Niklas Affolter

Max Glick

Pavlo Pylyavskyy

Sanjay Ramassamy

Vector-relation configurations and plabic graphs

Keywords: Pentagram map, plabic graphs, dimer model

We study a simple geometric model for local transformations of bipartite graphs. The state consists of a choice of a vector at each white vertex made in such a way that the vectors neighboring each black vertex satisfy a linear relation. Evolution for different choices of the graph coincides with many notable dynamical systems including the pentagram map, Q-nets, and discrete Darboux maps. On the other hand, for plabic graphs we prove unique extendability of a configuration from the boundary to the interior, an elegant illustration of the fact that Postnikov's boundary measurement map is invertible. In all cases there is a cluster algebra operating in the background, resolving the open question for Q-nets of whether such a structure exists.

Introduction

The dynamics of local transformations on weighted networks plays a central role in a number of settings within algebra, combinatorics, and mathematical physics. In the context of the dimer model on a torus, these local moves give rise to the discrete cluster integrable systems of Goncharov and Kenyon [START_REF] Goncharov | Dimers and cluster integrable systems[END_REF]. Meanwhile, for plabic graphs in a disk, Postnikov transformations relate different parametrizations of positroid cells [START_REF] Postnikov | Total positivity, Grassmannians, and networks[END_REF] which in turn define a stratification of the totally non-negative Grassmannian.

The dimer model also manifests itself in many geometrically defined dynamical systems. We focus on projective geometry and draw our initial motivation from the pentagram map. The pentagram map was defined by Schwartz [START_REF] Schwartz | The pentagram map[END_REF] and related to coefficienttype cluster algebra dynamics in [START_REF] Glick | The pentagram map and Y-patterns[END_REF]. Gekhtman, Shapiro, Tabachnikov, and Vainshtein [START_REF] Michael Gekhtman | Higher pentagram maps, weighted directed networks, and cluster dynamics[END_REF][START_REF] Michael Gekhtman | Integrable cluster dynamics of directed networks and pentagram maps[END_REF] placed the pentagram map and certain generalizations in the context of weighted networks.Although considerable work in various directions of the subject has been undertaken, most relevant to our work is a further generalization termed Y-meshes [START_REF] Glick | Y-meshes and generalized pentagram maps[END_REF].

We propose a simple but versatile geometric model for the space of edge weights of any bipartite graph modulo gauge equivalence. The induced dynamics of local transformations includes in special cases pentagram maps, Q-nets, and discrete Darboux maps. This common generalization resolves a long standing question of how the pentagram map and Q-nets relate. Moreover, our systems come with a cluster dynamics, which is new in the Q-net case. Lastly, in the setting of plabic graphs we define a geometric version of the boundary measurement map and its inverse. In this language, properties of the boundary measurement map imply unique solvability of a certain family of geometric realization problems.

Let G be a planar bipartite graph with vertex set B ∪ W. For b ∈ B let N(b) ⊆ W denotes its set of neighbors. Fix a vector space V. A circuit is a subset S ⊆ V such that S is linearly dependent but each of its proper subsets is linearly independent. Gauge equivalence classes of vector-relation configurations serve as our main object of study. For the remainder of the introduction, we define both local transformations and a boundary measurement type map for such configurations, and present our main results.

Local transformations

Goncharov and Kenyon [START_REF] Goncharov | Dimers and cluster integrable systems[END_REF] study line bundles with connection on bipartite graphs. This model amounts to a choice of edge weights modulo a notion of gauge equivalence. They also define local transformations of the graph and edge weights which are closely related to Postnikov moves [START_REF] Postnikov | Total positivity, Grassmannians, and networks[END_REF] in weighted directed networks.

Let G = (B ∪ W, E) be a planar bipartite graph. Let (v, R) be a vector-relation configuration on G. We can define edge weights wt(bw) = ±K bw where K bw is the coefficient of v w in R b and the signs satisfy the Kasteleyn condition, namely the product of signs on edges around each 2m-gon face equals (-1) m-1 . A choice of Kasteleyn signs always exists and, in the plane, any two such choices are gauge equivalent (see e.g. [START_REF] Kenyon | Lectures on dimers[END_REF]).

Theorem 1.3. The above associates a line bundle with connection on G to each gauge class of vector-relation configurations satisfying the circuit condition. Moreover, the classical local transformations can be implemented on the vector-relation level via local geometric rules.

In Section 2, we define these local rules and also give geometric definitions of face weights which are the main gauge-invariant coordinates. We also explain the sense in which our model is a common generalization of the pentagram map and Q-nets.

Configurations on plabic graphs

Plabic graphs are a family of finite planar graphs widely used in the study of positroids and the totally non-negative Grassmannian. Restricted to this case, our model provides a new take on plabic graphs and much of the surrounding theory.

A plabic graph is a finite planar graph G = (B ∪ W, E) embedded in a disk with the vertices all colored black or white. We assume throughout that G is in fact bipartite and that all of its boundary vertices are colored white. An almost perfect matching of G is a matching that uses all internal vertices (and some boundary vertices). Assume always that G has at least one almost perfect matching.

Fix for the moment a plabic graph G = (B ∪ W, E). Let M = |B|, N = |W|, and let n be the number of boundary vertices. As all boundary vertices are white that leaves Nn internal white vertices. Number the elements of B and W respectively 1 through M and 1 through N in such a way that the boundary (white) vertices are numbered 1 through n in clockwise order. We will sometimes use these numbers in place of the vertices, e.g. writing v i for the vector of a configuration at the white vertex numbered i. Let k = N -M. Each almost perfect matching uses exactly nk boundary vertices.

We make some modifications to our model to better cater to plabic graphs and related geometry. First, the natural ambient dimension is k = N -M so we simply fix as our vector space V = C k . We also require that the boundary vertices v 1 , . . . , v n span V. The main result given these assumptions, which in isolation is rather striking, is that a configuration is uniquely determined up to gauge by its boundary vectors. Moreover, the boundary vectors can be viewed as columns of a k × n matrix [v 1 • • • v n ] and the subset of the Grassmannian arising this way is dense in the associated positroid variety. Theorem 1.4. Fix a plabic graph G with all the preceding conventions and notation.

1. If (v, R) is a vector-relation configuration on G satisfying the circuit condition then A = [v 1 • • • v n ] lies in the positroid variety Π M where M is the positroid of G.
2. Suppose G is reduced. For generic A ∈ Π M , the columns v 1 , . . . , v n of A can be extended to a vector-relation configuration on G that is unique up to gauge at internal vertices. In particular, each internal vector is determined up to scale.

In Section 3.1 we review the definitions of the positroid M, the variety Π M ⊆ Gr k,n , and the reducedness condition for plabic graphs.

Example 1.5. Consider the plabic graph G in Figure 1. It has k = 3 and n = 6. In fact, the associated positroid is the uniform matroid, meaning all 3 element subsets of {1, . . . , 6} are included. So the positroid variety is the whole Grassmannian Gr 3,6 . In short, the boundary vectors v 1 , . . . v 6 ∈ C 3 of a configuration can be generic. Now suppose v 1 , . . . v 6 ∈ C 3 are given and consider the possibilities for the internal vector u. The lower black vertex forces u, v 1 , v 2 to be dependent while the top black vertex forces u, v 4 , v 5 to be dependent. If the v i are generic then u must lie on the line of intersection of the planes v 1 , v 2 and v 4 , v 5 . Hence u is determined up to scale. The other two black vertices have degree 4. It is always possible to find a linear relation among 4 vectors in C 3 , so there are no added conditions imposed on u.

Relation to previous work

Our model of vector-relation configurations has substantial precedent in the literature. In fact, a main selling point of our specific formulation is that it is versatile enough to tie into previously studied ideas in a variety of areas. We outline some of the relevant previous work here for the interested reader's convenience.

In the plabic graph setting, Lam's relation space [START_REF] Lam | Totally nonnegative Grassmannian and Grassmann polytopes[END_REF]Section 14] is in a sense dual to our model. In light of this connection, Lam's main result [START_REF] Lam | Totally nonnegative Grassmannian and Grassmann polytopes[END_REF]Theorem 14.6] demonstrates that the operation of boundary restriction as in our Theorem 1.4 agrees up to sign with the boundary measurement map [START_REF] Postnikov | Total positivity, Grassmannians, and networks[END_REF]. Our only claims to originality in Theorem 1.4 are the elegance and the elementary geometric nature both of the formulation and of our proofs. Another related model is provided by Postnikov [START_REF] Postnikov | Positive grassmannian and polyhedral subdivisions[END_REF]. Both [16, Section 14] and [START_REF] Postnikov | Positive grassmannian and polyhedral subdivisions[END_REF] are attempts to put on more mathematical footing on-shell diagrams [START_REF] Arkani-Hamed | Scattering amplitudes and the positive grassmannian[END_REF], and in fact physicists have informed us that vector-relation configurations are one way they think about said diagrams.

In the case of the dimer model on the torus, special cases and other hints of vectorrelation configurations appear in the following settings: higher pentagram maps [START_REF] Michael Gekhtman | Higher pentagram maps, weighted directed networks, and cluster dynamics[END_REF][START_REF] Michael Gekhtman | Integrable cluster dynamics of directed networks and pentagram maps[END_REF], discrete differential geometry [START_REF] Bobenko | Discrete differential geometry[END_REF], and the inverse spectral transform [START_REF] Kenyon | Planar dimers and Harnack curves[END_REF][START_REF] Fock | Inverse spectral problem for GK integrable system[END_REF]. We close this section by noting that the full version [START_REF] Affolter | Vectorrelation configurations and plabic graphs[END_REF] of this extended abstract contains full proofs of all results as well as investigations in a number of related directions.

Vector-relation configurations

In this section we develop the theory of vector-relation configurations on general planar bipartite graphs as in Definitions 1.1 and 1.2. To that end, let G = (B ∪ W, E) be a planar bipartite graph. We will denote a vector-relation configuration on G by (v, R) (or sometimes just v for short) where v = (v w ) w∈W and R = (R b ) b∈B .

For each w ∈ W, let P w denote the projection of v w ∈ V to the corresponding projective space PV. Proposition 2.1. If (v, R) satisfies the circuit condition then the points (P w ) w∈W uniquely determine the gauge class of the configuration.

We work on the level of the point configurations (P w ) w∈W for the remainder of this section. They are characterized by the property that for each b ∈ B the set {P w : w ∈ N(b)} is a circuit in projective space (e.g. 4 coplanar points, no 3 of which are collinear). Note that for any partition of a circuit into two subsets, there is a unique point on the intersection of the affine hulls of the subsets. Fix b ∈ B. Let F be a face containing b and let w 1 , w 2 be the neighbors of b along F. Define P(F, b) to be the point

P(F, b) = P w 1 , P w 2 ∩ {P w : w ∈ N(b) \ {w 1 , w 2 }} (2.1)
where • denotes an affine span. Figure 2 depicts the local transformations that we consider on bipartite graphs and sets notation for some of the graphs' vertices. We extend each transformation to a map- ping of point configurations. The first move is urban renewal which operates on a quadrilateral face F of the original graph. Define points at the new white vertices w 1 , w 2 to be P w It is a direct verification that the dynamics on the points P w correspond to classical local transformations on edge weights as claimed by Theorem 1.3.

b 1 w 1 w 2 b 2 b 1 w 1 w 2 b 2 w ′ 1 w ′ 2 w w ′ 1 w ′ 2 w 1 w k w k+1 w k+l w 1 w k w k+1 w k+l w ′
For a line bundle with connection on G, the basic gauge invariant functions are the monodromies around individual faces. These face weights evolve under local transformations via Y-pattern dynamics [START_REF] Fomin | Cluster algebras[END_REF] of the cluster algebra associated to the dual quiver. As such, the same formulas apply to our geometrically defined systems. Moreover, the face weights have simple projectively invariant formulas. Suppose points P 1 , . . . , P 2k in an affine space are given with the triples {P 1 , P 2 , P 3 }, {P 3 , P 4 , P 5 }, . . . , {P 2k-1 , P 2k , P 1 } all collinear. The multi-ratio (called a cross ratio for k = 2 and a triple ratio for k = 3) of the points is [P 1 , . . . , P 2k ] = P 1 -P 2 P 2 -P 3

P 3 -P 4 P 4 -P 5 • • • P 2k-1 -P 2k P 2k -P 1 .
Each individual fraction involves 3 points on a line and is interpreted as a ratio of signed distances. Example 2.4. The pentagram map [START_REF] Schwartz | The pentagram map[END_REF] inputs a polygon in the plane with vertices A i and outputs the polygon with vertices

B i = A i-1 , A i+1 ∩ A i , A i+2 .
Let G be an infinite square grid and associate an A i to each white vertex of G in the manner suggested by Figure 3. The two new points produced by urban renewal at F are precisely B 3 and B 4 .

In fact, urban renewal at half the faces (in a checkerboard pattern) followed by some degree 2 vertex removals results in a full application of the pentagram map.

Example 2.5. A Q-net [START_REF] Doliwa | Multidimensional quadrilateral lattices are integrable[END_REF] is a map from Z 3 to 3-space such that each primitive square maps to four coplanar points. Sufficient initial data for a Q-net lives on a so-called stepped surface which when projected to a plane looks like an infinite tiling by regular rhombi. A point of the Q-net lives at each vertex of the tiling and the four points associated to each rhombus are coplanar. This setup can be modeled by a bipartite graph G with a white vertex at each vertex of the tiling, a black vertex in the center of each rhombus, and edges connecting them in the natural way. A certain sequence of four urban renewals studied in [START_REF] Kenyon | Dimers and circle patterns[END_REF] realizes one step of Q-net dynamics which on the tiling level amounts to flipping three rhombi inside a regular hexagon.

3 Configurations on plabic graphs

Background on positroid varieties

We now return to the plabic graph case. The proof of Theorem 1.4 utilizes a significant amount of the theory of positroid varieties. We begin by reviewing the relevant material, generally following [START_REF] Muller | The twist for positroid varieties[END_REF] and [START_REF] Lam | Totally nonnegative Grassmannian and Grassmann polytopes[END_REF]. Given a k × n matrix A, and J ⊆ {1, . . . , n} of size k, let ∆ J (A) denote the maximal minor of A using column set J. The totally nonnegative Grassmannian is the set of A ∈ Gr(k, n) for which ∆ J (A) is real and nonnegative for all J. The matroid of any

A ∈ Gr(k, n) is M = {J : ∆ J (A) = 0}.
A positroid is a set of k-element subsets of {1, . . . , n} that arises as the matroid of a point in the totally nonnegative Grassmannian. We also denote a positroid by M even though this is a more restrictive notion than a matroid. Let M be a positroid. For j = 1, . . . , n, consider the column order j < j + 1 < . . . < n < 1 < . . . < j -1. Let I j be the lexicographically minimal element of M relative to this order. The collection of sets (I 1 , . . . , I n ) is called the Grassmann necklace of M. The positroids index a decomposition of the complex Grassmannian by open positroid varieties Π • M , defined as intersections of cyclic shifts of Schubert cells encoded by (I 1 , . . . , I n ). The positroid variety Π M is defined to be the Zariski closure of Π • M . In order to give quicker definitions, we fall back on the literature. Theorem 3.1 (Knutson-Lam-Speyer [START_REF] Knutson | Positroid varieties: juggling and geometry[END_REF]). The positroid variety Π M is a closed irreducible variety defined in the Grassmannian by

Π M = {A ∈ Gr k,n : ∆ J (A) = 0 for all J / ∈ M}.
Let G = (B ∪ W, E) be a plabic graph. Following our conventions, all boundary vertices are white. Let n be the number of boundary vertices and k = |W| -|B|. An almost perfect matching is a matching in G that uses all internal vertices. Hence it is a matching of B with W \ J where J ⊆ {1, . . . , n} (identified with the boundary vertices) and |J| = k. The positroid of G, denoted M G is the set of J that arise this way as the unused vertices of an almost perfect matching. An arc in a plabic graph is a direct path connecting the midpoints of two edges which meet at a vertex and are also part of a common face. The arc looks something like a circular arc centered at the common vertex, and it is oriented clockwise (resp. counterclockwise) if the vertex is white (resp. black). Near a boundary vertex an arc is also allowed to begin or end at a nearby boundary point. A strand is a maximal oriented path that decomposes into arcs. The collection of strands together is called the alternating strand diagram of the graph. One example of such a diagram is given in Figure 4.

Say a plabic graph G is reduced if its alternating strand diagram has the following properties:

• All strands start and end at the boundary.

• No strand has a self intersection unless it corresponds to a degree 1 boundary vertex whose neighbor is also degree 1.

• No pair of strands have a pair of intersections that they encounter in the same order.

If G is reduced then there are exactly n strands, one starting near each boundary vertex. Call j the number of the strand starting near vertex j. The strands divide the disk into regions. For each j, add a label of j to each region lying to the left of strand number j as it is traversed. Each region corresponds naturally to a white vertex, black vertex, or face of G. Use the notations S w , S b , and S F to denote the set of labels received by the region in the strand diagram associated to w, b, or F (see the right of Figure 4 for an example).

Recasting the main result

The remainder of this section is devoted to outlining the proof of Theorem 1.4, and we begin by reformulating this result. Let G = (B ∪ W, E) be a plabic graph with all the notation of Section 1.2. Recall in this setting we only allow gauge transformations at the internal vertices. Let C • G denote the space of gauge equivalence classes of vector-relation configurations on G satisfying the circuit condition modulo the action of GL k (C).

If (v, R) ∈ C G then by assumption v 1 , . . . , v n span V = C k . The v i are defined up to a common change of basis so

A = [v 1 • • • v n ] is a well-defined point of Gr k,n . We use Φ to denote the map Φ : C • G → Gr k,n taking (v, R)
to A, and we call Φ the boundary restriction map. In this language Theorem 1.4 can be stated as follows.

Theorem 3.3. Let G be a reduced plabic graph.

1. The map Φ is injective.

2. The Zariski closure of the image of Φ is the positroid variety Π M .

The reconstruction map

In this subsection, we discuss the proof of Theorem 3.3. Specifically, we define a rational map Ψ : Π M → C • G which turns out to be the inverse of Φ. As Ψ has the effect of reconstructing the entire configuration from just the boundary vectors, we term it the reconstruction map.

Fix A ∈ Π M and in fact fix a particular matrix representative so that the columns v 1 , . . . , v n of A all live in V. Let H j ⊆ V denote the linear span of {v i : i ∈ I j \ {j}}. 1. Each H j is a hyperplane.

2. The set {H j : j ∈ S F } of k hyperplanes are in general position for each face F.

3. Each L w is a line. We now have our definition of the reconstruction map Ψ : T G → C • G , namely it maps A to the configuration given by Proposition 3.6. Clearly Φ • Ψ is the identity. In plainer terms we have existence of an extension of generic A ∈ Π M to a full configuration. The following result establishes uniqueness.

L w = H 3 ∩ H 6 = v 4 , v 5 ∩ v 1 , v 2 .
One obtains a full configuration Ψ(A) by picking v w ∈ L w . On the other hand, Example 1.5 explains that any extension must have v w on this line verifying Proposition 3.7 in this case.

Definition 1 . 1 .Definition 1 . 2 .

 1112 A vector-relation configuration on G consists of choices of • a nonzero vector v w ∈ V for each w ∈ W and • a non-trivial linear relation R b among the vectors {v w : w ∈ N(b)} for each b ∈ B. In particular, each set {v w : w ∈ N(b)} must be linearly dependent. If each set {v w : w ∈ N(b)} is a circuit, say the configuration satisfies the circuit condition. Consider a vector-relation configuration on a graph G as above and suppose λ = 0. The gauge transformation by λ at a black vertex b ∈ B scales the relation R b by λ (and keeps all other vectors and relations the same). The gauge transformation by λ at a white vertex w ∈ W scales v w by 1/λ and scales the coefficient of v w by λ in each relation in which it appears to compensate. Two vector-relation configurations are called gauge equivalent if they are related by a sequence of gauge transformations.

6 uFigure 1 :

 61 Figure 1: A plabic graph corresponding to the open cell in Gr(3, 6)

Figure 2 :

 2 Figure 2: Combinatorial local moves on planar bipartite graphs.

FFigure 3 :

 3 Figure 3: A portion of the bipartite graph whose vector-relation dynamics coincide with the pentagram map.

Proposition 2 . 3 .

 23 Let F be a face with boundary cycle w 1 , b 1 , w 2 , b 2 , . . . , w m , b m in clockwise order. In terms of the points P w , the face weight of F equals Y F = (-1) m-1 [P w 1 , P(F, b 1 ), P w 2 , P(F, b 2 ), . . . , P w m , P(F, b m )] -1 .

Figure 4 :

 4 Figure 4: The alternating strand diagram for a plabic graph (left) and the associated labeling by sets of the faces and vertices of the graph (right)

Proposition 3 . 2 .

 32 Let F be a face of G, b ∈ B, and w ∈ W.• |S F | = k, |S w | = k -1, and |S b | = k + 1.• If b and w are on the boundary of F then S w ⊆ S F ⊆ S b .

Lemma 3 . 4 .

 34 For generic A ∈ Π M :

Proposition 3 . 5 .Proposition 3 . 6 .

 3536 Let b ∈ B and choose nonzero vectors v w ∈ L w for each neighbor w of b. Then these v w satisfy a unique linear relation up to scale, and this relation has all coefficients nonzero. Let A ∈ Π G be generic. Then there exists a unique configuration (v, R) ∈ C • G such that Φ(v, R) = A and v w ∈ L w for all w ∈ W.

Proposition 3 . 7 .

 37 Let (v, R) ∈ C • G and suppose A = Φ(v). Then (v, R) = Ψ(A).Example 3.8. Consider the plabic graph G in Figure4. As previously discussed, G corresponds to the uniform matroid in Gr 3,6 , and it follows that I j = {j, j + 1, j + 2} with indices modulo 6.Given A = [v 1 • • • v 6 ] then, H i = v i+1 , v i+2. The unique internal white vertex w has S w = {3, 6}, so

  The other two moves split a vertex into two new vertices of the same color, which are then attached by a new degree two vertex of the opposite color. If the vertex being split is a white vertex w, them put the same point P w 1 = P w 2 = P w at each new white vertex. Lastly, if the vertex being split is black there is a new degree two white vertex w . Put P w = P w 1 , . . . , P w k ∩ P w k+1 . . . P w k+l .

	Proposition 2.2. The above definitions yield valid point configurations on the graph resulting
	from each move.

1 = P(F, b 1 ) and P w 2 = P(F, b 2 ).

Acknowledgements

We thank Lie Fu, Rick Kenyon, and Kelli Talaska for many helpful conversations.

* N. A. was supported by DFG SFB/Transregio 109 "Discretization in Geometry and Dynamics" † P. P. was partially supported by NSF grants DMS-1148634 and DMS-1351590. ‡ S.R. was partially supported by the ANR-18-CE40-0033 project DIMERS.


Romain Lhermerout

Kristina Davitt

Controlled defects to link wetting properties to surface heterogeneity †

Small-scale heterogeneities have long been understood to give rise to contact angle hysteresis. More recently, the question of how they influence contact line dynamics has generated interest. Models that express the hysteresis or dynamics in terms of defect properties exist but have yet to be conclusively tested by experiment. Here we produce heterogeneous surfaces and thoroughly characterize individual defects. Precise dynamical measurements allow us to conclude that there is no evidence of a thermally-activated dynamics due to their presence, but that the hysteresis scales with their concentration and the square of their height, as predicted by some simple models.

Introduction

Defects or inhomogeneities on a solid surface affect how a liquid front moves over the surface. This is most readily seen in the contact angle hysteresis: the angle that the edge of a drop makes with the underlying surface depends on the direction in which the drop is moving. This arises from the ability of defects to pin the threephase contact line, thus necessitating a minimum force to overcome the pinning and for the drop to move. In their pioneering work, Joanny and de Gennes developed a simple model to express the balance between the pinning force of a individual chemical or topographical defect and the elastic restoring force of the deformed contact line and then related it to the macroscopic contact angle hysteresis in the case of non-interacting defects 1 . Although originally formulated for scales beyond that where van der Waals interactions are important, this model has recently been shown 2 to describe the pinning on nano-defects with pinning energies of the order of k B T . Further models and numerical studies 3 have investigated collective effects arising from the interaction between defects. Although hysteresis is readily observed, only a few experimental studies have attempted to show that model predictions are consistent with properties of the individual, microscopic or nanoscopic defects 2,[4][5][6][7][8] , or exploited its variation with concentration to investigate collective effects 9,10 . Systematic experiments to determine the scaling on defect parameters are lacking and which, if any, models are applicable to any given real surface remains therefore to be fully tested.

Contact angle hysteresis is easily observed in everyday phea Laboratoire de Physique Statistique (LPS), UMR 8550 CNRS, ENS, Univ. Paris Diderot, Sorbonne Université, 24 rue Lhomond, 75005, Paris, France. E-mail: kristina.davitt@lps.ens.fr † Electronic Supplementary Information (ESI) available:

[details of any supplementary information available should be included here].

See DOI: 10.1039/cXsm00000x/ nomena, such as the asymmetry of a drop slowly sliding down a vertical wall or stuck inside a straw. How defects affect the full dynamics of the contact line is less evident to the naked eye. This can be characterized by measuring the variation of the contact angle with the velocity of the contact line. A few experiments [11][12][13][14] have attempted to quantify how measurable surface properties affect the obtained dynamics. The task is not simple as the motion of the contact line on the background reference surface (i.e. without additional defects) exhibits its own dynamics, and in addition, experiments may access multiple dynamical regimes, each of which may be influenced differently by the presence of defects. For example, at high capillary number one expects viscous effects to be important, whereas at low velocities the motion may be dominated by thermally activated pinning and depinning on molecular or nanoscale imperfections. Indeed, even without explicitly added defects, it is an outstanding question of how to describe the full range of dynamics. Some attempts have been made to account for both viscous dissipation and activated dynamics 8,15,16 , including by ourselves 17,18 , where we propose a unified description of these regimes and extend the static model of Joanny and de Gennes to account for thermal activation. Although the situation is not yet clear, consensus is that -provided the energy barriers to motion created by defects are small enough-the slope of the dynamics in the activated regime will reflect characteristics of the defects. To answer the question of how defects affect the dynamics, it is therefore necessary to have an understanding of the dynamics on the reference surface, and then to compare the regime at lowest capillary numbers to that on surfaces containing well-characterized defects.

In this paper we produce and characterize purely topographical defects with continuously varying size, shape and density and measure the dynamics for each with several different liquids. We use a reference surface that presents a minuscule background hys-teresis and whose dynamics has been understood 19 . We find that the presence of defects has no effect on the slopes of the dynamics, even for defects with estimated energy barriers of the order of 100 k B T . The presence of defects only separates the advancing (A) and receding (R) branches of the dynamics, i.e. controls the hysteresis, which we define here as

H = γ (cos θ R -cos θ A ) , (1)
where γ is the liquid-vapor interfacial tension and the angles are measured at a low enough velocity such that viscous effects are negligible. We find that H varies linearly with the density of defects for all cases studied, from which an energy dissipated per defect can be determined; here ranging from 100 to 10 5 k B T . Further, for the spherical cap and Gaussian-shaped defects used here, H is found to scale as the square of the defect height. First we follow previous studies in comparing to the original model based on strong independent defects 1 and find that experimental results agree well with the simple scaling derived from this model, even in the case of weak and dense defects, that is, outside of the range of validity of the model. We suggest that the physical mechanism responsible for the observed hysteresis has therefore not been fully understood and call for further numerical work to simulate the hysteresis on surfaces covered with defects of known properties (such as the defect shape and concentrations measured herein) to evaluate the importance of other mechanisms such as collective effects or the existence of a force threshold even for independent weak defects.

Experiment

Obtaining defects

The purpose of this study is to link macroscopic wetting properties -the dynamics and hysteresis-to the heterogeneity of the underlying solid surface at microscopic scales. It is therefore necessary to have control over individual defects. They should be large enough that they are easily characterized by standard microscopy tools, but small enough that their associated energy barriers are comparable to thermal energy such that it is conceivable to measure a regime of thermally-activated motion, if it is indeed present. This is a question of resolution in the measurement of the dynamics.

Activated motion is the contact line moving from one local energy minimum to another by thermal fluctuations and can be modeled by an Arrhenius process. In the simplest form, it yields a velocity v (or capillary number Ca) that depends exponentially on the distance of the cosine of the dynamic angle from the equilibrium one:

|Ca| = ηv γ = Ca 0 exp γλ 2 2k B T cos θ -cos θ eq , ( 2
)
where η is the liquid viscosity, γ the liquid-vapor interfacial tension, k B is Boltzmann's constant and T is the temperature. Here λ is referred to as the activation length and Ca 0 as the transition capillary number 18 , which has also sometimes been related to the thermal attempt frequency 20 . If the dynamics is truly activated, then the activation length can be found from the slopes of exper-imental curves plotted on a semi-logarithmic scale (for example, see Fig. 2):

d log |Ca| d cos θ = γλ 2 2k B T ln 10 . (3)
In the experiments we describe below, a variation of about 0.01 • over one decade in velocity can be detected. A typical equilibrium contact angle in this study is θ eq ≈ 15 • . Assuming that the energy barrier to motion created by a single defect is of order γλ2 , then eqn 3 gives a maximum energy barrier of E b ≈ 5.10 4 k B T in order for an activated regime to be detectable. For simple liquids and assuming that λ is of the order of the defect dimension this implies that defects should be of the order of 100 nm or smaller.

We have achieved this by sintering silica nanospheres at different temperatures to obtain topographical bumps with different heights and slopes. The procedure has been borrowed from Ref. [ 21] and is only briefly outlined here. A piece of silicon wafer is cleaned in piranha solution then activated in oxygen plasma (30W, Harrick Plasma PDC-002) for 3 min. It is then immediately placed in a 1 mg ml -1 solution containing PEI polymer (polyethylenimine, Sigma-Aldrich) and left for 1 h. The adsorbed PEI layer is charged oppositely to the spheres; it adheres the spheres to the surface and minimizes the clustering of particles that occurs when a particle-containing liquid is dip-coated on a surface. After rinsing under deionized water, the sample is then stirred in a diluted solution of silica nanospheres. Although the stirring time and concentration of the nanosphere solution control the number density of spheres that are deposited in a predictable way [START_REF] Huwiler | [END_REF] , we nonetheless measure the density of beads on each sample by counting them under AFM. We count sufficiently many to ensure that the counting error is typically less than 3%. Unless otherwise stated, the nanospheres used in this study have a nominal diameter of 80 nm (Klebosol 30HB50K, AZ Electronic Materials, France). Samples are then placed in an oven (Nabertherm L5/13/B180) that ramps the temperature up in 2 h and holds it for another 2 h at final temperatures between 1080 • C and 1180 • C. This destroys the PEI layer, causes a thick silicon oxide to form on the bare parts of the wafer, and sinters the silica nanospheres, as illustrated in Fig. 1. The final height h 0 and shape profiles of the sintered beads are obtained by AFM. The resulting surface is thus chemically homogeneous (silicon oxide) but has topographical bumps of known shape and density. Figure 1 shows a typical distribution and example shapes of the obtained defects. We have verified that the molecular coating described below has a negligible effect on the size and shape of the defects formed from sintered 80 nm nanospheres 23 .

The reference surface

A reference surface that exhibits hysteresis indicates that there are heterogeneities present even in absence of explicitly added nanospheres. This raises the specter of competition or of collective behavior between the two classes of defects and complicates the interpretation of which microscopic parameters control the hysteresis and dynamics. To minimize this possibility, we aim to use a reference surface with minimal intrinsic hysteresis. We have therefore deposited a PDMS pseudo-brush on the silicon ox- ide surface, as illustrated in Fig. 1. PDMS pseudo-brushes have been reported to show exceptionally low hysteresis and they are very easily produced 24,25 . We have recently shown 19 that the contact line dynamics of good solvents, such as alkanes, on such surfaces is dominated by a visco-elastic dissipation that depends on the polymerization index of PDMS used to create the brush, with hydrodynamic dissipation kicking in at capillary numbers above roughly Ca ≈ 10 -7 . In the work reported here, we exclusively use PDMS with a polymerization index of N=126, which yield pseudo-brush layers about 4 nm thick. For decane on such a reference surface we find a contact angle hysteresis of the order of 0.1 • for velocities below 10 µm s -1 (red curve in Fig. 2), which is better than the hysteresis observed on other molecular-scale coatings such as carefully produced self-assembled monolayers of thiols or silanes, while being much easier to obtain reproducibly. The hysteresis with water is roughly 10 • at 10 µm s -1 or nearly 2 • at the minimum velocities we can attain.

Wetting measurements

We use a custom-built dip-coating apparatus to measure contact line dynamics. The principle is straightforward, and exhaustive details have been published elsewhere 26 . As illustrated in the inset of Fig. 2, the sample is plunged into, or retracted from, a liquid bath at a velocity controlled by either a motorized or piezo-driven stage. The dynamic contact angle θ is obtained by examining the capillary rise z cap under microscope and from the following relation:

z cap = γ ρg √ 1 -sin θ ( 4
)
where ρ is the liquid density and g is the standard acceleration due to gravity. Here we report measurements made with decane, hexadecane or water. The liquid properties used in this study are given in Table 1. Since one aim of this work is to examine the possible existence of thermally-activated motion, it is necessary to measure the dynamics down to the lowest possible velocities. Care has been taken to minimize vibrations, temperature variation and evaporation, which are key to obtaining reliable dynamics at velocities down to as low as 1 nm s -1 . We are able to reach a precision of 0.01 • in relative variations of the contact angle and about 1 • in the absolute angle. The contact angle hysteresis, defined in eqn 1, is determined here from the dynamic advancing and receding angles measured at a velocity of 10 µm s -1 . 3 Results and analysis Fits over the lowest four decades in velocity to eqn 3 yield activation lengths λ of order 100 nm for both the reference and the surface with defects. However, in the case of the former curve, in a previous study 19 we showed that the contribution coming from hydrodynamic dissipation is negligible below approximately 100 µm s -1 and that the remaining dynamics can be entirely attributed to a visco-elastic dissipation in the thin PDMS layer. In other words, there is no evidence of thermal activation on the ref- 0.98 0.97 0.96 0.95 0.94 Fig. 2 Contact line dynamics measured on the reference surface (red symbols) and on a surface covered with defects (blue symbols) using decane. The defects were sintered at 1080 • C and have a concentration of n = 0.8 µm -2 . Error bars shown on the data point taken at 10 µm s -1 correspond to the uncertainty in the absolute angle. This is also the velocity at which the hysteresis is defined. Inset illustrates the capillary rise near a solid surface and the quantities defined in eqn 4.

Dynamics

erence surface. There is no detectable difference in slope when defects are added, from which we conclude that there is no evidence for activation over the defects either. One may ask if the defects and their energy barrier to motion are indeed small enough to generate a detectable thermal activation. As described in section 3.3 below, much shallower defects have also been produced, which, consequently, have a much lower energy barrier. Dynamics obtained in this case † show a much smaller hysteresis such that the two dynamical curves nearly overlap on the scale of Fig. 2. There is still no detectable change in the low-velocity slope.

Hysteresis and defect concentration

We prepared a series of samples with different defect concentrations for nanospheres sintered at two different temperatures. Three liquids were used for each: decane, hexadecane and water. Figures 3a andb show the hysteresis obtained from the separation of the advancing and receding branches of the dynamics at 10 µm s -1 . There is a fundamental question of where the usual quasi-static definition of the hysteresis is located on an experimental dynamical curve, a notion that we have recently attempted to clarify 18 . Here, however, since the slope of the dynamics does not change with the presence of defects, the precise choice of the velocity at which the hysteresis is defined is of little consequence: it can potentially add an constant offset to the absolute H but does not change the variation of H (from sample to sample), which is what we will use in the following.

Highly sintered spheres result in nearly Gaussian-shaped defects with a width of 2d = 130 nm, as seen in the AFM profiles in Fig. 1c. Figure 3a shows that the hysteresis scales linearly with the defect concentration n for the two alcanes, up to the highest concentrations tested. This corresponds to very dense coverage since the defects are separated by about 200 nm, a distance comparable to their width. The data for water is not shown as it was too noisy to extract a reliable slope. Indeed, it should be noted that the full range of hysteresis here corresponds to less than 1.5 • and that the ability to measure such small variations due to the defects is in part due to the small and reproducible intrinsic hysteresis of the background surface, particularly when used with good solvents such as alkanes. The hysteresis measured with water is shown for barely sintered spheres in Fig. 3b, and still found to scale with n up to the highest concentrations. In this case, the detailed profile cannot be obtained by AFM due to tip effects and undercut, but we can approximate it by a spherical cap. This shape of defect pins the contact line more strongly and thus the hysteresis is significantly larger. The hysteresis also scales with n for the two alcanes (not shown in Fig. 3b but included in the summary Fig. 6), but the maximum concentrations tested were lower since the surfaces became perfectly wetting at concentrations below 10 µm -2 . Figure 3b shows that the advancing and receding angles both move away from the intermediate, equilibrium contact angle as the concentration increases. This has been observed in other studies with topographic defects [4][5][6] and can be interpreted by the fact that such defects have a bipolar force that is more wettable on one half and less wettable on the other (see Fig. 5 andAppendix A). In the case where the more and less wettable halves are nearly symmetric, then the force averaged over the surface is roughly null. This is in contrast to chemical defects, which usually have either a positive or negative force and therefore on average make the surface more or less wettable [START_REF] De Jonghe | [END_REF] .

A hysteresis that scales with the concentration has been interpreted as defects acting independently to pin the contact line. If this is the case, the slopes in Fig. 3 give the energy dissipated as the contact line jumps over a single defect. The values of H/n are given in each figure. For the case of barely sintered spheres and decane, the slope is ≈ 4.10 4 k B T , which is of the same order as the energy barrier estimated in Section 2.1. The slopes are at least an order of magnitude smaller in the case of the sintered spheres and still no change in the dynamics is observed.

Hysteresis and defect shape

Next, we prepared a series of samples with nominally the same defect concentration, but sintered at 10 different temperatures. Defect profiles were obtained by AFM for each sample. Figure 1c shows 2 examples. A mean defect height ranging from h 0 = 4.2 ± 0.6 nm to 74.1 ± 1.0 nm was determined from the detailed image analysis of typically 20 defects per sintering temperature. AFM yields accurate height data, but profile information can require interpretation due to possible tip-convolution effects and the inability to image undercut structures such as the bottom half of a sphere. These effects are most important for barely sintered defects. For final defect heights below 30 nm, the measured profiles are well described by a Gaussian

z = h 0 exp - x 2 + y 2 2d 2 , ( 5
)
where we refer to 2d as the width and h 0 as the height. The maximum slopes are less than 20 • and the convolution with the AFM tip of radius order 5 nm introduces only a small error in the overall defect width, which ranges from 80 to 200 nm.

The hysteresis was measured on each surface using two alcanes, as before. Knowing that H ∝ n from the experiments described above, an energy dissipated per defect was found for each shape by subtracting the reference substrate hysteresis and normalizing by the concentration n = 5.9 ± 0.2 µm -2 . Figure 4 shows the dissipated energy as a function of the defect height. It ranges from 10 5 k B T down to 250 k B T . In order to interpret the dependence on the shape, we first briefly recall the ideas that lead to the scaling expression for the hysteresis that is often quoted in experimental papers.

In a static picture, the shape that a contact line takes in the vicinity of a single defect depends on the balance between the pinning force of the defect and the elastic restoring force of the contact line. In the picture of Joanny and de Gennes, hysteresis arises because the force balance is multistable: there are multiple, locally stable energy minima and the specific configuration that the contact line adopts depends on the direction in which it is moving. They developed expressions for both forces applicable to a single defect and propose a graphical solution to the balance, like that shown in Fig. 5a. Figure 5b shows the sequence of solutions for an advancing and for a receding contact line, and the shaded area is the total energy dissipated as the contact line moves back and forth over one defect. Assuming that defects act independently, this area is H/n. They introduced the notion of weak and strong defects to refer to defects that are monostable and those that are multistable. Here we use the terminology very strong to signify that the defect force is at some points much much steeper than the elastic restoring force (in the coordinates plotted in Fig. 5). In this limit the shaded areas are nearly triangular, and consequently the hysteresis scales as

H = n F max d 2 k , ( 6
)
where F d is the defect force and k is the Hookean spring constant of the contact line. The spring constant can be written 1

k = γ sin 2 θ eq ln (L/d) , ( 7
)
where d is of order of the size of the defect and L is a large scale cutoff often taken as the capillary length or the size of a drop. Equation 6has often been used to interpret experimental data 2,[4][5][6]8,31 . In the case of Gaussian defects, the maximum defect force can be written (see Appendix A)

F max d = √ 2πγe -1/2 sin θ eq h 0 . ( 8
)
According to eqn 6 the hysteresis therefore scales as

H = 2πe -1 ln (L/d) γh 0 2 n. ( 9
)
The same dependence, with a different numerical prefactor, can be found for spherical-cap shaped defects.

For these shapes, the defect width does not enter into the estimate for H when assuming very strong defects. So, tip convolution errors do not enter. Returning to Fig. 4, a hysteresis that scales as the square of the defect height agrees with the experimental data. The full range of defect sizes are shown here, spanning nearly 3 orders of magnitude in H/n, with the most sintered spheres yielding 250 k B T .

Hysteresis summary

Equation 9 also gives a prediction for the scaling with the other parameters: the concentration of defects and the liquid. Figure 6 shows the experimental hysteresis (rescaled by the surface tension and with the hysteresis of the reference surface subtracted) as a function of nh 0 2 for all liquids, defect shapes and concentrations used in this study.

In the case of the defect concentration, in addition to the direct proportionality to n, the large-scale cutoff L can depend on the average spacing between defects 1 . Since we find H ∝ n for any given shape, we may argue that we are not sensitive to this variation, which appears as a logarithmic factor. It would nonetheless be perilous to attempt to extract the ln (L/d) from Fig. 6 since it depends highly on the prefactor of the scaling relation.

In the case of varying the liquid, both γ and θ eq are changed.

For strong defects, the angle dependence of the defect force and of the elastic force cancel, leaving a hysteresis proportional to the surface tension. It is already clear in Fig. 3a -which shows the normalized hysteresis-that there subsists a weak dependence on the contact angle, which is roughly 15 • in the case of decane and 33 • for hexadecane. This is seen again in Fig. 6 where hexacane data lies consistently above decane data. Such a liquiddependence has been observed by others 6 , who proposed the possibility of the emergence of a correlation length along the contact line and collective effects. We return to collective effects below, but here we note that an alternative possibility valid even for defects acting independently: as can be seen from geometrical arguments, for defects that are multistable but are not very strong according to the terminology introduced in the previous section, the shaded areas in Fig. 5 cannot be approximated by triangles and therefore depend on the details of the defect force F d and not just on F max d as is assumed in eqn 6. In this case, the hysteresis still scales with γ but its dependence on θ eq is not trivial and cases where it is not monotone can be found.

Discussion

The original aim of this study was to examine the relationship between the individual defects on a solid and the wetting properties on that surface. Despite having chosen the size range of defects with the explicit aim to measure thermal activation, we found no measurable effect on slope of the low-velocity dynamics with any of the defects tested here. We propose two possible explanations. First, the visco-elastic dynamics on the PDMS surface treatment might dominate the dynamics and hide a regime of thermal activation. This idea can be tested by repeating this study with a different reference surface, one that still has a small intrinsic hysteresis. We have tried liquid-phase deposited organosilanes 32 , and initial measurements have not found any difference in low-velocity slopes with or without defects of the same size as used in this study. A similar question arises however: do the thermally-activated dynamics already present on the silane dominate any possible thermal activation over additional defects? In other words, which scale is selected when there are multiple scales to the disorder? Another possibility is that the energy barriers to motion are much larger than estimated here, for example, if defects act collectively to pin the contact line and incremental motion thus requires overcoming a number of defects at a time.

We found that H scales linearly with n, which we have thus far discussed in the context of independent defects. Given the close proximity of defects for the high surfaces coverages used in this study, it is reasonable to expect that the range of action of neighboring defects should overlap. It has long been understood that collective effects can play a role in the hysteresis and one can ask if this linearity necessarily implies that defects act independently or if it can also be found for interacting defects.

In their simulation, Crassous and Charlaix focused on random, strong defects and found a sub-linear dependence 3 , which had been observed by others 9 . There is no evidence of sub-linearity in our present study. The data in Crassous and Charlaix also crosses into the regime of weak defects, but no scaling is explicitly given. By definition, a single weak defect does not pin the contact line, however, a collection of random defects that are individually weak can also act collectively. The contact line exhibits a coherence length (or Larkin length) over which a number of defects act cooperatively to pin the line and multi-stability emerges where there was none for a single defect. Provided that the coherence length does not exceed the capillary length, Robbins and Joanny showed that there is always a hysteresis, even for vanishingly weak defects 33 . Their expression reduces to a linear dependence on defect concentration 34 , and in particular gives the same scaling as eqn 9 for the case of Gaussian defects (see Appendix B). Therefore, Fig. 6 shows that experimental data follows this scaling, but this scaling does not discriminate which regime applies to the present study, ie. strong independent or weak collective pinning.

An advantage of the work presented here is that we have access to detailed defect information and can modify it to access a wide range of shapes, and, potentially, regimes. For example, we can attempt to answer the question whether individual defects are weak or strong by using the shape information to esti- 6 Measured increase in rescaled hysteresis due to the presence of defects as a function of the scaling of eqn 9 for all of the combinations of defect density and shape tested here (nearly unsintered (circles), varying degrees of sintering (stars) and the very sintered shape already shown in Fig. 3a (triangles)) and three different liquids (decane (red), hexadecane (green) and water (blue)).

mate the defect force (see eqn 12) and compare it to the effective spring constant (eqn 7). For the smaller defects used (such as the blue curve in Fig. 1c) the defects are not very strong. Given the plausible range of the ln (L/d) factor it is not certain that they are even strong at all, but may indeed be at the limit of being weak. Measuring a contact angle hysteresis on individually weak defects raises the question of the physical origin of this hysteresis: is it due to a multistability that emerges due to collective pinning, as outlined above, or is the problem still monostable and the hysteresis is evidence of a force-threshold that is present in a force-driven situation, as we recently found in the case of an effective energy landscape 18 ?

Conclusions

Wetting hysteresis -and more recently dynamics-in the presence of disorder or defects has been studied for some decades now. However, experiments linking the measurable surface properties and the wetting properties were lacking. By controlling the shape and concentration of defects and using a reference surface with well-characterized dynamics, we have made a step along this path. We find a simple scaling of the hysteresis but argue that it does not allow discrimination between models, thus illustrating that there is still considerable room to improve our understanding of which limits or physical conditions drive hysteresis for a given real actualization of the disorder. For example, are the individual defects weak or strong, do they pin the line independently or collectively, and is the problem monostable or multistable? In the case of dynamics, we show that there is no evidence of thermal activation on added defects, which we suggest may be due to another length scale emerging to determine the activation length. Numerical simulations that match the experimental defect properties are needed to further address these questions. One could thus control the transition of weak to strong pinning, investigate the passage from a single defect to a density of randomly-spaced defects and look for signatures of collective behavior such as the dependence on n and avalanches, in addition to simulating thermal-activation.

Appendix A

Here we justify the expression for the defect force in eqn 8. We begin by recalling that the local force on the contact line arises from the fluctuating part of the spreading parameter. For topographical defects and in the case of small slopes, this can be approximated as 1 h(x, y) = -γ sin θ eq ∂ z ∂ y ,

where z(x, y) is the surface profile, y is the direction of motion and x is transverse to this (parallel to the unperturbed contact line). We are interested in the total force integrated along the contact line, which can be approximated as 1

F d = - ∞ -∞
γ sin θ eq dz dy y=ψ(x) dx.

Here ψ (x) indicates that the y coordinate is to be evaluated at the distorted contact line position. In other words, the problem is recursive since the force depends on the precise position of the contact line and the position of the contact line depends on the force. Following Joanny and de Gennes, we assume that the integral is dominated by the maximum force, when the deformation is near its maximum and the y coordinate is denoted y m . For the Gaussian shape defined in eqn 5 the force can then be written solid and show that in our case it gives the same scaling as eqn 9.

They find that the threshold force for depinning the contact line is

H ∼ h 2 γ sin 2 θ eq , ( 13
)
where h is the r.m.s. of the fluctuating part of the spreading parameter, given in eqn 10 for topographical defects. For Gaussian defects, and provided that the defects are not overlapping, this can be written as h = π/2γ sin θ eq h 0 √ n, yielding

H ∼ γh 0 2 n. (14)

Fig. 1

 1 Fig. 1 (a) Schematic illustration of the procedure to fabricate controlled defects. (b) SEM images of the produced surfaces illustrating (left, scale bar 1 µm) the homogeneity in coverage even for high concentrations (here n = 35 µm -2 ) and the shapes (right, scale bars 100 nm) with little (1080 • C) or considerable (1164 • C) sintering. (c) Example height profiles of sintered defects obtained by AFM. Red and blue lines are fits to eqn 5 for sintering temperatures of 1144 • C and 1164 • C respectively. The former curve has been shifted upwards for clarity.

Figure 2

 2 Figure 2 shows the contact line dynamics measured on the reference surface in the absence of any additional defects and the dynamics on a surface with nearly 1 nanosphere/µm 2 . Here the defects are barely sintered (1080 • C, h 0 = 74.1 nm) and the liquid is decane. The dynamics are shown on a logarithmic scale, emphasizing the lowest velocities, where thermal activation may play a role.Fits over the lowest four decades in velocity to eqn 3 yield activation lengths λ of order 100 nm for both the reference and the surface with defects. However, in the case of the former curve, in a previous study19 we showed that the contribution coming from hydrodynamic dissipation is negligible below approximately 100 µm s -1 and that the remaining dynamics can be entirely attributed to a visco-elastic dissipation in the thin PDMS layer. In other words, there is no evidence of thermal activation on the ref-

Fig. 3

 3 Fig. 3 Hysteresis as a function of the defect concentration (a) for heavily sintered defects with the profile shown in Fig. 1 (1164 • C, h 0 = 6.3 nm) and (b) for nearly unsintered defects (1080 • C, h 0 = 74.1 nm). For the latter, the open symbols indicate the advancing and angles. Three different liquids are used: decane (red), hexadecane (green) and water (blue). The linear slopes are indicated and give the energy dissipated per defect, H/n.

Fig. 4 Fig. 5

 45 Fig. 4 Energy dissipated per defect as a function of the defect height as measured by AFM, for decane (red) and hexadecane (green). The dashed black line indicates the power law h 0 2 .

  Fig.6 Measured increase in rescaled hysteresis due to the presence of defects as a function of the scaling of eqn 9 for all of the combinations of defect density and shape tested here (nearly unsintered (circles), varying degrees of sintering (stars) and the very sintered shape already shown in Fig.3a(triangles)) and three different liquids (decane (red), hexadecane (green) and water (blue)).

Table 1

 1 Liquid properties at 20 • C

	liquid	γ (mN/m)	ρ 10 3 kg m -3	η (mPa s)	θ ( • )
	decane	23.83 [27]	0.7300 [28]	0.92 [28]	15
	hexadecane	27.47 [27]	0.7733 [28]	3.34 [28]	33
	water	72.88 [27]	0.9982 [29]	1.0016 [29]	106

| 1-9

| 1-9

2d 2 . (12)This gives the bimodal defect force illustrated in Fig.5. The maximum force occurs at y m = d and yields eqn 8.

Acknowledgements

The authors gratefully acknowledge discussions with Etienne Rolley, Bruno Andreotti and Eugene Choi and technical assistance from the ENS support staff: José Palomo, Michael Rosticher, Carlos Domingues Gonçalves and José Da Silva Quintas. This work was funded by the ANR grant REALWET.

Notes and references

Appendix B

Here we recall the results of Ref. 33 for the hysteresis due to collective pinning of a contact line over a randomly disordered

Conflicts of interest

There are no conflicts to declare.