Thomas Leduc

Kevin Hartwell

The Automatic Classification of Urban Open Space by a Pattern-matching Method of the Viewshed at Intersections

Keywords: Shape recognition, isovist field, pattern matching, Fourier transform, intersection

This research focuses on the automatic classification of small urban fragments through a morphological analysis of cognitivist inspiration. The recognition algorithm is performed on observer-centric forms, constructed through the use of visibility assessment techniques over a series of individual points of view. These tools are 1) the isovist for its capacity to delineate and synthesize the visual properties of the immediate viewshed from a point, and 2) the automatic construction of a typology of intersection patterns. The aim is to assimilate the forms of the theoretical intersection patterns to those extracted from the isovist field generated by a group of strategically placed points. Three different matching methods are proposed, and the significance of the parameters needed for optimal calibration are discussed.

Introduction Non-uniform representations of urban junctions

A comparison of the linear road data from Web Map service providers reveals that the treatment of intersections (nodes in the urban graph) is not a uniform procedure, thus resulting in multiple interpretations. As an introductory illustration, the intersection of the Lecesne, Eyries, du Bastion and Delavigne Streets in Le Havre, France (49°29'36.4589", 0°7'0.0235") is one of many such crossroads (figure 1). As can be seen, this zone is interpreted either as consisting of an eastern threebranch intersection and a western five-branch intersection (Bing Maps, figure 1a), a unique point with six branches (OpenStreetMap, Mappy, figure 1b), three distinct points (Google Maps, figure 1c), or two intersections with five branches for the southern point and three branches for the northern point (Géoportail, figure 1d).

A satellite view of the zone (figure 1e) reinforces the perceived complexity: a great number of concurring roads, a juxtaposition of acute and obtuse angles, atypical corner buildings creating mask effects, unconventional crosswalk positions, etc. The "ground truth" itself defies simple determination and consistent interpretation. This divergence in representation exhibits the lack of consensus concerning the interpretation of this complex urban object. Although some dissimilarities may stem from differing linear road extraction methods, much of the issue concerning intersection point placement and characterization also lacks proper formalization. This variation has repercussions on cartographic representation, urban graph analysis, accurate positioning and wayfinding processing.

As the difference between "correct" and "incorrect" representations of a given intersection are difficult to distinguish, we may rather ponder how to consistently read and represent complex urban junctions.

Embracing the subjective quality of this urban object, our objective is to classify intersections according to their impact on their in situ cognitive repercussions.

From street patterns and building patterns to open space patterns

Within the framework of map generalization, whose objective is to group and simplify more or less complex geographical forms and phenomena, various studies exist that aim to automate the detection and simplification of junctions in road networks. The existing processes are based on a thinning of the road network, coupled with edge deletion and contraction mechanisms to reduce the complexity within the junction [START_REF] Mackaness | Automating the Detection and Simplification of Junctions in Road Networks[END_REF]. Their objective is clearly to produce schematic and meaningful visualizations in cartography.

In the same vein, we can mention the attempts of [START_REF] Zhou | A comparative study of various strategies to concatenate road segments into strokes for map generalization[END_REF], who produced a "comparative study of various strategies to concatenate road segments into strokes for map generalization". This abstraction level is sometimes useful for better understanding the structure of a road network.

Aforementioned studies should be distinguished from that on the accurate road centerline detection from satellite imagery in digital photogrammetry [START_REF] Cao | Automatic Road Centerline Extraction from Imagery Using Road GPS Data[END_REF]. As [START_REF] Wang | A New Approach to Urban Road Extraction Using High-Resolution Aerial Image[END_REF] points out, the most recent approaches incorporate knowledge-based methods that complement the extractions that were previously based on spectral and shape features without texture information. Some authors propose to couple the use of vector road data to remote sensing technology in satellite image detection to connect broken road segments. In most of these analyses, the levels of detail used are coarse and the objective aims to reconstruct and guarantee the topology of the street network.

This centerline-based approach does not handle large open spaces, as for instance squares. Thus, as can be seen from figure 2, the skeleton of a T-junction (figure 2a) is almost identical to the skeleton of an intersection leading to a square (figure 2b). It is de facto sometimes inadequate to reflect the complexity of some urban places. A number of these studies employ or mention the notion of "pattern" which corresponds to a repeating structural, spatial or temporal feature referring to a composition, configuration or constitution [START_REF] Marshall | Streets & Patterns[END_REF]. For example, [START_REF] Zhang | Building pattern recognition in topographic data: examples on collinear and curvilinear alignments[END_REF][START_REF] Zhang | A Multi-Scale Residential Areas Matching Method Using Relevance Vector Machine and Active Learning[END_REF] have worked on "building pattern recognition" from topographic data by combining the results of multiple pattern-recognition algorithms.

In our present study, we question the representation of streetspace, the 2D open space that fits between buildings and forms the essential connective tissue of the city. As [START_REF] Marshall | Streets & Patterns[END_REF] points out, "streetspace forms the basic core of all urban public spaceand by extension, all public spaceforming a contiguous network or continuum by which everything is linked to everything else". This particular contiguity, the "plenum of the urban space" [START_REF] Couclelis | People Manipulate Objects (but Cultivate Fields): Beyond the Raster-Vector Debate in GIS[END_REF], renders the delineation and identification of localized objects such as intersections difficult. In contrast, [START_REF] Teller | A Spherical Metric for the Field-Oriented Analysis of Complex Urban Open Spaces[END_REF] considers that atomist space conceives of the urban space as a reference system, where objects are identified and defined by their clear and stable limits. These clearly definable objects may be transformed (through rotation, symmetry, translation, etc.) whilst retaining their identity.

The isovist as a solution to delineate the void between buildings

We have chosen the isovist field [START_REF] Benedikt | To take hold of space: isovists and isovist fields[END_REF] which consists in immersing a theoretical panoptic view at a point into the urban numerical space. This point and its associated view constitute a translation of the perception of the microscale streetscape features that affect the in situ experience of the open space (Fisher-Gewirtzman, 2018). Essentially, isovists describe local geometrical properties of spaces with respect to individual observation points with an equal discretion for all possible view directions by forming a 2D horizontal slice of a pedestrian's viewshed of the surrounding space.

In the absence of any opaque surfaces, the limit of the isovist is an arbitrary artificial horizon. By giving form to an area around a point in the continuous field strictly defined by the radius of the isovist and the neighboring building footprints, the isovist may constitute an atomist reduction of the urban plenum. This appealing concept offers an intuitively attractive analytical framework of a spatial environment, as it provides a description of the space from the point of view of individuals as they perceive, interact and move through it [START_REF] Turner | From isovists to visibility graphs: a methodology for the analysis of architectural space[END_REF].

Positioning viewpoints

Intuitively but also for reasons of convenience and availability of dataset, we chose to analyze the road network and locate these viewpoints -the viewsheds generating pointsat the road junctions. Nevertheless, as we will discuss later in this document, one must recognize that with such a solution, point placement is sometimes not optimal and that there is often a surplus of points for a single intersection area. Alternative solutions exist, such as skeletonization [START_REF] Sarradin | Comparing Sky Shape Skeletons for the Analysis of Visual Dynamics along Routes[END_REF], which make it possible to minimize this number of points.

Visual patterns and geospatial patterns [START_REF] Steiniger | Enabling pattern-aware automated map generalization[END_REF] describes a "visual pattern" as a subjective, immersed recognizable form, and a "geo-spatial pattern" as an objective form constructed with full knowledge of the terrain. [START_REF] Ai | A shape analysis and template matching of building features by the Fourier transform method[END_REF] noticed that traditional spatial analysis concentrates on geospatial patterns, for example geometric, topologic or semantic information, but rarely relies upon cognition-related information, which would be here represented by visual patterns.

The isovist form, its central point of view (figure 3a, junction in road network) and resulting viewshed (figure 3b), hold the same characteristics as a "visual pattern" as defined by Steiniger. To classify these visual patterns, we posit that information extracted from the isovist mimics a synthetic, exterior view of the studied terrain. Such similarities allow us to compare them to a series of objective intersection patterns (figure 3c). Such objective patterns can be associated with "geo-spatial pattern", as they are created with specific domain knowledge, independent of any specific point of view. In other words, we may assimilate visual patterns to geo-spatial patterns by understanding the in situ visual repercussions of spatial configurations. Our objective is therefore to match the cognition-related information (visual patterns captured in situ) with geospatial patterns that have been arbitrarily predefined.

After presenting the isovist as a solution for the atomization of plenum space, the decomposition of its resulting form into a function of radial distances is explained, followed by the methodology for the automatic construction of reference patterns. After the description of the three pattern recognition methods, three case studies are then presented illustrating the impact of the urban morphologies on the results obtained. Case study results allow us to draw conclusions regarding the parameters affecting the quality of the process.

Methods

Isovist form as a function of radial distances

The isovist is a visibility polygon, meaning all segments from the generation or view point to all other points within the shape lie entirely within the polygon. Taking advantage of this star-shaped property, we can derive a shape signature based on the generation point and its correspondent boundary. This dimensionality reduction solution limits the study of the form to that of its contour which is sampled at their intersection with a series of equally-angled segments (rays) originating from the center point.

This angular abscissa sampling solution allows us to transform the 2D-polygonal surface into a simple 1D-function of a real variable that, at a given azimuth (θk), returns the associated ray length (rk). Let n be the number of rays, using complex number formalism and Euler's number (e), the k-ray can be represented as:

𝑟 𝑘 𝑒 𝑖𝜃 𝑘 = 𝑟 𝑘 𝑒 𝑖 2𝜋𝑘 𝑛
Since this shape signature is only dependent on the location of the viewpoint and the points on the boundary, it sits in a local coordinate system and is thus invariant to translation in space. The matchmaking between the isovist form and a reference pattern is therefore independent from the viewpoint's localization on the map.

The radial "sweep" is systematically operated from 0° to 360° starting full East, in the counterclockwise direction. This means the function of radial distances is dependent on the local orientation of the intersection.

Pattern generation

The form-generating model that we have developed for the construction of our reference (or geo-spatial) patterns is entirely automatic, based on the principles of permutation of a "word", that "word" being the mechanism that allows us to encode the intersection.

Based on this formal definition of words, we may generate the figure 3c, representing all possible intersection forms having at most 8 branches separated by steps of 360/8=45°. Patterns equivalent with regards to the mirror effect share the same IDs.

A limit of eight branches was selected, as multiples of four include the most frequent angularity (90°) which a multiple of 5 for example would not. The number of possible branches exponentially increases the number of patterns, which reduces the distinguishability (i.e., distance) between patterns with relatively superfluous differences, greatly complicating the manual assessment. As complementary information, one can notice that with a maximum of 4 branches (resp. 8, 12, and 16), one may generate 13 different patterns (resp. 35, 351, and 4115).

Patterns were created by casting the rays corresponding to their encoding, then buffering the resulting multilinear geometry. The width parameter describes the buffer breadth used during the pattern construction. This has ulterior consequences on pattern matching for methods using the Fourier Transform, as it dictates the quantity of uninterrupted open space considered for a single branch. If a branch is much larger than the arbitrary width, Method 2 identifies it as two branches. Inversely, if the chosen width is significantly larger than the real-world branch, it might be overlooked. Optimal widths for our chosen case studies are 14m (New York) and 10m (Paris, Le Havre).

Pattern matching process: methodology overview

The strategies set in place to judge of the similarity are based on different mathematical principles. The first method uses the turning angle function's mean vector to reorient shapes and conduct a linear comparison (figures 4b and 4f). The second methods translate our function into the frequency domain and compares Fourier modules (figures 4c and 4g). Finally, the third method distinguishes the angles at which ray lengths reach their maximum (figures 4e and 4h). Method 1: Rectification of pattern and isovist orientations using the mean vector Using the previously established formalism, we may express the mean of all vectors linking the generation point to each contour as:

1 𝑛 ∑ 𝑟 𝑘 𝑒 𝑖𝜃 𝑘 𝑛 𝑘=1 = 𝑟̅ 𝑒 𝑖𝜃 ̅
The form's general orientation given by the mean vector allows us to "rectify" the viewshed (figure 4b). To do so, we position each ray length (i.e. rotation from the Eastmean alignment) and subtract the azimuth of the mean vector's using the following formula:

{𝑟 𝑘 𝑒 𝑖(𝜃 𝑘 -𝜃 ̅ ) } 𝑘∈{1,… ,𝑛}
If the form is perfectly symmetrical (as for the patterns #20, 43, 45, 63, and 80), the corresponding mean vector is null. We then decide not to apply the rectifying process.

The objective of this rotation is to rectify the studied form by aligning it full-East on the "azimuth of origin" for a direct linear comparison of the visual and geo-spatial patterns.

Method 2: Comparison of the frequency functions using the Discrete Fourier Transform Method2, using the Fourier Transform traditionally used for time-dependent signals, incorporates a periodic complex-valued function composed of regular angular intervals and their decomposition into a set of sinusoidal functions.

Because the function of radial distances is discrete (due to the implemented underlying ray-casting), we have applied the Python Numpy Fast Fourier Transform (FFT). FTT properties include completeness (it is an invertible and linear transform), periodicity, and shift capability. All three of these properties are essential to describe nearly equivalent shapes, if identically oriented before or after rotation.

To bypass the effects of a form's orientation, we exclude information relative to the angular distribution of the rays by omitting phase, and only consider the distribution of ray lengths given by the FFT modules (figure 4c).

Method 3: Comparison of mean angular distances between peaks

The objective of the third method is to produce a sequence of angular distances between the extrema of the form, which are determined by a threshold. Method 3 first evaluates the mean azimuth of the adjacent ray set qualified as extremum. In a second phase, the angular distance between two adjacent azimuths of the mean extrema are calculated, starting east and in a counter-clockwise order. In figure 4d, the application of Method3 gives the sequence [91.4°, 178.4°, 90.3°], as the Northern branch is recognized first, followed by the Western and Southern branches. The pertinence of the results, based on threshold value, is therefore dependent on the aforementioned ray length and the threshold ratio.

Measures of similarity and pattern pairing

After deriving the quantitative information from the set of real-world and auto-generated patterns, we assess the level of similarity between the two sets. The pairing consists in identifying the reference form whose distance is smallest from the real-world form (figure 5b). After testing multiple distance methods (Manhattan, Chebyshev, DTW, Canberra), we opted for the L2 Euclidian distance which yielded best results for all three methods. This pairing necessarily relies on the technique used in the previous step. It is therefore possible to assess the certainty of a result relative to other results found by the same procedure, but there is, as of yet, no suitable solution to compare the certainty of each method in cases of conflicting matches.

Results: Comparison of Use Cases

Three use cases (figure 6) with contrasting morphologies were analyzed. The first one is an area of about 4.6 km² located in Borough Park, Brooklyn, New York. The second use case is a 2 km² wide area located in Le Havre, France, and includes the rebuilt City Center as well as the North East Danton borough. At last, Paris 9 th borough, France, is an area of about 2.2 km². To assess the reliability of the different methods, each node of each real urban graph was preliminarily manually assessed. This manual classification is called "ground truth".

The morphology of the streetscape influences the difficulty in assigning an objectively "best pattern", for several reasons. Firstly, considering different distances from the center point may yield different results. To remedy this ambiguity, we have built a short range ground truth and a long range ground truth assessment representing the possible intersection pattern at different distances. Furthermore, the angularity of certain real intersections may be ambiguously similar to more than one pattern, so much so that differing pattern assignments may be considered equally correct. Finally, the width of certain streets or plaza openings are also a cause for ambiguous manual assessment, as explained further on in this paper.

As we may notice (see table 1), the first use case (New York borough) is relatively unambiguous, composed mainly of right-angled four-way intersections. Three diagonal roads disturb the orthogonal grid, forming unconventional intersections and plazas. As such, short range and long range assessments are undifferentiated (i.e. we consider that ray length has no influence on expected results).

In contrast, the manual assessment of the second use case (Paris's 9 th borough) was much more challenging. Ambiguous intersections were found to be much more prevalent due to the complex morphology. We therefore manually assessed each node considering the immediate (short range) and slightly more distant (long range) streetscape and assigned two possible correct results in 64 out of 335 cases at a short range scale, and 74 out of 335 at a long range scale.

Finally, the Le Havre borough, from which the Lecesne intersection originates, may be considered as an "in-between", as the Southwestern section is composed of an orthogonal grid, and the Northeastern is much more complex. We will notice though, that the orthogonal portions of the streetscape present very large open spaces. As such, the delimitation between "street" and "built environment" is much more ambiguous than the first two use cases. This caused us to consider nearly 37.4% of the intersections as ambiguous at a short range scale and 30.6% at a long range scale. Comparing Use Case Results

The three use cases reveal contrasting results for each automatic assessment method. We may first notice that the less complex the urban fabric is, the better the detection rates are (figure 7): 78% success rate for New York (Method 2), 59% for Le Havre (Method 2), and 53% for Paris (Methods 2/3). Furthermore, success rates by ray length (figure 8a) suggest that New York did indeed not need two different scales of evaluation: maximum success rates remain stable for all methods regardless of the distance between center point and viewshed limit. In contrast, when considering the short range interpretations of the other two use cases, success rates for short range interpretations tend to drop and long range success rates tend to rise after a certain ray length. We may notice though, that Le Havre's long range success rates stagnate whilst Paris's long range success rates present a dip at longer ray lengths. This means that some of Paris's intersections change pattern more than twice when increasing ray length, whilst two scales of interpretation are sufficient for Le Havre. As an example, the junction node #193 in Paris is similar to pattern #30 when the radius length is 30m, to pattern #45 when the radius length is 50m and to pattern #50 when the radius length is 100m (figure 9a). In addition, the threshold at which long range interpretations become more pertinent than short range interpretations is smaller for Paris than for Le Havre. Furthermore these complex urban fabrics include many curved streets, dead ends, or kiosks which limit isovists' lines of sight. All these obstacles justify the evolution of the success rate curbs at the long range scale. Thus the peak of the long range interpretations ratio at 70-80m (decreasing or increasing the length leads to a deterioration in performance) in Paris 9 th borough is clearly due to the fact that the corresponding intersections are tightly clustered and their interpretations vary greatly with ray length. Interpretations are modified when another intersection enters the line of sight, or when longer ray lengths prematurely end, such as in close-knit, three-way intersections or convoluted spaces such as winding roads.

We can also notice that the ground truth is even questionable. Thus, in New York and for the used dataset, node #87 (figure 9b) has been empirically assimilated to pattern #45 (X-junction). By means of Method 2, we detected it as pattern #43 (right angle crossroads). This "wrong" detection is relative insofar as an ambiguity remains related to the chosen angular distance sampling (which cannot be lower than 45°). As an exception to this rule, the Lecesne intersection presented at the beginning of the paper (figure 1) requires a single point with a viewshed of 55 meters for the six branches to be taken into account. A smaller viewshed requires at least two intersection (i.e. generation) points to cover the entire area.

Comparing Method Results

We may first notice that Method 1 yields unsatisfactory results for nearly all terrains, and more especially on the urban fabric composed mainly of right-angled four-way intersections (New York), regardless of ray length. This may be explained by the radial symmetry of many intersections, leading to a null mean vector and therefore the absence of proper rotation, strongly limiting the use of this method. Nonetheless, Method 1 yields satisfactory results on nodes that are not symmetrical.

Method 2 yields the more stable and convincing results on all three terrains. It is able to differentiate streets from large open spaces (plazas), but is sensitive to point position and building geometry such as uneven street widths or curbed streets. Method 3 is quite effective on the very regular urban fabric (New York) and the complex urban morphology (Paris), but does not differentiate between streets and plazas, giving a single mean direction for the opening of the plaza rather than recognizing that the plaza offers multiple directional choices, as Method 2 do. For example, in the case presented in figure 9c, Method 2 will return "Pattern 47" whereas Method 3 will return "Pattern 22". This explains the rather feeble success rates for Method 3 in Le Havre on a short range scale, which presents a nearly "Corbusean" morphology, with large open spaces separating individual buildings rather that a clear distinction between street and building blocks.

In an effort to reduce calculation times, different numbers of rays were tested. It was found that for all three terrains, at least 64 rays were necessary for the correct approximation of the form, this number growing with ray length so as to keep an acceptable distance between each ray extremity. Otherwise, small branches or buildings may go unnoticed. Inversely, a very high ray count creates noisy signatures, which interferes with all three methods.

Discussion: Unconventional or Ambiguous Intersections

As shown in figure 9d, the detection of junction whose shape is rather evasive nevertheless misleads Method 2. Such junctions, full of ambiguities, probably require an appropriate post processing. In addition, we have already mentioned that using official road dataset may present several disadvantages. Point placement is sometimes inconsistent, there is often a surplus of points for a single intersection area, and points may not be ideally placed (which leads to errors in the interpretation of the junction as exhibited in figure 9e). The use of a skeleton would reduce the necessary datasets to the single buildings layer. By building the road graph according to the built environment, we hope it will make for fewer, better and more consistently placed points considering the overall visibility of each street from the intersection.

For a further analysis of point location on expected results, we have pinpointed five intersections in the Parisian borough and compared the results of the graphs from the Real Urban Graph (RUG, figure 10a) with OpenStreetMap (OSM, figure 10b) road data and a network created by skeletonization (SKEL, figure 10c).These five intersections are here described by their position: three Northern intersections, including the Center and East intersections which are small and relatively unambiguous, and two southern intersections (east and west) which, like the Northwestern intersection, cover a much larger area and carry a varying number of points in all three networks. The most Northwestern, Southwestern, Southeastern junctions (but also to a lesser extent the Northeastern one) are obviously ambiguous and therefore hard to interpret. Due to misplacement and proliferation, viewpoints have incomplete views of the entire intersection; their interpretation is therefore incomplete yet coherent considering their local visibility. Identifying unique pattern that correctly agglomerates the information for the entire zone would be a more satisfactory result.

A closer look at the maps, let us think that the viewpoint offering the most complete and appropriate information appears to be the most centered one, such a vantage location would logically offer a simultaneous perspective on multiple branches.

The table presented in figure 10d lists, for each data layer, for each intersection and for each node, the corresponding isovist area. This listing is sorted by decreasing area value. In an effort to join the results of all three graphs, we noticed that the isovists with the largest areas at each intersection were those that gave the most pertinent results (figure 10e). This presumption that intrinsic isovist characteristics may indicate correct point placement needs to be further tested. If this hypothesis were confirmed, we could imagine an empirical research method from the optimal point of view (covering the most of the intersection studied). This heuristic would exploit some simple shape indicators such as area, perimeter or number of concavities.

Conclusion

We have presented three methods to consistently symbolize the open streetscape at urban intersections, based on the comparison of auto-generated patterns and isovist signatures. We have further provided guidelines for the most informative symbolization of the intersection of streets and plazas. We have defined a complex intersection as a concave space where all inlets are not visible at once, and provided orientations towards a way of partitioning such a space. Our case studies have uncovered a correlation between the area's urban morphology and the optimal isovist parameters needed to recognize the correct pattern.

An isovist creates a distinct separable entity to which we may give an identity, including the visual pattern associated with the rest of the atomized plenum. We have found multiple criteria for optimal isovist generation and pattern recognition. Most importantly, ray length and point placement are the two most determining factors in the quality of the results, along with angular distance sampling and the size of the pre-constructed patterns. The values of these four parameters are entirely dependent on the area's urban morphology. We may strongly presume that similar terrains will require similar parametrization.

In the future, legitimate distance measures for direct comparison of the three methods should be determined, to better identify which of the three methods consistently offers more suitable patterns. The method-based differences between matched patterns, as illustrated with differences at corner inlets of plazas, may be resolved by the addition of a new set of geo-spatial patterns taking into account the representation of large openings.

Finally, the question of point placement for the optimal visibility of the most outlets is still open. Several possibilities come to mind, but our research indicates that the centroid of the intersection's kernel, which would lead to the isovist with the largest area, may be an interesting path to pursue. The amount of convex areas within an intersection would then indicate the minimum number of points needed to fully describe the surrounding intersections.

Figure 1 :

 1 Figure 1: The Lecesne intersection as seen by: a) Bing; b) OpenStreetMap; c) Google Maps; d) Géoportail; e) Aerial View.

Figure 2 :

 2 Figure 2: The detection of road intersections by the skeleton method does not take into account large open spaces, as for instance squares.

Figure 3 :

 3 Figure 3: The isovist (b) created from a point of view (a) is similar to a purely geometric form (c).

Figure 4 :

 4 Figure 4: To compare the original signatures of the isovist (a) and the pattern (e), we use three methods: Method 1 -the rectification of the orientation using the mean vector (b and f), Method 2 -the modules of the Fast Fourier Transform (c and g), and Method 3the permutation of the mean angular distribution (d and h).

Figure 5 :

 5 Figure 5: The distance from the actual shape (a) to each of the 35 patterns is evaluated. The pattern selected is the one that minimizes these distances (b).

Figure 6

 6 Figure 6: a) Map of Borough Park (New York), b) Map of Paris 9 th Borough, c) Map of Le Havre (France).

Figure 7 :

 7 Figure 7: Box plots regarding the pattern matching assessment on the three studied districts.

Figure 8 :

 8 Figure 8: Results of the pattern matching assessment on Borough Park (a), Paris 9 th Borough (b and d), and Le Havre (France, c and e).

Figure 9 :

 9 Figure 9: Examples of spatial configurations exhibiting special features.

Figure 10 :

 10 Figure 10: Real Urban Graph (a), OpenStreetMap (b), and Skeleton (c) intersection points as characterized by Method 2 at a line of sight of 45m and with 64 rays. The result of selecting the point at each intersection with the greatest isovist area (e).

Table 1 :

 1 Ratio of ambiguous intersections at both short range and long range scales.

		Number of	Short range scale	Long range scale
		intersections
	New York borough 297		2.7%
	Paris's 9 th borough 335	19.1%	22.1%
	Le Havre borough	444	37.4%	30.7%

Acknowledgements

The authors would like to thank Michel Leduc, Honorary Professor at the University of Le Havre, France, for his helpful comments and suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.


Sylvain Franiatte

Philippe Tordjeman
email: philippe.tordjeman@imft.fr

Thierry Ondarçuhu
email: thierry.ondarcuhu@imft.fr

Wetting at the nanoscale: molecular mobility induced by contact line forces

In this paper, we study the interaction of a contact line with molecules physically adsorbed on a surface. We developed specific atomic force microscopy (AFM) experiments where a nanoneedle attached at the extremity of the cantilever is dipped in a liquid droplet. The motion of the contact line at the extremity of the meniscus formed depends on the presence of topographical and chemical defects at the surface of the nanoneedle. The analysis of the force measured by AFM based on a capillary model, allows to distinguish the effects of topographical and chemical defects and to monitor minute changes of surface properties. Using six different liquids and five tips, we show that the change of the surface properties of one nanoneedle results from the adsorption of airborne molecules when the tip is left in air and their desorption by the moving contact line. The desorption rate is found to depend only on the number of dipping cycles, and is not influenced by the velocity or the liquid properties. A model based on the estimation of capillary and adsorption energies confirms a capillary desorption mechanism in agreement with the experimental results. Finally, we demonstrate that three distinct desorption mechanisms may be at play. Interestingly, using a deliberate 1 contamination with large hydrocarbon molecules we show that the capillary desorption studied in this paper can be used to clean surfaces.

Introduction

It is well established that the wetting properties of a sessile drop are fixed by the topographic and chemical properties of the substrate. [START_REF] De Gennes | Wetting: statics and dynamics[END_REF][START_REF] Bonn | Wetting and spreading[END_REF] Recently, mechanisms involving the surface modification by a coupling between substrate and liquids have been considered. Substrate surface may adapt to a liquid by minimizing the interfacial energy. This can be achieved by different processes such as chemical groups reorientation, molecular ordering, swelling, diffusion or adsorption... [START_REF] Butt | Adaptive wetting-adaptation in wetting[END_REF] These phenomena are mainly considered at the solid-liquid interface. However, specific interactions can also play at the drop edge [START_REF] Tadmor | Open Problems in Wetting Phenomena: Pinning Retention Forces[END_REF] and modify the macroscopic wetting behavior. An example of such mechanism is the formation of a ridge on soft solid by elastocapillarity. [START_REF] Andreotti | Statics and dynamics of soft wetting[END_REF] The capillary force at the contact line which deforms the solid may also induce local molecular changes and modify the pinning of the sessile drop. The interplay between the substrate modification and the dynamics of the drop is an open question in wetting science, which requires new specific experiments at the molecular scale. The extreme sensitivity of the contact line motion to surface defects leading to contact angle hysteresis has been studied for a long time. [START_REF] Ramos | Contact angle hysteresis on nano-structured surfaces[END_REF][START_REF] Ramiasa | The influence of topography on dynamic wetting[END_REF][START_REF] Perrin | Defects at the nanoscale impact contact line motion at all scales[END_REF] Both types of defects, topographical and chemical, 9,10 lead to pinning and distortion of the contact line which greatly reduces the spreading and mobility of macroscopic droplets. [START_REF] Joanny | A model for contact angle hysteresis[END_REF] It was shown that a molecule monolayer [START_REF] Silberzan | Silanation of silica surfaces. A new method of constructing pure or mixed monolayers[END_REF][START_REF] Lim | Surface-treatment effects on organic thin-film transistors[END_REF] or an atomic thick coating [START_REF] Ondarçuhu | Wettability of partially suspended graphene[END_REF] modifies the wettability of the underlying substrate whereas nanometric asperities can make a surface super-hydrophilic or super-hydrophobic. [START_REF] Checco | Robust superhydrophobicity in large-area nanostructured surfaces defined by block-copolymer self assembly[END_REF]16 Contamination by particles or chemical species from the liquid or atmosphere also governs the wetting properties in many natural and industrial systems. [START_REF] Okorn-Schmidt | Particle cleaning technologies to meet advanced semiconductor device process requirements[END_REF][START_REF] Li | Effect of airborne contaminants on the wettability of supported graphene and graphite[END_REF][START_REF] Forrest | Influence of surface contamination on the wettability of heat transfer surfaces[END_REF] Since molecules adsorption energies and capillary energies are both of the order of a few kT, a possible action of the contact line on the surface, at molecular scale, requires much attention. Though it is known that the capillary force at the contact line can move particles [START_REF] Sharma | Detachment of colloids from a solid surface by a moving air-water interface[END_REF][START_REF] Aramrak | Detachment of deposited colloids by advancing and receding air-water interfaces[END_REF][START_REF] Naga | How a water drop removes a particle from a hydrophobic surface[END_REF] or align DNA [START_REF] Bensimon | Stretching DNA with a Receding Meniscus: Experiments and Models[END_REF] or carbon nanotubes, [START_REF] Gerdes | Combing a carbon nanotube on a flat metal-insulator-metal nanojunction[END_REF] this issue has never been addressed experimentally at the molecular scale.

Atomic force microscopy (AFM) allows to study the dynamics of a nanomeniscus at the nanometer scale. [START_REF] Yazdanpanah | Micro-Wilhelmy and related liquid property measurements using constant-diameter nanoneedle-tipped atomic force microscope probes[END_REF][START_REF] Delmas | Contact angle hysteresis at the nanometer scale[END_REF][START_REF] Mortagne | Dynamics of anchored oscillating nanomenisci[END_REF][START_REF] Guo | Direct measurement of friction of a fluctuating contact line[END_REF] Recently, we developed an AFM method which relies on the dipping of nanoneedle attached to an AFM cantilever in a liquid while monitoring the capillary force. [START_REF] Franiatte | Molecular Desorption by a Moving Contact Line[END_REF] In these AFM experiments, the nanoneedle fixed at the extremity of the cantilever is repeatedly dipped in a droplet. The evolution of the capillary force captures minute changes of nanoneedle surface properties. On the basis of this work, we demonstrated that a nanomeniscus which sweeps the needle surface induces the local desorption of molecules and modify the wetting properties of the surface. Interestingly, specific experiments based on partial tip dipping unambiguously demonstrate that the desorption of physically sorbed molecules is induced by the capillary force exerted at the contact line. An important result is that the total removal of contaminants is governed by a characteristic number N * of dipping cycles which does not depend on velocity and liquid properties.

In the present paper, we detail the interaction of a moving contact line with physically adsorbed species. We first detail the whole experimental results of wetting at the nanoscale.

With the help of a model of wetting of a heterogeneous surface, we understand the effects of topographical and chemical defects and show the existence of a molecular desorption mechanism which modifies locally the nanowetting. The change of the surface properties of the nanoneedle by adsorption of contaminants is also considered in the model and verified experimentally. As an application of this work, we demonstrate the efficiency of a moving contact line to clean surfaces at the molecular scale, even in the case of a deliberate contamination.

Experimental section

The experiment is sketched in Fig. 1. An AFM tip ended by a nanoneedle is dipped in and withdrawn from a liquid droplet. This "dipping cycle" is realized at constant velocity V and the deflection of the cantilever is measured using an AFM. The measured capillary force F is related to the contact angle θ of the liquid on the tip surface through F = 2πRγcosθ where R is the radius of the tip and γ the liquid surface tension. In this study, we consider six different model fluids, for which the main characteristics are given in the Table 1: ethylene glycol (EG), tetraethylene glycol (4EG), undecanol (Und), hexadecane (Hx), glycerol (Gly) and deionized water (W). Except for water, the liquids are purchased from Sigma-Aldrich (Merck) and are used as received. The choice of this series of liquids allows us to assess the effects of surface tension, viscosity and saturation vapor pressure. Indeed, among these liquids, the viscosity varies over 3 orders of magnitudes, the vapor pressure over 5 orders of magnitude and the surface tension by a factor of 2.6. Sessile drops of liquids of about 4 mm are deposited on a 1" silicon wafer (Siltronix) cleaned in acetone, isopropanol and water just before AFM experiments (Fig. 1c).

In the case of undecanol and hexadecane, the wafer is hydrophobized using 1H,1H,2H,2H, perfluorodecyltrichlorosilane (ABCR, Germany) to avoid the complete spreading of the liquids. Due to evaporation issue which induces a continuous decrease of the liquid height during measurement, droplets of water are produced at the extremity of a teflon capillary connected to a micro-injector (Narishige IM300, Japan) that controls the fluid pressure and therefore the droplet radius (Fig. 1d).

A JPK Nanowizard 3 AFM is operated in contact mode to measure the force curves when a nanoneedle is dipped in a drop at constant velocity comprised between 3 to 200 µm/s. The cantilever stiffness is characterized by thermal noise using the deflection sensitivity derived from contact mode experiments on a silicon wafer substrate. [START_REF] Burnham | Comparison of calibration methods for atomic-force microscopy cantilevers[END_REF] Note that the stiffness of all cantilevers used (k ∼ 2 N/m) is at least forty times larger than the effective spring constant of the interface, which is of the order of γ/2. [START_REF] Dupré De Baubigny | Shape and effective spring constant of liquid interfaces probed at the nanometer scale: Finite size effects[END_REF] We record between 4000 and 12000 points for each curve (1 point/nm). The frequency of acquisition of the AFM (200 kHz) is therefore sufficient even for the larger velocity used (200 µm/s). The raw data are analyzed with home-made Python ® codes.

Two types of tips are used: commercial NaugaNeedle tip (NN) made in Ag 2 Ga alloy (Nauganeedle, USA) and homemade tips (Si1 to Si4) carved at the end of a conventional silicon AFM probe (OLTESPA, Bruker) using a dual beam FIB (1540 XB Cross Beam, Zeiss). [START_REF] Mortagne | Dynamics of anchored oscillating nanomenisci[END_REF] The characteristics of the 5 tips used in this study are presented in Table 2. The nanofibers have radius in the range 60-500 nm and length of several microns. The experiments presented below are based on the measurement of force curves when a nanoneedle is dipped in and withdrawn from the droplet. Figure 2 compares the force curves obtained for a given tip (Si3) and four different liquids. Each curve represents the force F variation as a function of the immersion depth h (see Fig 1) for a same tip velocity V = 50µm/s. When approaching the interface from above, the tip enters in contact with the liquid at h = 0 and the meniscus is formed. As h increases, the contact line advances on the tip surface and the force characterizes the "advancing" branch of the curve. At a given h ∼ 1 -1.3µm the tip is retracted. In a first step, the force increases linearly while h decreases. In this stage, the contact line is pinned and the contact angle varies from advancing to receding contact angles which induces an increase of capillary force. The negative slope of this branch gives the meniscus stiffness. [START_REF] Dupré De Baubigny | Shape and effective spring constant of liquid interfaces probed at the nanometer scale: Finite size effects[END_REF] In a second step, the contact recedes on the tip until it reaches the end and the meniscus breaks. Damped oscillations of the cantilever are observed once the tip is released.

Considering the force curve for glycerol (Fig 2a), we observe that the advancing and receding curves are characterized by similar features: two large peaks followed by smaller ones are measured in both branches. The fact that we observe the same peaks in the two parts of force curves, which can be linked by a line with a slope equal to the meniscus stiffness, demonstrates that the defects are of topographic nature and not chemical one. [START_REF] Joanny | A model for contact angle hysteresis[END_REF][START_REF] Mortagne | Dynamics of anchored oscillating nanomenisci[END_REF] Some differences between advancing and receding curves may be attribute to asymmetric defects that pin the contact line more strongly in one direction than in another. Hence, the force curve realized with glycerol indicates that the nanoneedle exhibits two "big" topographic defects followed by smaller ones, leading to the contact angle hysteresis. These features appear with all liquids, although the magnitude of the force depends on the surface tension of the liquid and on the contact angle. In particular, for undecanol (Fig 2d), due to the small surface tension of this liquid, advancing and receding curves are nearly superimposed and the two large peaks are smoother (as discussed in ref [START_REF] Delmas | Contact angle hysteresis at the nanometer scale[END_REF] ). Finally, the fact that the same characteristics of the curves remains after thousands of dipping cycles in each liquid is an indication of the cleanliness of the experimental conditions and the robustness of the results.

We realized two types of experiments called I and II. The first one consists of 100 successive dipping cycles at a constant tip velocity in ambient conditions in order to monitor the evolution of the nanoneedle surface properties with the number of cycles N . With that aim, all force curves are recorded and analyzed by plotting the force associated with one peak (one defect) as a function of N . In the second experiment, we perform 10 successive series of 100 dipping cycles separated by various times t p during which the tip is left in air. Here, we investigate the effects of airborne contaminants and their adsorption dynamics on the tip surface. These two experiments are realized for all the liquids and 5 velocities to understand the mechanism of desorption of airborne molecules induced by a moving contact line.

After proving that airborne molecules are responsible of the nanoneedle contamination, we also developed a specific experiment with a deliberate contamination by exposition of the tip to hexadecane vapor wich is detailed in the Desorption Mechanism section.

Capillary force model for a cylinder with topographical defects

The nanoneedle fabrication process always leads to the presence of topographical defects.

These defects are responsible for force peaks and hysteresis in the capillary force measurements. Since these nanodefects are difficult to characterize by standard microscopy methods, it is useful to settle our analysis of experimental data on a simple model.

We consider a cylinder along the z axis for which the radius is a function r(z) of the vertical position. This cylinder is partially dipped in a liquid. We note h the distance from the basis of the cylinder to the flat liquid surface reached beyond the capillary length l c , and z CL the distance from the contact line to the extremity of the cylinder (see Fig. 1c). The height of the meniscus is given by H = z CL -h. In our experiment the capillary force F is measured as a function of the immersion depth h. The model allows to compute F (h) from the following equations:

32 F = 2πγr(z CL ) cos(θ v (z CL ))
, where θ v is the angle between the liquid interface and the vertical axis z; this angle can be expressed as a function of the constant equilibrium contact angle θ 0 of the liquid on the cylinder,

θ v (z CL ) = θ 0 -arctan r ′ (z CL ); the height
H of the meniscus is given by: 33 H(z

CL ) ∼ = r(z CL ) cos θ v (z CL ) ln 4lc r(z CL )(1+sin θv(z CL ) -C
, where C ≃ 0.57721 is the Euler constant. These equations are solved using Python ® code for a given r(z). For the sake of simplicity, we use a sine function with small amplitude b and wavelength λ for the small scale roughness and Gaussian functions to mimic i blemishes characterized by an height A i , a width σ i at different positions

z i : r(z) = r 0 + b sin(2πz/λ) + i A i exp[-(z -z i ) 2 /2σ 2 i ].
We assessed the validity of this modelling on the curves of Fig. 2 which were obtained for the same tip in four different liquids. As discussed in the previous section, the qualitative analysis of the force curves reported on Fig. 2 allows to infer that the tip surface exhibits 2 large topographical defects, followed by several smaller ones. We therefore define a r(z) with 6 successive defects, 2 large ones and 4 smaller ones modelled by Gaussian functions with A = 20, 30, 2, 2, 2, 10 nm and σ = 80, 80, 2, 2, 2, 2 nm. In addition we superpose a sinusoidal function with amplitude b = 0.2 nm and wavelength λ = 5 nm to mimic the small scale roughness of the tip after fabrication process and a parabolic variation of the radius r 0 with z to reproduce the change in tip section. The force curves computed by the model on such a tip for the four liquids (glycerol, ethyleneglycol, tetra ethyleneglycol, undecanol) used in Fig. 2 are reported in Fig. 3. For each liquid, the value of θ 0 is varied over five degrees to evidence the influence of the contact angle on the force curves.

The comparison of the computed and experimental curves is fairly good, in particular given the fact that the model considers axisymmetric defects, which is probably not the case for real tips. The overall shape of the curves are reproduced with the 2 large peaks and the smaller ones, with force values close to the experimental ones. The hysteresis between advancing and receding curves is also found to decrease when the contact angle decreases, reaching a very small value for undecanol. The model also predicts well the position of the peaks and the (negative) height h at which the meniscus breaks. This shows that the analysis of the force curves allows a description of the tip surface topography with a nanometric precision. On Fig. 3 are also reported the variation of the force with a small increase of the contact angle θ 0 on the surface (from red to blue). It shows that a decrease of θ 0 leads to an increase of the force on both advancing and receding branches. As expected, the force increase is less pronounced in the part of the curves for which the contact line is pinned on a defect (for example 0 ≲ h ≲ 0.5µm in the receding branch). It also leads to a translation of the peaks position along the h axis. This is due to the fact that the meniscus height H varies with θ 0 . The measurement of capillary force therefore allows to monitor minute changes of surface properties, in particular for liquids with large surface tensions.

F [nN] Und 5°-10°U nd 5°-10°U nd 5°-10°U nd 5°-10°U nd 5°-10°F
igure 3: Simulated force curves using the expression of r(z) given in the main text for the tip profile. The liquid name and the range of θ 0 values used in the calculations (decreasing from blue to red) are reported as legends.

Experimental results

In Fig. 4 are plotted the typical results of experiment I (100 successive dipping cycles) for six liquids and tips, at different tip velocities V (see caption). In this representation we selected several force curves at different numbers N of wetting cycles to show the evolution of the tip surface properties. The number N of wetting cycles increases from blue to red force curves. Compared to the initial curve, we observe a global enhancement of the capillary force both in the advancing and receding branches when N increases, for all liquids except for hexadecane. Interestingly, the details of the advancing and receding curves does not show any evolution which suggests that the observed evolution when N increases is due to a change of the surface energy of the needle without any modification of its topography, as modelled in the previous section (Fig. 3). This fundamental result is general for all liquids (except for hexadecane) and all experimental conditions. On the other hand, the amplitude of change varies with liquid nature and is, in the limit of uncertainties, independent of tip type and V . For hexadecane no measurable evolution is observed. In order to characterize quantitatively the force variation, we select one defect in the advancing (receding) branch and measure the associated minimum (maximum) force F as a function of N . Figure 5 represents the relative variation of the force ∆F/F ∞ = (F -F 0 )/F ∞ as a function of N for different liquids and tips, where F 0 and F ∞ are the values of F for N = 0 and N = 100, respectively. All curves show that the relative force increases and reaches a plateau after about 20 dipping cycles. We also observe that the data obtained with NN tip (Fig. 5b) are more accurate than with Si tips (Fig. 5a). This is due to the larger tip radius of NN (Table 2) which improves the signal-to-noise ratio. The results plotted in logarithmic scale show that the amplitude of the force change strongly depends on the nature of the liquid (typically from 30 % for glycerol and water down to 0.4 % for undecanol). However, the dynamics as a function N seems to be similar.

-1 0 1 2 3 h [ m]
∞ = (1 -F 0 /F ∞ )(1 -e -N/N * ).
The experimental data can be fitted by a simple exponential equation

F = F ∞ + (F 0 - F ∞ )e -N/N *
where N * is the characteristic number of cycles required to reach a stationary force. For all liquids and tips we find N * = 14 ± 5. This result points out that the physical properties of the liquid does not play any role in the value of N * . Given the large range of vapor pressures and viscosities for the liquids used (Table 1), this rules out any effect of evaporation or viscous stress on the observed phenomenon. The measurements of N * at different V also show no clear evolution except at low velocity value, i.e. V <25 µm. This point is discussed below.

A typical result of experiment II is presented in Fig. 6. This curve is composed of 12 successive series of experiments I, separated by various contamination times t p during which the tip is left in air. The values are fixed in order to study the influence of t p and verify the reproducibility of the experiments. The first series corresponds to an unknown value of t p since it is the first one after mounting the tip on the cantilever holder. The other series are obtained for t p values between 1 and 31 s. As expected, for each series we recover the behavior described above with an increase of the force until a plateau F ∞ . Figure 6 shows that F ∞ may slightly differ due to the high sensitivity of AFM experiments. However, in the limit of experiments accuracy, we find that the values of F ∞ and N * are similar for all series.

We observe that the values of F 0 decreases significantly with t p . For short t p = 1 s values, F 0 is close to F ∞ . If we compare the results of all series, we check that F 0 is fixed by t p .

Indeed, the second series corresponding to t p = 8 s is characterized by a value comparable to those of 6 th (t p = 10 s) and 7 th (t p = 14 s) ones. In this work, we realized about 30,000 dipping force curves for each tip. The results for one liquid, one tip and one velocity can be presented by plotting a 3D map F (h, N ) which gathers, in color code, all F (h) force curves for all the N dipping cycles. An example of such a map is given in Fig. 8 (left) for the experiment II of Fig. 5. The real interest of this map is to give a comprehensive view of the wetting property evolution for the whole tip (all defects). This map can be understood by plotting on the same figure, the F (N ) curve for the topographical defect corresponding to h ≃ 0.3µm (defect 1). The map displays clearly the series of eight experiments I. We observe that the force variation linked to the change of θ 0 for the topographical defect 1 (Fig 5b) is coupled to a shift in h that results from a change of meniscus height H, as pointed out in the Model section (Fig. 3). Hence, the F (h, N ) maps display the evolution of the capillary force with the number of dipping cycles for the whole tip wetted surface. In this representation, the series are clearly identified.

Desorption mechanism

The experiments reported above demonstrate that AFM allows to monitor minutes changes in the wetting properties of a solid surface with a good time resolution. This is possible thanks to a precise understanding of the force curves shape which allows to disentangle the effects of topography and chemistry of the tip. The results based on the analysis of more than 30,000 force curves for each tip show a good reproducibility which allows to draw robust conclusions about the adsorption of airborne contaminants and their desorption by a moving contact line. [START_REF] Franiatte | Molecular Desorption by a Moving Contact Line[END_REF] In this section, we discuss more quantitatively the different mechanisms at play in this processes. Note that we study only the adsorption and desorption of weakly adsorbed contaminants. Chemically bonded molecules may also be present from the beginning and throughout the whole experiments, and are insensitive to capillary forces.

The adsorption stage is characterized by two parameters extracted from the force curves, namely the characteristic adsorption time τ and the relative change of force ϵ max obtained at saturation when the tip is left for a long time in air. As expected, the value of τ = 7 ± 4 s is independent of the experimental conditions. This value is much shorter than the typical timescale of contamination of surfaces by airborne hydrocarbon molecules which are of the order of tens of minutes. [START_REF] Li | Effect of airborne contaminants on the wettability of supported graphene and graphite[END_REF] However, a recent study performed on graphite points out that a rapid contamination by strongly adsorbed contaminants occurs just after exfoliation. [START_REF] Salim | Airborne contamination of graphite as analyzed by ultra-violet photoelectron spectroscopy[END_REF] In Fig. 9, we report the values of ϵ max which greatly increase with the liquid surface tension.

For undecanol, the force change is of the order of 0.5 % whereas it reaches more than 30 % for glycerol and water. In order to interpret this result, we assume a surface with contact angle θ 0 , partially covered with a surface fraction Φ of hydrophobic molecules characterized by a contact angle θ d for a full coverage (Φ = 1). Following the approach proposed by Cassie and Baxter, [START_REF] Cassie | Wettability of porous surfaces[END_REF] the equilibrium contact angle θ on the composite surface is given by cos θ = cos θ 0 + Φ(cos θ dcos θ 0 ) which leads to a maximum relative force variation

ϵ max = Φ max cos θ 0 -cos θ d cos θ 0
where Φ max is the maximum fraction of the tip surface covered by contaminants. The values of θ 0 are extracted from the experiments whereas the values of θ d are retrieved from the literature as the contact angle values of the liquid on paraffin which mimics a surface fully covered (Φ = 1) by hydrocarbons, [START_REF] Kamusewitz | The relation between Young's equilibrium contact angle and the hysteresis on rough paraffin wax surfaces[END_REF][START_REF] Jańczuk | Some remarks on the components of the liquid surface free energy[END_REF][START_REF] Shepard | Surface roughness as related to hysteresis of contact angles. III. The Systems paraffin-ethylene glycol-air, paraffin-methyl cellosolve-air and paraffinmethanol-air[END_REF][START_REF] Good | A comparison of contact angle interpretations[END_REF] assumed as the main contaminants. On Fig. 9 are reported in green the estimated values for all liquids for a value of Φ max = 7 %. It reproduces fairly well the large range of evolution of contact angle as a function of the liquid nature. The value of Φ max corresponds to a maximum coverage of one molecule per nm 2 if we assume short hydrocarbon chain molecules. [START_REF] Millet | Atmospheric volatile organic compound measurements during the Pittsburgh Air Quality Study: Results, interpretation, and quantification of primary and secondary contributions[END_REF] This is comparable with the density of single silanol groups [START_REF] Nawrocki | Silica surface controversies, strong adsorption sites, their blockage and removal. Part I[END_REF] which may play as adsorption sites. This calculation allows to estimate the characteristic distance between molecules ξ ≃ 1 nm which will be used in the model. The experiments I allow to demonstrate that the continuous sweeping of a surface by the contact line is able to desorb locally contaminant molecules. This process is quantified by the characteristic number of cycles needed to remove these molecules and recover a clean surface.

The contaminant surface coverage Φ defined above follows the relationship Φ = Φ max e -N N * .

Interestingly, the value of N * is found to be independent of the liquid used (except for hexadecane for which no evolution is observed). We plot in Fig. 10, the average value of N * for all tips and liquids as a function of the tip velocity V . No evolution is clearly observed except a small decrease for small velocities, leading to a value N * = 14 ± 5 independent of experimental conditions. This rules out any temporal process, such as a diffusive desorption of the molecules in the liquid or any action of viscous stress (in particular at the contact line)

that would lead to a velocity dependence. Moreover, the value of the capillary number in our experimental conditions Ca = ηV /γ ∼ 10 -4 -10 -8 confirms the fact that capillary effects at the contact line are predominant in the desorption mechanism. The decrease of N * for small

V values can be attributed to recontamination. To evaluate this effect, we estimate the time T spent by the tip in air between two cycles. It is of the order of T = L a /V where L a , the average length of the cycle during which the tip is in air, is of the order of L a ∼ 1 -3µm.

For large velocities, the time T is very short compared to the adsorption time τ leading to a constant desorption rate with no possibility of contaminant adsorption but it becomes comparable for small velocities. Two antagonistic effects are then at play, pure desorption characterized by N ∞ and adsorption characterized by T /τ . Assuming exponential evolution for both mechanisms, we get:

Φ = Φ max 1 - 1 -e -1/N ∞ e T /τ -e -1/N ∞ 1 -e -N ( 1 N ∞ + T τ ) , (1)
which defines an effective number of cycles given by: We plot on Fig. 10, in red solid line, the result of Eq. 10 with τ = 7 s, N ∞ = 16 and L a = 3µm. The overall evolution is well described and validates the fact that the decrease of N * at small velocity is due to a recontamination effect. If we neglect this effect, an intriguing results of our experiments is that the desorption of molecules is governed by the number of times the contact line sweeps the solid surface. Again considering the contaminant molecules as short hydrocarbon chains (propane, butane), their adsorption energy on a silica surface is of the order of E ads ∼ 10kT . [START_REF] Funk | Adsorption kinetics and dynamics of small organic molecules on a silica wafer: Butane, pentane, nonane, thiophene, and methanol adsorption on SiO2/Si(111)[END_REF][START_REF] Zhao | Carbon dioxide adsorption on polyacrylamideimpregnated silica gel and breakthrough modeling[END_REF] This energy barrier which prevents a desorption by thermal energy can be decreased by the capillary energy E cap . This quantity can be estimated by considering the elastic energy stored in the deformation of a contact line caused by the presence of contaminant molecules. According to [START_REF] Joanny | A model for contact angle hysteresis[END_REF] , these weak heterogeneities leads to a capillary energy per molecule given by

1 N * = 1 N ∞ + L a V τ . (2
E cap = γξ 2 cos θ d -cos θ 0 2θ 0 2 × ln L ξ
, where ξ ∼ 1 nm is the distance between molecules as deduced above and L is the tip perimeter. As reported in [START_REF] Franiatte | Molecular Desorption by a Moving Contact Line[END_REF] , the value of E cap is of the order of 7 -7.5kT for all liquids. This value is comparable to contaminants physical adsorption energy and may therefore lower the desorption energy barrier. On the contrary, it is not sufficient to remove molecules chemically adsorbed on the surface (with energies of the order of 100 kT).

While the velocity does not affect the desorption mechanism characterized by a constant N * value, the tip velocity governs the dynamics of the contact line. In our experimental conditions where Ca = ηV /γ ∼ 10 -4 -10 -8 , viscous dissipation is negligible. However, non linear effects arise from the pinning of the contact line on topographical and chemical defects through thermally activated processes. [START_REF] Blake | Kinetics of liquidliquid displacement[END_REF][START_REF] Guan | Asymmetric and speed-dependent capillary force hysteresis and relaxation of a suddenly stopped moving contact line[END_REF] The model presented above that computes the static angle θ v when the contact line moves on a topographical defect allows to simulate the experimental force curve. This model is purely deterministic and depends simply on the equilibrium contact angle θ 0 . Our experiments show that the molecular mobility affects θ 0 , through adsorption/desorption mechanisms. Yet, θ 0 also depends on the dipping velocity. The experimental F (V ) curves (Fig. 11) follow a logarithmic trend, in good agreement with the model. In particular, we note that the curves for F 0 and F ∞ have the same slope nkT /λ. The value of λ extracted from the slope λ ≃ 1.1 nm which is similar to ξ calculated independently from the value of Φ max . However, the similar slopes of the F 0 (V ) and F ∞ (V ) indicate that the ratio n/λ is constant and is therefore not affected by the desorption mechanism. Thus, contaminant molecules are not the main source of dissipation.

On the other hand, they change in the model the barrier height E 0 . Note that λ value is also of the same order of magnitude as the small scale roughness scale used in the topographical model.

In order to get more insight on the desorption mechanism, we analysed the F (h, N ) maps which allow following the evolution of the whole tip surface upon continuous dipping.

The analysis of the whole set of data evidences three main mechanisms associated to the molecular desorption by the moving contact line:

-the sweeping of the tip by the contact line induces a global change of contact angle by a progressive desorption of contaminants (Fig. 8) on the whole surface of the tip as discussed above.

-a bistable behavior is sometimes observed when the contact line passes a particular topographical defect (Fig. 12 at h ≃ 0.18µm). In this case, the depinning force alternates randomly between two close values as N increases. The force curves associated with the two states are reported in Fig. 12(a-c). It shows that the depinning of the contact line can occur at two well defined positions. Interestingly, the probability of occurrence of these two states evolves with N : for small N values, the "weak" pinning site is preferred [Fig. 12(c)] and progressively, as N increases, the two sites are equiprobable [Fig. 12(b)] and finally the "strong" pinning site becomes predominant [Fig. 12(a)].

The intermittency then disappears after about 1000 cycles. This bistable dynamics is linked to a local change of wettability at the depinning point when the contact line recedes. One possibility to explain the bistability is to consider that a contaminant can oscillate in position around the depinning point and is finally desorbed. -a specific experiment of deliberate contamination was achieved by exposing the tip Si4 to a vapor of hexadecane by letting it a few microns above a hexadecane droplet overnight. The map corresponding to 10,000 dipping cycles (experiment I) in glycerol at 50µm/s is reported in Fig. 13. We observe a same trend of force evolution F (N ) as previously described. Yet, the force increases over a much larger number of cycles (N * ≃ 500) which is associated with an important level of contamination with large molecules (hexadecane). In this case, the map shows a particular behavior characterized by sudden increase or decrease of force, along topographical defect lines. These large changes in force value may be attributed to the appearance and removal of contaminant clusters which are formed by accumulation of hexadecane on defects by the moving contact line. Example of such clusters are pointed out by arrows on the map of Fig. 13. Note that, when left in air, the clean surface obtained at the end of this process is recontaminated by airborne molecules which leads to a N * ≃ 15 value similar with the one obtained in the studies above (not shown). This is consistent with the fact that main volatile organic compounds in air are short hydrocarbon chains. Yet, the particular case of hexadecane contamination clearly demonstrates the efficiency of capillary desorption induced by a meniscus even for large molecules.

Conclusions

We developed AFM experiments in order to study the effect of blemishes on the wetting properties of a needle at the nanoscale. We realized about 30,000 dipping cycles per tip at different velocities with 6 liquids with a large range of properties. We obtained very robust results which show that a moving contact line is able to desorb contaminant molecules physically adsorbed on the surface. In our experiments, the adsorption energy of contaminants and the capillary energy are of the same order of magnitude. Two types of experiments permit to characterize both the adsorption and desorption of molecules. In our experimental conditions, we measure a contamination time of the order of τ = 7 ± 4 s, much shorter than the one observed in the literature. The desorption is characterized by a constant number of dipping cycles N * independent of velocity and liquid properties but influenced by the nature of contaminants. We identified three mechanisms of desorption: one for which the contaminant molecules are simply swept by the moving contact line; a second for which the molecules are trapped and oscillate in position at the top of a topographical defect; and a third for which molecules of contaminants form clusters which are displaced by the moving contact line.

All the experimental results are analyzed on the basis of a topographical model which allows us to disentangle the contributions of topographical and chemical defects. The AFM experiments at different velocity show that a thermally activated kinetic mechanism governs the dynamics of the contact line at the nanoscale. According to the kinetic model, the change of the surface energy by contaminant desorption modifies the apparent potential barrier. Before or after the decontamination, the dissipation processes occur at the scale of 1 nm and limit the contact line dynamics on the rough surface. One evident application of this work is based in the development of cleaning method by capillary force when clean surface at the molecular scale are required. The patterning of surface at the molecular level could also be achieved by controlling the dynamics of a moving contact line.

Figure 1 :

 1 Figure 1: (a)-(b) SEM images of the NN tip and of a nanoneedle fixed at the extremity of the silicon tip, respectively; (c) sketch of the experiment with a zoom on the nanomeniscus for definition of variables; (d) image of a glycerol sessile droplet; (e) image of a water droplet resting at the end of a small capillary connected to a micro-injector.

Figure 2 :

 2 Figure 2: Force curves obtained with the Si3 tip at 50 µm/s in glycerol (a), ethylene glycol (b), tetra ethylene glycol (c) and decanol (d).

  For undecanol, due to small contact angles values the variations are much smaller.

Figure 4 :

 4 Figure 4: Force curves obtained in water at V = 200 µm/s (a), tetraethylene glycol at V = 50 µm/s (b) (both with the NN tip), glycerol (Si4 tip at 50 µm/s) (c), ethylene glycol (Si3 tip at 50 µm/s) (d), undecanol (e) and hexadecane (f) (both obtained with the NN tip at V = 50 µm/s).

Figure 5 :

 5 Figure 5: Force variation ∆F/F ∞ as a function of N for: (a) three different liquids (W, Gly, 4EG) with the Si3 tip at V = 200µm/s; (b) three different liquids (G, 4EG, Und) with the NN tip at V = 50µm/s. Red curves are fits of the data with ∆F/F ∞ = (1 -F 0 /F ∞ )(1e -N/N * ).

Figure 6 :Figure 7 :

 67 Figure 6: Force F measured on one peak of the advancing branch of the NN tip in Gly at V = 50µm/s as a function of N for 12 series of 100 dipping cycles. The values of the delay time t p between series are 8, 1, 1, 6, 10, 14, 16, 17, 21, 25, 31 s, respectively.

Figure 8 :

 8 Figure 8: Force F as a function of the height h and the dipping cycle number N at the advancing for the left part of the figure. Force F measured on one peak of the advancing for the right part. These data have been obtained with the NN tip in glycerol at V = 50 µm/s, as for Figure 6.

Figure 9 :

 9 Figure9: ϵ max as a function of surface tension γ. The round black points are obtained with our data and their error bars refer to statistical uncertainties (95% of confidence). The triangular green points are obtained thanks to the literature data and their error bars refer to the uncertainties given in the literature and an uncertainty of 5 • on the results of our contact angle calculation.

Figure 10 :

 10 Figure 10: N * vs. V measured experimentally. The error bars refer to the standard deviations of the measurements. The red curve is a plot of the N * vs. V from Eq. 2 which takes into account the kinetics of contamination.

Figure 11

 11 Figure 11 displays the velocity dependence of the capillary force in a semi-log representation.The data are obtained by repeating experiment I for dipping velocities comprised between 3 and 200 µm/s. The curve compares the force F 0 obtained on the first cycle for each velocity, on one topographical defect, and F ∞ averaged over the last 10 cycles of the series. As an example, the inset of Fig.11show the advancing force curve for 6 different velocities at N = 90. These data can be interpreted on the basis of a simple kinetics model[START_REF] Blake | Kinetics of liquidliquid displacement[END_REF] where the "jump" contact line frequency on a distance characteristic of surface defects λ depends on the contact line force F as: ν = ν 0 exp [-(E 0 -λ∆F )/kT ] where E 0 is the energy barrier height, ν 0 the attempt frequency characteristic of the liquid and ∆F = F (V ) -F (V = 0) which is negative when the contact line advances. Considering that V ≃ νλ, the force dependence on velocity is given by ∆F = n λ [kT ln(V /V 0 ) + E 0 ] where n = 2πR/λ is the number of defects on the needle perimeter.

Figure 11 :

 11 Figure11: Force F measured on one peak of the advancing branch of the Si1 tip in Gly as a function of the velocity V . The round points are the first measures of a 100 dipping cycles series for each velocity. The square points are the means of the last ten measures and the error bars associated refer to the statistical uncertainties (95% of confidence). Red curves are logarithmic with a slope of 4.6. The inset shows the forces measured on the advancing branch at different velocity from refers to V = 3 µm/s (blue curve) to V = 200 µm/s) (brown curve).

Figure 12 :

 12 Figure 12: Force curves at some dipping cycles as indicated above the curves (a,b,c) and force F as a function of the height h and the dipping cycle number N at the receding (d). These data have been obtained with the Si3 tip in glycerol at velocity V = 50 µm/s.

Figure 13 :

 13 Figure 13: Force F as a function of the height h and the dipping cycle number N at the advancing obtained with the Si4 tip in glycerol at velocity V = 50 µm/s.

Table 1 :

 1 Relevant properties of the liquids used : volumic mass ρ, viscosity µ, surface tension γ and saturation vapor pressure at 25 °C P sat .

	Liquid	ρ (kg/m 3 ) µ (mPa.s) γ(mN/m) P sat (bar)
	Glycerol (Gly)	1260	910	63	0.025
	Tetraethylene glycol (4EG) 1120	50	46	7
	Ethylene glycol (1EG)	1110	30	50	13
	Undecanol (Und)	832	13.8	29	0.4
	Hexadecane (Hx)	770	3	27	0.22
	Water (W)	997	1	72	3170

Table 2 :

 2 Relevant properties of the tips used: quality factor Q, cantilever spring constant k, resonance frequency in air f 0 and radius R.

	Pointe Q	k (N/m)	f 0 (kHz) R (nm)
	Si1 Si2 Si3 Si4 NN	120 ± 3 1.08 ± 0.03 61.7 184 ± 11 1.91 ± 0.12 68.2 239 ± 11 3.06 ± 0.14 79.8 197 ± 30 2.13 ± 0.32 70.6 362 ± 32 3.86 ± 0.34 74.4	60 to 125 nm. 60 nm 70 nm 67.5 nm 310 to 530 nm
	All the AFM experiments are realized in a gray room maintained at a constant temper-
	ature of 23 ± 1 °C and a relative humidity comprised between 55 and 65 %.