represented in Figure 3-b: the local minimum at the CB state, shown on top
with the solution of Equation (3), and the global W minimum reported in the
bottom panel, in which is compared with the solution of Equation (4). Results
are in good agreement for the CB state, both concerning the contact angle and
the number of intruded cavities. On the other hand, contact angles are not in
perfect agreement for the W state, due to the local pinning at pillars, which
is not captured in Equation (4), see below for details. On the other hand, the
model predicts correctly that W is the global free energy minimum and the free
energy values are in reasonable accord, see the _Supporting Information_.

Figure 3: Homogenised model vs $\Delta\mathcal{F}$. (a) Side view of
configuration at the minimum of $\Delta\mathcal{F}$ for surface S1 in blue.
Red dashed line is the numerical solution of the homogenised model for the CBI
state, Equation (3). (b) Top: side view of a sessile droplet on surface S3 at
the local minimum of $\Delta\mathcal{F}$ (point I in Figure 2) and the
solution of Equation (3). Bottom: global minimum of $\Delta\mathcal{F}$ in the
W state (point VII in Figure 2). The blue solid line is the numerical solution
of the model in Equation (4) for the WI state. (c) View from below of the 3D
configuration of the droplet placed on S3 at two filling levels: point IV in
Figure 2-a (maximum) and point VII (global minimum); colors correspond to the
height of liquid in each lattice site.

Besides the fair agreement at the global minimum, the main difference between
the two approaches is that the simulated free energy profile presents several
local minima which are not accounted for in any homogenised model. This
difference is particularly important for the W state, which helps to explain
the difference in the global W minimum. Indeed, an important simplification
introduced in the model (4) is that the W state is achieved by _homogeneous_
wetting: below the droplet base, there is a perfect cylinder filling the
cavities; individual wetting of pillars is thus disregarded; the cost to
create the interface between the liquid and air is also neglected in the
model. However, simulations show that the infiltration of the substrate is not
homogeneous, that pinning at individual pillars may occur, and that the
interface between the liquid and the gas below the droplet is quite rough, as
exemplified in Figure 3-c for one minimum and one maximum: the configurations
are particularly far from being a cylinder when $\Delta\mathcal{F}$ is at a
maximum.

The nontrivial shape of the droplet in contact with the surface drove us to
investigate the contribution to the free energy of each interface. We propose
a putative free energy defined as: $\Omega=\Delta
P\,V_{0}+\sigma_{\scriptscriptstyle\rm GL}A_{\scriptscriptstyle\rm
GL}+\sigma_{\scriptscriptstyle\rm SL}A_{\scriptscriptstyle\rm
LS}+\sigma_{\scriptscriptstyle\rm SG}A_{\scriptscriptstyle\rm SG}$, where
$\Delta P$ can be interpreted as the Laplace Pressure, $\Delta
P=-\alpha\sigma_{\scriptscriptstyle\rm GL}$ with $\alpha$ being the mean
curvature of the drop and $A_{ij}$ are the interfaces between two different
phases ${ij}$ (solid, liquid, and gas). We then use the Young equation
$\sigma_{\scriptscriptstyle\rm SG}=\sigma_{\scriptscriptstyle\rm
SL}+\sigma_{\scriptscriptstyle\rm GL}\cos\theta_{\scriptscriptstyle\rm Y}$ to
rearrange the terms and note that the total area of the substrate
$A_{\mathrm{tot}}=A_{\scriptscriptstyle\rm LS}+A_{\scriptscriptstyle\rm VS}$
is constant. The contribution of liquid-gas surface area
$A_{\scriptscriptstyle\rm GL}$ is split in two parts, one corresponding to the
surface of the spherical cap $A_{\scriptscriptstyle\rm
GL}^{\scriptscriptstyle\rm C}$, and other to the part below the droplet,
$A_{\scriptscriptstyle\rm GL}^{\scriptscriptstyle\rm B}$. We can then write
the difference in free energy with respect to the reference one
$\Omega_{\mathrm{ref}}=\sigma_{\scriptscriptstyle\rm SG}A_{\mathrm{tot}}$:

$\Delta\Omega=\sigma_{\scriptscriptstyle\rm GL}\left(A_{\scriptscriptstyle\rm
GL}^{\scriptscriptstyle\rm C}-\alpha V_{0}~{}+~{}A_{\scriptscriptstyle\rm
GL}^{\scriptscriptstyle\rm B}-A_{\scriptscriptstyle\rm
LS}\cos\theta_{\scriptscriptstyle\rm Y}\right).$ (5)

Interestingly, the surface tension $\sigma_{\scriptscriptstyle\rm GL}$
factorises and only geometrical quantities appear in the parenthesis on the
right hand side, together with the contact angle
$\theta_{\scriptscriptstyle\rm Y}$.

Figure 4-a compares, for surface S3, the free energy $\Delta\mathcal{F}$
obtained in the MC simulations via Equation (2) with the putative free energy
$\Delta\Omega$ always computed from MC simulations but by measuring the
geometrical quantities in Equation (5). In particular, the surface areas
$A_{ij}$ and the mean curvature $\alpha=2/R$ are measured from droplet
configurations along the converged string. Since the free energy
$\Delta\mathcal{F}$ is computed by integration, it is known up to a constant
and thus can be freely displaced along the $y$-axis. The only free parameter
left to compare $\Delta\Omega$ to $\Delta\mathcal{F}$ is
$\sigma_{\scriptscriptstyle\rm GL}$, which should be of the order of the
coupling parameter $J_{\scriptscriptstyle\rm ij}$.

Figure 4-a shows $\Delta\mathcal{F}$ and $\Delta\Omega$ using
$\sigma_{\scriptscriptstyle\rm GL}=1$, which is the same value used for
$J_{\scriptscriptstyle\rm ij}$. The two quantities show similar trends and are
in semi-quantitative agreement for all three substrates analysed in this work
(see the _Supporting Information_ for the other two substrates).
Interestingly, the ruggedness observed in $\Delta\mathcal{F}$ is also present
in $\Delta\Omega$.

In Figure 4-b each geometrical term in Equation (5) is shown separately and
compared with $\Delta\mathcal{F}$. Note the different scales and units for
areas (right axis) and $\Delta\mathcal{F}$ (left axis). Most of the
interfacial areas show a smooth dependence on $f$, the exception being
$A_{\scriptscriptstyle\rm GL}^{\scriptscriptstyle\rm B}$ that presents some
maximum and minima which is thus a good candidate to explain the roughness of
the free energy connected to wetting.

Figure 4: $\Delta\mathcal{F}$, $\Delta\Omega$ and areas as a function of $f$
for S3. (a) Comparison of $\Delta\mathcal{F}$ and $\Delta\Omega$. (b) Each
geometrical term in Equation (5) is shown separately and compared with
$\Delta\mathcal{F}$ in blue. (c) $\Delta\mathcal{F}$ in blue and
$A_{\scriptscriptstyle\rm GL}^{\scriptscriptstyle\rm B}$ is shown in black.
The red curve is a linear fit of $A_{\scriptscriptstyle\rm
GL}^{\scriptscriptstyle\rm B}$ and the red vertical lines indicate the area
variations discussed in the main text. Dashed lines denote the maxima of
$\Delta\mathcal{F}$.

In Figure 4-c we show $A_{\scriptscriptstyle\rm GL}^{\scriptscriptstyle\rm B}$
and $\Delta\mathcal{F}$ to assess the importance of the former in the free
energy. Vertical dashed lines correspond to maxima of $\Delta\mathcal{F}$. We
observe variations in the curve of $A_{\scriptscriptstyle\rm
GL}^{\scriptscriptstyle\rm B}$ that are indicated by vertical red lines. The
typical size of these variations is $\Delta A_{\scriptscriptstyle\rm
GL}^{\scriptscriptstyle\rm B}\approx 1000L^{2}$. In terms of dimensionless
energy, this corresponds to $\Delta A_{\scriptscriptstyle\rm
GL}^{\scriptscriptstyle\rm B}\sigma_{\scriptscriptstyle\rm GL}\approx 1000$.
This energy variation is comparable to the free energy barrier for S3, $\Delta
A_{\scriptscriptstyle\rm GL}^{\scriptscriptstyle\rm
B}\sigma_{\scriptscriptstyle\rm GL}\approx H^{R}$, which suggests that the
term relative to the interface between liquid and gas below the droplet plays
an important role in generating local minima in the free energy. For S1,
instead, abrupt variations of $A_{\scriptscriptstyle\rm
GL}^{\scriptscriptstyle\rm B}$ are not sufficient to generate local minima,
due to the steep slope of $\Delta\mathcal{F}$ vs $f$, see the _Supporting
Information_.

In the previous section, we mentioned that local minima of $\Delta\mathcal{F}$
correspond to pinning of the drop at the pillars edges, which gives rise to
several possible minimal configurations characterised by different numbers of
pillars. To connect this picture with the observation that variations of
$A_{\scriptscriptstyle\rm GL}^{\scriptscriptstyle\rm B}$ correlate with
variations of $\Delta\mathcal{F}$, we suppose that the wet domain below the
droplet can be approximated by a cylinder of height $h$ and increasing radius
$B$, such that $A_{\scriptscriptstyle\rm GL}^{\scriptscriptstyle\rm B}=2\pi
hB$. This approximation does not take into account the roughness of
$A_{\scriptscriptstyle\rm GL}^{\scriptscriptstyle\rm B}$ shown in Figure 3-c,
but is reasonable in the case of minima. From the volume differences between
neighbouring minima in Figure 4 we can thus compute the jump in droplet radius
$\Delta B$ in the cylindrical approximation, $\Delta B\approx\Delta
A_{\scriptscriptstyle\rm GL}^{\scriptscriptstyle\rm B}/(2\pi h)\approx 16L$.
The estimated value of $\Delta B$ corresponds to the typical size of a cavity
$\mathrm{d}=\mathrm{w}+\mathrm{a}$, which is plausible and supports the idea
that local minima correspond to jumps of the droplet front across discrete
numbers of pillars.

To summarize, we propose that the global minimum of the free energy
corresponds to configurations that minimise the total interfacial energy for a
droplet of fixed volume. Local minima, instead, occur in correspondence of
abrupt variations of the liquid-vapor interface below the droplet connected
with the overcoming of individual pillars.

###  3.3 Minima of the free energy and contact angle hysteresis

We now evaluate the free energy barrier sizes that separate the local minima
and the consequences on the metastability of the substrates. In Figure 2-d,
the barriers $H^{R}$ and $H^{L}$ are defined. From Figure 2-a, one measures
for S1 typically $H^{R}\in(3000,9000)$ and $H^{L}\in(0,100)$ showing that,
even when the system is initialised in a W-like state, it rapidly evolves
towards the CB minimum. For substrate S3 both barriers vary typically in the
range $H^{R}$ $\approx$ $H^{L}\in(370,3700)$, which in physical units are
between $9\times 10^{-12}$J and $9\times 10^{-11}$J, assuming $L=1~{}\mu$m.
This is much higher than the thermal energy $k_{B}T\approx 4.1\times 10^{-21}$
J at ambient temperature: thermal fluctuations are not sufficient to drive the
system from one minimum to the other. Only other larger sources of energy can
move the drop away from local minima, e.g., mechanical vibrations.

When the system is prepared with some generic initial condition, it will fall
in the closest minimum and remain trapped there; this was verified by running
unrestrained MC simulations from several points along the curves in Figure
2-a. Only for the S1 the system always returned to the superhydrophobic CB
state, even though two shallow minima were identified numerically; this can
either be due to the numerical accuracy of the free-energy profiles or to the
size of the barriers, which is so low that the fluctuation imposed by the
effective temperature of the Monte Carlo simulations are enough to bypass
them. In other words, the free energy profiles in Figure 2 help understanding
the origin of contact angle hysteresis in contact angle measurements, which
can be rationalised in terms of a rough free energy landscape with multiple
minima, separated by large free energy barriers. Furthermore, the importance
of the initial conditions becomes apparent, which correspond to the
preparation of the sessile drop experiment.

Finally, the profiles account for the superhydrophobic properties of the CB
state, which are connected to the existence of a single minimum, i.e., with
low hysteresis. On the other hand, the presence of multiple W minima explains
why CAH is so pronounced in this wetting state and its stickiness [6, 49, 48].
Figure 5 shows the apparent contact angle $\theta_{\scriptscriptstyle\rm C}$
of the droplet measured in MC simulations together with $\Delta\mathcal{F}$
for substrate S3. The fact that there is a basin of Wenzel states and that
each minimum has a different contact angle allows us to define the contact
angle hysteresis $\theta_{\rm H}$ related to the wet W state as the difference
in $\theta_{\scriptscriptstyle\rm C}$ of the configuration associated to the
first and the last local minimum of $\Delta\mathcal{F}$, see Figure 5. Table 1
summarizes the values $\theta_{\rm H}$ for the three substrates together with
their roughness ratio $r$. We measured an increase of $\theta_{\rm H}$ when
$r$ decreases, which is in line to what was previously observed [50].

Figure 5: Wetting free energy and apparent contact angles as a function of liquid filling for substrate S3. The scale on the left corresponds to $\Delta\mathcal{F}$ and, on the right, to $\theta_{\scriptscriptstyle\rm C}$. The difference between the $\theta_{\scriptscriptstyle\rm C}$ associated to the first and last local minimum (indicated as black circles) is defined as the $\theta_{\rm H}$. Substrate | roughness ratio, $r$ |  $\theta_{\rm H}$, (in ∘)
---|---|---
S1 | 3 | 0
S2 | 2.2 | $24^{\circ}$
S3 | 1.8 | $36^{\circ}$
Table 1: Contact angle hysteresis $\theta_{\rm H}$ and roughness ratio $r$ for
the considered substrates.

##  4 Discussion: modeling rough wetting

We have identified by free energy simulations that some pillared substrates
present several local minima separated by high barriers, while others present
only one minimum. Incidentally, for tall/tightly packed pillars only the
superhydrophobic CB state is possible, while for more sparse pillars multiple
local wet minima arise. The latter surfaces thus display a behavior which is
strongly dependent on the initial conditions: if a droplet is deposited on a
substrate at a random configuration, it would accommodate in a wetting state
correspondent to the closest minimum which can be either superhydrophobic or
sticky. The present findings also imply that theories that predict a single W
state cannot be complete [2].

We measured all apparent contact angles $\theta_{\scriptscriptstyle\rm C}$ of
the droplet in configurations correspondent to physical minima identified in
our simulations for the three substrates and compared them with theoretical
models, see Figure 6. Squares corresponds to the superhydrophobic CB minimum
and circles to wet W states. Lines are solutions of the classical Cassie-
Baxter and Wenzel models, whose apparent contact angles are given by:

$\displaystyle\cos\theta_{\scriptscriptstyle\rm C}^{\textrm{CB}}$
$\displaystyle=$ $\displaystyle\phi_{\scriptscriptstyle\rm
S}\cos\theta_{\scriptscriptstyle\rm Y}-(1-\phi_{\scriptscriptstyle\rm S}),$
(6) $\displaystyle\cos\theta_{\scriptscriptstyle\rm C}^{\textrm{W}}$
$\displaystyle=$ $\displaystyle r\cos\theta_{\scriptscriptstyle\rm Y}\;.$ (7)

For the case where the global minimum is CB, which happens for substrate S1,
Figure 6 shows that the prediction of the Cassie-Baxter model is almost
quantitative. The CB model also has a reasonable agreement with simulated
$\theta_{\scriptscriptstyle\rm C}$ in cases where it is only a local minimum,
which happens for substrates S2 and S3. The agreement with CB model
deteriorates as the pillar distance increases, which can be explained by the
increasing curvature of the menisci suspended among pillars, not accounted for
in the classical CB model.

In principle, no direct comparison can be made between the W model and the MC
results, mainly because the model predicts only one state, while simulations
demonstrate the existence of multiple wet states. However, a fair comparison
can be made considering only the global minimum, which is compatible with the
minimisation procedure used in all homogenised models. It is seen that the W
model prediction is far from the measured contact angle and, for S2, it even
predicts a contact angle higher than the actual CB one. When the finite size
of the droplet is taken into account, as done in Equation (3) and (4), the
solution for the CB does not change but the W curve shifts to smaller values.
This simple correction captures the overall trend with interpillar distance
and improves the agreement with the global minima, although it is not
quantitative. However, we remark that the finite size is not enough to account
for the existence of multiple minima, which is due to pinning at individual
pillars and is crucial to account for contact angle hysteresis.

Figure 6 confirms that the extent of CAH has the following trend
$\mathrm{S}_{1}<\mathrm{S}_{2}<\mathrm{S}_{3}$, which suggests that the more
favorable the W state, the higher $\theta_{\rm H}$. This trend is related to
the number of local minima and to the facility of wetting the bottom of the
surface for short pillars, but what limits the total number of minima still
remains an open question.

Figure 6: Apparent contact angle computed at physical minima in Figure 2 as a
function of the distance between pillars. Squares are
$\theta_{\scriptscriptstyle\rm C}$ of the CB state, circles for the W minima;
the open circles denote the global W minimum. Orange lines are the solution
for the homogenised models: dot-dashed for CB, Equation (3), and dotted for W,
Equation (4). Black lines are the predictions of the classical models: dot-
dashed for CB, Equation (6), and dotted for W, Equation (7).

##  5 Conclusions

In this work, we have used a combination of the cellular Potts model [25, 33]
and the string method [26, 27, 28] to compute numerically the free energy
$\Delta\mathcal{F}$ of a droplet with a fixed volume placed on surfaces
decorated with pillars. The computed free energy landscape for wetting the
surface is found to be rough, with one minimum corresponding to the
superhydrophobic Cassie-Baxter state; depending on the pillar spacing,
multiple local minima can exist, associated with different wet Wenzel states.
Free energy barriers typically larger than the thermal energy are found, whose
origin can be traced back to pinning of the liquid-gas interface below the
droplet at individual pillars. Wenzel minima are characterized by an
increasing number of filled cavities, corresponding to different apparent
contact angles. This scenario accounts for the strong contact angle hysteresis
of the W state(s), in which the final wetting state depends on the initial
condition.

We have compared the apparent contact angles obtained in our _in silico_
experiments with the predictions of simple models, including the classical
Cassie-Baxter [4] and Wenzel [3] ones. Results showed that the Cassie-Baxter
model has a good prediction capacity, which can be improved by considering the
curvature of menisci overhanging surface textures. On the other hand, the
prediction of the apparent contact angle cannot be made by simple models in
the W state effectively hindering the deduction of surface characteristics
from $\theta_{\scriptscriptstyle\rm C}$ measurements. Actually, the problem of
predicting $\theta_{\scriptscriptstyle\rm C}$ in the W state seems ill-posed
and the relevant challenge is to assess its contact angle hysteresis, which
can be achieved only by detailed models.

An interesting and long-standing question is the possibility of predicting CAH
[13, 14, 16], which was possible for individual surfaces within our approach.
It is found that the wider the pillars are spaced, the larger CAH. Our
analysis suggests that larger drops would have liquid enough to fill more
cavities and thus the free-energy profile would show even more local minima.
What limits the total number of filled cavities and how these local minima
connect to CAH is however not clear and deserves to be investigated.

Finally, our findings suggest that, for a complete characterization of the
wetting properties of a substrate, it is necessary to repeat sessile drop
experiments several times for different initial conditions. Moreover, the fact
that free energy barrier between minima are typically much larger than thermal
fluctuations suggests that mechanical vibrations of the substrate are
necessary to drive the droplet across different minima. The amount of
mechanical energy necessary to reach different minima need to be evaluated by
computing the barrier sizes, which is possible with the methodology proposed
in this work.

Supporting Information

The supporting information describes the algorithm used to find the
thermodynamic wetting state for droplet placed on the pillared surface. Its
also presents the criteria to adjust parameter $\kappa$ and description of the
videos available. Then there is a discussion of the comparison between the
free energy obtained from the simulation with the Cassie-Baxter and Wenzel
models. Finally the areas of the components for the proposed analytical free
energy are presented.

Acknowledgements

M. Silvestrini and C. Brito thanks CAPES and CNPq for financing this work.
This work has been supported by the project “Understanding and Tuning FRiction
through nanOstructure Manipulation (UTFROM)” funded by MIUR Progetti di
Ricerca di Rilevante Interesse Nazionale (PRIN) Bando 2017 - grant 20178PZCB5.
This research is part of a project that has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No. 803213).

## References

  * [1] T. Young,  _Philos. Trans. R. Soc. London_ 1805, _95_ 65\.
  * [2] D. Quéré,  _Annu. Rev. Mater. Res._ 2008, _38_ 71\.
  * [3] R. N. Wenzel,  _Ind. Eng. Chem._ 1936, _28_ , 8 988.
  * [4] A. Cassie, S. Baxter,  _Trans. Faraday Soc._ 1944, _40_ 546\.
  * [5] R. E. Johnson Jr, R. H. Dettre,  In _Contact Angle, Wettability, and Adhesion_ , volume 43, 112–135. ACS Publications, 1964.
  * [6] M. Nosonovsky, B. Bhushan,  _Nano Lett._ 2007, _7_ 2633\.
  * [7] L. Gao, T. J. McCarthy,  _Langmuir_ 2007, _23_ , 7 3762.
  * [8] A. Marmur, E. Bittoun,  _Langmuir_ 2009, _25_ , 3 1277, pMID: 19125688.
  * [9] G. McHale,  _Langmuir_ 2007, _23_ , 15 8200.
  * [10] H. Y. Erbil, C. E. Cansoy,  _Langmuir_ 2009, _25_ , 24 14135.
  * [11] H. Y. Erbil,  _Surf. Sci. Rep._ 2014, _69_ , 4 325.
  * [12] H. Y. Erbil,  _Colloids Interfaces_ 2021, _5_ , 1 8.
  * [13] J. Joanny, P.-G. De Gennes,  _J. Chem. Phys._ 1984, _81_ , 1 552.
  * [14] M. Reyssat, D. Quéré,  _J. Phys. Chem. B_ 2009, _113_ , 12 3906.
  * [15] C. Semprebon, S. Herminghaus, M. Brinkmann,  _Soft Matter_ 2012, _8_ , 23 6301.
  * [16] A. Giacomello, L. Schimmele, S. Dietrich,  _PNAS_ 2016, _113_ , 3 E262.
  * [17] J. Crassous, E. Charlaix,  _EPL_ 1994, _28_ , 6 415.
  * [18] M. Yan, T. Li, P. Zheng, R. Wei, Y. Jiang, H. Li,  _Phys. Chem. Chem. Phys._ 2020, _22_ , 21 11809.
  * [19] W. Li, A. Amirfazli,  _J. Colloid Interface Sci._ 2005, _292_ , 1 195.
  * [20] H. Kusumaatmaja, J. M. Yeomans,  _Langmuir_ 2007, _23_ , 11 6019.
  * [21] F. Schellenberger, N. Encinas, D. Vollmer, H.-J. Butt,  _Phys. Rev. Lett._ 2016, _116_ , 9 096101.
  * [22] N. A. Patankar,  _Langmuir_ 2010, _26_ , 10 7498.
  * [23] W. Ren,  _Langmuir_ 2014, _30_ , 10 2879.
  * [24] S. Bonella, S. Meloni, G. Ciccotti,  _Eur. Phys. J. B_ 2012, _85_ , 3 1.
  * [25] F. Graner, J. A. Glazier,  _Phys. Rev. Lett._ 1992, _69_ , 13 2013.
  * [26] E. Weinan, W. Ren, E. Vanden-Eijnden,  _Phys. Rev. B_ 2002, _66_ , 5 052301.
  * [27] L. Maragliano, A. Fischer, E. Vanden-Eijnden, G. Ciccotti,  _J. Chem. Phys._ 2006, _125_ , 2 024106.
  * [28] E. Weinan, W. Ren, E. Vanden-Eijnden,  _J. Chem. Phys._ 2007, _126_ , 16 164103.
  * [29] D. Dan, C. Mueller, K. Chen, J. A. Glazier,  _Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys._ 2005, _72_ 041909\.
  * [30] L. R. de Oliveira, D. M. Lopes, S. M. M. Ramos, J. C. M. Mombach,  _Soft Matter_ 2011, _7_ 3763\.
  * [31] D. M. Lopes, L. R. de Oliveira, S. M. M. Ramos, J. C. M. Mombach,  _RSC Adv._ 2013, _3_ 24530\.
  * [32] V. Mortazavi, R. M. D’Souza, M. Nosonovsky,  _Phys. Chem. Chem. Phys._ 2013, _15_ 2749\.
  * [33] H. C. M. Fernandes, M. H. Vainstein, C. Brito,  _Langmuir_ 2015, _31_ , 27 7652.
  * [34] R. Xu, X. Zhao, L. Wang, C. Zhang, Y. Mao, L. Shi, D. Zheng,  _RSC Adv._ 2021, _11_ , 3 1875.
  * [35] D. M. Lopes, J. C. Mombach,  _Braz. J. Phys._ 2017, _47_ , 6 672.
  * [36] Y. Zhang, W. Ren,  _J. Chem. Phys._ 2014, _141_ , 24 244705.
  * [37] G. Pashos, G. Kokkoris, A. G. Boudouvis,  _Langmuir_ 2015, _31_ , 10 3059.
  * [38] A. Giacomello, S. Meloni, M. Müller, C. M. Casciola,  _J. Chem. Phys._ 2015, _142_ , 10 104701.
  * [39] A. Shahraz, A. Borhan, K. A. Fichthorn,  _Langmuir_ 2012, _28_ 14227\.
  * [40] M. Silvestrini, C. Brito,  _Langmuir_ 2017, _33_ , 43 12535.
  * [41] M. Amabili, S. Meloni, A. Giacomello, C. M. Casciola,  _J. Phys. Chem. B_ 2018, _122_ , 1 200.
  * [42] A. Tinti, A. Giacomello, Y. Grosu, C. M. Casciola,  _PNAS_ 2017, _114_ , 48 E10266.
  * [43] P. Tsai, R. Lammertink, M. Wessling, D. Lohse,  _Phys. Rev. Lett._ 2010, _104_ , March 116102.
  * [44] A. Tinti, A. Giacomello, C. M. Casciola,  _Eur. Phys. J. E: Soft Matter Biol. Phys._ 2018, _41_ , 4 52.
  * [45] E. S. Savoy, F. A. Escobedo,  _Langmuir_ 2012, _28_ , 7 3412.
  * [46] E. S. Savoy, F. A. Escobedo,  _Langmuir_ 2012, _28_ , 46 16080.
  * [47] M. Amabili, A. Giacomello, S. Meloni, C. M. Casciola,  _Phys. Rev. Fluids_ 2017, _2_ , 3 034202.
  * [48] M. Callies, D. Quéré,  _Soft Matter_ 2005, _1_ , 1 55.
  * [49] M. Sbragaglia, A. M. Peters, C. Pirat, B. M. Borkent, R. G. H. Lammertink, M. Wessling, D. Lohse,  _Phys. Rev. Lett._ 2007, _99_ , October 156001.
  * [50] J. Wang, Y. Wu, Y. Cao, G. Li, Y. Liao,  _Colloid Polym. Sci._ 2020, _298_ 1107\.


1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{\infty}(\mathbb{R}^{2},w_{L}^{1/q})}\\\
=\sup_{x\in\mathbb{R}^{2}}\|\tilde{G}_{R,N}^{m}(x,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{\infty}^{-\kappa}(\mathbb{R}^{2},w_{L}^{1/q})}$ (C.26)

up to a multiplicative constant $C$, which depends on $m$. The first equality
above follows from the fact that for $F\in
C(\mathbb{R}^{2}\times\mathbb{R}^{2})$ it holds
$\sup_{x\in\mathbb{R}^{2}}\sup_{y\in\mathbb{R}^{2}}F(x,y)=\sup_{y\in\mathbb{R}^{2}}\sup_{x\in\mathbb{R}^{2}}F(x,y)$.
By Theorem A.5 (B), since $q>2/\kappa$, the above expression is bounded by

$\sup_{x\in\mathbb{R}^{2}}\|\tilde{G}_{R,N}^{m}(x,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{q}(\mathbb{R}^{2},w_{L}^{1/q})}\leq\sup_{x\in\mathbb{R}^{2}}\|\tilde{G}_{R,N}^{m}(x,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{q}(\mathbb{R}^{2},w_{R}^{1/q})}=\sup_{\mathrm{x}\in\mathbb{S}_{R}}\|G_{R,N}(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})^{m}\|_{L_{q}(\mathbb{S}_{R})}\\\
=\sup_{\mathrm{x}\in\mathbb{S}_{R}}\|G_{R,N}(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{mq}(\mathbb{S}_{R})}^{m}\leq
C\sup_{\mathrm{x}\in\mathbb{S}_{R}}\|G_{R,N}(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L^{(mq-2)/mq}_{2}(\mathbb{S}_{R})}^{m}\\\ =C(4\pi
R^{2})^{-m/2}\,[\mathrm{Tr}(G_{R,N}(1-\Delta_{R})^{(mq-2)/mq}G_{R,N})]^{m/2}.$
(C.27)

The first bound above is true because $R\geq L$. The second bound is a
consequence of the Sobolev embedding stated in Lemma A.9, since $q\geq 2/m$.
The first of the bounds from the statement of the lemma follows now from Lemma
B.1 applied with $N^{\prime}=1$ and $\kappa^{\prime}=2/mq$. To prove the
second of the bounds we use exactly the same strategy as above with $m=1$ and
the operator $G_{R,N}$ replaced by $(1-K_{R,N})G_{R}(1-K_{R,N})$. ∎

## Acknowledgments

The financial support by the grant ‘Sonata Bis’ 2019/34/E/ST1/00053 of the
National Science Centre, Poland, is gratefully acknowledged.

## References

  * [1] S. Albeverio, S. Kusuoka, _The invariant measure and the flow associated to the $\Phi^{4}_{3}$ quantum field model_, Ann. Sc. Norm. Super. Pisa Cl. Sci., (5), 20, 1359–1427, (2020), [arXiv:1711.07108].
  * [2] S. Albeverio, S. Kusuoka, _Construction of a non-Gaussian and rotation-invariant $\Phi^{4}$ measure and associated flow on $\mathbb{R}^{3}$ through stochastic quantization_, [arXiv:2102.08040].
  * [3] F. Arici, D. Becker, C. Ripken, F. Saueressig, W.D. van Suijlekom, _Reflection positivity in higher derivative scalar theories_ , J. Math. Phys. 59, 082302, (2018).
  * [4] N. Barashkov, F. C. De Vecchi, _Elliptic stochastic quantization of Sinh-Gordon QFT_ , [arXiv:2108.12664].
  * [5] J.C.A. Barata, C. D. Jäkel, J. Mund, _The $P(\varphi)_{2}$ Model on de Sitter Space_. Memoirs of the American Mathematical Society, 281, 1389, (2023).
  * [6] W. Beckner, _Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality_. Annals of Mathematics, 138, 213–242, (1993).
  * [7] H. Brezis, P. Mironescu, _Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces_ , Journal of Evolution Equations, 1(4), 387–404, (2001).
  * [8] G. Da Prato, A. Debussche, _Strong solutions to the stochastic quantization equations_ , Ann. Probab. 31(4), 1900–1916, (2003).
  * [9] G. Da Prato, J. Zabczyk, _Ergodicity for infinite dimensional systems_ , Cambridge University Press, (1996).
  * [10] G. F. De Angelis, D. de Falco, G. Di Genova. _Random fields on Riemannian manifolds: a constructive approach_ , Commun. Math. Phys. 103, 297-303, (1986).
  * [11] J. Dimock, _Markov quantum fields on a manifold_ , Rev. Math. Phys. 16(02), 243–255, (2004), [arXiv:math-ph/0305017].
  * [12] D. E. Edmunds, H. Triebel, _Function spaces, entropy numbers, differential operators_ , Cambridge University Press, (1996).
  * [13] S. N. Ethier, T.G. Kurtz, T, _Markov Processes: Characterization and Convergence_ , Wiley, (2005).
  * [14] R. Figari, R. Høegh-Krohn, C.R. Nappi, _Interacting relativistic boson fields in the de Sitter universe with two space-time dimensions_ , Commun. Math. Phys. 44, 265–278, (1975).
  * [15] J. Glimm, A. Jaffe, T. Spencer, _The Wightman axioms and particle structure in the $P(\Phi)_{2}$ quantum field model_, Ann. Math., 100, 585–632, (1974).
  * [16] J. Glimm, A. Jaffe, _Quantum Field Theory and Statistical Mechanics: Expositions_ , Birkhäuser, (1985).
  * [17] J. Glimm, A. Jaffe, _Quantum Physics. A functional integral point of view_ , Springer, (1987).
  * [18] M. Gubinelli, M. Hofmanová, _A PDE construction of the Euclidean $\Phi^{4}_{3}$ QFT_, Commun. Math. Phys., 382, 1–75, (2021), [arXiv:1810.01700].
  * [19] F. Guerra, L. Rosen, B. Simon, _The $P(\phi)_{2}$ Euclidean quantum field theory as classical statistical mechanics_, Ann. Math., 101, 111–259, (1975).
  * [20] M. Hairer, R. Steele, _The $\Phi^{4}_{3}$ measure has sub-Gaussian tails_, J. Stat. Phys., 186, 38, (2022), [arXiv:2102.11685].
  * [21] C. D. Jäkel, J. Mund, _The Haag-Kastler Axioms for the $P(\varphi)_{2}$ Model on the De Sitter Space_. Ann. Henri Poincaré 19, 959–977, (2018).
  * [22] C. Muscalu, W. Schlag, _Classical and Multilinear Harmonic Analysis_ Cambridge University Press, (2013).
  * [23] E. Nelson, A quartic interaction in two dimensions, in _Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965)_ , 69–73, M.I.T. Press, (1966).
  * [24] D. Nualart, _The Malliavin calculus and related topics_ , Springer-Verlag, (2006).
  * [25] K. Osterwalder, R. Schrader, _Axioms for Euclidean Green’s functions. II_ , Commun. Math. Phys., 42, 281–305, (1975).
  * [26] G. Parisi, Y. S. Wu, _Perturbation theory without gauge fixing_ , Sci. Sinica, 24(4), 483–496, (1981).
  * [27] B. Simon, _The $P(\Phi)_{2}$, (Quantum) Field Theory_, Princeton University Press, (1974).
  * [28] S. J. Summers, _A Perspective on Constructive Quantum Field Theory_ , [arXiv:1203.3991].
  * [29] H. Triebel, _Theory of Function Spaces_ , Birkhäuser, (1983).
  * [30] H. Triebel, _Theory of Function Spaces II_ , Birkhäuser, (1992).
  * [31] H. Triebel, _Theory of function spaces III_ , Birkhäuser, (2006).
  * [32] N. Varopoulos, L. Saloff-Coste, T. Coulhon, _Analysis and Geometry on Groups_ , Cambridge University Press, (1992).


Common Information Dimension

Osama A. Hanna^†^*, Xinlin Li^†^*, Suhas Diggavi^† and Christina Fragouli^†

^†University of California, Los Angeles

Email:{ohanna, xinlinli, suhasdiggavi<EMAIL_ADDRESS>Equal contribution.This work was supported in part by the NSF grants 1705077, 2221871 and 2007714, and the Army Research Laboratory grant under Cooperative Agreement W911NF-17-2-0196.

The exact common information between a set of random variables $X_1,...,X_n$ is defined as the minimum entropy of a shared random variable that allows for the exact distributive simulation of $X_1,...,X_n$. It has been established that, in certain instances, infinite entropy is required to achieve distributive simulation, suggesting that continuous random variables may be needed in such scenarios. However, to date, there is no established metric to characterize such cases. In this paper, we propose the concept of Common Information Dimension (CID) with respect to a given class of functions $\mathcal{F}$, defined as the minimum dimension of a random variable $W$ required to distributively simulate a set of random variables $X_1,...,X_n$, such that $W$ can be expressed as a function of $X_1,\cdots,X_n$ using a member of $\mathcal{F}$. Our main contributions include the computation of the common information dimension for jointly Gaussian random vectors in a closed form, with $\mathcal{F}$ being the linear functions class.

§ INTRODUCTION

§ NOTATION AND DEFINITIONS

§ CID AND RCID FOR JOINTLY GAUSSIAN RANDOM VARIABLES

[1]
S. Watanabe, “Information theoretical analysis of multivariate correlation,”
IBM Journal of research and development, vol. 4, no. 1, pp. 66–82,

[2]
P. Gács and J. Körner, “Common information is far less than mutual
information,” Problems of Control and Information Theory, vol. 2,
no. 2, pp. 149–162, 1973.

[3]
A. Wyner, “The common information of two dependent random variables,”
IEEE Transactions on Information Theory, vol. 21, no. 2, pp. 163–179,

[4]
A. J. Bell, “The co-information lattice,” in Proceedings of the Fifth
International Workshop on Independent Component Analysis and Blind Signal
Separation: ICA, vol. 2003, 2003.

[5]
R. Ahlswede and I. Csiszár, “Common randomness in information theory and
cryptography. i. secret sharing,” IEEE Transactions on Information
Theory, vol. 39, no. 4, pp. 1121–1132, 1993.

[6]
——, “Common randomness in information theory and cryptography. ii. cr
capacity,” IEEE Transactions on Information Theory, vol. 44, no. 1,
pp. 225–240, 1998.

[7]
U. M. Maurer, “Secret key agreement by public discussion from common
information,” IEEE transactions on information theory, vol. 39,
no. 3, pp. 733–742, 1993.

[8]
I. Csiszár and P. Narayan, “Common randomness and secret key generation
with a helper,” IEEE Transactions on Information Theory, vol. 46,
no. 2, pp. 344–366, 2000.

[9]
H. Hotelling, “Relations between two sets of variates,” Breakthroughs
in statistics: methodology and distribution, pp. 162–190, 1992.

[10]
I. Csiszár and J. Körner, Information theory: coding theorems for
discrete memoryless systems.1em plus 0.5em minus 0.4emCambridge University Press, 2011.

[11]
O. Shayevitz, “On rényi measures and hypothesis testing,” in 2011
IEEE International Symposium on Information Theory Proceedings.1em
plus 0.5em minus 0.4emIEEE, 2011, pp. 894–898.

[12]
L. Yu, V. Y. Tan et al., “Common information, noise stability, and
their extensions,” Foundations and Trends® in
Communications and Information Theory, vol. 19, no. 2, pp. 107–389, 2022.

[13]
Y. Li, M. Yang, and Z. Zhang, “A survey of multi-view representation
learning,” IEEE transactions on knowledge and data engineering,
vol. 31, no. 10, pp. 1863–1883, 2018.

[14]
M. Wu and N. Goodman, “Multimodal generative models for scalable
weakly-supervised learning,” Advances in Neural Information Processing
Systems, vol. 31, 2018.

[15]
Y. Shi, B. Paige, P. Torr et al., “Variational mixture-of-experts
autoencoders for multi-modal deep generative models,” Advances in
Neural Information Processing Systems, vol. 32, 2019.

[16]
T. M. Sutter, I. Daunhawer, and J. E. Vogt, “Generalized multimodal ELBO,”
in International Conference on Learning Representations, 2021.
[Online]. Available: <https://openreview.net/forum?id=5Y21V0RDBV>

[17]
C. Zhang, Z. Han, H. Fu, J. T. Zhou, Q. Hu et al., “Cpm-nets: Cross
partial multi-view networks,” Advances in Neural Information
Processing Systems, vol. 32, 2019.

[18]
G. R. Kumar, C. T. Li, and A. El Gamal, “Exact common information,” in
2014 IEEE International Symposium on Information Theory.1em
plus 0.5em minus 0.4emIEEE, 2014, pp. 161–165.

[19]
P. W. Cuff, H. H. Permuter, and T. M. Cover, “Coordination capacity,”
IEEE Transactions on Information Theory, vol. 56, no. 9, pp.
4181–4206, 2010.

[20]
H. S. Witsenhausen, “On sequences of pairs of dependent random variables,”
SIAM Journal on Applied Mathematics, vol. 28, no. 1, pp. 100–113,

[21]
W. Liu, G. Xu, and B. Chen, “The common information of n dependent random
variables,” in 2010 48th Annual Allerton Conference on Communication,
Control, and Computing (Allerton).1em plus 0.5em minus 0.4emIEEE, 2010, pp. 836–843.

[22]
K. B. Viswanatha, E. Akyol, and K. Rose, “The lossy common information of
correlated sources,” IEEE Transactions on Information Theory,
vol. 60, no. 6, pp. 3238–3253, 2014.

[23]
G. Xu, W. Liu, and B. Chen, “A lossy source coding interpretation of wyner’s
common information,” IEEE Transactions on Information Theory,
vol. 62, no. 2, pp. 754–768, 2015.

[24]
C. T. Li and A. El Gamal, “Distributed simulation of continuous random
variables,” IEEE Transactions on Information Theory, vol. 63, no. 10,
pp. 6329–6343, 2017.

[25]
L. Yu and V. Y. Tan, “On exact and $\infty$-rényi common informations,”
IEEE Transactions on Information Theory, vol. 66, no. 6, pp.
3366–3406, 2020.

[26]
D. L. Donoho, “Compressed sensing,” IEEE Transactions on information
theory, vol. 52, no. 4, pp. 1289–1306, 2006.

[27]
E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from
incomplete and inaccurate measurements,” Communications on Pure and
Applied Mathematics: A Journal Issued by the Courant Institute of
Mathematical Sciences, vol. 59, no. 8, pp. 1207–1223, 2006.

[28]
Y. Wu and S. Verdú, “Rényi information dimension: Fundamental limits
of almost lossless analog compression,” IEEE Transactions on
Information Theory, vol. 56, no. 8, pp. 3721–3748, 2010.

[29]
A. Rényi, “On the dimension and entropy of probability distributions,”
Acta Mathematica Academiae Scientiarum Hungarica, vol. 10, no. 1, pp.
193–215, 1959.

[30]
H. S. Witsenhausen, “Values and bounds for the common information of two
discrete random variables,” SIAM Journal on Applied Mathematics,
vol. 31, no. 2, pp. 313–333, 1976.

[31]
S. Satpathy and P. Cuff, “Gaussian secure source coding and wyner's common
information,” in 2015 IEEE International Symposium on Information
Theory (ISIT).1em plus 0.5em minus 0.4emIEEE, 2015, pp.