Eric Andres
email: eric.andres@univ-poitiers.fr

Lydie Richaume
email: lydie.richaume@etu.univ-poitiers.fr

Gaelle Largeteau-Skapin
email: gaelle.largeteau.skapin@univ-poitiers.fr

Digital Surface of Revolution with Hand-Drawn Generatrix Digital Surface of Revolution

Keywords: digital surfaces, implicit functions, surface of revolution

In this paper we present a simple method to create general 3D digital surfaces of revolution based on a 2D implicit curve of revolution (therefore not limited to a circle) and a hand-drawn generatrix. Our method can handle any sequence of Euclidean 2D points, that represents a curve, as generatrix. One can choose the topology of the surface that may have 1-tunnels, 0tunnels or no tunnels with applications in 3D printing for instance. An online tool that illustrates the method is proposed.

Introduction

Surfaces of revolution (as a particular class of sweep surfaces), are an important class of surfaces in computer modeling [START_REF] Salomon | Curves and surfaces for computer graphics[END_REF]. It allowed early on, with limited user interface capabilities, to design common 3D objects based on 2D forms (a 2D circle and 2D generatrix): pen, glass, ower pot, chess piece, . . . Moreover, fundamental surfaces such as spheres, cylinders, torii, . . . can be dened as surfaces of revolution. A surface of revolution is dened by a 2D revolution curve (classically a circle) and a curve as prole, most often an explicit function, called generatrix. In this paper we are interested in digital surfaces of revolution with a free-shaped generatrix and an implicitly dened curve of revolution with some control on the topological properties of the digital surface. The digital surface of revolution is dened by a ake digitization method which is dened as all the akes centered on digital points cut by the continuous surface of revolution.

In our DGCI 2016 paper [START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF], we proposed a ake based simple digital surface of generation tool with a generatrix limited to explicit functions y = g(z) (see section 1.1 and section 2.3 for more details). This was somewhat limiting. One of the motivations for our work on digital surfaces is to propose exible generation tools for the general public and a local artist, Aurélie Mourier, who works with us. Classically for an explicitly dened generatrix y = g(z), for each z value, there is one and only one y value. In the present paper, the generatrix can be any curve represented by an ordered sequence of 2D continuous points, therefore there can be more than one y value for a given z. This may correspond to a hand-drawn curve which was the main motivation for this work (see section 4.1), an open or closed sampled parametric curve (section 4.2), or a 2D digital curve (section 4.3).

In order to handle such generatrices, we devised a surface of revolution generation algorithm with the following steps (see section 3.4):

The generatrix can be any nite and open sequence of points. The generatrix is divided into monotonic explicit or horizontal point sequences. Each generatrix section can then be digitized independently. This works because our method is based on morphological type digitizations: we ensure that way that the union of the digitizations is the same as the digitization of the unions. For horizontal generatrix sections (see section 3.3) we simply need the minimum and maximum y value in the sequence.

For monotonic explicit sequences of points dening a section of the generatrix, the sequence is recomputed in order to have a sequence of y for integer and half-integer z values (see section 3.1.1, section 3.1.2 and section 3.2.1). The ake digitization method we are using only requires to compute y values for integer and half-integer z values of the generatrix. This allows a direct O(1) constant time look-up (see section 3.1.2) for the generatrix curve values. For all (x, y, z) ∈ Z 3 in the generation window, for each generatrix section, we test if the corresponding ake centered in (x, y, z) is cut by the continuous surface of revolution. If it is, the voxel of coordinates (x, y, z) belongs to the digital surface. There are however some diculties that need to be handled. A monotonic generatrix section may be dened only on a nite z domain and thus only partially cut some akes. We introduce in section 3.2.2 and section 3.2.3 the notion of cropped ake and cropped adjacency ball that can handle such partial cuts. Without this, there could be holes at the junction of two generatrix sections in the surface since the end of one monotonic sequence will typically be the beginning of the next monotonic or horizontal generatrix section. Note that the generatrix curve does not have to be a closed curve.

To illustrate our method, we propose an online tool (that can be found at: http://xlim-sic. labo.univ-poitiers.fr/demonstrateurs/DSoR_ Generator/?lang=en), that was used for most of the examples in this paper, in particular for a 3D printed chess queen (see Figure 9). After some basic notations and the analytical implicit surface digitization method in section two, we present, in section three, how we handle an explicit nite or an horizontal generatrix. In section four we propose some results, limitations, an algorithm and we conclude and provide perspectives in section ve.

Previous work

There are relatively few papers that have dealt specifically with the problem of dening or generating digital surfaces of revolution. N. Stolte et al. have dened a generation method for digital implicit surfaces with cylindrical and spherical coordinates [START_REF] Stolte | Novel techniques for robust voxelization and visualization of implicit surfaces[END_REF][START_REF] Yongsheng | Robust voxelization based ray tracing of implicit surfaces[END_REF]. Their goal was not so much to dene a generation method for digital surfaces as to propose a visualization method for implicit surfaces. Their method is limited to supercover type surfaces (tunnel-free surfaces) that are dened with cylindrical and spherical coordinates. More recently, G.Kumar and P.Bhowmick [START_REF] Bhowmick | Digital circularity and its applications[END_REF][START_REF] Kumar | Wheel-throwing in digital space using number-theoretic approach[END_REF] proposed a generation algorithm for digital surfaces of revolution for the design of virtual pottery. Their idea was to superpose 2D digital annuli since a rasterized horizontal slice of height one of a surface of revolution corresponds to a 2D digital annuli. The authors worked with a classical notion akin to the Bresenham circle [START_REF] Bresenham | A linear algorithm for incremental digital display of circular arcs[END_REF], dened only for integer radii and center. The main drawback is that concentric Bresenham circles of increasing radii leave points that do not belong to any circle. The authors therefore determine what they call absentee voxels, to ll-up the 6-connected holes in the digital surface [START_REF] Bera | On the characterization of absentee-voxels in a spherical surface and volume of revolution in Z 3[END_REF]. The method for lling the holes with absentee voxels is however very complicated and limited to circular curves of revolution. One way around this problem could have been to use Andres circles [START_REF] Andres | Discrete circles, rings and spheres[END_REF][START_REF] Andres | The discrete analytical hyperspheres[END_REF][START_REF] Andres | Analytical description of digital circles[END_REF] that do ll space and are dened for arbitrary center and radii. Their method denes only naive type surfaces (2-tunnel free surfaces). It can be note however, that they allow a general form of generatrix akin to the type of generatrix we are proposing in the present paper.

A surface of revolution can be dened as an implicit surface [START_REF] Salomon | Curves and surfaces for computer graphics[END_REF]. This was the basis for our DGCI 2016 paper [START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF], where we proposed an approach based on a morphology type digitization method for implicit nD surfaces [START_REF] Toutant | Implicit digital surfaces in arbitrary dimensions[END_REF][START_REF] Andres | Digital Analytical Geometry: How Do I Dene a Digital Analytical Object?[END_REF] (generalizing the approach of Laine [START_REF] Laine | A topological approach to voxelization[END_REF]). The digital surface of revolution presented in [START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF] was dened by an explicit function y = g(z) as generatrix and a 2D implicit function r(x, y) = 0 as curve of revolution (not limited to 2D circles). The method is very simple and straight-forward to implement. Furthermore, under some regularity constraints [START_REF] Toutant | Implicit digital surfaces in arbitrary dimensions[END_REF], the digitization method oers a control over the digital surfaces' topology by dening k-tunnel free digital surfaces. This allows the design of digital surfaces adapted to specic applications: for visualization purposes, the thinnest, so called naive surface, 2-tunnel free surface might be adequate. For 3D printing however, one might want to choose 2-connected, so called supercover, tunnel free surfaces. The present paper is an extension of this last paper.

Notations, basics and recalls

Let {e 1 , . . . , e n } denote the canonical basis of the ndimensional Euclidean vector space. Let Z n be the subset of R n that consists of all the integer coordinate points. A digital (resp. Euclidean) point is an element of Z n (resp. R n ). We denote by x i the i-th coordinate of a point x, that is its coordinate associated to e i . In 3D, for the sake of simplicity, we will often refer to the classical xyz coordinate system corresponding to {e 1 , e 2 , e 3 }. A digital (resp. Euclidean) object is a set of digital (resp. Euclidean) points. The Chebychev norm of a vector x is dened by x ∞ = max (|x i |). The Manhattan norm of a vector x is dened by

x 1 = |x i |.
For all k ∈ {0, . . . , n -1}, two integer points v and w are said to be k-adjacent or k-neighbors, if for all i ∈ {1, . . . , n}, |v i -w i | ≤ 1 and n j=1 |v j -w j | ≤ n -k. In the 3-dimensional space, the 0-, 1-and 2-neighborhood notations correspond respectively to the classical 26-, 18-and 6-neighborhood notations.

A k-path is a sequence of integer points such that every two consecutive points in the sequence are kadjacent. A digital object E is k-connected if there exists a k-path in E between any two points of E. A maximum k-connected subset of E is called a k-connected component. If the complement of a digital object E, Z n \ E admits exactly two k-connected components C 1 and C 2 , i.e. there exists no k-path joining integer points of C 1 and C 2 , then E is said to be k-separating in Z n .

Let ⊕ be the Minkowski addition, known as dilation, such that A ⊕ B = ∪ b∈B {a + b : a ∈ A}.

Adjacency norms

The adjacency norms have been introduced in [START_REF] Toutant | Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties[END_REF]. Every digital adjacency relationship can be associated to a norm called an adjacency norm dened as follows: Denition 1 (Adjacency norms [START_REF] Toutant | Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties[END_REF]) Let n be the dimension of the space. Let k be a positive integer lower than n. The k-adjacency norm [•] k is dened as:

∀x ∈ R n , [x] k = max x ∞ , x 1 n -k .
The name adjacency norms stems for the following property:

Lemma 1 (Digital adjacency and adjacency norms [START_REF] Toutant | Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties[END_REF]) Let v and w ∈ Z n . Then, v and w are

k-adjacent i [v -w] k ≤ 1. Let B [•] k ( 1 2
) be the ball of radius 1 2 under the norm [•] k . The associated distance is denoted by d k . It is easy to see that the 0-adjacency norm corresponds to the Chebychev norm and the (n -1)-adjacency norm to the Manhattan norm.

Implicit surface digitization

In this paper we are considering morphological digitization schemes [START_REF] Andres | Digital Analytical Geometry: How Do I Dene a Digital Analytical Object?[END_REF][START_REF] Toutant | Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties[END_REF][START_REF] Heijmans | Morphological discretization[END_REF][START_REF] Heijmans | Morphological sampling[END_REF]. A summary on this type of digitization schemes and how they can be used to dene digital objects can be found in [START_REF] Andres | Digital Analytical Geometry: How Do I Dene a Digital Analytical Object?[END_REF]. The basic idea is to say that a digital object is dened as the digital points inside a Minkowksy sum between a structuring element E and the, to be digitized, continuous object S: Fig. 1 (1,2) 2D adjacency balls

B [•] 0 ( 1 2 ) and B [•] 1 ( 1 2 ) ; (3,4,5) 3D adjacency balls B [•] 0 ( 1 2 ), B [•] 1 ( 1 2 ) and B [•] 2 ( 1 2 ).
Denition 2 (Morphological Digitization) Let us consider an object S and a structuring element E both dened in R n . The morphological digitization of S by E is dened by :

D E (S) = (S ⊕ E) ∩ Z n = {v ∈ Z n : (v ⊕ E) ∩ S = ∅} . Denition 3 (Adjacency Digitization [12, 2, 15]) The k-adjacency digitization is the D B [•] k ( 1 2 ) morpho- logical digitization.
There are two fundamental properties that can almost immediately be deduced from this denition: Proposition 1 (Adjacency Norm Digitization Properties) [START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF][START_REF] Heijmans | Morphological discretization[END_REF][START_REF] Heijmans | Morphological sampling[END_REF][START_REF] Andres | The supercover of an m-at is a discrete analytical object[END_REF]

D E (S ∪ T ) = D E (S) ∪ D E (T )
; if S is a (n-1)-dimensional object, then D B [•] k ( 1 2 ) (S)
is locally k-tunnel free [START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF][START_REF] Toutant | Implicit digital surfaces in arbitrary dimensions[END_REF][START_REF] Andres | Digital Analytical Geometry: How Do I Dene a Digital Analytical Object?[END_REF][START_REF] Toutant | Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties[END_REF] (see Figure 3). The rst of these properties is very interesting when it comes to modeling. The second means that we have some type of control on the topology of a digitized surface (see Figure 3). The problem is that the Minkowsky sum (or the intersection of a surface and a adjacency ball) of a continuous object with an adjacency ball can be dicult to compute (see [START_REF] Toutant | Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties[END_REF] for the complicated equations describing the adjacency ball digitization of 3D spheres). In order to simplify the analytical characterization, and subsequently the generation algorithm, of a digital 3D surface, we introduced the notion of adjacency akes in [START_REF] Toutant | Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties[END_REF] and used those to propose a digitization method for digital implicit surfaces [START_REF] Toutant | Implicit digital surfaces in arbitrary dimensions[END_REF]. An adjacency ake can be described as the union of anite number of straight segments joining the opposite vertices of a k-adjacency ball (see Figure 2). Denition 4 (Adjacency Flake [START_REF] Toutant | Implicit digital surfaces in arbitrary dimensions[END_REF][START_REF] Toutant | Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties[END_REF]) Let 0 ≤ k < 3. A 3D k-adjacency ake, F k is dened by: Instead of an adjacency digitization, with an adjacency ball as structuring element, we are considering a digitization with the corresponding adjacency ake for implicitly dened surfaces. Let S be an implicit surface S = x ∈ R 3 : f (x) = 0 which separates space into one (or several) region(s) where f (x) < 0 and one (or several) region(s) where f (x) > 0:

F k = λu : λ ∈ 0, 1 2 , u ∈ {-1, 0, 1} 3 , 3 i=1 |u i | = 3 -k .
Denition 5 (Flake Digitization [START_REF] Toutant | Implicit digital surfaces in arbitrary dimensions[END_REF][START_REF] Andres | Digital Analytical Geometry: How Do I Dene a Digital Analytical Object?[END_REF][START_REF] Toutant | Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties[END_REF]) The k-Flake digitization of a 3D surface S is dened by

D F k (S) = v ∈ Z 3 : (v ⊕ F k ) ∩ S = ∅ .
When the surface S veries a regularity condition, meaning S is r-regular [START_REF] Stelldinger | Towards a general sampling theory for shape preservation[END_REF] with r >

√ 3 -k + √ 3 /2
[12], then the k-ake digitization can be analytically characterized by considering only the vertices of the kake. Let us introduce such a digitization scheme: Denition 6 (Simple Analytical Flake Digitization [START_REF] Toutant | Implicit digital surfaces in arbitrary dimensions[END_REF])

A k (S) = v ∈ Z 3 : min{f (x) : x ∈ (v ⊕ F k )} ≤ 0 and max{f (x) : x ∈ (v ⊕ F k )} ≥ 0 .
When the surface veries the regularity conditions previously mentioned then A k (S) = D F k (S) and thus veries all the properties of morphological digitizations and adjacency ake digitizations. Otherwise there are some dierences that may in some cases create topological problems [START_REF] Toutant | Implicit digital surfaces in arbitrary dimensions[END_REF] (see section 4.5 and Figure 11). This is largely compensated by the fact that A k (S) is easy to construct while D F k (S) may not. Figure 3 illustrates the dierences between the three digitizations A 2 , A 1 and A 0 . Let us mention one natural-seeming property that is missing with the way we have dened the adjacency akes: a digitization inclusion property. For the adjacency balls, we have

B [•] 2 ( 1 2 ) ⊂ B [•] 1 ( 1 2 ) ⊂ B [•] 0 ( 1 2 )
which means that for a given Euclidean object S, ). This is not the case for adjacency balls as we have dened them since F 2 F 1 F 0 . One way around this is simply to dene a sequence of akes that have the inclusion property: Denition 7 (3D Matryoshka Flakes) Three dimensional Matryoshka akes M k are dened by: 2).

D B [•] 2 ( 1 2 ) (S) ⊆ D B [•] 1 ( 1 2 ) (S) ⊆ D B [•] 0 ( 1 2 ) (S
M 2 = F 2 , M 1 = M 2 ∪ F 1 and M 0 = M 1 ∪ F 0 (Figure
The implicit surfaces digitizated with the M 1 and M 0 Matryoshka akes still have the same topological properties than the one digitized with the adjacency akes but now we have D M2 (S) ⊆ D M1 (S) ⊆ D M0 (S). This can be used to solve some problems with non regular surfaces (see Figure 11).

Digital surface of revolution with explicit generatrix [2]

Classically, a surface of revolution is dened by the rotation of a curve (the generatrix) around a straight line (the axis). The axis is most often simply the z-axis of a classical xyz-axis system. In our previous work [START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF], we considered the generatrix as an explicit function y = g(z), dened for all z ∈ R. The horizontal slice at height z of the surface of revolution is classically a circle (the curve of revolution) of radius g(z). The generatrix acts as an homothetic function. It is easy to see that this denition corresponds to an implicit 3D surface where the curve of revolution is not limited to a circle: Denition 8 (Implicitely dened Euclidean Surface of Revolution [START_REF] Salomon | Curves and surfaces for computer graphics[END_REF][START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF])

S(g, r) = (x, y, z) ∈ R 3 , r x g(z) , y g(z) = 0 , where r(x, y) = 0, (x, y) ∈ [-1, 1] 2 ⊂ R 2
, is an implicitly dened curve of revolution that separates the window [-1, 1] 2 into regions of opposite signs, and

∀z ∈ R, g(z) > 0.
The implicit surface of revolution can be digitized into a Digital Surface of Revolution with an innite explicit generatrix using the A k -digitization. This corresponds to the DGCI 2016 paper [START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF].

Digital surface of revolution generation algorithm

In this section we are going to explore our proposed method of digital surface of revolution generation with a hand-drawn generatrix. The particularity of a handdrawn generatrix is that we cannot assume anything particular about the generatrix: it may correspond to an innite explicit function (dened on the whole height interval of denition of the surface. Section 2.3), to anite explicit curve (section 3.2.1), to an horizontal curve (section 3.3), or to the union of some or all of those cases (section 3.4). The input for the algorithm is a curve of revolution based on an implicit 2D function dened in a window [-1, 1] 2 and an ordered sequence of Euclidean points (z i , y i ), 1 ≤ i ≤ n, typically following the order in which the generatrix has been drawn and representing a sampled curve g(z). In this case there can be several values g(z) for a given z. The function is not necessarily explicit anymore. In order to generate the digital surface of revolution, we will dene anite window with, in particular, a generation domain [z min , z max ] , (z min , z max ) ∈ Z 2 , along the z-axis. For what follows we will consider only one (connected) hand-drawn curve but it is easy to expand the method to work with several dierent generatrix curves as, by essence of morphological digitizations each surface of revolution corresponding to each generatrix curve can be digitized independently. The dierent steps of the digital surface of revolution generation algorithm are presented in the algorithm 1. Let us detail the dierent aspects of this algorithm: The generatrix acts as an homothetic factor to the curve of revolution. For a generatrix dened as a sequence of 2D points representing a sampled curve g(z), there can be more than one such value for a given z. In order to handle this, the generatrix is divided into a set of strictly monotonic (in z, increasing or decreasing) or horizontal sequences. The end point of one sequence is duplicated as the starting point of the next sequence. The considered morphological digitization methods mean we can simply digitize each sequence separately.

Now, let us remark that we need to compute a value g(z) at the vertices of akes that have only integer and half-integer coordinates (with the exception of cropped akes that will be introduced in the following section 3.2.2). Each strictly monotonic sequence (not the horizontal sequences) is replaced by a sequence with interpolated y values for integer and half-integer z values. One can use any kind of interpolation function to compute a y value for each integer and half-integer z value. For all of the following gures and for the online program, we simply used a linear interpolation scheme between appropriate points of the initial sequence. For each strictly monotonic sequence, we need to keep the starting and ending points that dene the domain of denition of the nite explicit generatrix (see subsection 3.2.1). For the horizontal sequences, the points with minimum and maximum y are the only two points we need (see subsection 3.3).

Example

Let us give a simple example of this recomputation process:

Let us start with the generatrix dened by the following sequence of (z i , y i ) points: {(16.9, 12.15), (17.8, 13) The rst and the last of those sequences are strictly monotonic: one with increasing z values and one with decreasing z values which is reversed to allow a uniform treatment for all monotonic sequences. The second sequence corresponds to an horizontal sequence (in z). The rst strictly monotonic is interpolated and replaced by: {(16.9, 12.15), (17., 12.24), (17.5, 12.72), (18., 13.7), (18.1, 14.05)};

The horizontal sequence is replaced by: {(18.1, 14.05), (18.1, 18.95)}. We do not need the other values; The last strictly monotonic sequence is interpolated and replaced by: {(16.4, 22), (16.5, 21.86), (17., 21.14), (17.5, 20.22), (18., 19.16), (18.1, 18.95)}.

Direct access to a monotonic sequence value

The interesting thing about recomputing a strictly monotonic sequence in such a way is that given the z value, one can immediately determine the place in the sequence in a O(1) constant time look up: let us say that we have a monotonic sequence {(z 1 , y 1 ), (z 2 , y 2 ), . . . , (z n , y n )}: the index in the list to return is:

If z < z 2 then the index to return is 1 ; if z > z n-1 then the index to return is n ; otherwise the index to return is 2z -2z 1 + 2. Where x is the ceiling of x (i.e. the smallest integer greater than x). For example, in the last monotonic sequence in the previous example, for z = 17.5, we have 2 * 17.5 -2 * 16.4 + 2 = 4. This allows a direct access to the value we are looking for. The formula should be modied according to the starting index.

Handling strictly monotonic generatrix sequences

A rst simple case comes when the strictly monotonic sequence is dened on the whole z generation domain: Let us assume now that we have a strictly monotonic sequence of Euclidean points {(z 1 , y 1 ), . . . , (z n , y n )} with z 1 < . . . < z n and a generation domain interval [z min , z max ] along the z-axis. If z 1 < z min and z n > z max , we are in the case where the generatrix dened by the sequence of points can be handled as an innite explicit generatrix just as we did in [START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF] with the appropriate interpolation and resampling as presented in subsection 3.1. It gets more complicated when this is not the case.

Explicit nite generatrix

Let us now consider a strictly monotonic sequence of Euclidean points {(z 1 , y 1 ), . . . , (z n , y n )} with z 1 < . . . < z n , and a generation domain interval [z min , z max ] along the z-axis such that z 1 > z min or z n < z max . This case needs to be handled specically because a digitization of the surface of revolution by a digitization scheme A k supposes that we are able, for all points in 1 2 Z 3 , to compute a value for r x g(z) , y g(z)

where r is the implicitly dened curve of revolution and g is the interpolated generatrix function. In the nite case, when testing a voxel, it is possible that one or several vertices of the k-ake fall outside the denition domain of the generatrix: g(z) is not dened for such a vertex. While it seems logical that the voxel is not part of the digital surface if g is not dened for any of the vertices of a ake, when some vertices of the ake are inside the definition domain of the generatrix and some not then the voxel may be simply partially cut (see 2D illustration in Figure 5.2-8) and not simply crossed by the continuous surface of revolution (see 2D illustration in Figure 5.1). Simply ignoring the vertices for which we have no value can lead to many wrongly discarded voxels and lead to holes at the junction of consecutive generatrix sequences.

To get these values, we can choose substitute points, to better adjust to the domain of the function. For this we introduce cropped akes and cropped adjacency balls. These points are actually easy to choose since the domain is restricted along only one coordinate: the z-axis.

Cropped akes

There are two ways of considering alternate ake vertices. The rst method consists simply in considering only the parts of the ake that are inside the domain. The considered vertices for the computations are the endpoints of the cropped ake line segments. For the sake of clarity, Figure 5 shows all the dierent congurations in 2D for an A 2 digitization (it is easy to extend to A 1 and A 0 ). An example, Figure 5.9 shows the conguration of Figure 5.2 in 3D.

Cropped adjacency ball

The second method consists in considering the vertices of the cropped adjacency ball which the k-ake approximates. There are a few more points for the A 1 and A 2 digitizations, but the results are much closer to what we would get by computing the intersection between the surface and the adjacency balls.

Again, for the sake of clarity, Figure 6 shows, in 2D, which points should be computed in every possible conguration for the A 2 digitization. Figure 6.9 shows the corresponding 3D conguration to Figure 6.2. Fig. 5 The dierent congurations that can occur in the digitization process when cropping akes. Fig. 6 The dierent congurations that can occur in the digitization process when cropping balls.

Handling a horizontal generatrix

Let us now consider an horizontal sequence {(z 0 , y 1 ) , . . . , (z 0 , y n )}. As shown in the example of section 3.1.1, we only keep the minimum and maximum of the y values: y min = min (y 1 , . . . , y n ), y max = max (y 1 , . . . , y n ). An horizontal generatrix would be dened by z = z 0 , z 0 ∈ R on an interval y ∈ [y min , y max ] ⊂ R. Such a generatrix, associated to a circle as the curve of revolution, would result in an annulus as the generated surface. To determine if a voxel is part of this annulus, we simply need to know if it is inside the large circle (radius = y max ) and outside the small one (radius = y min ). This works of course exactly in the same way for any implicitly dened curve of revolution. We can test this simply using the A k digitization. The main diculty is the same as in 3.2.1: it will most likely be impossible to compute the value corresponding to any vertices, since only one value is possible for z. Using the same solution as before we will get a slice of adjacency ball (or a slice of ake) dening the substitute points we should use.

Additional considerations

We have now all we need to generate the surface (see algorithm 1). Since the basis of the generation is a morphological type digitization scheme, Proposition 1 allows us to generate each single sequence independently as described in the previous sections. The most immediate approach is simply to test if each voxel, in a 3D nite generation window, belongs to the surface of revolution of a generatrix point sequence. One can also determine a starting point as seed and generate the surface by neighborhood propagation. This is the main generation method proposed in our online tool. This is usually faster but there is no guarantee that the surface of revolution is formed by only one connected component or that it is not composed of all the voxels of the generation window. In terms of complexity, we test each voxel of the 3D window where we generate the digital surface of revolution, and this for each of the sequences in the set of generatrix point sequences.

For a generation window of size n 3 with a set of m generatrix point sequences, the worst case complexity would be O(m.n 3 ) with no absolute limit on m (one can think of a generatrix based on a function such as z = sin(1/y)). There could be some improvement implemented here in future works. There are some obvious optimizations that help during the generation: if a point (x, y, z) is part of the digital revolution surface for one generatrix point sequence, there is no need to test further if it belongs also to the digital surface of another generatrix point sequence. Moreover, each generatrix point sequence is dened on some interval z min ≤ z ≤ z max . We only need to test the voxels (x, y, z) verifying z min -0.5 ≤ z ≤ z max + 0.5.

Results

In this section we present results of our digital generation algorithm with various ways to obtain the point list for the generatrix.

Hand-drawn generatrix and online creation tool

The rst way to create a generatrix as a list of points is to consider a hand-drawn curve dened by a sequence of 2D Euclidean points. This was the main incentive for this work: proposing a natural method to create a digital surface of revolution. We developed an online digital surface creation tool that can be found at the following address:

http://xlim-sic.labo.univ-poitiers.

fr/demonstrateurs/DSoR_Generator/?lang=en. A screenshot of the interface with an example of handdrawn generatrix can be seen in Figure 10. Some other examples of surfaces generated by the online tool are presented in Figure 14. In Figure 9 we show the work of a local artist, Aurélie Mourier, who used our online tool to create a set of chess pieces. We printed out the White Queen piece in two halfs so as to show that it is an actual surface we are creating (see Figure 9). In this case, the surface is based on the M 0 Matrioshka ake in order to have a robust 2-connected tunnel free surface.

Parametric generatrix

An alternative way of generating a list of points as generatrix is to sample a parametric function. You will nd two examples in Figure 7. An adaptative sampling might be useful depending on the curve [START_REF] De Figueiredo | Adaptive sampling of parametric curves[END_REF].

Digital curve as generatrix

A generatrix (or a set of generatrices) can also be extracted from a digital image as a (set of) digital curve(s). This can be done in many dierent ways. The main problem here is extracting an ordered list of digital points which depends on the type of digital curves present in the image. The main idea for handling such digital curves is to decompose them into a set of digital line segments and then taking the continuous analog of those digital line and regenerate the y values for integer and half-integer z values. Other types of interpolation schemes can of course be used as well. The important thing here is not simply to consider the set of digital points as input to the algorithm otherwise the surface may not look smooth. In Figure 8, we decomposed a digital curve into line segments with a regular line segment recognition method [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF]. For each digital line of analytical equation 0 ≤ ax -by + c < max (|a| , |b|), we considered the continuous line ax -by -c -|b| /2 = 0 or ax -by -c -|a| /2 = 0 depending on the orientation. A little care has to be taken for the end points of the dierent digital line segments. If the intersection of two consecutive line segments lies outside of the digital starting/end point of the respective line segment then a little patch function has to be added (see [START_REF] Breton | Towards an invertible euclidean reconstruction of a discrete object[END_REF]) or one has to use an adapted line recognition where this problem does not occur [START_REF] Sivignon | Discrete analytical curve reconstruction without patches[END_REF]. This creates a piecewise dened generatrix. Figure 8 shows a digitized sinusoid that we have decomposed into digital line segments. In this example, the curve of revolution is a circle. A digital generatrix decomposed into digital straight segments (left) and the resulting revolution surface (right) using the unit implicit circle as revolution curve.

Crenelated digital surfaces

As shown in Figure 12.2, the digitization of an implicit surface can lead to a crenelated surface edge. Figure 12.2 has been obtained with the A 2 digitization (with the direct mathematical approach presented in [START_REF] Andres | Digital Surfaces of Revolution Made Simple[END_REF]) of the implicitly dened surface shown in Figure 12.1. This proves that this is not a side eect of our present approach, which, of course, may also create crenelated surfaces. The explanation for this crenelation can be seen in Figure 12.3: when, locally, the generatrix is dened on a nite domain, for instance for z ≤ z 0 , for some z 0 ∈ R, then the corresponding surface of revolution might only cut a disconnected set of adjacency balls on layer z = z 0 + 1 2 . This leads to a crenelated result.

Limitations

The main limitation of this digital revolution surface generation method is that the analytical digitization may miss some points. This may happen if the surface is not r-regular [START_REF] Stelldinger | Towards a general sampling theory for shape preservation[END_REF] with r > [START_REF] Toutant | Implicit digital surfaces in arbitrary dimensions[END_REF]. As mentioned in section 2.2, if the continuous surface of revolution is not r-regular, the digitization where we consider only the vertices of the akes is not equivalent to the one with the whole akes and the topological properties may be lost. Figure 11.1 illustrates this with a curve that crosses a voxel but the vertices of the ake are all on the same side of the implicit curve of revolution. The voxel is therefore wrongly discarded from the digitization result and disconnections or holes appear. This is classically dealt with interval arithmetics [START_REF] Du | Interval arithmetic recursive subdivision for implicit functions and constructive solid geometry[END_REF] but interval arithmetics works only for digitizations corresponding to F 0 -adjacency akes. One way to reduce the occurrence of such problems is to use Matryoshka akes as illustrated in Figure 11. In gure 11.3 we have a 0-adjacency ake digitization with a lemniscate as curve of revolution (Figure 11.2). The surface is disconnected. The use of the Matryoshka M 0 -ake (gure 11.4) solves the problem. Of course, there will always be cases where the topological properties can not be guaranteed. As a concluding remark, let us note that a ake digitization does not guarantee and optimal k-tunnel free surface even though the regularity conditions are met. There can be simple points with regard to the k-tunnel freeness property. Ensuring topological optimality is still an open question.

√ 3 -k + √ 3 /2

Conclusion and perspectives

In this paper we proposed a simple algorithm to generate a large class of surfaces of revolution based on an im- A hand-drawn generatrix and the obtained surface of revolution using the unit implicit circle as revolution curve. Online tool can be found at : http://xlim-sic.labo.univpoitiers.fr/demonstrateurs/DSoR_Generator/?lang=en . plicit 2D curve as curve of revolution and a hand-drawn generatrix. Any ordered sequence of points representing a 2D curve can be used to dene a generatrix. Given some regularity conditions for the surface, we control the topology of the resulting digital surface: the type of tunnels that appear in the surface can be dened, which allows to generate surfaces that are adapted for Fig. 12 1. An implicit 3D surface. 2. Its A 2 digitization. 3. Only some adjacency balls are cut by the surface. various applications (rendering, 3D-printing, etc.). We propose an online tool that illustrates the proposed method and that can be used to create surfaces and export them into various formats including 3D printing formats. The online tool can be found at the following address: http://xlim-sic.labo.univ-poitiers.fr/ demonstrateurs/DSoR_Generator/?lang=en. We set up an imgur animated gif repository for some of our digital surface creations : http://imgur.com/a/eDFbY. Let us be noted that the repository holds not only digital surfaces of revolutions but also swept tubes and may contain examples of future extensions. The methods we propose are not limited to digital surfaces of revolution and can be adapted to more general types of surfaces. For example the generatrix can be used as central axis for the revolution curve (see the left of Figure 13): Denition 9

S(g, r) = x, y, z ∈ R 3 , r (x, y -g(z)) = 0 .

We can also combine an homothetic function h(z) and two translation functions t(z), u(z) for the center of the revolution curve (see the right of gure 13): Denition 10 S(h, t, r) = x, y, z ∈ R 3 , r x h(z)

-u(z), y h(z) -t(z) = 0 . This will be handled in a forthcoming work. One of the next steps will be to consider hand-drawn curves of revolution which will allow further control of the types of surfaces that can be generated.

Hand-Drawn Generatrix

Curve of Revolution Digital Surface of Revolution

Fig. 2 F

 2 Fig. 2 F k Adjacency and M k Matryoshka akes in 3D.

Fig. 3

 3 Fig. 3 Illustration of the topological properties with A 2 , A 1 , A 0 digitizations of a simple surface of revolution.

Fig. 4

 4 Fig. 4 Illustration of the ake digitization for dierent type of generatrices.

Algorithm 1 :

 1 Surface of revolution digitization input : A List Gen of 2D Euclidean points as generatrix, an implicit 2D function Rev as curve of revolution, a connexity conex, and a digitization window W (x min , xmax, y min , ymax, z min , zmax). output: A list V of voxel belonging to the digitization of the surface of revolution. begin Initialize the list V as empty; GenParts=DivideAndRecompute(Gen) (* see section 3.1*); foreach digital point P (x,y, z)∈ W do foreach P art in GenParts do if IsHorizontal(Part) then if inHorizontal(Rev,conex,x,y,z,P art)then Add the voxel to V ; else if inMonotonic(Rev,conex,x,y,z,P art) then Add the voxel to V ; return V; 3.1 First step of the Algorithm: divide and recompute each generatrix section

Fig. 7

 7 Fig. 7 Two examples of surfaces of Revolution with a parametric curve as generatrix and a circular curve of revolution.

Fig. 8

 8 Fig. 8

Fig. 9

 9 Fig. 9 Set of chess pieces and example of 3D printed White Queen. First row: the generatrices; second row: the revolution curves; third row: the resulting surfaces; last row: several views of the 3D printed White Queen.

Fig. 10

 10 Fig. 10

Fig. 11 1 .

 1 Fig. 11 1. Case where ake misses a point. 2. Lemniscate curve of revolution 3. Disconnected 2-connected surface. 4. 2-connected surface with Matrioschka M 0 -ake.

Fig. 13

 13 Fig.[START_REF] Andres | Digital Analytical Geometry: How Do I Dene a Digital Analytical Object?[END_REF] Two examples of some other surfaces we can build with our algorithm.

Fig. 14

 14 Fig. 14 Examples of hand-drawn generatrix, implicit curve of revolution and the corresponding digital surface of revolution.

  An implicit function Rev, a connexity conex, a voxel center Vx,Vy, Vz and a Euclidean point list P art which is a strictly monotonic increasing part of the generatrix . output: A boolean that is True if the voxel is in the digitization of this part of the surface (Rev × P art) and false otherwise.

	Algorithm 2: inMonotonic(see section 3.2.1)
	begin
	Compute the ake vertices of the voxel for the chosen
	connexity (* see section 3.2.3 and 3.2.2 *);
	Compute the values Lvalues by applying the
	revolution function on these ake vertices using the
	homothetic factor found in P art;
	if min(Lvalues)<0 and max (Lvalues>0) then
	return(TRUE);
	else
	return(FALSE);

input :

  An implicit function Rev, a connexity conex, a voxel center Vx,Vy, Vz and a Euclidean point list P art which is an horizontal part of the generatrix . output: A boolean that is True if the voxel is in the digitization of this part of the surface (Rev × P art) and false otherwise.

	Algorithm 3: inHorizontal (see section 3.3)
	begin
	if The horizontal P art crosses the voxel
	(|P artz -Vz| ≤ 0.5) then
	Compute the ake vertices of the voxel for the
	chosen connexity (* see section 3.2.3 and 3.2.2 *);
	Compute the values Lvalues by applying the
	revolution function on these ake vertices using
	the homothetic factor found in P art;
	if min(Lvalues)<0 and max (Lvalues>0) then
	return(TRUE);
	else
	return(FALSE);

input : else return (F ALSE);

Acknowledgement: This work has been supported by the CPER 2015-2020, NUMERIC Program and FEDER-FSE MODEGA Project of the Nouvelle-Aquitaine Region, France. we would like to thank Aurélie Mourier, http://www.aureliemourier.net/, a local artist, that worked with us on the online tool and on creating the chess pieces.


Juan Víctor Perales-Rondón

Jose Solla-Gullón

Enrique Herrero

Carlos Manuel Sánchez-Sánchez

Enhanced catalytic activity and stability for the electrooxidation of formic acid on lead modified shape controlled platinum nanoparticles

Keywords: shape-controlled Pt nanoparticles, Pb adsorption, Pb modified electrodes, formic acid oxidation, SECM. Highlights

High catalytic activity for formic acid oxidation reaction (FAOR) is demonstrated in Pb modified shape-controlled Pt nanoparticles (NPs). Cyclic voltammetry is used to follow the effective modification of Pt NPs by Pb. Octahedral shaped Pt NPs (having a (111) preferential surface structure) modified by Pb are proved the most active electrocatalyst studied towards FAOR and display a catalytic activity of c.a. 7 mA cm -2 at 0.5 V in 0.1 M formic acid solution. This current density represents an enhancement factor 29.5 with respect to the unmodified shaped Pt NPs and this is 2.7 and 2.3 times higher than that found on Tl θ /100-Pt NPs and Sb θ /111-Pt NPs, respectively, some of the most active electrocatalysts based on adatoms modified Pt NPs reported so far. This outstanding activity is displayed at maximum Pb coverage and also confers a wide electrocatalyst stability over the entire potential range studied. FAOR is also studied using scanning electrochemical microscopy (SECM) by the micropipette delivery/substrate collection (MD/SC) working mode as a preliminary rapid test to identify active electrocatalysts. In particular, the remarkable activity enhancement exhibited by a Pt ultramicroelectrode (100 µm diameter) modified by Pb is rapidly imaged by SECM providing preliminary catalyst performance information. Thus, this technique emerges as a suitable and fast method to test, and in some cases quantify, catalyst activity for reactions of interest in fuel cell applications.

Introduction

Formic acid oxidation reaction (FAOR) represents a very important reaction in electrocatalysis, since it can be used as a model in fundamental studies for other small organic molecules (SOMs) such as methanol or ethanol [START_REF] Koper | Fuel Cell Catalysis: A Surface Science Approach[END_REF]. Moreover, formic acid has been proposed as a fuel for direct liquid fuel cells (DLFCs), which can be used as small power supply [START_REF] Rice | [END_REF]3] in electronic devices, since FAOR presents fast oxidation kinetics, safety in the manipulation and less fuel crossover through the ionic exchange membrane than other fuels such as methanol [4]. In a direct formic acid fuel cell (DFAFC) [5], the thermodynamic cell potential is 1.428 V, which makes it more interesting than hydrogen fuel cells from an available energy point of view. Nevertheless, the overpotential for the FAOR at present needs to be improved in order to reach commercial applications.

Pt is one of the most studied metals in electrocatalysis [START_REF] Koper | Fuel Cell Catalysis: A Surface Science Approach[END_REF]. In particular, FAOR on Pt electrodes has been widely studied over the last decades due to the high activity of this metal for the oxidation of different SOMs. Since FAOR has probably the simplest oxidation mechanism among all different SOMs, a deep understanding of the FAOR mechanism on Pt should be very useful for other important electrocatalytic oxidation reactions. It is well accepted that FAOR on Pt electrodes follows two different reaction pathways [6][7][8]. One of then, the socalled indirect via, involves the formation of CO on the electrode surface, which acts as a poison intermediate. The other pathway is known as the direct via, and implies the formation of an active intermediate, which is immediately oxidized into CO 2 . On the other hand, it is also well-known that FAOR is a surface sensitive reaction [9,10]. Studies on Pt single crystal electrodes (Pt(hkl)) have allowed a better comprehension of some of the elementary steps in FAOR, proving that Pt(100) is the most active electrode for both paths, while Pt(111) is least active one, in spite of the fact that the formation of CO is almost negligible on this electrode [11].

The modification of the surface chemical composition on the Pt(hkl) electrodes is one of the most widely employed methods to increase the catalytic activity for the FAOR. This strategy is mainly based on the incorporation of different adatoms, which can be either metals or semi-metals, on the surface of the Pt(hkl) electrodes. This epitaxial adsorption and deposition of a submonolayer of adatoms on a foreign metal substrate is usually performed either by

irreversible adsorption at open circuit potential or by underpotential depostion (UPD). These two interesting methods to modify noble metals may generate surface alloys in some cases [12]. In the case of modified Pt electrodes, the UPD method [13] is based on the electrodeposition of an adatom monolayer from a solution containing the adatom dissolved as a cation at potentials significantly less negative than that for the bulk electrodepositon of the adatom [14]. The main difference between UPD and irreversible adsorption methods is the fact that irreversible adsorbed adatoms remain stable on the Pt surface in the absence of the adatom cation in solution [START_REF] Koper | Fuel Cell Catalysis: A Surface Science Approach[END_REF][15][16][17][18][19], but on the contrary, UPD adatoms are not stable on the Pt surface unless the solution contains the adatom cation in low concentration. Moreover, irreversible adsorption allows achieving adatom coverages independently of the applied potential, as far as it remains within the adatom electrochemical stability limits. In addition, this approach also avoids the problem of accuracy in the coverage quantification that appears when the UPD method is used, due to its dependency with the applied potential and solution composition. The well accepted positive effect of the presence of some adatoms on the electrocatalytic activity of Pt electrodes towards FAOR is visualized by an evident shift to lower potential values on the onset oxidation potential and/or by increasing current densities of the oxidation reaction. In this sense, it is proposed that adatoms may act following three main different mechanisms; i) the third body effect in which the foreign adatom modifies the reaction mechanism by steric interference, since it blocks specific adsorption sites on the Pt surface preventing CO formation [20], ii) the electronic effect [21,22], where the modification of the Pt electronic structure due to the presence of foreign adatoms enhances the activity of the surface and iii) the bi-functional effect [23], in which the adatom and the Pt surface sites have distinct roles in the oxidation mechanism [24].

Over the last decades, adatoms such as bismuth (Bi) [25,26], arsenic (As) [27], antimony (Sb) [28], palladium (Pd) [29] and lead (Pb) [19,30,31] adsorbed on Pt(hkl) electrodes has been studied, showing an important improvement in the FAOR activity, by following at least one of the proposed mechanisms mentioned above. Nowadays, the next challenge is to transfer all that knowledge from single crystal electrodes to nanoparticles (NPs) with a preferential surface structure. In this sense, the role of some adatoms such as Bi [32], Sb [33] and Tl [34] modifying shape-controlled Pt NPs have been already reported. Particularly, Bi adatom has shown a significant enhancement in the activity of the (111) preferential Pt NPs towards FAOR [35], which agrees with previous studies using Bi modified Pt single crystal electrodes [36]. Some new approaches in the study of FAOR on modified Pt NPs electrode also include trimetallic systems [37] and graphene-Pt NPs hybrid materials [38].

Despite the number of adatoms already studied modifying Pt NPs, there are still some of them untested. This is the case of Pb adatom on shape-controlled Pt NPs for FAOR, which is studied herein. However, Pb has been already used in Pt-Pb bimetallic alloys [39][40][41] to improve conventional spherical Pt NPs activity towards FAOR, as well as intermetallic Pb-Pt catalysts [42,43]. The main goal of this article is to address that question in order to explore the feasibility of using Pb modified shape-controlled Pt NPs as catalyst for fuel cell applications. The use of conventional Pb or PtPb NPs in real electrochemical filter press reactors either for electroorganic synthesis [44] or DFAFCs [45] have been already demonstrated in the literature.

Finally, the use of novel electroanalytical techniques for studying the reaction mechanism in electrocatalytic reactions provides new tools for electrocatalyst interrogation [46][47][48][49][50]. Among of them, the scanning electrochemical microscopy (SECM) [START_REF] Bard | Scanning Electrochemical Microscopy[END_REF], which is based on the steady-state diffusion controlled regime provided by either an ultramicroelectrode (UME) or a micropipette, has already demonstrated its utility for studying FAOR [START_REF] Jung | [END_REF][53][54]. However, no SECM imaging studies have been devoted to prove the activity improvement of adatoms modifying Pt electrodes in FAOR. So far, only a voltammetric study of FAOR on Bi modified Pt UME has been reported [55]. In the present work, we use SECM images to provide some additional information in the comparison of the activity for FAOR on pristine Pt and Pb modified Pt (Pb-Pt).

Experimental section

Pb modified shape controlled Pt NPs (Pb θ /Pt NPs)

Three types of shape-controlled Pt NPs dispersed in water with surface area to volume ratios of 0.03 cm 2 µL -1 for Cubic 100-Pt NPs and 0.017 cm 2 µL -1 for octahedral 111-Pt NPs and for quasi-spherical poly-Pt NPs were used. The chemical synthesis of those NPs was carried out as previously described in detail [58]. The Pt NPs morphology/shape was confirmed by transmission electron microscopy (TEM and HRTEM). TEM experiments were performed with a JEOL JEM-2010 microscope and a JEOL 3010 microscope working at 200 kV and 300 kV, respectively. The procedure to get all the TEM images is explained in detail elsewhere [59].

The average particle size of the cubic and octahedral NPs was around 9 nm, whereas the size of the quasi-spherical NPs was around 4 nm. Before being modified with Pb, the Pt NPs were electrochemically characterized and a CO adsorption-stripping experiment was performed in order to clean the Pt NPs surface, as has been previously reported [57,58]. In brief, this procedure consists in blocking the Pt NPs electrode surface by bubbling CO in solution for about 30 s. After this time the electrode is kept at potential 0.1 V for about 15 min, while the solution is bubbled with Ar(g) in order to remove any CO traces in solution.

Finally, the CO adsorbed on the Pt NPs electrode surface is stripped and

θ 𝑃𝑏 = 1 -θ H = 𝑞 𝐻 0 -𝑞 𝐻 𝑃𝑏 𝑞 𝐻 0 (1)
where 𝑞 𝐻 0 and 𝑞 𝐻 𝑃𝑏 are the hydrogen desorption charges for the pristine and Pb modified Pt electrodes respectively, calculated after the subtraction of the double layer contribution [34,35]. However, as it was already described for Sb [33] and Tl [34] modified Pt NPs, the characterization of θ Pb is not possible at high Pb coverage, since the decrease in the hydrogen adsorption/desorption peaks shown in the hydrogen UPD region is compensated by the appearance of a new peak associated with a surface Pb oxidation/reduction signal from the Pb adsorbed on the gold substrate electrode, preventing the quantification of high Pb coverages by this procedure. Thus, the maximum Pb coverage obtained on each type of Pt NPs studied here is not quantified and it is identified as θ max . Then, the three types of Pb modified Pt NPs are named: i) Pb θ /100-Pt NPs, ii) Pb θ /111-Pt NPs and iii) Pb θ /poly-Pt NPs.

Scanning Electrochemical Microscopy (SECM)

The Figure 1 shows a much higher net oxidation current appearing when the micropipette scans above Pb-Pt than on Pt electrode at both studied potentials. In fact, the current collected at 0.3 V on Pt is so low that a dashed circle has been added into the image to mark the exact Pt electrode location. Thus, these facts clearly point out that the current collection for FAOR (difference between the maximum current at the electrode location and the background current in the SECM image at a given tip-substrate distance [64]) is significantly higher in the case of Pb modified Pt electrode. Table 1 summarizes the collected current for Pt and Pb-Pt electrodes at both potentials exhibiting values from 1.1 to 34.8 nA.

Moreover, the catalytic enhancement factor, which in this case is defined as the ratio between the collected current on Pb-Pt electrode versus that collected on Pt electrode at the same potential and tip-substrate distance, is calculated and included in Table 1.

Characterization and stability of Pb modified shape controlled Pt NPs.

Once the FAOR activity enhancement on modified Pb-Pt electrodes has been demonstrated by preliminary rapid SECM imaging, it is necessary to transfer those results into Pt NPs in order to gain insight into the catalytic mechanism and to envisage the potential feasibility of using Pb modified Pt NPs as electrocatalysts for real DFAFCs.

Figure 2 shows some representative TEM images of the different Pt nanoparticles employed in this work. The results obtained are similar to those previously reported [58]. In figure 2a Pt nanoparticles with octahedral and tetrahedral shapes can be easily distinguished. These shapes would suggest the presence of a {111} preferential orientation. On the other hand, figure 2b shows Pt nanoparticles with a preferential cubic shape thus suggesting a preferential Figure 4 shows the electrochemical characterization of all three types of shape controlled Pt NPs modified with different coverage of Pb in sulfuric acid solution. The voltammetric profiles without Pb modification (θ Pb = 0) agree with those reported already in the literature for shaped controlled Pt NPs [68]. Then, for the three types of shape controlled Pt NPs modified with different amount of Pb, the charge associated within the hydrogen adsorption and desorption region on Pt clearly decreases when the amount of Pb is increased. The Pb coverage is estimated using equation [START_REF] Koper | Fuel Cell Catalysis: A Surface Science Approach[END_REF]. Nevertheless, the peak appeared around 0.22 V when the amount of Pb is significantly increased (θ Pb = max), which is due to a redox process associated to Pb adsorbed on the Au polycrystalline surface [65,69], prevents a proper Pb coverage quantification. For this reason, the maximum Pb coverage is not quantified (θ max ).

The stability of the Pb adlayer on Pt NPs before (solid plots) and after (dash dotted plots) FAOR is shown in Figure 5 at both, high (θ max , red plots) and low coverage (black plots, θ Pb /100-Pt = 0.27 and θ Pb /111-Pt = 0.31), in sulfuric acid solution 0.5 M. At low Pb coverage, the Pb deposition is more unstable and the characteristic hydrogen features are clearly identified after FAOR experiment (dash dotted black plot), due to the partial dissolution of Pb [70]. A similar behavior is found when increasing the number of electrochemical cycles that reach 0.8 V as a positive cycle limit (not shown). On the contrary, at maximum Pb coverage, the modified electrode profile before and after FAOR remains stable, which may suggest the formation of a chemically stable Pb surface alloy on Pt when high coverages are reached, which avoids its rapid dissolution and confers more durability to the catalyst. Moreover, the current density at potentials below 0.30 V is almost negligible in the positive potential scan in all three types of pristine Pt NPs shown in Figure 6 (see Table 2), which is due to CO ads poisoning effects. However, this behavior is particularly important in the pristine 100-Pt NPs (figure 6b, solid black plot), which perfectly agrees with previous Pt(hkl) studies, which demonstrated that (100) domains are the most active for the CO formation during FAOR [22].

FAOR on

Above 0.3 V, the current density on pristine Pt NPs increases slowly until ~0.80

V. In contrast, the negative potential scans (dashed black plots in Figure 6) show significantly higher current densities producing an important hysteresis in all three cases because the CO is removed from the electrode surface at 0.8 V. Figure 6 also shows a significant increase in the oxidation current densities when Pb is present on the surface of all three types of Pt NPs, even at low Pb coverages (θ Pb < 0.2), reaching a maximum at ~0.50 V. Moreover, this activity enhancement is displayed in both positive and negative scan directions, exhibiting a very small hysteresis when Pb is present on the Pt NPs surface. In fact, for Pb θ /poly-Pt NPs there is not hysteresis between both scans, whereas in the cases of Pb θ /100-Pt NPs and Pb θ /111-Pt NPs, this effect is somehow more visible, being always the current in the positive scan larger than in the negative one. This behavior is completely different from that observed in unmodified Pt NPs and indicates suppression of CO formation when Pb is present on the Pt NPs surface.

Furthermore, there is a shift of the onset potential for the FAOR with respect to the unmodified Pt NPs towards less positive potentials (~0.15 V) in all three cases. More importantly, a significant current increase is observed for Pb θ /111-Pt

NPs and particularly at maximum Pb coverage (6718 µA cm -2 at 0.5 V).

Table 2 summarizes the density current values reached at 0.3 V and 0.5 V in the positive scan direction for each type of shape controlled Pt NP at θ Pb maximum. Additionally, in order to achieve a proper comparison with other adatoms already studied in the literature, comparable results for FAOR using Sb [33] modified Pt NPs and Tl [34] modified Pt NPs are also included in Table 2.

That comparison points out that Sb θ /111-Pt NPs and Tl θ /100-Pt NPs exhibit a little bit larger current densities than any of the Pb modified Pt NPs at 0.3 V (1.2 and 1.1 times higher than Pb θ /111-Pt NPs, respectively). Nevertheless, the maximum current reported for FAOR on Pt NPs modified by adatoms at higher potentials (at 0.5 V) is the one exhibited by Pb θ /111-Pt NPs (2.7 times higher than on Tl θ /100-Pt NPs and 2.3 higher than on Sb θ /111-Pt NPs). In addition to this, all three types of Pb modified Pt NPs exhibit higher current densities than Sb or Tl modified Pt NPs at 0.5 V, independently of the preferential Pt surface structure. Furthermore, the modification by Pb provides the highest average current comparing all 3 adatoms and considering the average of all three types of NPs for each case, both at 0.3 and 0.5 V, since the reaction is less sensitive to the surface structure of the NPs, which also supports the possible formation of a chemically stable Pb-Pt surface alloy [71].

Figure 7 shows the catalytic enhancement factor for FAOR as a function of applied potential calculated from the corresponding positive-sweep voltammetric profiles displayed in Figure 6. This factor of normalization is defined as the ratio between the current densities of Pb modified Pt NPs at maximum Pb coverage and that obtained in the corresponding pristine Pt NPs. It can be seen that, despite Pb θ /111-Pt NPs is the most active electrode comparing absolute values of current density, the normalized activity is very similar to that of Pb θ /100-Pt NPs. However, Pb θ /poly-Pt NPs show a lower increase in the activity. In fact, the maximum normalized activity is reached around 0.37 -0.40

V, and the catalytic enhancement factor is 29.5 and 28.5 for Pb θ /111-Pt NPs and Pb θ /100-Pt NPs, respectively, and 18 for Pb θ /poly-Pt NPs. This fact could be related with a NPs size effect. Since poly-Pt NPs are significantly smaller than the others, they have a larger fraction of edge atoms on their surface, which affords a smaller catalytic enhancement factor. Thus, a Pb adatom deposited neighboring to an edge will have a lower amount of neighboring Pt surface atoms, but the increase in activity is proportional to the number of Pt-Pb ensembles created by the adatom deposition, that is, to the number of Pt neighboring the Pb adatom [20]. It is clear that the deposition of Pb close to an edge will result in an increase of activity lower than that measured when the adatom is deposited in the middle of a large domain. However, despite a lower increase, the shape of the curve in figure 7 is the same in all three cases, which indicates that the actuation mechanism is the same in all Pb modified Pt NPs.

Furthermore, no difference is observed when comparing the catalytic enhancement factor at 0.3 V calculated from the current collected within the SECM images (13, see Table 1) and the one calculated from the positive-sweep voltammetric profiles in Figure 6a for Pb modified poly-Pt NPs (13.5, see Table 2), since that potential fits into the charge transfer control regime. In contrast, the same comparison at 0.5 V, a potential at which diffusion may play a role in the kinetics, denotes a clear underestimation of the catalytic enhancement factor when it is calculated from SECM images, mainly due to the tip-substrate distance used in figure 1, which does not allow a 100% collection efficiency.

Figures 8 and9 show the chronoamperometric experiments for 600 s at 0.3 and 0.5 V, respectively, obtained with all three types of Pb modified Pt NPs.

Chronoamperometric experiments not only provide information about the catalytic enhancement at any specific potential, but also it can give some light about the stability of the catalytic activity vs. time. For this reason, they may be used to calculate the deactivation rate, which is defined as (j 60s -j 600s ) / j 60s and it is reported in Table 3. In particular, figure 8 shows that all three types of Pb modified Pt NPs provide similar current density values (aprox. 400 µA cm -2 , see

Table 3) at maximum Pb coverage after 600 s, which implies that there is not a specific surface structure effect in the activity when modifying by Pb at this potential, as was also suggested by the voltammetric behavior. However, when the cronoamperometric experiment is performed at 0.5 V (figure 9), the Pb θ /111-Pt NPs at maximum Pb coverage clearly exhibits the highest current density value after 600 s (4190 µA cm -2 , see Table 3), showing also a moderate deactivation rate (15%, see Table 3) similar to the one already reported for Sb θ /111-Pt NPs at optimum Sb coverage [33].

FAOR mechanism discussion for Pb modified shaped controlled Pt

NPs.

As it is well known, HCOOH can yield adsorbed CO by a dehydration reaction [72]. This process has specific surface site requirements [20,73], which makes the reaction susceptible of control by adding adatoms, which can act blocking some surface Pt ensemble sites and hindering in this way the CO formation. From the voltammetric profiles shown in Figure 6 it is clear that the CO adsorption is hindered by the presence of Pb on the Pt NPs surface, at low and high coverage, because of a third body effect [26,74]. This is more evident by the loss of the hysteresis between the positive and negative potential sweep at each case compared with the unmodified Pt NPs.

As in the case of other electropositive adatoms such as Bi, Sb or Te, their presence on Pt promotes the adsorption of the formate on the adatom in a configuration which facilitates the cleavage of the C-H bond with almost no activation energy on a neighboring Pt site, justifying a bifunctional mechanism by the Pb adatom. This is well documented and demonstrated in recent contributions [26,74]. This behavior is certain assuming only the formation of a submonolayer of adatom on the electrode surface. In parallel, the presence of Pb hinders the dehydration reaction of HCOOH [75]. Thus, the net result of both mechanism of action is that the formation of CO is almost completely inhibited and much higher oxidation currents are reached by the direct pathway.

Conclusions

Pb a From reference [33] b From reference [34] Table 3. Chronoamperometric activities at 600 s and deactivation percentage at 0.3 and 0.5 V for each shape controlled Pt NPs modified by Pb calculated by comparing the current density in the corresponding chronoamperometry at 60 s and 600s.

  converted into CO 2 by running a CV at slow scan rate (20 mV s -1 ), which reaches a high enough positive potential (0.8 V vs RHE). Then, each type of Pt NPs was electrochemically characterized in 0.5 M H 2 SO 4 solution before and after the irreversible adsorption of Pb adatoms on its surface. The Pt NPs active surface area was determined by the charge involved in the hydrogen UPD region (between 0.05 V and 0.50 V for 100-Pt NPs; 0.05 V and 0.45 V for poly-Pt NPs, and 0.05 V and 0.65 V for 111-Pt NPs) after the subtraction of the double layer [57]. Then, Pb adatoms were irreversibly adsorbed on each type of Pt NPs outside the electrochemical cell by simple immersion of Pt NPs at open circuit potential within dilute solutions of 10 -3 ~ 10 -5 M Pb(ClO 4 ) 2 in 0.5 M HClO 4 and following a common procedure for the irreversible adsorption of different adatoms on Pt [60]. After this, the electrode was rinsed with ultrapure water (except for the maximum coverage case, where the electrodes were not rinsed before being immersed into the solution) and transferred to an electrochemical cell (different to the one used for Pt NPs electrochemical characterization) to perform the electrocatalytic test for FAOR in deaerated 0.1 M HCOOH and 0.5 M H 2 SO 4 solution. By varying the adatom concentration in solution as well as the electrode immersion time, different Pb adatom coverages can be reached on the Pt NPs surface. Finally, the Pb coverage (θ Pb ) on the Pt NPs surface was quantified by following the decrease of the charge involved in the hydrogen adsorption or desorption peaks present in the hydrogen UPD region of the corresponding cyclic voltammetry, according to the equation:

  working electrode for SECM experiments was a 100 µm diameter pristine Pt UME and the same electrode after in-situ surface modification by irreversible adsorption of Pb (3 min immersed in a diluted Pb(ClO 4 ) 2 solution) (Pb-Pt). SECM images were carried out using the micropipette delivery/substrate collection (MD/SC) mode [53] of SECM using a CHI 910B microscope (CH Instruments) and a three-electrodes configuration cell at room temperature. The micropipettes used to deliver the HCOOH in the vicinity of the studied electrodes presented an opened diameter of ca. 15-20 µm and were loaded with a mixture 50:50 (v/v) DCE:HCOOH. They were fabricated by pulling borosilicate capillaries with O.D.: 1.5 mm, I.D.: 1.0 mm and length 90 mm using a laserbased puller P-2000 from Sutter Instrument Co. The inner walls of the micropipettes were made hydrophobic following a well-established silanization protocol [61, 62] by filling them with octyltriethoxylsilane overnight and finally, drying them exhaustively. For MD/SC SECM imaging [53, 63], a gold wire, 0.5 mm diameter, was used as counter electrode and a commercial Hg/Hg 2 SO 4 (K 2 SO 4 sat.) electrode within a Luggin capillary as reference, although all potential values have been referred in the article to RHE (E (Hg/Hg2SO4) = +0.64 V vs RHE). The electrochemical cell employed was built in Teflon with a 2 mm diameter aperture, where the Pt UME perfectly fitted facing up and was held at a potential value positive enough to produce the electrochemical oxidation of HCOOH at the electrode surface. An electrochemical pre-treatment for removing the CO ads accumulated at the electrode surface was performed before each SECM image. This consists in holding the electrode potential constant at 0.8 V for 5 s. The micropipette was located at a constant tip-substrate distance in Z (d = 50 µm) facing the Pt UME and the MD/SC SECM images were obtained recording the substrate current while the micropipette was scanned on the X-Y plane in an argon saturated 0.5 M H 2 SO 4 solution. The tip scan rate was 75 µm s -1 , using increments of 15 µm each 0.2 s.

3. Results and discussion 3 . 1 .

 31 FAOR activity enhancement on modified Pb-Pt electrodes by SECM imaging.

Figure 1

 1 Figure 1 displays several SECM images for FAOR on pristine and Pb-

  {100} orientation. Finally, spherical Pt nanoparticles with a poly-oriented surface can be observed in figure2c. It is worth noting that before preparing the Pb modified Pt NPs, a careful verification of the current contribution provided by the substrate electrode used as a support and current collector should be conducted.In order to carry out that verification, the Pb surface modification (maximum coverage) of an Au disk electrode was performed. The voltammetric profiles of Pb modified Au disk (black plot) and Pb modified 111-Pt NPs supported on Au disk electrode (red plot) are displayed in figure3. The main feature observed in both curves corresponds to a sharp anodic peak around 0.22 V, which is associated with Pb UPD/dissolution at different crystal faces of the Au substrate[65]. This is evident in figure3awhen comparing the response of 111-Pt NPs on Au substrate (red plot) and on glassy carbon (GC) substrate (blue plot), where the main difference corresponds to the sharp peak displayed at 0.22 V only present in the Au substrate case. A broad redox signal extended from 0.3 V to 0.8 V only appears when the electrode contains 111-Pt NPs, suggesting Pb redox processes on the Pt surface taking place at more positive potentials than on Au[19,31,66,67]. Furthermore, the presence of 111-Pt NPs in the electrode causes an important raise in the apparent double layer capacitance current. These signals are associated to the Pb redox processes and are only well-observed for high coverages. Finally, figure 3b exhibits the response of Pb modified Au and bare Au electrodes for the FAOR, showing no activity enhancement when the Au electrode is modified by Pb. This fact rules out any relevant effect of the Au substrate electrode in the electrocatalytic behavior displayed by the Pt NPs studied.

  Pb modified shape controlled Pt NPs.

Figure 6

 6 Figure 6 presents the positive scan direction of CVs for all three different

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. TEM pictures for (A) octahedral, (B) cubic and (C) quasi-spherical Pt NPs.

Figure 3 .

 3 Figure 3. (a) Voltammetric profiles in argon saturated 0.5 M H 2 SO 4 solution for Pb

Figure 4 .

 4 Figure 4. Voltammetric profiles: (a) Pb θ /poly-Pt NPs, b) Pb θ /100-Pt NPs and c) Pb θ

Figure 5 .

 5 Figure 5. Voltammetric profiles: (a) Pb θ /100-Pt NPs and b) Pb θ /111-Pt NPs at low

Figure 6 .

 6 Figure 6. Voltammetric profiles for FAOR. Solid line, positive scan direction and

Figure 7 .

 7 Figure 7. Catalytic activity comparison between the three types of pristine and Pb-

Figure 8 .

 8 Figure 8. Chronoamperometric curves recorded at 0.3 V vs RHE for catalytic

Figure 9 . 5 M H 2

 952 Figure 9. Chronoamperometric curves recorded at 0.5 V vs RHE for catalytic

  Figure 1.

Figure 4 .

 4 Figure 4.

Figure 9 .

 9 Figure 9.

2.1. Chemicals

	1,2-dichloroethane	(CH 2 Cl-CH 2 Cl,	DCE)	anhydrous	99.8%,
	octyltriethoxysilane 97.5%, lead (II) perchlorate hydrate Pb(ClO 4 ) 2 •H 2 O
	(Aldrich ® 99.995%), H 2 SO 4 (Merck KGaA Suprapur ® 96%), HCOOH (Merck
	KGaG 98 %), polyethylene glycol dodecyl ether (Brij ® 30), n-heptane (ACROS
	Organics, Analysis), sodium borohydride (Aldrich ® 99.99%), sodium
	polyacrylate (Aldrich ® ), K 2 PtCl 4 (Aldrich ® 99.99%), H 2 PtCl 6 •6H 2 O (Aldrich ®
	37.50%), HCl (Aldrich ® 37%), NaOH (Merck, Pellets for analysis). All solutions
	were prepared with ultrapure water (18.2 MΩ cm, Purelab Ultra system, Elga-
	Vivendi).
	2.2

. Electrochemical experiments 2.2.1. Voltammetric and chronoamperometric studies

	are referred to RHE. CVs for Pt surface area quantification were carried out in a
	deaerated 0.5 M H 2 SO 4 solution. CVs and CAs for studying the electrocatalytic
	activity for FAOR were carried out in deaerated 0.1 M HCOOH and 0.5 M
	H 2 SO 4 solution. An electrochemical pre-treatment for removing the CO ads
	accumulated at the electrode surface from previous FAOR experiments was
	performed before starting each CA. This consists in holding the electrode
	potential at 0.8 V for 5 s. The potential window where it is possible to work
	avoiding structure modifications on the shape-controlled Pt NPs surface is
	between 0.05 V and 0.8 V [56, 57].
	Cyclic voltammetry (CV) and cronoamperometry (CA) were performed
	using a conventional electrochemical glass cell of 3 electrodes at room
	temperature and a potentiostat-galvanostat AUTOLAB PGSTAT302N (Metrohm
	Autolab B. V.) or a potentiostat CHI 760E. The working electrode was prepared
	by adding 4 µL of each type of Pt NPs (2 µL for 100-Pt NPs) coating a 3 mm
	diameter gold (Au) disc electrode, which is used as a NPs support and current
	collector, and allowing the water solvent to be evaporated before entering the
	electrode in the cell. The sample volume of 4 µL was selected after performing a
	calibration curve that proved no diffusion limitations for FAOR due to the
	thickness of that layer of NPs deposited on the gold disc electrode. Each Pt NPs
	suspension was sonicated for 2 min before being used. A gold wire was used as a
	counter electrode, and a reversible hydrogen electrode (RHE) within a Luggin
	capillary was employed as a reference electrode. All potentials used in this paper

  modified shape controlled Pt NPs represents a highly efficient method for improving the catalytic activity for FAOR of Pt NPs by reaching catalytic enhancement factors as high as 29.5. In particular, Pb θ /111-Pt NPs at θ max reaches 2.7 times higher current density than Tl θ /100-Pt NPs[34] and 2.3 higher than Sb θ /111-Pt NPs[33] at 0.5 V in 0.1 M HCOOH and 0.5 M H 2 SO 4 solution, so far, two of the most active catalysts reported for FAOR under low reactant concentration conditions (0.1 M). It should be also highlighted that all three types of Pb modified Pt NPs studied at maximum Pb coverage exhibit higher current density at 0.5 V than Tl θ /100-Pt NPs and Sb θ /111-Pt NPs, which could suggest the formation of a Pb surface alloy on the Pt surface at high Pb coverage and an important bifunctional effect leading the FAOR mechanism towards the direct reaction pathway and suppressing the CO formation by a third body effect.

	Moreover, at maximum Pb coverage, the modified electrode voltammetric profile
	remains stable after FAOR in a wide potential range, which confers a remarkable
	durability to the Pb modified Pt electrodes and opens up its application in
	diffusion gas electrodes for fuel cells. Finally, it should be pointed out that the
	SECM imaging by the MD/SC mode emerges as a suitable and fast method to
	perform preliminary catalytic tests, and in some cases quantify, the catalyst
	activity for reactions of interest in fuel cell applications, such as FAOR,
	providing a fast method to test in advance the effect of a specific adatom
	adsorbed on metallic electrodes.

Table 1 .

 1 Current collected on the SECM images and catalytic enhancement factor comparing those currents for Pt and Pb-Pt electrodes at 0.3 and 0.5 V.

	Applied	I collected (nA)	I collected (nA)	Catalytic
	potential (V)	Pt	Pb-Pt	enhancement factor
	0.3	1.1	14.3	13
	0.5	8.7	34.8	4

Table 2 .

 2 Density current at 0.3 and 0.5 V from the positive-sweep voltammetric profile performed at 0.02 V s -1 in 0.1 M HCOOH and 0.5 M H 2 SO 4 solution for each type of shape controlled Pt NPs modified at maximum coverage by Pb and Tl and at optimum coverage by Sb. Their corresponding catalytic enhancement factors are also included. They are all calculated in comparison with the current densities on pristine Pt NPs reported here.

		j at (0.3 V) / µA cm -2	Catalytic enhancement factor at 0.3 V	j at (0.5 V) / µA cm -2	Catalytic enhancement factor at 0.5 V
	poly-Pt NPs	46.6		341
	100-Pt NPs	32.7		210
	111-Pt NPs	41.7		372
	Pb θ /poly-Pt NPs	628	13.5	2975	8.7
	Pb θ /100-Pt NPs	553	16.9	3770	18
	Pb θ /111-Pt NPs	642	15.4	6718	18
	a Sb θ /poly-Pt NPs	520	9	2150	6.3
	a Sb θ /100-Pt NPs	260	8	1070	5.1
	a Sb θ /111-Pt NPs	820	19.7	2880	7.7
	b Tl θ /100-Pt NPs	712	21.8	2447	11.7
	b Tl θ /111-Pt NPs	425	10.2	1670	4.5

Acknowledgements

This work has been financially supported by the MICINN (Spain) (projects CTQ2013-44083-P and CTQ2013-48280-C3-3-R), Generalitat Valenciana (project PROMETEOII/2014/013, FEDER) and CNRS (project Défi Instrumentation aux limites 2015).

5.