Robin Roussel
email: robin.roussel.15@ucl.ac.uk

Marie-Paule Cani

Jean-Claude Léon

Niloy J Mitra

Exploratory Design of Mechanical Devices with Motion Constraints

Keywords: computational design, design space, constrained exploration, drawing machines, mechanical motion, fabrication

Mechanical devices are ubiquitous in our daily lives, and the motion they are able to transmit is often a critical part of their function. While digital fabrication devices facilitate their realization, motiondriven mechanism design remains a challenging task. We take drawing machines as a case study in exploratory design. Devices such as the Spirograph can generate intricate patterns from an assembly of simple mechanical elements. Trying to control and customize these patterns, however, is particularly hard, especially when the number of parts increases. We propose a novel constrained exploration method that enables a user to easily explore feasible drawings by directly indicating pattern preferences at different levels of control. The user starts by selecting a target pattern with the help of construction lines and rough sketching, and then finetunes it by prescribing geometric features of interest directly on the drawing. The designed pattern can then be directly realized with an easy-to-fabricate drawing machine. The key technical challenge is to facilitate the exploration of the high dimensional configuration space of such fabricable machines. To this end, we propose a novel method that dynamically reparameterizes the local configuration space and allows the user to move continuously between pattern variations, while preserving user-specified feature constraints.

We tested our framework on several examples, conducted a user study, and fabricated a sample of the designed examples.

INTRODUCTION

Toy of the Year in 1967, the Spirograph is a simple-to-use family of interlocking cogs and teethed rings allowing to draw a great variety of patterns. Although many other mechanical drawing tools preceded and followed it (see Fig. 1), this modest set of gears has marked a generation, and remains one of the most well-remembered today. As a product of the relationship between art and technology, drawing machines are still popular across artists [START_REF] Nolan | James Nolan Gandy[END_REF], enthusiastic inventors [START_REF] Freedman | Cycloid Drawing Machine | leafpdx[END_REF], and makers [START_REF] Torms | 3d printēts spirogrāfs[END_REF]. The simplicity of the mechanical parts involved makes them easily fabricable with modern personal fabrication devices, which in turn open the door to a level of customization leading to new and fascinating patterns. Beyond this goal, the inverse problem of finding the machine tracing out a specific trajectory has numerous applications (see e.g. Coros et al. [START_REF] Coros | Computational Design of Mechanical Characters[END_REF]).

Designing such machines, however, is particularly challenging. First, many mechanical devices transform an input rotation into a more complex cyclic output by combining oscillations of different periods and amplitudes. To produce a closed end-effector curve, the radii of mating gears (or equivalently, the number of teeth) need to have rational ratios. It is easy to enforce this constraint by restricting radii to natural numbers; the size of the pattern can still be controlled by a global scaling factor. The downside is that the design space becomes much more complex to explore: as the period is governed by modular arithmetic between radii, the visual output can radically change from one value to the next. Furthermore, the number of design parameters obviously increases with the number of parts. While this greatly enriches the space of possible curves, manually refining a design becomes difficult with as little as three continuous parameters. Indeed, nonlinearities make the influence of each control hard to grasp, and each one possibly influences the bounds of the others, making the space harder to explore.

In this paper, we propose a constraint-based exploration framework to design complex mechanical trajectories by interacting directly with the output pattern. In contrast to previous work [START_REF] Bächer | LinkEdit: Interactive Linkage Editing Using Symbolic Kinematics[END_REF], we focus on: (i) highly structured curves, which would be tedious to edit point by point, and (ii) allowing the continuous exploration of local design variations, rather than recomputing a new solution after each curve edit. Indeed, the latter has the disadvantage that modifications made in one place of the pattern may result in unexpected changes somewhere else. Our method, on the other hand, allows the user to define visual preferences and explore the resulting constrained subspace.

Our exploration workflow consists in a coarse-to-fine definition of visual preferences that progressively refine the choice of curves. First, as an entry point into the design space, the user draws a coarse sketch that defines the global properties (e.g., order of rotational symmetry) and appearance of the desired pattern. After selecting an initial curve among suggestions proposed by the system, changes can be made via sliders within a domain that respects the feasibility constraints of the corresponding mechanism. When one slider is moved, the bounds of the others are automatically updated. As a key interaction, the user can define visual preferences directly on the drawing. These take the form of special points on the curve that can be constrained according to their geometric properties. The user can then explore local variations closest to these specifications via new handles that are automatically generated. Once the user is satisfied, the shape of the mechanical parts is automatically generated and exported for laser cutting fabrication (see Fig. 2). Technically, we enable the above key interaction with a novel dynamic reparameterization method that locally samples the high dimensional configuration space of a given mechanism, measures closeness to the user-defined preferences, approximates the closest subspace, and exposes new parameters to navigate this subspace.

We evaluated the effectiveness of our design tool on several test scenarios, conducted a user study, and fabricated several physical prototypes able to draw patterns created by the users. Overall, we found that dynamic reparameterization allowed users to reliably make meaningful fine scale adjustments to their pattern designs.

This paper extends the previous conference work [START_REF] Roussel | SPIROU: Constrained Exploration for Mechanical Motion Design[END_REF] by significantly extending the pattern retrieval step (including a novel technique to reduce the search space), providing new figures, a more detailed explanation of the fabrication process, and a virtual extension to a mechanical character to demonstrate the generality of our method.

RELATED WORK

Drawing machines have a long history in mathematics [START_REF] Hall | Trochoids, Roses, and Thorns -Beyond the Spirograph[END_REF], art [START_REF] Ernst | -7. Young Man Intrigued by the Flight of a Non-Euclidean Fly[END_REF], and as toys. Before the computer era, they were the only way to accurately draw certain curves, with applications in architecture, astronomy, engineering, etc. [START_REF] Garcia | DrawingMachines[END_REF]. While simulating such machines is nowadays relatively easy, the inverse problem of mapping an arbitrary end-effector trajectory to a reasonably simple mechanism remains a challenge. One of the most fundamental results in this regard is Kempe's universality theorem [START_REF] Kempe | On a General Method of describing Plane Curves of the nth degree by Linkwork[END_REF], which states that for any arbitrary algebraic plane curve, a linkage can be constructed that draws the curve. The constructive method proposed by Kempe, however, produces mechanisms with so many links that they are impossible to fabricate in practice. There have been many endeavors since then; to cite only one, Liu et al. [START_REF] Liu | Design of Mechanisms to Draw Trigonometric Plane Curves[END_REF] recently proposed a method to reproduce trigonometric plane curves with either Scotch yoke mechanisms or serial chains. Robotic arms and CNC machines have not made this problem obsolete, since there are situations where size is an issue (e.g. MEMS or nano applications), or electric power is unavailable or undesirable.

The following paragraphs discuss related advances in computational design and fabrication, with different application settings, both for inverse modeling and design exploration.

Computational design from target motion. In the context of automata design, researchers have investigated replicating target motion using an arrangement of mechanical parts in a classic instance of inverse problem setup. The general approach involves sampling configurations from a library of components to retrieve a local arrangement of parts, and then refining the shape parameters using a gradient-descent based optimization to fit the target motion. For example, Zhou et al. [START_REF] Zhu | Motion-guided Mechanical Toy Modeling[END_REF] and Coros et al. [START_REF] Coros | Computational Design of Mechanical Characters[END_REF] design automaton characters, while Ceylan et al. [START_REF] Duygu Ceylan | Designing and fabricating mechanical automata from mocap sequences[END_REF] design automata to replicate mo-cap sequences.

In more interactive design settings, Umetani et al. [START_REF] Umetani | Pteromys: Interactive Design and Optimization of Free-formed Free-flight Model Airplanes[END_REF] design paper airplanes based on their predicted flight dynamics; Thomaszewski et al. [START_REF] Thomaszewski | Computational Design of Linkage-based Characters[END_REF] use global optimization to design linkage-based characters; while Bächer et al. [START_REF] Bächer | LinkEdit: Interactive Linkage Editing Using Symbolic Kinematics[END_REF] develop a system to support interactive editing of fabricable linkages. Recently, Ion et al. [START_REF] Ion | Metamaterial Mechanisms[END_REF] took a different approach by generating 3D-printable microstructures that are able to transmit movement through shearing of their constitutive cells.

The main goal of the above efforts is to either approximate a given motion, or indirectly edit a target automata or linkage-based kinematic chain. Formulating these problems as optimizations suggests that the user already knows what she wants, and simply wishes to obtain a specific motion as easily as possible. While some parts of our method appear similar (e.g. the rough pattern retrieval step), what we propose is a framework to support exploration and progressive refinement, which had not yet been achieved in this context. More formally, instead of trying to map a specific curve to a point in design space, our method maps a set of curve constraints to a design subspace. These approaches complete rather than oppose each other: even in the first case, there is interest in being able to explore local design alternatives, as it might reveal solutions that the user had not thought of.

Guided exploration of valid designs. In a broader context of design exploration, researchers have studied various properties in order to optimize a shape based on its intended usage. Examples include shape optimization based on stability considerations [START_REF] Prévost | Make It Stand: Balancing Shapes for 3D Fabrication[END_REF], or updating object shape for desired moment of inertia for object spin [START_REF] Bächer | Spin-it: Optimizing Moment of Inertia for Spinnable Objects[END_REF], adaptively adjusting object parts for better reinforcement and strength [START_REF] Lu | Build-tolast: Strength to weight 3d printed objects[END_REF][START_REF] Zhou | Worst-case Structural Analysis[END_REF], designing hollow chambers for desired acoustic behavior [START_REF] Bharaj | Computational Design of Metallophone Contact Sounds[END_REF], zero-waste furniture design [START_REF] Koo | Towards Zero-Waste Furniture Design[END_REF], or modeling elastic behavior of foam microstructures for procedurally generating them for target material properties [START_REF] Martínez | Procedural Voronoi Foams for Additive Manufacturing[END_REF].

Closer to our concerns, Umetani et al. [START_REF] Umetani | Guided exploration of physically valid shapes for furniture design[END_REF] proposed a furniture modeling system that actively guides the user to navigate valid regions of the design space; Bokeloh et al. [START_REF] Bokeloh | An algebraic model for parameterized shape editing[END_REF] and Yumer et al. [START_REF] Mehmet Ersin Yumer | Semantic Shape Editing Using Deformation Handles[END_REF] developed modeling systems that preserve high-level structural and semantic relations in edited 3D models, while Koo et al. [START_REF] Koo | Creating works-like prototypes of mechanical objects[END_REF] proposed the use of functional specifications to map user prescriptions to constrained modeling for 'works-like' prototypes of furniture. We have been inspired by the work of Shugrina et al. [START_REF] Shugrina | Fab Forms: Customizable Objects for fabrication with Validity and Geometry Caching[END_REF], who precompute the domain defined by fabricability and functionality constraints to expose sliders with valid ranges to the user. In our work, however, additional constraints are defined by the user at runtime directly on the drawing, and new sliders are automatically generated to explore the resulting constrained space. Recently, Guerrero et al. [START_REF] Guerrero | PATEX: Exploring Pattern Variations[END_REF] proposed efficient local approximations to enable exploration of pattern variations by dropping different constraints in the input patterns. However, the method is not suitable for variations that are additionally required to be physically realizable by drawing machines. As in these works, we aim to ease the exploration of the feasible space, but apply this to the problem of exploring the curve patterns produced by a mechanism, instead of exploring designs of the mechanism itself.

Additionally, in a classical constrained modeling setup, design from geometric constraints specifications has also been studied. Ivan Sutherland's Sketchpad [START_REF] Sutherland | Sketchpad: A Man-machine Graphical Communication System[END_REF] pioneered CAD software as the first graphical interface to support geometric constraints. Analyzing and solving such constraints have been tackled by Fudos et al. [START_REF] Fudos | A graph-constructive approach to solving systems of geometric constraints[END_REF] in a general setting, and Sitharam et al. [START_REF] Sitharam | How the Beast really moves: Cayley analysis of mechanism realization spaces using CayMos[END_REF] in the context of mechanisms. Theoretically, even determining parameter bounds characterizing the (constrained) solution domain remains a known difficult topic [START_REF] Barki | Re-parameterization reduces irreducible geometric constraint systems[END_REF]. Instead, in this work, we seek a general approximation strategy supporting interactive design space exploration for an intuitive workflow.

OVERVIEW

Mechanical drawing machines typically are arrangements of cogs and parts, with an end-effector that traces out intricate 2D patterns. Each such machine physically realizes an algebraic expression connecting the machine part parameters to the output drawing. This tight coupling between the parameters and the resultant pattern variations makes the designers' task of exploring the design space very challenging. Specifically, while on one hand modifying a single parameter may cause several simultaneous changes (e.g., twisting and scaling), on the other hand a single desired change often requires synchronous manipulation of multiple parameters. Our goal is to decorrelate these variations. Rather than trying to find the best possible separation (which tends to be subjective or contextdependent), our goal is to allow users to define their own visual constraints or invariants in the drawing space, so that other variations can be explored independently.

There are two main technical challenges to tackle: first, mechanisms are often described by a relatively high number of parameters (3-8 in our examples), both continuous and discrete, and whose valid domain is implicitly defined by a set of non-linear constraints; second, mapping invariants in the drawing to a corresponding parameter subspace cannot be done analytically in the general case, as the relation between parameter changes and drawing changes is very complex.

We address these challenges with a two-step workflow (see Fig. 2). The first step, described in Sec. 4, consists in selecting an appropriate machine by defining global pattern characteristics and providing a coarse sketch. This step notably allows to assign and fix all discrete parameters. During the second step, described in Sec. 5, local continuous variations can be explored while dynamically specifying visual invariants. Our key contribution is twofold: identify a set of recurring geometric regularities involving relevant feature points that can be tracked as the drawing changes (Sec. 5.1), and a novel local approximation method that allows to explore the subspace where such regularities appear (Sec. 5.2).

We evaluate our method in several ways (see Sec. 6). First, we demonstrate a number of cases where our invariants allow meaningful changes in the drawing (Sec. 6.1). Second, we validated the feasibility of our drawing machines by fabricating several prototypes (Sec. 6.2). Lastly, we conducted a user study to assess the Let us now define the notation used in the rest of the paper. Each type of machine is described by:

• A set of design parameters p = (p d , p c ) (the concatenation of discrete and continuous parameters), evolving in a design space D implicitly bounded by a system of algebraic constraints C(p) (typically nonlinear). • A simulator that works out the trajectory D p (t), t ∈ [0, T ] (where T is the period) traced by the end point. It also provides a time series of the positions and orientations of each machine part for visual inspection. • A representation R p of the mechanism geometry at different levels of detail (coarse for visual checking, detailed for export and fabrication). In this paper, 'configuration space' or 'parameter space' refers to D, while 'curve space' or 'drawing space' denotes the Euclidean 2D space of the trajectory.

PATTERN RETRIEVAL

The first step of the design workflow is an inverse problem: finding the parameter combination solution of

min p∈D ′ d(D p , S) (1)
where S is a spline fitted to the user's sketch (Section 4.1), D ′ is a subspace of D computed from the features of S (Section 4.2), and d is a measure of dissimilarity between curves (Section 4.3). Since the patterns produced by drawing machines are most often abstract, intricate, and generally tedious to sketch precisely, the result may only coarsely match the user's intent. Our goal in this step is to find a value for the discrete parameters (i.e. the radii), so that the remaining continuous space can be explored. Once S and D ′ are computed, a set of N b best matching candidates D i p is retrieved via brute force comparison of S to drawings sampled in D ′ . Redundant drawings are avoided (to a certain extent) by making sure that the radii are coprimes (i.e. they do not share a divisor other that 1). This is enforced by sampling values from the Farey sequence (without the first term), which can be computed with linear complexity in the number of terms [START_REF] Routledge | Computing Farey Series[END_REF]. Continuous parameters, on the other hand, are naively sampled (within feasible bounds) using grid search. The N b best matches are presented to the user, who can thus choose the best p (see Fig. 3).

Sketching and spline fitting

The user first draws a rough sketch. Construction lines can be used to pre-set the order of rotational symmetry: the pen strokes are then automatically symmetrized (see Fig. 4).

Our input is therefore a sequence of N s disconnected, noisy strokes (represented as polylines) that we want to turn into a smooth closed spline. Obviously, if N s = 1 we can fit the spline directly. Otherwise, our goal is to obtain the shortest closed path that runs through all strokes. This amounts to a variant of the traveling salesman problem (TSP), where part of the path is already fixed. We start with the integer program formulation of Dantzig et al. [START_REF] Dantzig | Solution of a Large-Scale Traveling-Salesman Problem[END_REF], and make the following modification. Let the 2N s ends of each stroke be the vertices V of a complete undirected graph. The set of edges E in this graph corresponds to the connections between each end. In order to force the connection between two ends of the same stroke, we define the coefficient of each edge as

c i j =
-k if i and j belong to the same stroke,

d i j otherwise, (2)
where d i j is the Euclidean distance between the vertices i and j, and k is a positive constant. For k high enough, it is always more advantageous for the solver to connect the two ends of a given stroke together, while still having the freedom of choosing the connection order. We use the Gurobi solver [START_REF]Gurobi Optimizer Reference Manual[END_REF] for this optimization, which relies on a linear-programming based branch-and-bound algorithm. We obtain an ordering of the strokes as well as the points within these strokes (given by the order of the ends). Lastly, we concatenate the strokes and fit a closed cubic B-spline S with a user-defined smoothing factor. We note that while the TSP is NP-hard, in practice the number of strokes tends to be small; therefore, in all of our examples, the processing time of the entire step was around 0.01s.

Reducing the search space

To prevent the grid search from becoming prohibitively expensive, we prune the search space with a number of empirically determined rules. First of all, the degree or rotational symmetry, if there is such a symmetry, is always equal to the reduced radius r 1 of the leading gear G 1 ('reduced' meaning that all radii have been divided to become coprimes). Second, the frequency spectrum of the pattern is a useful source of information regarding the other radii. Indeed, each pattern is a result of the combination of periodic movements produced by the gears. Our strategy starts by computing the discrete Fourier transform (DFT) of a target curve to discover the dominant frequencies, before translating them in terms of gear radii.

First, we sample N f equally spaced parameter values in [0, 1] and compute the corresponding points along the target spline S. For convenience, we write these 2D points as complex numbers x n ∈ C, and compute the DFT X = F {x} with

X k = N f -1 n=0 x n • e -2π i N f kn ∀k ∈ [0 . . N f -1]. ( 3
)
The DFT is, by definition, a sampling of the discrete-time Fourier transform (DTFT), which is itself a continuous function of frequency. Each value X k of the DFT corresponds to a frequency f k given by

f k = f s N f k ∀k ∈ - N f 2 . . N f 2 -1 , (4)
where f s is the signal sampling frequency. The different interval for k compared to Eq. 3 is not a problem since X is periodic over this range. Moreover, by construction, our input curve spans exactly one period T . Therefore, f s = N f /T and Eq. 4 becomes

f k = k T ∀k ∈ - N f 2 . . N f 2 -1 . ( 5
)
Our goal is to find the equation of motion closest to the input curve; therefore, T is a priori unknown. As we will see, however, we do not need to determine its actual value. The middle graph in Fig. 5 illustrates the (f k , |X k |) mapping, also called frequency spectrum, for a given curve (where we arbitrarily set T = 1). As we can see, for such a clean curve only a few frequencies dominate, symmetrically distributed around the fundamental frequency f 1 . Two of the simplest machines (Spirograph and Cycloid Drawing Machine) present a second dominant frequency f 2 , while the Hoot-Nanny even displays a third dominant frequency f 3 (both f 2 and f 3 being accompanied by their harmonics). We postulate that the number of such frequencies increases by one for each new pair of mating gears in the system (unless of course they cancel each other out). In practice, however, f 3 may be a multiple of f 2 , in which case it is indistinguishable from f 2 's harmonics; this is a case where our analysis would only partially reduce the search space.

|X k | autocorr(|X k |) f i
Figure 5: Fourier analysis of a Hoot-Nanny drawing. The radii are (4, 3, 2), and the highest peaks of Fourier spectrum happen at frequencies (-3, 1, 3). One can easily check the validity of Eq. 6.

To compute the radii of the N G gears from the frequencies f i , we have experimentally found that the radius of gear G i satisfied

r i = lcm(n 1 , . . . , n N G ) n i with n i = T | f i | if i = 1, T | f i -f 1 | ∀i ∈ [2 . . N G ], (6)
where lcm() is the least common multiple, and n i is the number of rotations of G i over one period. From the frequency formula (Eq. 5), it is clear that T cancels out; therefore, we can simply set T = 1 from the start, thus obtaining integer frequencies.

Eq. 6 can be verified in the case of the epitrochoid (i.e. the curve produced by a Spirograph with one gear rolling outside the other), whose Cartesian equation of motion is

x(t) y(t) = r R (q + 1) cos(t) sin(t) -d cos ((q + 1) t) sin ((q + 1) t) t ∈ [0, T ] (7)
with q = r F r R , where r F and r R respectively denote the radii of the fixed and rolling gears, and d is the distance from the pen hole to the center of the gear. Since Eq. 7 is conveniently written as a Fourier series, and q + 1 > 1 whatever the radii, it is clear that

f 1 = 1 2π and f 2 = q + 1 2π . (8)
Taking the ratio,

f 1 f 2 = 1 q + 1 ⇐⇒ q = f 2 -f 1 f 1 ⇐⇒ r F f 1 = r R (f 2 -f 1 ). (9)
Since r F and r R are supposed coprimes,

r F f 1 = r R (f 2 -f 1 ) = lcm(f 1 , f 2 -f 1 ), (10)
which is indeed a specific case of Eq. 6. The proof for other machines involves much more complex equations of motion, but we can intuitively understand why it still works: different linkages only affect the amplitude of the oscillations, and not their frequencies; the latter only depend on the radii of mating gears. Therefore, for instance, the frequency spectrum of a Hoot-Nanny can be seen as a sum of epitrochoids. The challenge, then, is to determine the f i from the spectrum of S. First, we can safely set the f 0 peak (constant component) to 0, effectively centering the curve. Second, while f 1 is the highest peak for the (Elliptic) Spirograph, this is not necessarily true for other machines (see Fig. 5). The spectrum, however, always tends to be symmetric; therefore, we perform an autocorrelation of the Fourier peaks (i.e. a discrete convolution of the peaks with themselves) to make the fundamental detection more robust. If the machine has only two gears, determining f 2 is easy: it is the next maximal peak. The case of the Hoot-Nanny is more complex, as we saw that its frequency spectrum appears to be the sum of its two gear matings spectra considered separately. This has two consequences. First, f 2 and f 3 can only be distinguished if they are coprimes (which happens when the corresponding radii are also coprimes); otherwise their harmonics overlap. While all three radii are constrained to share no common divisor, any pair of them taken separately still can, which is why f 2 and f 3 are not necessarily coprimes. Second, their order in the spectrum remains ambiguous no matter what: the next highest peak after f 1 can correspond to r 2 or r 3 , while the Hoot-Nanny curves are sensitive to the order of r 2 and r 3 .

From a practical point of view however, the goal of this step is only to reduce the search space, not to completely determine the radii. If the order of r 2 and r 3 is uncertain, we test both combinations; if the amplitude of a peak is too low, we leave the corresponding radius undetermined and sample it along the continuous parameters during grid search. Any pruning of the search space is good to take, since the complexity is combinatorial in the number of parameters. Using a sampling density of 4 values per dimension for all machines in our dataset, our experiments showed that curve retrieval was faster by an order of magnitude when at least one parameter value was fixed.

Curve dissimilarity measure

Let us consider two curves C A and C B represented as polylines with N A and N B vertices respectively. In the previous version of this method [START_REF] Roussel | SPIROU: Constrained Exploration for Mechanical Motion Design[END_REF], curves were compared by normalizing and aligning them, then computing their respective distance fields F A and F B defined as

∀x ∈ R 2 F * (x) = inf x ′ ∈ C * ∥x ′ -x∥, (11)
and finally computing the symmetric distance

d F (C A , C B ) = max 1 N A N A i=1 F B (x A i ), 1 N B N B i=1 F A (x B i ) . (12)
In words, each side of the max tells us, on average, how close a point from one curve is to the other curve. Intuitively, d F quantifies how curves differ in terms of the 'density' of strokes in a region of space. From a practical point of view, this metric accepts a wide range of inputs, as each curve could be a collection of polylines, or even a binary image obtained from a picture. On the other hand, this metric is not as precise as e.g. Procrustes analysis [START_REF] Ian | Statistical shape analysis[END_REF] or the discrete Fréchet distance [START_REF] Eiter | Computing Discrete Frechet Distance[END_REF], two classic curve dissimilarity measures. The former requires N A = N B , but inherently normalizes and registers the curves with linear complexity in N A and N B , while the latter has (sub-)quadratic complexity. Having the same number of vertices is not difficult in our new method, since we can resample S to match the size of D p . Therefore, we adopt a symmetrized Procrustes distance as our new measure, defined as

d P (C A , C B ) = min N i=1 ∥x A i -x B i ∥ 2 , N i=1 ∥x A i -x B N -i+1 ∥ 2 , ( 13
)
where

N = N A = N B .
Here, one curve is compared to the other with its vertices taken in increasing, then decreasing order. This is because the regular Procrustes distance depends on the order of the sampling, and we do not know if S and D p are drawn in the same direction. Three comparisons between the distance field metric and the Procrustes distance are given Fig. 6. Both metrics were used to explore the same dataset of candidate curves and retrieve the best matching one. We observe that while the distance field metric is able to capture the general aspect of the curve, it is insensitive to other aspects such as the arc length because of its image based nature. Thus, for example, the trefoil knot in the middle is evaluated as similar despite looping twice. Furthermore, our timing measurements (see Table 1) show that the Procrustes distance between curves at a resolution of 1024 points is more than 20 times faster than the distance field metric between curves rasterized at a resolution of 512 by 512 pixels.

CONSTRAINED EXPLORATION

Once the the discrete parameters of the machine are fixed, the user can focus on fine tuning the continuous parameters. We note that an intuitive system should allow the user to edit different features of the drawing as independently as needed. This is not always possible: the smaller the number of degrees of freedom, the harder it is to prevent several changes from happening at the same time. For instance, a drawing machine with a single continuous parameter would not benefit from our system. Conversely, as the number of parameters increases, so does the extent to which modifications can be decorrelated. However, the exact combination of parameters that allows a constrained change is generally complex to determine, as it requires to either solve a system of non-linear equations, or to resort to manual trial-and-error. Hence, our goal is to efficiently identify, abstract, and expose the space of valid machine configurations subject to the specified constraints.

We allow the user to specify visual preferences as geometric properties that should stay fixed when a change is made. Note that this is different from handle-based deformation as the user indicates what shouldn't change during editing, rather than a specific target change. Then, our system computes a new parameter space that incorporates the previous machine-specific and global constraints with the new shape invariant(s). The resulting space can be explored via sliders, whose bounds are dynamically updated after each modification. The user can subsequently add more invariants, which further constrain the solution space until no remaining degree of freedom is left.

We first introduce the shape invariants that are supported and how they are dynamically computed and tracked (Section 5.1). Then, we propose a local reparameterization method that enables the user to intuitively explore the resulting invariant space in the form of desirable pattern variations (Section 5.2).

Pattern invariants

The curves generated by drawing machines are often highly structured and can be described at several levels of detail. If we attempt to decompose such a shape, the smallest discernible element is the point. However, not all points are perceptually equal: some have particular properties that make them stand out, such as intersection points and curvature maxima (see Fig. 7). We call them Points of Interest (PoI). Such points have generic attributes, such as Euclidean coordinates in curve space, and one (or two) associated time (or arclength) values. They also display properties which are specific to their type, such as the angle made by curve tangents at an intersection point, or the value of the curvature at the maximum (see Fig. 8).

Next, we define Relations of Interest (RoI), as relations that hold either between a PoI and an external object (e.g., a PoI lying on a geometric primitive), or between a group of PoIs (e.g., the distance between two PoIs). Any relation that can be expressed as an algebraic equation involving one or more features of one or more PoIs can be implemented in the system. While higher-level entities could also be considered, such as edges between PoIs, cells formed by edges, or even envelopes formed by sequences of PoIs, we currently only support PoIs and RoIs, as they are easier to compute and track in the parameter space. We note that the computation of the invariant subspace is agnostic of the nature of the features defined by the user.

Selection and computation. During the interactive session, the

PoIs closest to the user's mouse are highlighted. Selecting one (or two) of them opens a menu allowing the user to choose a feature to freeze.

For generality, we compute the PoIs on a discretized curve output by the simulation, instead of solving for them analytically. Curvature maxima are straightforward to obtain, as discrete curvature on polyline vertices is easy to compute. Finding the self-intersections of a polygon, however, is more involved, as the naive algorithm (testing every pair of segments) has a complexity that is quadratic in the number of sides. This problem has been extensively studied and several methods (essentially sweep-line based) have been proposed. We use the Bentley-Ottmann algorithm, whose complexity is O((n + k) log(n)), with n line segments and k crossings.

Tracking. Key to our approach, PoIs must be tracked as continuous parameter values change: in other words, when considering two patterns relatively close in the parameter space, we need to establish correspondences across the PoIs allowing us to quantify how much a specific PoI property has changed between two curves, and therefore, to build an invariant space.

Given two drawings D and D ′ and a reference PoI π D r on D, a naive criterion for such correspondence is to superimpose both drawings and take the closest PoI in D ′ . In some configurations however, several PoIs can overlap each other, leading to ambiguities. This search can be made more robust by considering proximity in terms of the arc length (see Fig. 9):

π D ′ r arg min Λ(π D r ) -Λ(π D ′ i ) , (14)
where Λ(π ) gives the arc length (or pair of arc lengths) of π and i indexes the PoIs in D ′ . This is especially true in the case of drawing machines where the tracer needs to make a full turn before coming close again to the same area. Moreover, it should be noted that the matching PoI does not always exist: some intersections or curvature maxima are only present in a limited range of parameter values. Therefore, we define a distance threshold σ P oI between the reference PoI and its match, and discard curves for which this limit is exceeded. This threshold can also be used to make the search more efficient: indeed, candidate PoIs in other curves need only be computed in the circle of radius σ P oI centered at the reference PoI.

Exploring the invariant space

Once the desired pattern invariants have been selected by the user, the challenge is to explore the resulting constrained parameter space. In the general case, the invariant space is difficult to determine analytically. Therefore, we opt for a sample-based local linear approximation (see Fig. 10 for an illustration). In terms of interaction, our algorithm aims to provide the user with new sliders that allow interactive exploration. Since the approximation we use is only linear, regular re-projections and reapproximations of the invariant subspace are required. We perform them each time a slider is released after a move, which is preferable to continuous updates for two reasons: it ensures interactivity and allows reversible changes, ie. give the user the ability to come back continuously to a previous design while keeping the button pressed. We now describe our approach for sub-space approximation.

We consider an n-dimensional continuous parameter space implicitly bounded by several machine-intrinsic constraints, containing a point p 0 associated with the initial drawing D 0 . For simplicity, we assume a single user-defined invariant expressed by

d F i F (π D 0 r ), F (π D i r ) = 0 ( 15
)
where D i is the drawing associated to a neighboring point p i , F is the feature of interest (real-or vector-valued), and d F i is the Euclidean distance in the corresponding feature space.

First, we sample neighboring points p i , within the feasible continuous parameter domain, taking them on a grid whose resolution is adapted dimension-wise to the length of the feasible range. We instantiate the associated drawings D i , and track the corresponding PoI π D i r . We define the invariance score as

S i exp(-d F i ). ( 16
)
We will use these scores as weights for the regression of the solution space. Before that, we filter the samples to keep only a fraction of the highest weights. We assume, given the locality of the neighborhood, that the resulting domain is convex and not disjoint.

Then, to perform the regression, we use a Weighted Principal Component Analysis (WPCA) centered on the starting point. Since the weights are our invariance scores, this algorithm provides a basis of vectors ordered by decreasing contribution to the invariant space. A local basis can therefore be taken as the first m Principal Components, where m is the dimensionality of the invariant subspace. It is important to note that m cannot simply be deduced from the number of algebraic constraints, which are not necessarily independent. In other words, some constraints may be redundant, either between themselves or with the intrinsic constraints of the mechanism.

Pt of interest p 1 p 2

Figure 10: Illustrating the invariant space with 2 continuous parameters. The user identifies a PoI directly on the curve and specifies its desired invariant (here: position). Our system then locally samples the parameter space, evaluates an invariance score (shown as a color map), and performs a linear regression on the sample values. Sweeping the regressed solution (cyan line) amounts to exploring desirable curves (e.g., maroon curve).

In order to determine the dimensionality of the resultant space, we first make sure that it is not reduced to a singleton by checking the number of samples with a sufficiently high invariance score (superior to σ inv = 0.9). If less than two points are found, we consider that the system is over-constrained and invite the user to remove one invariant. The WPCA gives us the proportion of variance explained by each Principal Component. Defining v 1 r el as the highest relative variance in the set, we keep all components whose proportion is superior to σ var = 0.1v 1 r el . Each axis of the resulting subspace is mapped to a slider shown to the user. If no component is filtered out, we consider that all the invariants were redundant with the intrinsic constraints, and hence keep the original parameterization.

Next, we compute the bounds of the resultant solution space. Since the approximation is local, we do not need to allow too wide an amplitude around the starting position. Since each Principal Component is normalized, we put coarse bounds at -2 and 2. Even then, the intrinsic constraints may impose tighter bounds along some dimensions, which depend on the value of the other parameters; therefore, they need to be re-computed every time a slider is moved. We formulate this as a sequence of non-linear constrained optimization problems: for each parameter, with the other parameters held fixed, we successively find its minimal and maximal values. Please note that this optimization only uses the intrinsic constraints of the system, which do not require a simulation or the evaluation of PoIs (see supplementary document for details). Further, since we assumed that the local neighborhood was convex and connected, we expect a single range of possible values within the coarse bounds.

We are now ready to present the user with a set of sliders that can be moved while respecting the invariants. Once a slider is released, we update our model accordingly. First, we project the current position back onto the solution space, by finding the point closest to this position that maximizes the invariance score. Then, we recompute a local approximation of the solution space, following the procedure that has just been described.

In addition, we make the system more intuitive to use by ensuring that the sliders have a temporally consistent visual effect on the drawing. Indeed, re-approximating the invariant subspace may typically result in the Principal Components flipping or rotating (see Fig. 11). Flipping can be easily resolved by comparing the old and new principal directions pairwise -since their order is preservedand flipping them back if necessary. Rotation of principal directions, which typically happens when the spread is symmetrical (Fig. 11c), can be avoided by projecting the previous local basis onto the new one, and normalizing the resulting vectors. This ensures a consistent behavior of the sliders throughout the exploration.

RESULTS AND DISCUSSION

Our database of mechanisms contains four parametric models whose specifics are given in the supplementary document. Table 2 summarizes the main characteristics of these machines. While the Spirograph, the Cycloid Drawing Machine and the Hoot-Nanny are motivated by existing drawing machines, the elliptic Spirograph was designed by the authors to experiment with non-circular gears.

Various patterns designed with our system are shown in Figs. 2, 10, 12 and 13. Constrained exploration results are provided in Fig. 12 and in the accompanying videos, where we compare slider manipulation in the chosen design space to the corresponding changes happening to the base parameters. We further evaluated our method in two ways: first, we ensured that it produces mechanically functional machines by fabricating prototypes; second, we conducted a user study to assess the intuitiveness of our system compared to a forward simulator.

Constrained exploration results

We demonstrate examples of curve invariants for each row in Fig. 12. They were kept voluntarily simple to emphasize the effect of a given constraint (see the accompanying videos for more complex examples). Let us discuss each of these experiments. • Fixed point. The user fixed the location of the selected PoI.

On the left (CDM), the interior boundary was pulled in, while keeping the external arc fixed. On the right (HN), the cusp point is fixed, while increasing the symmetric lobes. • Fixed curvature. The user fixed the curvature at the selected PoI. In the left example (ES), the center was pulled in, while maintaining the PoI's curvature. In the right example (CDM), the central part was reduced and rotated, while maintaining the PoI's curvature. • Fixed intersection angle. The user fixed the angle between tangents at the selected intersection point. In the left example (ES), the center was pulled in while preserving tangency between the curve segments (i.e., zero angle). In the right example (CDM), the loop size was changed, while keeping the inter-curve intersection angle (and symmetry). • Moving along radial line. The user restricted the movement of the PoI along a radial line. On the left (CDM), the center was closed in while keeping the global orientation. On the right (HN), the central part was pulled in and the curvature at the cusp was changed, while keeping the original orientation. • Fixed distance between 2 PoIs. The user fixed the distance between 2 selected PoIs. In the left example (ES), the external boundary size was maintained, while pulling the petals closer together. In the right example (ES), the size of the petals was held fixed, while pulling them apart. • Multiple specifications: In these examples, multiple constraints were specified on selected PoIs. On the left (CDM), the asymmetry was changed while keeping the global orientation and curvature of petals. On the right (HN), the petals were made more ornamental while preserving their curvature and restricting movement along radial line.

Precise modeling and fabrication

We fabricated several examples of machines (see Fig. 13):

• the elliptic Spirograph, an easy to fabricate two-parts mechanism that we used to validate the first invariants; • the Hoot-Nanny, which demonstrates our ability to manage devices with a wider range of parts and connectors.

Our general principle during the fabrication process was to lasercut the precision-critical, horizontal parts, and to 3D-print the remaining custom connectors, which notably ensure the transmission of movement and support the different layers of flat components. While the vector files given to the laser cutter are automatically generated by a script, the 3D-printed components were designed by hand using CAD software, requiring to adjust tolerances to help the machine run smoothly.

One challenge encountered during fabrication was the design of gear profiles. Such profiles are usually not represented in CAD software, as they would unnecessarily make the geometric model more complex; moreover, these pieces are traditionally manufactured with normalized shaper cutters. Laser cutters, on the other hand, require a precise geometric model as input. Therefore, we implemented a procedural generation of involute gear profiles (which optimize the transmission of torques, see Fig. 14), for both circular and elliptical gears. The latter, which is less common, was derived from a method by Bair [START_REF] Biing | Computerized tooth profile generation of elliptical gears manufactured by shaper cutters[END_REF].

Pictures of some of the fabricated examples are given Fig. 2 and 13. Demonstration of their usage is given in the main supplementary video.

User study

We conducted a user study with 8 participants to validate the efficiency of a mode of exploration based on visual constraints. We chose to focus on an important premise of our method -the fact that defining visual preferences can help navigating the configuration space more easily -rather than trying to evaluate the entire pipeline. This choice allowed to focus on the core contribution of constrained exploration, and made user sessions reasonably short in time and easier to compare. We defined the following protocol. Each user session was divided into four pattern-editing tasks. In each of these tasks, the candidate was asked to transform an initial curve A into a target curve B, using sliders, in less than two minutes. The set of target patterns was the same for all users, while initial patterns were randomly generated for each new session. The editing operation had to be performed twice: once with the basic machine parameters (subtask 1), and once with parameters corresponding to a predefined visual invariant (subtask 2). The interface was kept minimal, has shown in Fig. 15 left. In order to focus solely on the efficiency of the parameterization, we designed both subtasks to be as close as possible interaction-wise. First, the same number of sliders was exposed each time (despite our method allowing to reduce this number), and the order in which the subtasks successively appeared was randomized. Second, the predefined invariant was not shown to the user. Lastly, we presented the re-projection and re-approximation process as a little "helper" which could be called by pressing the spacebar, triggering a change in the curve and in the behavior of the sliders. This "helper" had a negligible effect in the base case: a dummy waiting time was triggered (inferior to the time required by the true "helper"), and a tiny perturbation was added to the sliders. This managed to make both versions completely indistinguishable for all users. At the end of the session, candidates were presented with a table displaying their results (see Fig. 15 right). For each task, they were asked to rate the similarity with the target pattern between 0 and 5.

Results are given for two metrics (total time and perceived dissimilarity) in Fig. 16. With comparable times, candidates were in most cases able to reach a final result perceptually closer to the Figure 15: Left: interface for a subtask of the user study (target pattern in grey). Right: summary sheet presented to the user in order to rate the results (each column is respectively the target pattern, and results of subtask 1 and 2 in an arbitrary order). target curve. The slightly higher times in our case can be attributed to the re-approximation step, which could take up to three times longer than the dummy step defined for the base case. This could, however, be reduced with a more efficient implementation. Moreover, additional time-independent metrics, namely the total number of slider moves and the total Euclidean distance travelled in the parameter space (given as supplementary material), demonstrate that our parameterization was more efficient.

Lastly, we note that this study only partially validates the efficiency of our method, as candidates were not allowed to choose their own invariants (which would have required a longer familiarization time). Therefore, the intuitiveness of the Points of Interests and associated invariants has not been assessed. Moreover, an editing task with a specific target does not exactly correspond to the exploration scenario we envisioned for this method; it is, however, easier to evaluate quantitatively.

Discussion

Our method presents several limitations, which open the way for future developments:

• The curve metric used for pattern retrieval, while efficient, does not necessarily reflect perceived proximity between drawings. A possible improvement could be to train a featurebased curve metric with a perceptual study, as done by Coros et al. [START_REF] Coros | Computational Design of Mechanical Characters[END_REF], while being aware that perception tends to be application dependent, and may not be as general. • Our naive grid-based local sampling method is combinatorial in the number of parameters, which allowed to keep interactive rates up to only six continuous parameters in our single-threaded Python implementation; we note, however, that computing samples and PoIs could be done in parallel, and that a subset of the most significant parameters can be preselected before applying our method.

• Bounded sliders are straightforward to implement, but they lack a clear meaning in terms of visual effect on the drawing. Possible improvements include adding intuitive visual clues beside each slider, or more advanced controls. • Lastly, transforming an abstract mechanical model into a fabricable assembly remains a tedious task, as many physical aspects that are neglected (gear backlash, defaults in 3D-printed parts, frictions and instabilities) may end up impairing the final drawing quality. This notably explains why we were not able to reliably build machines with a higher number of gears: while our method is completely able to handle more complex devices virtually, the accumulation of physical errors made the resulting prototypes too unstable to produce drawings of satisfying quality.

The specific application domain presented in this paper, while interesting from an educational and artistic point of view, could be seen as limited in terms of practical value. The core concepts of our method, however, are not bound to drawing machines: they remain valid for more general mechanisms, as illustrated in Fig. 17. Following our previous work [START_REF] Roussel | SPIROU: Constrained Exploration for Mechanical Motion Design[END_REF], we selected one of the elementary mechanisms from Coros et al.'s paper [START_REF] Coros | Computational Design of Mechanical Characters[END_REF] and added it to our system. Connecting a 'leg' to the end-effector (animated with inverse kinematics), we can imagine a situation where a user wishes to have the foot kicking a ball at the maximum of curvature (therefore deciding to fix its position) while exploring the various trajectories taken by the foot to reach this position.

Furthermore, designing cyclic motions is also relevant in industrial settings such as assembly lines, where the available constraints on a point of interest could be extended to speed and force, with no change needed in the rest of the pipeline.

Lastly, this method should be applicable to a wider range of generative design systems with a set of continuous and discrete parameters as input, assuming access to a reasonably fast forward simulation (or procedural generation) leading to an output having measurable (and desirable) invariants.

CONCLUSION

We presented a framework for exploring and fabricating drawing machines. The user can directly select among different machines along with their parameter settings using high-level scribbles, and then refine the retrieved drawing pattern by specifying constraints on dynamically computed feature points.

The main idea is to locally sample the design space and regress to the subspace that best preserves user-specified constraints on Points of Interest in the drawing. We linearize the space using a weighted PCA and expose the desirable region of the design space to the user. The user can simply navigate the solution space using an intuitive slider interface. We tested our setup on several classical drawing machines, designed various patterns using it, and fabricated a few prototypes to demonstrate the effectiveness of the approach.

In the future, we would like to extend our framework in different ways. An important next step would be to support interactive topological changes to machine configurations and allowing users to seamlessly transition across such variations directly by sketching curves and indicating suitable invariants. Another interesting extension would be to support 3D space curve drawing machines which would be relevant for recently introduced 3D doodle pens. Finally, we plan to investigate how our dynamic reparameterization approach can be used in other contexts of design exploration where analytically solving for and characterizing valid solution spaces is too expensive and impractical.

Initial curve New curve Pt of interest

Figure 1 :

 1 Figure 1: Some examples of drawing machines: (a) Spirograph, (b) Hoot-Nanny, (c) Harmonograph, (d) Cycloid drawing machine.

Figure 2 :

 2 Figure 2: Overview of our design workflow. The user first selects a mechanically feasible drawing by providing a rough sketch (a), and is then able to interactively explore local alternatives (b) by defining visual constraints directly on the pattern (here, the cusp position). The resulting machine is automatically exported to laser cutter profiles for fabrication. (c)

Figure 3 :

 3 Figure 3: Starting from a user sketch (left), we retrieve the best matching patterns in the database -lower value indicates a better candidate.

Figure 4 :

 4 Figure 4: Examples of splines fitted to input sketches. Each sketch required only 1-4 strokes that were automatically symmetrized.

Figure 6 :

 6 Figure 6: Comparison of best matching curves for a given input spline (top row), respectively obtained with the distance field metric (middle) and the Procrustes distance (bottom).

Figure 7 :

 7 Figure 7: Example of Points of Interest in a drawing.

Figure 8 :

 8 Figure 8: Types of Points of Interest (PoI) and associated invariants supported by our system.

Figure 9 :

 9 Figure 9: Using unambiguous features to discriminate between Point of Interests. Although the red intersection point is close to the others, we can still differentiate it using the pair of arc lengths values at the crossing.

Figure 11 :Figure 12 :

 1112 Figure 11: Ensuring temporal consistency. Since the Principal Components may flip during a slider move from (a) to (b), or rotate when the user explores a near position (c), the axes of the new linear approximation are flipped or rotated when necessary, within the subspace of interest, to remain as consistent as possible with the original principal directions.

Figure 13 :

 13 Figure 13: Examples of fabricated prototypes: (a) Elliptic Spirograph with two curves drawn using a different elliptic gear; (b) and (c) Instances of the Hoot-Nanny. Drawn patterns on the right.

Figure 14 :

 14 Figure 14: Different possible gear profiles. Profile (a), although more complex to generate, maximizes torque transmission.

Figure 16 :

 16 Figure 16: User study results (mean and standard deviation bars). "BAS" and "INV" respectively denote the base and invariant-space parameterizations.

Figure 17 :

 17 Figure 17: Extension to a mechanical character: a leg kicking a soccer ball.

Table 1 :

 1 Sampling and computation times for the three examples of Fig.6. There were 12248 samples before pruning.

				Time per sample (s)
	Input drawing	# samples	Sampling time (s)	Distance field	Procrustes distance
	Loops	376	0.85	0.0100	0.0006
	Trefoil	360	1.04	0.0154	0.0007
	Whirl	600	0.94	0.0225	0.0008
			Average	0.0159	0.0007

Table 2 :

 2 Mechanisms implemented in our system.

	Name	# exposed parameters (discrete+continuous)
	Spirograph (S)	2 + 1
	Elliptic Spirograph (ES)	2 + 2
	Cycloid Drawing Machine (CDM)	2 + 4
	Hoot-Nanny (HN)	3 + 5

ACKNOWLEDGMENTS

We are grateful to the anonymous reviewers for their valuable comments and suggestions. We also thank Aron Monszpart for proofreading the paper and Estelle Charleroy for helping with the video. This work was supported by the European Research Council Starting Grant SmartGeometry 335373 and Advanced Grant Expressive 291184, and gifts from Adobe. Prototypes were fabricated with the Equipex Amiqual4Home (ANR-11-EQPX-0002).


Katerina Soulantica

Justine Harmel

Adrien Berliet

Kassiogé Dembélé

Cécile Marcelot

Anne-Sophie Gay

Ovidiu Ersen

Sylvie Maury

Antoine Fécant

Bruno Chaudret

Philippe Serp

A seed-mediated approach for the preparation of modified heterogeneous catalysts

Keywords: colloids, nanocatalysis, supported catalysis, Fischer-Tropsch reaction, nanowires 10.1002/cctc.201701860

Colloidal chemistry is very efficient in providing nanocrystals of well controlled structural characteristics. While nanoparticles are essential components of heterogeneous catalysts, colloidal methods rarely employed for their preparation. We have employed a seed mediated growth approach for modifying the structural characteristics of a conventional cobalt-based Fischer-Tropsch catalyst. The Co particles of this catalyst can play the role of seeds for the overgrowth of shape and structure controlled cobalt nanostructures using a simple wet chemical method involving a molecular cobalt precursor and stabilizing agents. Thus, cobalt nanorods exhibiting the hexagonal compact structure were selectively grown on the immobilized Co particles of a reference Co/Al2O3-SiO2 catalyst. The as obtained catalyst shows a better stability than a reference catalyst for the Fischer-Tropsch reaction. The removal of most of the ligands from the nanorod catalyst allows improving the catalyst activity while maintaining its stability. This is a proof of concept concerning the implementation of wet chemistry nanoparticle synthesis for the modification of heterogeneous catalysts.

10.

Introduction

Heterogeneous catalysts employed in industrial reactions are multicomponent systems in which the catalytically active phase is in the majority of the cases inhomogeneous in terms of crystallite size, shape and sometimes crystal structure. Catalytic activity, selectivity and stability being affected by catalyst microstructure, the possibility to tune the size, the shape and the crystal structure of the catalytically active phase offers the opportunity to optimize its performances. [1] Wet chemical routes have recently progressed enough to allow a fine control of nano-object characteristics, including their chemical composition, which can be now finely tuned. [2] In this context, fine tuning of catalyst nanoparticles through wet chemical route syntheses could improve heterogeneous catalytic system performances through the introduction of specific characteristics to the catalyst, enabling the establishment of structure-performance relations in model catalysts and their subsequent translation into realistic systems. [3] The Fischer Tropsch Synthesis (FTS) is a catalytic process that converts synthesis gas (syngas, CO/H2), which can be obtained from various sources such as coal, natural gas, naphtha or biomass, into hydrocarbons and water. [4] This process attracts a lot of research attention as the products obtained consist in clean and high quality jet fuel and diesel. Currently, there are two FTS operating modes: high-and low-temperature FTS processes. [5] Both Fe and Co catalysts can be used in the lowtemperature FTS (200-240°C) process to produce linear longchain hydrocarbon waxes and paraffins in fixed bed or slurry reactors. [6] Cobalt catalysts are more active, present a longer lifetime, and have a simpler product slate of mainly paraffins and some α-olefins. The FTS is a reaction for which structure sensitivity has been demonstrated. Numerous studies have been devoted to rationalizing the influence of the microstructure of cobalt-based catalysts on their catalytic performances (activity/selectivity). [4c] Apart from the particle size that has been shown to play an important role, [5c, 7] the crystal structure might also influence the catalytic performances. This is especially relevant for catalyst systems based on cobalt, since cobalt crystallizes mainly in two structures: face centered cubic (fcc) and hexagonal close packed (hcp). Several works suggest that the Co-hcp structured catalysts present a higher activity. [8] Furthermore, activity is shape dependent since theoretical studies indicate that it is influenced by the crystal facet type exposed. [9] Some of us have reported the controlled synthesis in solution of purely metallic cobalt nanorods (NRs) crystallized in the hcp structure and exposing (11-20) facets. [10] They can be prepared by reduction under H2 of [Co{N(SiMe3)2}2(thf)] in the presence of lauric acid (LA) and hexadecylamine (HDA). Since Co(0) is the active catalyst phase, these NRs can be employed as FTS catalysts without any post-synthesis reduction step. In addition, thanks to their well-controlled shape, the exposure of a large majority of (11-20) facets could give information on the catalytic performances of these specific facets.

In addition to catalytic activity and product selectivity, catalyst deactivation is a crucial issue for industrial FTS catalyst development. The activity of Co-based FTS catalysts is deteriorated during their operation; the main suspected reasons,

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with varying contributions according to operating conditions, are: i) oxidation of the surface Co atoms, ii) Co-support interactions and formation of mixed compounds that are reducible only at high temperatures, iii) sintering, iv) refractory coke formation, v) loss of cobalt because of attrition, and (vi) hetero atom poisoning (e.g. sulfur). [8d,11] .Therefore, improving catalyst stability is a major concern among technology providers and plant operators. Different strategies, most of them related to catalyst operation, have been followed to improve Co-based catalyst stability. [4b, 12] However, considering the very long reaction time necessary to accurately evaluate catalyst stability under FTS conditions (typically 400-500 hrs), relatively few studies have been published so far on this critical aspect. [8d, 11c,d] We report herein the modification of a conventional nonpromoted Co/Al2O3-SiO2 FTS catalyst by growing anisotopic hcpcobalt nanostructures on the Co particles of the pristine catalyst, [13] and the remarkable stability of the resulting modified catalyst under FTS conditions in a slurry reactor. Wet chemistry methods have been employed for Co overgrowth from a solution containing a molecular Co(II) precursor. To the best of our knowledge this innovative approach has never been employed on heterogeneous catalysts.

Results and Discussion

The possibility to synthesize purely metallic Co NRs of homogeneous dimensions [10,14] incited us to use them as catalysts after immobilization on a conventional oxide support Al2O3-SiO2. However, the impregnation with pre-synthesized cobalt NRs did not produce a homogeneous distribution of the Co NRs inside the support. Figure S1 in the Supporting Information (SI) presents transmission electron microscopy (TEM) and scanning transmission electron microscopy-energy-dispersive Xray spectroscopy (STEM-EDX) results that show the accumulation of the Co NRs on the support surface. This is not surprising considering the NR shape and dimensions (length: 53 ± 5nm, diameter: 4.9± 0.9 nm), and the textural properties of the support (mean pore diameter = 9.2 nm). Alternatively, the NR synthesis was performed in the presence of the support. In contrast to the cobalt NRs, the cobalt precursor solution should easily diffuse in the support porosity, allowing the formation of NRs in the pores during the reduction step. However, heating a toluene solution containing [Co{N(SiMe3)2}2(thf)], LA and HDA (Co/HDA/LA = 1/1.7/1.2) at 110°C under H2 resulted in an extended nucleation of Co NRs and nanoparticles, mainly in the solution. (Figure S2 in SI). In addition, all attempts to separate the NRs formed in solution by washing the product resulted in the liberation of the Co nano-objects from the support, pointing towards an insufficient interaction of the Co with the Al2O3-SiO2 support. Thus, both impregnation and in situ formation of Co NRs on the Al2O3-SiO2 support were unsuccessful. The so called seeded growth (or seed-mediated growth) is a powerful strategy for the synthesis of a variety of nanocrystals. [15] Inspired by this approach, we took advantage of Co particles already immobilized on a support, i.e. Co particles of a conventional FTS catalyst prepared by a standard impregnation method, and used them as seeds on which hcp-cobalt NRs could overgrow from a solution containing molecular cobalt. Indeed, some of us have recently shown that purely hcp Co NRs and nanowires (NWs) can be grown from a solution containing [Co{N(SiMe3)2}2(thf)], LA and HDA, on crystallographically oriented metallic substrates, which play the role of seeds for the epitaxial Co growth. [16] These studies have demonstrated that the solution composition dictates the crystallographic structure and the shape of the Co nanostructures, and this independently of the substrate nature and crystallographic orientation. Indeed, hcp Co NRs/NWs have been obtained in all cases, their growth starting either from (111) fcc or (0001) hcp surfaces of various metallic films, and even on the surface of bulk metals. Thus, by controlling the solution composition, the Co overgrowth could take place anisotropically on the surface of ( 111) fcc or (0001) hcp 6-fold symmetry facets of the Co particles of the catalyst, which would play the role of seeds. [16b] To validate this approach, we first used a conventional 15% Co/Al2O3-SiO2 catalyst (SBET = 128 m 2 /g, pore volume = 0.32 mL/g, mean pore diameter = 10 nm) prepared as described in the experimental section, for which the particles presented a broad size distribution (Figure S3 in SI). In this catalyst, cobalt is crystallized mainly in the fcc structure, and the Co(0) amount after reduction is around 88% as estimated by temperature programmed reduction (TPR) analysis. This catalyst was introduced in a toluene solution containing [Co{N(SiMe3)2}2(thf)], HDA and LA with different Co/HDA/LA ratios. The [Co{N(SiMe3)2}2(thf)] amount was first fixed so that a total Co loading of 25% was obtained after complete reduction of the Co precursor in solution. Figure 1 presents the modified catalysts obtained by heating four mixtures of [Co{N(SiMe3)2}2(thf)], HDA, LA and 15% Co/Al2O3-SiO2 in toluene at 110°C under 3 bars of H2 for 24 hours. In these mixtures the Co/HDA/LA ratio has been varied, keeping the Co concentration constant. Although Co NRs can normally grow from a solution containing Co, HDA and LA in a molar ratio of 1/1.7/1.2 (Co/HDA/LA) [10b] in the absence of any support, it can be seen from Figures 1a-d that in the presence of the 15% Co/Al2O3-SiO2 catalyst it was necessary to increase the amount of ligands, and particularly that of LA, in order to obtain quantitatively anisotropic nano-objects (NRs/NWs). This is attributed to the adsorption of part of the ligands on the 15% Co/Al2O3-SiO2 catalyst, which modifies the Co/HDA/LA ratio in the solution and affects the shape of the growing nano-objects. Indeed, in a previous work and in the absence of any support we have shown that the Co/HDA/LA ratio is very important for the reaction outcome as far as shape and crystal structure are concerned. [17] , a high LA/Co ratio favoring the formation of anisotropically shaped hcp nanostructures. Thus, cobalt NRs/NWs on the pristine Co/Al2O3-SiO2 catalyst were selectively obtained for Co/HDA/LA molar ratio of 1/2/2.5 and 1/2/3. For the rest of this study, the ratio was fixed to 1/2/3, since it gives rise to an exclusive growth on the 15% Co/Al2O3-SiO2 catalyst, without any homogeneous nucleation of cobalt in solution. This was confirmed by TEM examination of the colorless supernatant obtained after reaction. Since the preexisting nanoparticles and the overgrowing from the solution metal are the same, we have no possibility to directly evidence that Co NRs/NWs overgrow only on the surface of the Co particles present on the parent Co catalyst and not on the Al2O3-SiO2 support. Therefore, we performed a blank experiment in which a solution of Co/HDA/LA of a 1/2/2 ratio was heated under H2 in the presence of bare Al2O3-SiO2, Co multipods grow only in solution while the color of the support remains white. We can therefore conclude that the NRs/NWs grow only on the surface of the Co particles of the pristine catalyst.

As described in the Experimental section, the Co seeds of the pristine catalyst were reduced prior to the Co NRs/NWs growth. We have seen that although it is possible to grow Co NRs on CoO, this is not possible on Co3O4 (Figure S4, SI). The difference in lattice parameters between hcp-Co (a = 4.057 Å), CoO (a = 4.260 Å) and Co3O4 (8.084 Å) is presumably at the origin of this result.

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Once the optimal 1/2/3 ratio fixed, and in order to compare to a well-studied reference catalyst, the initial cobalt loading was reduced so that the total loading after overgrowth corresponds to a realistic value for FTS. Thus, a conventional 8% Co/Al2O3-SiO2 catalyst was employed (SBET = 157 m 2 /g; Vp = 0.43 mL/g, average pore diameter = 9.8 nm, Co(0) crystallite size 12.7 nm), on which 7% w/w additional cobalt was deposited from the [Co{N(SiMe3)2}2(thf)] decomposition in the presence of HDA and LA. Figure 2a and 2b show respectively a TEM and a STEM-high angle annular dark field (STEM-HAADF) micrograph of the as obtained material. The NRs/NWs length measured by TEM is 52.6 ± 13.6 nm, the diameter being variable within the whole length. TEM observations of a sample obtained by ultramicrotomy indicate that the Co NRs grow both on the surface and in the porosity of the catalyst (Figure S5 in SI). However, it has to be noted that the shape of the Co nano-objects grown in the interior of the grains is not always anisotropic. SEM observations of the catalyst powder (Figure 2c-d) show that its granulometry is not affected by the growth of the Co NRs, and that the NRs/NWs are present on the catalyst grains. SEM-EDX analyses evidence that the cobalt is uniformly distributed on the surface of the catalyst particles (Figure S6 in SI). While quantitative analysis is not possible due to the interference of the support, the XRD diffractogram (Figure 3) of the 7%Co/8%Co/SiO2-Al2O3 catalyst shows the increase of the Co hcp phase, compared to the parent catalyst. Indeed, two hcp-specific peaks (48.7 and 55.5°, characteristic of ( 100) and ( 101

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Other signals centered around 52° and 61° are detected, corresponding to the presence of fcc-Co particles, presumably due to the seed Co particles of the 8%Co/SiO2-Al2O3. ICP analysis indicates 13% Co w/w in the final catalyst, and TGA analyses reveal that 8.4% w/w of organic ligands are present. The performances of the 7%Co/8%Co/SiO2-Al2O3 catalyst were monitored in the low-temperature FTS slurry process for 475 hours. For comparison, a 15%Co/SiO2-Al2O3 conventional catalyst was used as a reference. In Figure 4 we present the catalytic activities of these catalysts. An excellent stability can be noticed for the 7%Co/8%Co/SiO2-Al2O3 catalyst, which is not deactivated after 475h TOS. While the reference catalyst is initially more active than the 7%Co/8%Co/SiO2-Al2O3 modified catalyst, a deactivation process leads to an equalization tendency for the activities as compared to the 7%Co/8%Co/SiO2-Al2O3 catalyst. The TEM micrographs and XRD diagram of the 7%Co/8%Co/SiO2-Al2O3 catalyst after reaction are presented in Figure S8 in SI. While the anisotropic shape seems not to be retained, TEM observation of samples obtained by ultramicrotomy have shown that Co is present as nanoparticles of about 20 nm, accumulated in aggregates of 100-200 nm. Furthermore, XRD diffractograms of the catalyst before and after the catalytic run evidence that the small amount of CoO, which was present on the fresh catalyst, is reduced during the catalytic run with a simultaneous increase of the hcp Co component.

The catalyst synthesis through this colloidal method, which allows the Co overgrowth on the Co of the 8% Co/Al2O3-SiO2 precursor catalyst and makes use of organic stabilizers (LA and HDA), does not allow their complete elimination by simple washing. The presence of ligands on the surface of the catalytically active phase could be responsible for the reduction of the catalytic activity by blocking part of the catalyst active sites. [18] Several efforts in order to remove the excess of ligands have been performed. The best results were obtained by a heat treatment at 270°C for 20 minutes under a flow of Ar/H2 and under dynamic vacuum. This treatment decreases the amount of residual ligand to 3% instead of 8.4% for the as-made catalyst as determined by TGA, while conserving the hcp structure and the anisotropic shape (Figure S9 in SI for TEM and XRD). The thermally treated catalyst was also tested under similar conditions (Figure 4). The performances of all catalysts in terms of activity, selectivity and stability at 200 hours TOS are presented in Table 1. Interestingly, the thermally treated 7%Co/8%Co/SiO2-Al2O3 catalyst shows the best activity, that exceeds the one of reference catalyst after 200h TOS, combined to an excellent stability, which is one of the main concerns in industrial processes. While the available data do not allow drawing safe conclusions concerning structure-performance relationships, we propose that the high stability of the modified catalysts is probably connected to the absence of sintering as evidenced by the TEM examination of the spent catalyst. No large Co particles, which could have been formed by migration of small particles to the surface of the catalyst grains, [19] were observed (Fig. S8 in SI), whereas a Co extraction and sintering effect is clearly observed for the reference catalyst (Figure S10, SI).The CoO content of the spent catalyst seems to be reduced compared to the fresh one (Fig. S8), which points towards an increased stability towards oxidation. We do not detect carbide formation, which is also a classical deactivation pathway of cobalt catalysts in FTS. The presence of large quantities of hcp-Co less prone to form Co2C than fcc-Co [20] could be an explanation. The selectivity to undesired light products and in particular to methane can be ranked as follows: thermally treated 7%Co/8%Co/SiO2-Al2O3 > 7%Co/8%Co/SiO2-Al2O3 > reference catalyst 15%Co/SiO2-Al2O3. The selectivity to longer chains is decreased for the thermally treated sample. This tendency is not yet understood and further studies would be needed to rationalize it, however, it is possible that the presence of remaining ligands on the surface of the cobalt particles for the non-treated sample neutralizes unselective sites. It is worth mentioning that higher methane selectivity have already been observed, but not rationalized, for hcp-Co catalysts. [8e, 21] 10.1002/cctc.201701860

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Finally, it has to be noted that the anisotropically grown cobalt domains have not a constant diameter throughout their total length, but their diameter diminishes upon growing. This means that they do not adopt a perfect NR shape, therefore the exposed facets are not the (11-20) facets adopted by the regular NRs. [10] Furthermore, the anisotropic domains grow from Co seed-particle facets (the ones of the Co particles of the conventional catalyst) that are not homogeneous in size so different pods of the multipods are most likely enclosed by different facets. Therefore, any correlation of the catalytic performances of the modified catalysts to specific facets is not possible.

Conclusions

The present work is an example of the possibilities offered by colloidal chemistry methods for the synthesis of nano-objects in the domain of heterogeneous catalysis. We have shown that cobalt nano-objects of hcp crystal structure and anisotropic shape can be grown on the surface of Co particles of a conventional catalyst, from a solution containing a molecular Co precursor and organic stabilizers. While the initial activity of the modified catalyst is inferior to the one of the conventional catalystits stability is far superior. As the catalytic run progresses, the activity of the modified catalyst at least attains the one of the conventional catalyst. The urchin shape of the hcp-Co phase is not retained under the conditions employed for the FTS, however, its metallic state, and more precisely the hcp structure, remains intact. Furthermore, while the NR shape is not maintained in the spent catalyst, no sintering has been observed. The fact that the catalytically active phase remains well dispersed throughout the reaction, as well as the conservation of its metallic character are probably the reasons for the increased catalyst stability. Elimination of a great part of the native organic ligands increases catalyst activity, without compromising its stability. These results demonstrate that colloidal methods for the synthesis of metal nano-objects can be exploited, allowing adding and/or modifying specific structural characteristics of classical heterogeneous catalysts. Since the catalytic phase microstructure might determine catalyst reactivity, such modifications may allow improvement of several other catalytic systems.

Catalyst

Activity (mol CO converted/h/gCo) Selectivity (%) [a] Stability [b] 7%/8%Co-after heat treatment 0.24 0.9 14 13 68 6.15 [a] Expressed in weight percentage over the total product mass % w/w. [b] Expressed as the ratio of activity vs. time slope at 200h between reference and said catalyst.

Experimental Section

General methods

Co(NO3)2, 6 H2O was purchased from Alfa Aesar (purity 98%), silicaalumina from Sasol Gmbh. Hexadecylamine (HDA) was purchased from Aldrich (purity 98%), lauric acid (LA) from Acros (purity 99%), and the Co precursor [Co{N(SiMe3)2}2(thf)] [22] from NanoMeps. These products were introduced in a glove-box and used without any further purification. Toluene was purchased from Fischer Chemical (purity 99%), distilled and degassed by argon bubbling. It was then introduced in a glove box and kept under molecular sieves.

Catalyst preparation

Co/Al2O3-SiO2 catalyst

The conventional cobalt catalyst was prepared by incipient wetness impregnation of a Co(NO3)2, 6H2O aqueous solution on a powder silicaalumina Al2O3-SiO2 (Al2O3: 95%, SiO2: 5% -SBET = 171 m 2 .g -1pore volume = 0.52 cm 3 .g -1mean pore diameter = 9.2 nm, average particle size = 50µm). After the impregnation step, the solid was dried under air flow at 85°C during 5h and then calcined at 400°C during 4h. For loadings of 7% cobalt, only one step was necessary, whereas in the case of a catalyst containing 15% cobalt, the impregnation drying and calcination steps where carried out twice. The final oxide obtained was then reduced under pure hydrogen at 400°C during 16h and transferred in a solution of heptane without exposition to air. The catalyst obtained after activation will be called hereafter XCo/Al2O3-SiO2, X being the amount of cobalt in the solid.

7%Co/8%Co/SiO2-Al2O3 catalyst

For the growth of hcp Co on Co/Al2O3-SiO2 catalysts all experiments were performed under inert conditions, in the interior of a glove box or by using standard Schlenk techniques due to the air sensitivity of the metal precursors, as well as of the Co-particles. Typically, in a Fischer Porter bottle, a solution of [Co(N(SiMe3)2)(thf)] (646.2 mg, 1.4 mmol) in 2mL of toluene was added at room temperature to a solution of a mixture of LA (870.0 mg, 4.2 mmol) and HDA (700.2 mg, 2.8 mmol) in toluene (100 mL). Then 1.2 g of the 8%Co/SiO2-Al2O3 catalyst were added to the violet solution and let under stirring at room temperature for 10 minutes. The resulting mixture was heated at 110°C during 24 hours under 3 bars of hydrogen. At the end of the reaction, the mixture was transferred into the glove box. The black precipitate was separated from the supernatant colorless transparent solution, and washed three times with 200 mL of 10.1002/cctc.201701860

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toluene, and finally with 50 mL of pentane and dried (1,870 gr). For catalytic performance evaluation, the as-prepared catalyst was stored in n-C22 wax in order to protect it from reoxidation before testing.

The thermal treatment performed in order to reduce the organic content in the catalyst was realized under controlled atmosphere in a tubular oven.

The sample was heated under a flow of Ar/H2 (93%/7%) continuously pumped under vacuum to 250°C at a 20°C/min rate, then to 270°C at a 2°C /min, followed by a step of 20 minutes at 270°C, then the sample was left to cool down under gas flow, and transferred back into the glove box.

Catalyst characterization

Transmission Electron Microscopy (TEM) observations were carried out on a JEOL JEM 1400 and a JEOL JEM 2100F. The 70 nm ultra-microtomy slices were obtained by embedding the sample in an epoxy resin prior to the cutting 70 nm slices with a UCT ultramicrotome (Leica Microsystems).

The slices were deposited on TEM grids to allow the observations. FEG-SEM (Field Emission Gun-Scanning Electron Microscopy) observations were carried out on a JEOL 7800F. XRD analyses were performed on a Panalytical Empyrean (Co Kα = 1.789010 Å) diffractometer (Highscore software) and on a PANalytical X'Pert Pro (Cu Kα1 = 1.5406 Å) diffractometer (EVA software). ICDD database was used for the data interpretation. The ICP measurements were carried out at the analytical service of the LCC (Laboratoire de Chimie de Coordination) after digestion in nitric acid. The TGA analyses were performed on a TGA/DSC 1 Star System from Metler Toledo. The sample was heated at a 10°C/minute rate between 25°C and 500°C followed by a two hours step at 500°C.

Catalytic runs and product analysis

The Fischer-Tropsch reaction was carried out in a CSTR at 220°C, 20 bars, H2/CO = 2. A mass of 2 to 3g of catalyst was loaded in the reactor prefilled with n-C22 solvent in order to prevent re-oxidation of the catalyst. The conversion was maintained at 45% for all the catalysts by adjusting the inlet syngas flow in order to compare the selectivity at iso-conversion for all the solids.

The gas phase was analyzed by on-line gas chromatography. Analysis of permanent gases was performed using a Molecular Sieve column (5A) and a thermal conductivity detector. Hydrocarbons were separated in a PONA column and analyzed by a flame ionization detector. An external calibration of the chromatograph performed once a day. This gave access to CO conversion, activity (CO conversion x CO molar flow/Co mass), and selectivity in: CH4, CO2, light compounds (C2-C4), heavy compounds with more than 5 C atoms (C5+). A standard deviation for GC analysis of 3 % relative is considered. The total mass balance was calculated at each product collecting time by weighing liquid products (water and waxes) and measuring gaseous components, and kept between 97 % and 103%

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Figure 1 .

 1 Figure 1. Co nano-objects obtained by decomposition of [Co{N(SiMe3)2}2(thf)] on a 15% Co/Al2O3-SiO2 catalyst at different Co/HDA/LA ratios: a) Co/HDA/LA = 1/1.7/1.2; b) Co/HDA/LA = 1/2/2; c) Co/HDA/LA = 1/2/2.5; and d) Co/HDA/LA = 1/2/3. Scale bars: a) 100 nm, b-d) 200 nm.

  ) diffraction plans of hcp-Co respectively) are more intense in the modified catalyst. These conclusions are corroborated by TEM Selected Area Diffraction (SAED) observations, which show that the whole multipode is composed of a mixture of fcc-and hcp-Co, while the podes are composed of hcp Co (FigureS7in SI).

Figure 2 .

 2 Figure 2. Co nano-objects obtained by decomposition of [Co{N(SiMe3)2}2(thf)] on a 8% Co/Al2O3-SiO2 catalyst: a) TEM; b) STEM-HAADF image of a typical aggregate illustrating the shape and the localization of the anisotropic Co nano-objects on the support.; c) SEM overview of the modified catalyst and d) SEM micrograph of higher magnification (scale bars: a), b) = 200 nm, c) =100 m; d) = 100 nm).

Figure 3 .

 3 Figure 3. XRD diagrams of the SiO2-Al2O3 support (black), the 8%Co/SiO2-Al2O3 parent catalyst (red), and the 7%Co/8%Co/SiO2-Al2O3 catalyst (blue). The characteristic peaks of hcp-Co (green lines), fcc-Co (black lines) and Al2O3 (asterisks) are also marked.

Figure 4 .

 4 Figure 4. Catalytic activities versus TOS of the: 7%/8%Co catalyst (in blue), and the 7%/8%Co-catalyst after heat treatment (in green) and the 15%Co-SiO2-Al2O3 reference catalyst (in red) The catalytic evaluation marked #2 corresponds to a longer evaluation of the same reference catalyst 15%Co-SiO2-Al2O3 over a longer period, in order to demonstrate the reproducibility of the results.

  been grown on the cobalt particles of a conventional Fischer-Tropsch catalyst by a seededgrowth strategy, employed in the wet chemistry synthesis of nanoparticles. This modification stabilizes the catalyst J. Harmel, A. Berliet, K. Dembélé, C. Marcelot, A.-S. Gay, O. Ersen, S. Maury, A. Fécant, B. Chaudret, P. Serp,been grown on the cobalt particles of a conventional Fischer-Tropsch catalyst by a seeded-growth strategy, employed in the wet chemistry synthesis of nanoparticles. This modification stabilizes the catalyst. J. Harmel, A. Berliet, K. Dembélé, C. Marcelot, A.-S. Gay, O. Ersen, S. Maury, A. Fécant, B. Chaudret, P. Serp,*

Table 1 .

 1 Catalyst performances at 200 hours TOS.

Acknowledgements

J. Harmel thanks the Université de Toulouse (Nysfit project N°. 6013) and the Région Midi Pyrénées (Nysfit project N°. 13050269) for financial support. This study has been partially supported through the grant NEXT N°. ANR-10-LABX-0037 in the framework of the "Programme des Investissements d'Avenir".