F Brygo

A Semerok

J.-M Weulersse

P.-Y Thro

R Oltra

Laser ablation of a turbid medium: modelling and experimental results

Q-Switched Nd:YAG laser ablation of a turbid medium (paint) is studied. The optical properties (absorption coefficient, scattering coefficient and its anisotropy) of a paint are determined with a multiple scattering model (3-flux model), and from measurements of reflection -transmission of light through thin layers. The energy deposition profiles are calculated at wavelengths 532 nm and 1.064 µm. They are different from those described by a Lambert -Beer law. In particular, the energy deposition of the laser beam is not maximum on the surface but at some depth inside the medium.

The ablated rate was measured for the two wavelengths, and compared with the energy deposition profile predicted by the model. This allows to understand the evolution of the ablated depth with the wavelength: the more the scattering coefficient is higher, the more the ablated depth and the threshold fluence of ablation decrease.

Introduction

Several studies were devoted on the laser ablation of paint for decontamination processes in the nuclear industry [1 -7]. Nuclear dismantling faces the problem of paint removal on large surfaces of painted walls. The conventional methods for paint stripping on concrete walls are mainly based on mechanical grinder and lead to an important volume of aerosols and wastes. Laser ablation has been evaluated as a promising method for paint removal with a number of advantages: the method reduces considerably the waste volume as the removal of paint is selective, the ablated matter can be collected by aerosol aspiration/filtration, and the automation of the process can provide a higher personal safety.

Laser ablation is a complex phenomenon, involving the fast heating of the material and its ejection. The effects of the laser beam parameters on the ablation rate, as the fluence, the wavelength, the pulse duration and the repetition rate, have been studied and described in the literature [4 -8]. However, paints are complex materials, including polymer properties and oxides (pigments) properties, and the ablation process is not completely understood. This paper focuses on the first step of the laser-matter interaction, which is the energy deposition of the laser beam in the material. From the energy deposition in depth profile, the laser ablation properties, i.e. the threshold fluence, the ablated rate and the ablation efficiency, will be analysed.

Paint is a turbid medium. The multiple scattering of the light in the bulk leads to a particular energy deposition of the laser beam in the medium, which differs from the Lambert -Beer law. In particular, overheating inside the bulk were predicted [START_REF] Jacques | [END_REF] and measured [10]. The laser ablation efficiency and optical properties of a turbid medium were already compared [11] for sintered polytetrafluoroethylene. Other works are reported for the ablation of tissue [START_REF] Jacques | [END_REF][START_REF]Fundamentals and Applications[END_REF]. However, for paint, no study relates to the characterization of the energy deposition, compared to the ablation rate, despite that optical properties are characterized for studies in conservation [START_REF] Jones | [END_REF], on solar panels [14 -17], on the ageing of paint in space [18], or on the scattering of light by pigments [19,20].

Nd:YAG laser ablation of a grey epoxy paint, at wavelengths 532 nm and 1.064 µm, are presented in this paper. The absorption coefficient, scattering coefficient and its anisotropy are determined with a 3-flux model and from optical measurements of transmitted and diffuse flux. The measured parameters and the model allow calculating the energy deposition profile. The ablation rates are measured for the two wavelengths, and compared with these specific energy deposition profiles.

3-flux model

The 3-flux model [21,[START_REF]Wave propagation and scattering in random media[END_REF] is applied to describe the light propagation in a turbid medium and to determine the energy deposition profile. The fundamental principles of the model are presented below:

One considers the flux F 1 , representing the laser (collimated) beam travelling in the positive direction + z in a turbid medium. Within a differential distance dz, the collimated flux can be transferred in diffuse flux in the same direction (F 2 ) and in the opposite direction (F 3 ), and the diffuse flux F 2 and F 3 can be exchanged (cf. Figure 1). Five parameters are introduced: k, the absorption coefficient of the collimated flux; K, the absorption coefficient of the diffuse flux; S 1 , the scattering coefficient from the collimated flux to the diffuse flux in the direction + z; S 2 , the scattering coefficient from the collimated flux to the diffuse flux in the direction -z; and S, the scattering coefficient from a diffuse flux to the other.

The 3-flux equations are:

( ) ( ) ( ) 3 2 1 2 3 3 2 1 1 2 1 2 1 1 F S K SF F S dz dF SF F S K F S dz dF F S S k dz dF + - + = - + + - = + + - = (1)
These equations are one-dimensional, which imposes that the laser beam diameter is much larger than the thickness e of the medium, 1/k or 1/S, which represent characteristic absorption length and diffusion length respectively. Solutions to this set of equations are:

z 3 z 2 z 2 1 3 z 3 z 2 z 1 1 2 z 1 1 e ) 1 ( C e ) 1 ( C e A C F e ) 1 ( C e ) 1 ( C e A C F e C F σ σ - λ - σ σ - λ - λ - β + + β - + = β - + β + + = = (2) w i t h 2 1 S S k + + = λ ; ( ) S 2 K K + ± = σ ± ( ) 2 2 1 2 1 S S K SS A λ - σ λ + + + = ; ( ) 2 2 2 1 2 S S K SS A λ - σ λ - + + = ; S 2 K K + = β (3)
Coefficients C 1 , C 2 and C 3 are determined from the boundary conditions at z = e and z = 0. C 1 represents the laser flux penetrating into the medium. One imposes C 1 = 1. C 2 and C 3 are determined from the reflections at the interfaces. They are calculated as being:

) R R 1 )( R R 1 ( e ) R ) 1 ( 1 )( R ) 1 ( 1 ( ) R R 1 )( A A R ( e ) A R A )( R ) 1 ( 1 ( C 0 0 e e e 2
β + + - β β + + - β - - β + β + - β + β + β + + - β - + - - β + β + - = σ - σ - λ - ( ) ) R R 1 )( R R 1 ( ) R ) 1 ( 1 ( R ) 1 ( 1 e ) R ) 1 ( 1 )( R A A ( e ) A A R )( R R 1 ( C 0 0 e e 0 e
- β β + + - β - - β + β + - β + β + - β + β + - - - β + + - β = σ λ - σ (4)
The reflection coefficients for the diffuse flux at the interfaces z = 0 (R 0 ) and z = e (R e ) correspond to an average value of the reflection for angle of incidence between 0 and π / 2. The reflection of the collimated beam at the interface z = e is supposed negligible (< 5% in our case).

The measured reflection and transmission of the light through thin layers are distinguishable in 3 coefficients: the diffuse reflection R diffuse , the diffuse transmission T diffuse and the transmission of the collimated beam T collimated . The diffuse reflection represents only the bulk reflection. The surface reflections (e.g. the specular reflection) are not considered (one supposed with C 1 = 1 that the total laser beam energy penetrates in the medium). We have:

R diffuse = F 3 (0) -F 2 (0) (5)
T diffuse = F 2 (e) -F 3 (e) T collimated = F 1 (e)
The unknown parameters k, K, S, S 1 , S 2 can be resumed to 3 unknown parameters. The reference [21] showed that the absorption of the diffuse flux is close to twice the absorption of the collimated flux: K ≈ 2 k. This difference is related to a larger optical path of the diffuse flux compared to the collimated flux, according to the axis z. The coefficients S 1 and S 2 are in relation with the phase function p(cos θ) [START_REF]Wave propagation and scattering in random media[END_REF], which describes the angular properties of the scattering. The phase function is normalised as:

∫ - = θ θ 1 1 1 ) (cos d ) (cos p 2 1 (6)
with θ the angle between the direction of the incident beam and the scattered beam. By introducing the global scattering coefficient s, we have:

s = S 1 + S 2 . (7)
S 1 expresses the forward diffusion, i.e. the whole light scattered by a particle and propagating in the positive half space +z:

∫ θ θ = 1 0 1 ) (cos d ) (cos p 2 s S (8)
And, S 2 , for the negative half space:

∫ - θ θ = 0 1 2 ) (cos d ) (cos p 2 s S (9)
If we express the phase function as a series of Legendre polynomials P n (cos θ), as

∑ ∞ = θ = θ 0 n n n ) (cos P g ) (cos p ( 1 0 ) one gets: ...) 8 g 2 g 1 ( 2 s S 3 1 1 + - + = (11)
with g i calculated as:

∫ + - θ θ θ       + = 1 1 n n ) (cos d ) (cos P ) (cos p 2 1 n g (12)
Thereafter, one considers only the first order of the development in the equation ( 10), (i.e. parameter g 1 ), because the other terms are negligible compared to it, or are divided by a more important factor as seen in (11). The reference [21] showed that coefficients S and s are related as:

S = s (3 g 0 -g 1 ) / 4 (13)
Since the values of the three unknown parameters k, s, and g 1 , representing the absorbing coefficient, the scattering coefficient and its anisotropy, are determined, the energy deposition profile is calculated as:

3 2 1 KF KF kF ) z ( M + + = (14)
The energy deposition profile is related to the heat source term Q(x, y, z, t) usually introduced in the heat equation to calculate the temperature evolution of the material [START_REF]Laser Processing and Chemistry[END_REF], as Q(x, y, z, t) = (1-R) M(z) I 0 (x, y, t), with R the reflection at the surface. For non-scattering semi-infinite matter, the energy deposition profile is normalised as:

1 dz ) z ( M 0 = ∫ ∞ (15)
Relation ( 15) can't be used with the relation (14), because one part of the laser beam is reemitted from the surface as a diffuse reflection (coming from the backward flux F 3 ). For the relation (14), with a semi-infinite medium, one have:

diffuse 0 R 1 dz ) z ( M - = ∫ ∞ (16)
The parameters k, s and g 1 must be known to calculate the energy deposition profile. They can be determined from measurements of the diffuse reflection, diffuse transmission and collimated transmission of light through thin layers.

Measurement of the optical properties of the paint

Homogeneous thin layers of paint of a few tens of micrometers were realised to measure the diffuse reflection, diffuse transmission and collimated transmission of a laser beam. The thin layers were fabricated by "spin coating", on a transparent polycarbonate substrate (lexan). A characteristic profile of the thin layers is given on figure 2.

The diffuse reflection, collimated transmission and diffuse transmission are measured with an integrating sphere "Labsphere" and a sensor "Centronic TO5". The internal reflection coefficient of the integrating sphere was calibrated before the measurements. Two lasers are used for measurements: a He:Ne laser at λ = 543.5 nm, and a Nd:YAG laser at λ = 1.064 µm. Results are supposed to be the same at the wavelength λ = 532 nm and the wavelength λ = 543.5 nm. The 3-flux model parameters k, s and g 1 are fitted from the experimental results. At λ = 543.5 nm, the measurements and the theoretical transmissions and reflections as a function of the thickness of the paint are presented on figure 3, with the best fitted parameters s = 1.8×10 5 m -1 , k = 1.2×10 4 m -1 , g 1 = 1.45. In the same way, the transmissions and reflection as a function of the thickness of the paint are presented on figure 4 at λ = 1.064 µm, with the best fitted parameters s = 9×10 4 m -1 , k = 9×10 3 m -1 , g 1 = 1.8.

The scattering coefficient is twice more important at λ = 543.5 nm than at λ = 1.064 µm. This is due to the diameter of the scattering pigments (TiO 2 ). One can estimate the scattering coefficient of the medium by taking into account the mean diameter of the particles, measured to be close to 350 nm, and with the volume proportion ~ 0.8 %. The ordinary index of refraction of TiO 2 (n 0 ) is 2.66, and the extraordinary index of refraction (n e ) is 2.96 at 543 nm [START_REF]CRC materials science and engineering handbook[END_REF]. By assuming that TiO 2 is isotropic, its index of refraction is calculated as n TiO2 = (2n 0 + n e ) / 3 = 2.76 at 543 nm. In the same way, the index of refraction of TiO 2 is 2.57 at 1.064 µm. From these values, the scattering coefficient of the medium is calculated as the scattering coefficient of a single particle multiplied by its volume proportion in the material. The scattering coefficient of a single particle is calculated from the Mie theory [START_REF]Wave propagation and scattering in random media[END_REF], with the refraction index of the polymer measured as being close to 1.5 for both wavelengths. That gives s ≈ 1.6×10 5 m -1 for λ = 543.5 nm and s ≈ 7×10 4 m -1 for λ = 1.064 µm. These values should be understood as estimations, because of the complexity of the medium, but they are closed to the ones obtained from the measurements and the 3-flux model.

The absorption coefficient is nearly the same for the two wavelengths. It's due to the absorbing pigments (carbon particles) that are completely opaque to the laser radiation. Thus, no difference is expected between the two wavelengths, which is well verified from the measurements and the 3-flux theory.

Finally, figure 5 represents the energy deposition profile M(z) in a semi-infinite medium calculated from the 3-flux model and the previous optical parameters for the two wavelengths.

Laser ablation

The ablation of the paint was performed with Q-switched Nd:YAG lasers. The pulse duration is 80 ± 10 ns (FWHM), the repetition rate 20 Hz, and the wavelengths λ = 1.064 µm and λ = 532 nm. The spatial distribution of the beam intensity was homogenised by a multimode optical fibre. Details of the systems are presented in the reference [START_REF] Brygo | [END_REF]. The crater depths and spatial profiles are measured with a profilometer (MAHR) with a mechanical sensor (MFW-250). The depth resolution of the measurements is better than 1 µm. The crater spatial profiles are directly correlated with the beam profile (flat top, with the same diameter that the laser beam), as seen on figure 6. Thus, the presented crater depths are an average value of the depth measured in different zones of the craters.

The craters depth evolution with the fluence is presented on figure 7. The depth is proportional to the number of applied pulses in the whole studied range (3 -100 pulses). The threshold fluence is 1.7 J cm -2 at λ = 1.064 µm and 1.2 J cm -2 at λ = 532 nm. Over this fluence, the ablated depth increases until reaching a saturation. The maximum available depth is higher at λ = 1.064 µm (13 µm / pulse) than at λ = 532 nm (6.5 µm / pulse).

Discussion

The thermal conductivity of the paint was estimated as being D = 0.03 cm 2 s -1 [START_REF] Brygo | [END_REF]. The thermal diffusion length during the laser pulse Dt ≈ 0.5 µm is low compare to the optical penetration depth (~ 10 µm), which involves that the thermal diffusion during the pulse can be neglected. Thus, the ablation depth should be related to the profile of the energy deposition M(z) in the material. The ablation threshold fluence should depend on the laser energy deposited on the surface, and the ablated depth should depend on the optical penetration depth. Figure 5 shows that the maximum of the energy deposition is not located at the surface but inside the medium. By increasing the scattering coefficient from s = 0.9×10 5 m -1 (λ = 1.064 µm) to s = 1.8×10 5 m -1 (λ = 532 nm), the total quantity of energy absorbed in the medium ∫ ∞ 0 dz ) z ( M decreases, the penetration depth decreases, and the maximum of the energy deposition increases. This can be also seen on Figure 8 where the energy deposition in a medium with a constant absorption coefficient k = 1×10 4 m -1 is calculated for different scattering coefficient ranging from s = 1×10 4 m -1 to s = 1×10 6 m -1 (g 1 = 0). Table 1 compares the energy deposition in the paint with the maximum available ablation depth and with the threshold fluences, for the two wavelengths. The energy deposition profile is characterized by the penetration depth, defined as z M(0) / e , and by the position of the maximum of the energy deposition z Mmax (cf. notation on figure 5). z M(0) / e is the depth at which the energy deposition is equal to 1/e of the energy deposition at the surface. The value of the energy deposition is represented by the terms M(0) and M max . One notices that a proportional relation is found between these values and the ablation measurements: by decreasing the penetration depth of a factor ~ 1.9 between 1.064 µm and 532 nm, the ablation depth decreases by a factor 2. Moreover, by decreasing the value of the energy deposition at the surface (or close to the surface) by a factor ~ 0.6, the threshold fluence decreases of a factor ~ 0.7. The same proportionality is obtained by considering the position of the maximum of the energy deposition z Mmax and the value M max instead of z M(0) / e and M(0). The maximum available ablation depths are close to the depths of the maximum of the energy depositions z Mmax for both wavelengths.

Thus, one highlights that the maximum ablation depth is more important with a low scattering material, as well as the global absorbed energy. However, the maximum ablation efficiencies (i.e. the ablated volume per joule) are nearly the same for the two wavelengths. The ablation efficiencies are presented on figure 9. Because the craters have flat top profiles with the same diameter as the laser beam, the ablation efficiency (mm 3 J -1 ) is the depth per pulse, presented on figure 7, divided by the fluence. The maximum efficiency is close to η = 0.25 mm 3 J -1 , for the fluence F = 2.2 J cm -2 at 532 nm and η = 0.22 mm 3 J -1 for the fluence F = 3.5 -4.5 J cm -2 at 1.064 µm. These similar efficiencies can be well understood by the modelling of the laser beam energy deposition in the bulk: the increase of the scattering of the paint from the wavelength 1.064 µm to 532 nm decreases the ablation threshold at the same time that the ablated depth decreases. Of course, for industrial application of a decontamination process, the wavelength λ = 1.064 µm should be highlight, because the available power is higher than at λ = 532 nm, for similar cost.

Conclusion

Nd:YAG laser ablation of a turbid medium (paint) was studied at the wavelengths 1.064 µm and 532 nm. The paint scattering and absorption coefficients were determined with a 3-flux model and from measurements of optical transmission and reflection of light through thin layers. The energy deposition profiles were calculated with the obtained optical parameters and compared with the ablation depth and ablation threshold. Variation of the threshold fluence and maximum available ablation depth per pulse with the wavelength are well explained by the energy deposition profile. More particularly, one observes that a higher scattering coefficient decreases the total absorbed energy, decreases the penetration depth, and increases the maximum of the energy deposition. This leads to lower threshold fluence and lower ablated depth. Thus, for this paint, the ablation efficiencies are similar for the two wavelengths. Additional experiments and studies will be performed on the link between the absolute value of the ablation rate and the energy deposition profile. In particular, the influence of the overheating inside the bulk due to the multiple scattering will be studied in future work.

Table Table 1: Comparison between the energy deposition, the ablation depth and the threshold fluence for the two wavelengths.

Figures captionsFigure 1 :

 1 Figures captions

Figure 2 :

 2 Figure 2: Characteristic profile of a thin layer.

Figure 3 :

 3 Figure 3: Collimated transmission, diffuse transmission and diffuse reflection for various thicknesses at λ = 532 nm. Lines: 3-flux model with s = 1.8×10 5 m -1 , k = 1.2×10 4 m -1 , g 1 = 1.45. Symbols : Experimental results.

Figure 4 :

 4 Figure 4: Collimated transmission, diffuse transmission and diffuse reflection for various thicknesses at λ = 1.064 µm. Lines: 3-flux model with s = 9×10 4 m -1 , k = 9×10 3 m -1 , g 1 = 1.8. Symbols : Experimental results.

Figure 5 :

 5 Figure 5: Energy deposition profile in the paint at λ = 1.064 µm (s = 9×10 4 m -1 , k = 9×10 3 m -1 , g 1 = 1.8) and λ = 532 nm (s = 1.8×10 5 m -1 , k = 1.2×10 4 m -1 , g 1 = 1.45) from the 3-flux model.

Figure 6 :

 6 Figure 6: Typical crater profile, obtained at F = 4 J cm -2 , λ = 1.064 µm, 10 pulses.

Figure 7 :

 7 Figure 7: Ablated depth per pulse as a function of the fluence for λ = 1.064 µm and λ = 532 nm.

Figure 8 :

 8 Figure 8: Energy deposition profile from the 3-flux model with k = 1×10 4 m -1 , g 1 = 0, and for various s.

Figure 9 :

 9 Figure 9: Ablation efficiencies as a function of the fluence for the two wavelengths.

Figures Figure 1 Figure 6 :

 16 Figures

Figure 9

 9 Figure 9

Acknowledgment

This research is financially supported by AREVA-COGEMA. The authors would like to express their gratitude to Mr G. Decobert and Mr H. Masson for fruitful discussions.


Vincent Guillerm

Daniel Maspoch

Geometry mismatch and reticular chemistry: strategies to assemble metal-organic frameworks with non-default topologies

Keywords: metal-organic frameworks, reticular chemistry, rational design, geometry mismatch, self-assembly, porous materials

The past 20 years have witnessed tremendous advances in the field of porous materials, including the development of novel metal-organic frameworks (MOFs) that show great potential for practical applications aimed at addressing global environmental and industrial challenges. A critical tool enabling this progress has been reticular chemistry, through which researchers can design materials that exhibit highly regular (i.e. edge-transitive) topologies, based on the assembly of geometrically-matched building blocks into specific nets. However, innovation sometimes demands that researchers steer away from default topologies to instead pursue unusual geometries. In this review, we cover this aspect and introduce the concept of geometry mismatch, in which seemingly incompatible building blocks are combined to generate non-default structures. We describe diverse MOF assemblies built through geometry mismatch generated by use of ligand bend-angles, twisted functional groups, zigzag ligands and other elements, focusing on carboxylate-based MOFs combined with common inorganic clusters. We aim to provide a fresh perspective on rational design of MOFs and to help readers understand the countless options now available to achieve greater structural complexity in MOFs.

Introduction

Since the consecutive discovery of HKUST-1 1 and MOF-5 2 some 20 years ago, great efforts have been dedicated to understanding the basic principles that govern MOF assembly. 3 The final structure of a given MOF is now widely understood to be strongly influenced by the geometries of its constituent organic ligands and inorganic moieties (e.g. single metal ions, polynuclear clusters, infinite chains, etc.). Numerous design approaches such as use of molecular building blocks (MBBs) or secondary building units (SBUs) [4][5][6][7][8][9][10] have been combined with the mathematic discipline of topology [11][12][13] to synthesize myriad novel porous structures, 14 including MOFs that exhibit high performance in crucial environmental and industrial applications such as energy/gas storage separation, [15][16][17][18] waste/valuables removal, 19,20 water harvesting [21][22][23][24][25] and smart materials. [26][27][28] Concerted efforts by chemists and materials scientists have given rise to a new subfield of chemistry: reticular chemistry. The dedicated Reticular Chemistry Structure Resource (RCSR) database created in 2008 by O'Keeffe and coworkers now contains roughly 3,000 three-periodic nets (topologies). 29 One might imagine that there would be a limited number of ways to periodically assemble polygons and polyhedra and that all of these structures would have already been predicted. Surprisingly, this is not the case, even though there are already almost 100,000 MOF structures reported in the MOF subset of the Cambridge structural database. 30 Indeed, there are near-monthly reports of MOFs that exhibit topologies described as "novel", "previously unknown", "unprecedented" or "unique".

For instance, Jiang et al. discovered more than 120 novel topologies, 31 called merged nets. 9 Each of these has minimal transitivity, resulting from the merging of compatible nets. However, reticular chemistry is not limited to the discovery of novel 3D periodic nets: in fact, researchers have often discovered MOFs whose constituent clusters have never previously been reported as discrete molecules. Some of these clusters have high connectivity, which is a highly desired asset. [32][33][34][35] Indeed, the greater the connectivity within the building blocks, the lower number of possible structures attainable upon assembly and consequently, the lesser the chance of obtaining an undesired structure. For example, squares 36,[START_REF] Eddaoudi | Proc. Nat. Acad. Sci[END_REF] or tetrahedra can assemble into numerous structures (248 zeolitic nets known), 38 whereas triangles and rhombicuboctaedra (rco) can only assemble into one net the rht topology. [START_REF] Nouar | [END_REF] Accordingly, researchers have been seeking MBBs 10,32,34,[40][41][42][43][44][45][46] or supermolecular building blocks (SBBs) 4,32,[START_REF] Nouar | [END_REF]47,48 of high connectivity, with which to rationally construct MOFs. However, whereas most efforts are focused on joining together compatible building blocks to form MOFs of predicable structures and topologies, an opposite strategy would be to combine apparently incompatible building blocks into structures and topologies, which would provide an opportunity to learn about their behavior and access unprecedented materials. 32 Along these lines, scientists can employ less-symmetric organic ligands, with bend-angles, 34 introduce steric hindrance 49 or use zigzag-shaped ligands (transversal reticular chemistry). 50 These strategies all induce structural irregularity known as geometry mismatch.

Theoretically, geometry mismatch could be harnessed for any class of MOFs, such as azolated or phosphonated MOFs. However, in this review, we focus on carboxylate-based MOFs combined with common inorganic MBBs (i.e. those for which the synthetic conditions are well known). Given their abundance in the Cambridge Structural Database, 51 we consider the following clusters suitable for reliable analysis: M II (Cu, Zn, Co, etc.) paddle wheels, as in HKUST-1; 1 M III (Fe, Cr, Sc, Ga, Al, etc. ) trimers, as in MIL-100; 52 Zn tetramers, as in MOF-5; 2 and M III/IV (rare earths [RE], Zr and Hf) hexamers, as in UiO-66. 53 Thus, here we describe examples of MOFs and synthetic strategies that have pushed the envelope of reticular chemistry, challenged standard assembly rules and enabled high degrees of structural complexity, all thanks to the geometry mismatch generated through assembly of their (apparently incompatible) respective building blocks.

Topological tools in reticular chemistry: a precious but limited asset

Nets, nodes, augmented nets and vertex figures

In this section, we briefly highlight the role of topology in the standard design strategy for MOF synthesis, endeavoring to help novices understand the potential and limits of topology in structure design and prediction, through the relatively counterintuitive strategies that we describe in this review. In fact, reticular chemistry has always been associated with topology, and several articles from Delgado-Friedrich et al. have been a major source of inspiration for molecular architects. 11,12,[54][55][56][57] Targeting a specific MOF requires selection of at least one organic MBB and one inorganic MBB, each of which will have a specific connectivity (i.e. the number of other MBBs it will be linked to). Ideally, assembly of these building blocks would be limited to only one possible structure. The resultant three-periodic net should be as regular as possible, preferably edge-transitive (i.e. having only one type of edge bridging the nodes of the net), such that the risk of obtaining undesired structures is minimized. As an illustrative example, we consider the ideally 12-connected (12-c) Zr hexanuclear cluster and the 4-connected ligand, tetrakis(4-carboxyphenyl)porphyrin (tcpp). The expected topology would be an edge-transitive (4,12)-c net.

According to the RCSR database, three such nets exist: ith, ftw and shp (Figure 1a). To differentiate among these three nets and find the expected structure, one must inspect the corresponding augmented nets (net-a, Figure 1b), in which the vertices (nodes) are replaced with the corresponding vertex figures (polygons or polyhedra). Such inspection reveals that ith, ftw and shp differ from each other: thus, ith-a is the combination of icosahedra with tetrahedra; ftw-a comprises cuboctahedra linked by squares; and shp-a comprises hexagonal prisms and squares (Figure 1c). In parallel, the twelve extension points of the Zr cluster together form a cuboctahedron, whereas the four extension points of the tcpp ligand form a square. Therefore, formation of the nets ith or shp can be excluded.

Consequently, for a MOF with 12-c Zr cluster and a 4-c tcpp ligand, the expected topology is ftw (MOF-525, Figure 1d). 58 After illustrating (vide supra) the prime importance and power of topology as a design tool in reticular chemistry, we address here its relative weakness. Indeed, the simplification of a structure into its underlying net (topology) reduces MBBs as nodes, or for the case of ligands that are 2-connected, it is simply discarded and corresponds to the edges of the net. For instance, UiO-66 comprises 12-c Zr MBBs bridged by 2-c terephthalates; however, the corresponding fcu topology is uninodal (12-c net). Therefore, it is difficult to gain insight a priori on the possible effects of different ligand geometries (i.e. introducing a twist, an angle or an offset into a 2-c ligand) on the resulting structure or topology, apart from evidence that it might create geometry mismatch and thus, prevent formation of the default structure or topology (Figure 2). In the following sections, we aim to elucidate existing strategies and remaining challenges for the efficient design of MOFs with less regular, non-default topologies.

Angles, offset and twists in organic ligands: limitations of topology in MOF design

Linear, twisted ligands with steric hindrance: a cornerstone of MOF design

In this section, the default topologies are the edge-transitive sql net (4-c), in the case of 4-c paddle wheels, and fcu net (12-c), in the case of Zr/Hf/RE hexanuclear clusters ("UiO type cluster"). Both types of MOFs assemble by bridging linear ligands with coplanar carboxylate groups. Breaking this coplanarity by introducing steric hindrance creates geometry mismatch (Figure 2b). Until 2002, the majority of reported structures resulted from either isoreticular expansion of known MOFs, to create isoreticular MOFs (IRMOFs), 59 or from exploratory trial-and-error work. 1,[60][61][62][63][64] Then, the report of MOF-101, a Cu dicarboxylate, appeared as a cornerstone of rational MOF design in reticular chemistry (Figure 3). 49 To modify the orientation of the Cu paddle wheels, the coplanarity between the two carboxylic groups in the linear terephthalate (bdc) was broken, by employing a sterically hindered ligand (Br-bdc). This altered the directionality of the connectivity points by 90º, enabling formation of a 3-periodic structure with the nbo topology (Figure 3b), rather than the sql topology of MOF-2 (Figure 3a). 63 However, the twist angle does not need to be as large as 90º to affect the structure and topology of a MOF. For example, smaller twists in the ligands (28 o and 47 o ) will alter the directionality of the paddle wheels to a lesser extent than in MOF-101. Using this approach, Furukawa et al. formed MOF-604, which exhibits the cds topology. 36 This strategy has also been applied with longer ligands such as 2,2′dimethyl biphenyl-4,4′-dicarboxylate (Me 2 bpdc) 36 as well as with zirconium, enabling formation of Zr-based MOFs with the bcu topology, [65][66][67] rather than the fcu topology, which is commonly obtained when linear ligands are used. 40,53,[68][69][70][71] Lü et al. described a surprising example along these lines, having assembled the complex, chiral N,N′-di-(4-benzate)-1,2,6,7-tetrachloroperylene-3,4,9,10-tetracarboxylic acid diimide (pdi) with zirconium. 72 The dihedral angle between the two carboxylates (38°), and the unusual length and slight flexibility of the ligand, together generated the dia topology, rather than the fcu or bcu topologies. In this structure, the Zr clusters of Zr-PDI are capped by eight ligands, such that each cluster is connected to four others (i.e. two ligands each bridge two similar clusters). As a concluding example, we consider the zinc tetramer used to produce IRMOFs. 59 Interestingly, unlike in the case of M II paddle wheels (Figure 4a) and M III/IV hexanuclear clusters (Figure 4c), for IRMOFs, introducing a 90º twist into the organic ligands does not have any impact on the resulting connectivity directions of the cluster (Figure 4b). This is due to the octahedral connectivity of the zinc tetramer. To the best of our knowledge, this observation has not been reported elsewhere.

Bent ligands: the door to polymorphism

In this section, the default topologies are the edge-transitive sql net (4-c), in the case of 4-c paddle wheels, and fcu net (12-c), in the case of Zr/Hf hexanuclear clusters ("UiO type cluster"). Both types of MOFs assemble by bridging linear ligands with coplanar carboxylate groups. Breaking this linearity by introducing a bending angle (Figure 2c), through use of a bent ligand, creates geometry mismatch; consequently, the ligands that surround the same MBB end up having many distinct orientations.

Combining paddle wheels with bent ligands of various angles leads mainly to metal-organic polyhedra (MOPs) 73,74 and to 2D MOFs (sql and kgm). [START_REF] Eddaoudi | Proc. Nat. Acad. Sci[END_REF] For MOPs generated with square building units (paddle wheels or single metal ion), the influence of the bend-angle in the ligand is clear, enabling isolation of MOPs of various geometries and distinct organic/inorganic content (i.e. metal/ligand [M/L] content), which has already been well documented elsewhere. 4,[75][76][77][78][79][80] To illustrate the influence of bent ligands on MOF assembly, we have chosen hexanuclear Zr/Hf clusters as the representative inorganic MBB (Figure 5), owing to their usually high and adaptable connectivity (from 4 to 12). Introducing an angle into a dicarboxylic ligand (i.e. changing the directionality of its connectivity) can, depending on the bending angle, prevent the formation of the ubiquitous fcu-type framework (Figure 2c). Interestingly, at an angle of 160º, such as that in the ligand 9-fluoreneone-2,7dicarboxylate (fldc), found in DUT-122, the topology remains fcu (Figure 5a). However, to offset the effects of the bend-angle, the clusters exhibit an overall inclination of ca. 14º from their ideal orientation. 81 At smaller angles, such as ca. 150º, as in the case of the ligands dithienothiophene dicarboxylate (dttdc) and 4,4'-(2 H-1,2,4-triazole-3,5-diyl) dibenzoate (tadiba), the angle has a large influence on the outcome, leading to the reo-MOFs DUT-51 82 and JLU-MOF58, 83 respectively. Interestingly, using a shorter ligand of comparable angle, 2,5-thiophenedicarboxylate (tdc), leads to four distinct structures: three 8-c nets (the reo-MOF DUT-67 84 [Figure 5b] and its two polymorphs, the bon-MOF DUT-68 5c] 84 whose rare topology is also found in MOF-802, which is based on the ligand 3,5-pyrazoledicarboxylate (pzdc). 25 These examples highlight the great potential of bent ligands to promote structural diversity, by conferring the bent ligands around the Zr clusters with different orientations without necessarily altering the connectivity of the binding groups (Figure 5). An intriguing case is CAU-28 (Figure 5f), 86 which comprises two crystallographically distinct 2,5-furane dicarboxylate (fdc) ligands, one of which exhibits a 124º angle, four of them surrounding the Zr/Ce clusters and bridging them into a Kagomé lattice. The other ligands (four per cluster), which have a markedly smaller angle (ca. 103º), act as "pillars" between the layers along the z axis, to form an overall kag topology.

[Figure
No section on bend-angles would be complete without mentioning a bent ligand with a 120º angle. Surprisingly, and to the best of our knowledge, no one has yet reported assembly of a MOF from Zr clusters and the ubiquitous 120 o bent isophthalate ligand (m-bdc). However, Xie et al. did recently report combination of Zr clusters with either 4,4'-(benzene-1,3-diyl)dibenzoate (bdb) or 4,4'-(naphthalene-2,7-diyl)dibenzoate (ndb) to form the pcu-MOFs BUT-66 (Figure 5g) or BUT-67, respectively. 87 Although their clusters are 12-connected, these MOFs exhibit the same topology as in MOF-5: each Zr cluster is bridged to six others by pairs of ligands. Similarly to Xie et al., Wei et al. subsequently reported that combination of Gd hexanuclear clusters and tetrafluoroisophthalate (mbdc-F 4 ) generates a pcu-MOF. 88 Finally, Krause et al. evaluated an even smaller angle, of 90º, in their assembly of a MOF from Zr clusters and the ligand 9h-carbazole-3,6-dicarboxylate (cdc) into the 1D coordination polymer DUT-80 (Figure 5h). In this structure, the 8-c Zr clusters are quadruple-bridged to each other by cdc ligands to form "chains" of clusters, which can be connected together to yield the flexible MOF DUT-98 (vide infra). 89 g) pcu (BUT-66) topologies, as well as to 1D coordination polymers (DUT-80). Some organic rings are filled in yellow to better distinguish the ligand orientations around the clusters.

Zigzag ligands: transversal reticular chemistry

In this section, the default topology is the edge-transitive fcu net (12-c) with Zr/Hf hexanuclear clusters ("UiO type cluster"). Such MOFs assemble by bridging linear ligands with coplanar carboxylate groups. Breaking this linearity by introducing a transversal offset creates geometry mismatch (Figure 2d).

In addition to linear, twisted or bent ligands, another type of ditopic bridge, zigzag ligands (Figure 2d), is currently gaining attention in MOF assembly. 90 Examples of MOFs built with such ligands, in which the collinearity between chelating groups is broken, have been reported. [91][92][93] However, before the advent of transversal reticular chemistry, no clear focus had been made on the potential of their unique shape (Figure 6a). 50 By adding the transversal parameter of width (w) to the height (h) of the ligands, Guillerm et al. demonstrated that, along with making ligands taller or shorter, they could also stretch them transversally, thus breaking the collinearity of the binding carboxylates. In the case of Zr-based MOFs, this effect is reflected by generation of geometry mismatch (Figure 6b), as zigzag ligands do not match the perfect alignment of Zr clusters. Thus, assembling Zr clusters with zigzag ligands leads to bcu-MOFs, which can be regarded as fcu-MOFs with systematic, ordered defectsnamely, four missing ligands on each MBB (Figure 6c). 50 Guillerm et al. reported Zr-bcu-MOFs (Figure 6e-h) with trans, trans muconate (tmuc), 2,6-naphthalene dicarboxylate (26ndc), 2,2'-bipyridine-4 4'-dicarboxylate (22bipy44dc) or azobenzene-3 3'-dicarboxylate (azo33). 50 Additionally, Maurin, Serre and co-workers used simulations to predict that Zr clusters and zigzag succinate (suc) ligand would combine to form another MOF, Zr-bcu-suc (MIP-204, Figure 6d), whose assembly they subsequently confirmed experimentally. 94 Shortly after, by introducing steric hindrance by the mean of a sulfonic group on a 26ndc ligand, Nguyen et al.

showed the possibility to not only deviate from the 12-c fcu net to the 8-c bcu net, but also to assemble its polymorphic analog, the 8-c reo net. 95 The case of ligands with small widths, such as 26ndc and azobenzene-4 4'-dicarboxylate (azo44), is worth mentioning. Some of these act as linear ligands, as their disorder in the framework compensates for their small width values without affecting the cluster orientation or reducing the overall symmetry (Figure 7). 84,96 Multi-functionality has recently become a highly-desired property for MOFs, as it facilitates their selective postsynthetic modification. 97,98 Many multivariate MOFs have been reported, 9, 99-105 most of which contain randomly distributed ligands having similar shapes but carrying diverse functional groups. 103 Strategies to achieve ordered multi-functionality include ligand insertion/pore partition, 65,66,106 use of programmed pores 107 and assembly of merged nets 9 . Transversal reticular chemistry is taking crystal engineering to the next level, by enabling selective postsynthetic placement of ligands (i.e. functional groups) to create materials that mimic natural structures such as proteins and nucleic acids. For instance, Kim et al. reported ready functionalization of the MOF Zr-bcu-26ndc (Figure 8a) with tagged terephthalates to produce ordered, multivariate fcu-MOFs. 108 To complete the coordination of the Zr clusters in Zr-bcu-26ndc assembled through transversal reticular chemistry (vide supra), they filled the resultant unoccupied positions with their desired ligand (Figure 8b). Consequently, the channels in Zr-bcu-26ndc are transformed into the tetrahedral cages in Zr-fcu-26ndc/bdc.

Introducing a zigzag ligand into a MOF can influence the MOF's geometry without impacting its overall topology (Figure 9). For example, Chevreau et al. described the case of MIL-142B (Figure 9a), an Fe-nht-MOF constructed with 4,4'-biphenyldicarboxylate (44bpdc) and 4,4',4'',-benzene-1,3,5-triyl-trisbenzoate (btb), 99 whose framework is interpenetrated, yet whose transversal reticular analog, MOF-909 (Figure 9b), which contains azo33 instead of 44bpdc, comprises a single nht net. This is due to the distortion of the framework, preventing intergrowth of a second nht net. 109 Interestingly, Nguyen et al. reported that introduction of 50% of azo33 is sufficient to prevent its interpenetration (MOF-908).

Polytopic ligands with various angles and/or twists

Triangular ligands and cuboctahedral building blocks: geometry mismatch par excellence

In this section, there is no edge-transitive net to assemble regular triangles (equilateral, coplanar carboxylates) with 12-c Zr/Hf/RE hexanuclear clusters ("UiO type cluster"). Therefore, all existing structures arise from geometry mismatch, which explains the high variety of topologies (Figure 10). of unveiling novel clusters and topologies, which had yet to be discovered in rare-earth and MOF chemistries. 32 After determining the conditions to form the desired rare-earth hexanuclear cluster as a discrete entity, they used similar conditions, in combination with btb. Their system was highly versatile, as additional metal ions and carboxylates caused the hexanuclear cluster to spontaneously evolve into an 18-c, nonanuclear cluster, which was compatible with 3-c btb and gave rise to an unprecedented, minimal transitive, (3,18)-c net with gea topology (Figure 10a). Further studies on this net revealed its suitability for rational design of MOFs using supermolecular building blocks (SBBs), 4 which we discuss later in this review. 32 Subsequently, Eddaoudi and co-workers also studied the effects of varying the angles of 3-c ligands. 34 Using the 90° angle provided by the ligand 9-(4carboxyphenyl)-9H-carbazole-3,6-dicarboxylate (bcdc), they were able to unveil yet another nonanuclear cluster, a 12-c one that differs slightly from that observed in gea-MOF-1 (Figure 10a) and which led to formation of a MOF with a novel (3,12,12)-c net, aea topology (Figure 10b). In parallel, they demonstrated that reducing two branches of the btb ligands (e.g. by using the [1,1'-biphenyl]-3,4',5-tricarboxylate ligand [bptc]) generated another previously unknown topology, pek, and that in the resulting pek-MOFs (Figure 10c), a 12-c nonanuclear cluster coexisted with the classical hexanuclear cluster (albeit, with connectivity reduced to eight). Finally, by using the extended ligand 5-(4-carboxybenzyloxy)isophthalate (obi), they prepared the isoreticular analog pek-MOF-2. Inspired by these reports, Wang et al. recently discovered three novel MOFs in the RE/3-c ligand system: PCN-912 (Figure 10d), PCN-918 (Figure 10e) and PCN-909 (Figure 10f). They obtained these MOFs by reducing one branch of the btb ligand and introducing various levels of steric hindrance (using functionalized [1,1':3',1''-terphenyl]-4,4'',5'tricarboxylate ligands [R-tptc]). 43 They unveiled two novel highly-connected nets: the ( 6.1.2. The zirconium example. Under many synthetic conditions, zirconium can produce a similar hexanuclear cluster to that of rare earths (vide supra). 111,112 Since the first report of UiO-66 in 2008, 53 and given the success achieved by forcing competition between monotopic acids (as modulators) and polytopic acids in their crystallization, 40,70,71,113 Zr-MOFs have become some of the best-studied MOFs. 33,[114][115][116] These findings paved the way towards construction of MOFs with topologies other than fcu. For instance, in 2012, Morris et al. described assembly of Zr-MOFs with square ligands. They obtained the expected MOF-525 and MOF-535, which exhibit the edge-transitive ftw topology, and surprisingly, also obtained MOF-545, which shows another edge-transitive net, csq, whose hexanuclear cluster exhibits a connectivity reduced to eight (the first-ever reported example of this in a MOF). 58 Regarding equilateral triangular ligands with coplanar carboxylate groups, since they cannot be assembled with 12connected cuboctahedral shapes, the clusters instead reduce their connectivity to eight, to fit to edge-transitive nets having lower connectivity. This adaptability of the Zr clusters explains the generation of the two spn-MOFs MOF-808 25 (ligand: trimesate [btc]) and PCN-777 (ligand: 4,4′,4″-s-Triazine-2,4,6-triyl-tribenzoate [tatb], Figure 10g), 117 whose Zr clusters are six-connected with octahedral directionality. Similarly, Wang et al. reported that in presence of btb, Zr clusters also exhibit six-connectivity, albeit in hexagonal directionality, leading to formation of a kgd-MOF based on edge-transitive kgd layers that interpenetrate to form an overall 3D framework (Figure 10h). In this MOF, the carboxylates are slightly twisted (38º and 46º) relative to the ligand plane. 118 An example of a MOF in which the carboxylates show a greater twist angle (90º) was given by Wang and co-workers, who reported that Zr clusters reacted with 4,4',4''-(2,4,6-trimethyl-benzene-1,3,5-triyl)tribenzoate (Me 3 btb) adopt a connectivity of eight, with cubic directionality, to yield BUT-12 (Figure 10i), a MOF with the edge-transitive the topology. 119 Likewise, the use of other ligands sharing the geometry of Me (3,9)-c topology (sep) within the Zr/3-c ligand system, as illustrated in their discovery of BUT-39 (Figure 10l). 110 As we mentioned above, DUT-80 is a unidimensional structure constructed from 8-c Zr clusters and cdc, a ligand with a bend-angle of 90º. Krause et al. discovered that using this 1D motif with such an angle implemented in the bcdc ligand leads to an unprecedented (3,12)-c net: they obtained the llj-MOF DUT-98 (Figure 10j), which exhibits high structural flexibility. 89 Another possibility to obtain a novel (3,12)-c net was reported by Lee et al. in their construction of MOF-1004, which exhibits the sky topology, in which the geometry of the 12-c node is cuboctahedral. 120 To generate this net, they used the wide and slightly flexible ligand 4,4',4''-[benzene-1,3,5-triyltris(ethyne-2,1-diyl)]tribenzoate (bte), in which the angles between two branches is only 94º (compared to 120º in the ideal geometry of the ligand).

Interestingly, Sun and co-workers followed this prerequisite in the design of their ligand 4,4',4''-(1H-imidazole-2,4,5-triyl)tribenzoate (ittc), which they subsequently used to assemble UPC-158 (Figure 10k), an isoreticular analog with the sky topology. 121 Their work demonstrates that even such unexpected topologies can be rationally targeted, provided that the geometrical requirements are known.

Triangular ligands with square paddle wheel building blocks

In this section, the default topology to assemble regular triangles (equilateral, coplanar carboxylates) with paddle wheels (4-c) is the (3,4)-c edge-transitive tbo net. Modifying the coplanarity of the carboxylates, the angles or using ligands with unequal edge-lengths creates geometry mismatch (Figure 11).

Although the structural and topological variety of MOFs based on paddle wheels and triangular ligands is not as rich as that for hexanuclear clusters and these ligands (vide supra), we do consider some examples worthy of discussing here. Combination of a regular planar ligand such as trimesate (btc) with paddle wheels leads to HKUST-1, which exhibits the tbo topology; 1 however, obtaining extended reticular analogs is not trivial. Interestingly, using 4,4',4''-benzene-1,3,5-triyl-tris(benzoate) (btb) instead of btc leads to two MOFs that exhibit the pto topology, another edge-transitive net: MOF-14 64 (interwoven) and MOF-143 122 (single net, Figure 11b). This is due to the natural twist of the carboxylates in btb. Alternatively, using a fully planar and regular ligand such as 4,4',4"-s-triazine-2,4,6-triyl-tribenzoate (tatb) or 4,4',4''-(benzene-1,3,5-triyl-tris)benzene-4,1-diyltrisbenzoate (bbc), affords the expected tbo-MOFs PCN-6 (Figure 11a) 123 or MOF-399, 122 respectively.

Another approach to access unusual topologies from paddle wheels and triangular ligands is simply to elongate one or more ligand branches. For instance, Wong-Foy et al. employed biphenyl-3,4,5-tricarboxylate (bptc) to elongate one branch of btc, enabling them to assemble UMCM-150 (Figure 11c), a MOF resulting from the pillaring of Kagomé layers (kgl) by a rare Cu trimer to generate the agw topology. The layers are constructed from paddle wheels bridged by the isophthalic moieties of the bptc ligands, whereas the remaining benzoic moieties alternately point up or down from the layers, such that another paddle wheel cannot be incorporated, thus "forcing" formation of a 6-c Cu trimer. 124 Lim et al. reported that this Cu trimer could be substituted with other metal trimers (Co, Fe, Ni), 125 and Lu et al. found that the ligand could be further extended to obtain greater porosity. 126 Similarly, Schnobrich et al. used terphenyl tricarboxylate (tptc) to elongate two branches of the btc ligand, ultimately obtaining a UMCM-151 structure that contains paddle wheels only, albeit in an unpredictable, complex, pentanodal (3,3,4,4,4)-c net, fmj topology (Figure 11d). 127 In addition to varying the carboxylate orientation and/or elongating the ligand branches, another approach to obtain dramatically different topologies is to introduce a 90º angle between two branches of the ligand by using carbazole groups, such as those in 4,4′-(9-(4′-carboxy-[1,1′-biphenyl]-4-yl)-9H-carbazole-3,6-diyl)dibenzoate (bpcdc). For example, Stoek and co-workers rationally designed DUT-75 (Figure 11e) and DUT-76 (gee topology) by using the SBB approach to adopt overall ftw-type structures. 128 Exploiting the possibility to form carbazole-based MOPs with cuboctahedral shapes, 129 they simply linked the 12-extremities together through a paddle wheel.

Triangular ligands with 6-c trigonal prismatic building blocks

In this section, there is no edge-transitive net with which to assemble regular triangles (equilateral, coplanar carboxylates) with 6-c M III (Fe, Al, Cr, Ga, In, etc.) trimers. Given the high incidence of the moo net, which derives from the zeolitic mtn net, we can consider it to be the default net for these assemblies. Modifying the coplanarity of the carboxylates, the angles using ligands with unequal edge-lengths leads to geometry mismatch (Figure 12).

Although assembly of regular triangular ligands with 6-c trigonal prismatic MBBs cannot generate an edgetransitive net, various compatible nets with only two types of edges have been reported, including the ceq, dag, hwx, sit and ydq topologies. Surprisingly, most MOFs in this system adopt the extremely complex, decanodal moo topology of MIL-100. 52 This assembly had first been predicted by simulations 130,131 and can be explained by formation of super-tetrahedra that favor further self-assembly of MOFs into an overall zeolitic (mtn type) topology, as observed in the examples of MIL-100, 52 MIL-100-btb, 132 PCN-332, 133 PCN-333 133 and PCN-888 (Figure 12a). 134 Intriguingly, and similarly to other systems that we discuss in this review, not all assemblies of regular triangular ligands with 6-c trigonal prisms lead to the moo topology. For example, when Ibarra and co-workers assembled btb, which can yield moo MOFs, 132 with scandium trimers, they instead obtained the MOF Sc-btb (Figure 12b), which exhibits a different topology, (3,6)-c (not described in the RCSR database). 135 This is not surprising, given that the carboxylates from btb are naturally out of plane, as has been observed in other systems. 32,64,118,136 Additional examples of MOFs constructed from btb (PCN-260, Figure 12c), or amino-(PCN-261, Figure 12d) or hydroxyl-(PCN-261) functionalized btb, also derive from the moo net. In these MOFs, geometry mismatch again resides in the natural twist (3,3,3,6,6)-c net, xxx topology; for PCN-260 or forced twist (3,3,6)-c net, yyy topology; for PCN-261 and PCN-262 of the carboxylates. 137 Although it is difficult to know if it is a parameter governing the topology, or resulting from it, the angles formed by the ligands around the clusters vary significantly from one structure to another, and are far from the ca. 115° angle required to form the supertetrahedra necessary for the assembly of mtn/moo type MOFs (Figure 12). Another example of geometry mismatch in these systems is to replace btb with a ligand that has one shorter branch (e.g. tbab), such that upon assembly of the ligand with the metal trimer, the shorter branch length precludes formation of super-tetrahedra. In this sense, the assembly of tbab with Fe, Ga, In or Al trimers to yield Fe-MOF (Fe-fmz-tbab, Figure 12e) and SNNU-5, which exhibit a (3,6)-c net, fmz topology. 138, 139

Triangular ligands with 6-c octahedral building blocks

In this section, pyr is the edge-transitive net to assemble regular triangles with 6-c Zn tetramers ("MOF-5 type" cluster). They require equilateral ligands with highly twisted carboxylate groups of the ligand (i.e. relatively flexible branches). Non-equilateral ligands, ligands with lower coplanarity and ligands with less-twisted carboxylates all generate geometry mismatch (Figure 13).

There are several examples of regular triangular ligands being assembled with 6-c octahedral Zn tetramers to form pyr-MOFs, including MOF-150, constructed with 4,4',4"-tricarboxylate triphenylamine (tca) (Figure 13a); 140 MOF-155-J, built with 1-(3-amino-4-carboxyphenyl)-3-(4-carboxyphenyl)-5-(4-carboxynaphthalen-1-yl)-benzene (btb-mNH 2 ); 141 and MOF-950, made from benzene-1,3,5-tri-β-acryate (btac). 142 However, MOF-177 (Figure 13b), 136 built from btb, and its many (> 20) functionalized 141 or extended analogs, 143 are all based on the "queen of MOFs", the qom net. In these cases, preferential generation of qom over pyr is due to the weaker twisting of the carboxylates in the ligands used in the qom MOFs. However, contrary to a common assumption 136and as we explained in the previous paragraphthe carboxylates in the constituent btb ligands of MOF-177 are not coplanar.

Another MOF based on different carboxylate twists induced by the use of btb ligand with steric hindrance is MOF-156-J, which exhibits the rtl topology (Figure 13c). 141 In addition to modifying the level of twisting in the ligand carboxylates, other methods to confer systems with geometry mismatch might include altering either the ligand length 144,145 or the angle between the ligand branches 146 (i.e. using non-equilateral ligands). The fact that no structure built this way has yet been reported could be explained by the idea that these approaches would likely prevent formation of the Zn tetramer; however, their possible assembly cannot be ruled out.

Square/rectangle ligands with 4-c paddle wheels building blocks

In this section, the default topology to assemble square/rectangle ligands with 4-c paddle wheels, is the (3,3,4) -c fof net, derived from the edge-transitive nbo net (4-c). fof MOFs assemble by linking rectangle ligands with two coplanar dicarboxylate groups with a 120° angle (i.e. isophthalate). Breaking this coplanarity or modifying the dicarboxylate angle creates geometry mismatch (Figure 14). Note: this section does not cover examples with ligands based on "naturally tetrahedral" ligands (i.e. containing sp 3 carbon, adamantane central core, etc.).

Numerous MOFs constructed from di-isophthalic-based ligands and paddle wheels have been reported, including MOF-505 (Figure 14a), 147 and their default nbo/fof topology is widely described in the literature. 4,148,149 Their structures can be considered as ligand-to-ligand (L-L) pillaring of supermolecular building layers (SBLs). 4 The default SBLs are kgl, which are staggered in the fof net. Interesting non-default examples include PCN-12 (Figure 14b) 150 and ZJU- 25,151 in which bending of the central core of their respective ligands prevents staggered packing and instead, generates eclipsed pillaring to yield an ssa/sty topology. Another noteworthy example is NOTT-109 (Figure 14c), 152 in which a bulky naphthalene core precludes formation of the main cage characteristic of nbo/fof MOFs and instead, leads to formation of sql SBLs, which are pillared in an eclipsed fashion to form the ssb/stx topology. Researchers have built other pillared sql MOFs by using highly flexible ligands. 4,153 Expectedly, such MOFs may be subject to polymorphism, depending on the ligand conformation. 154,155 Thus, planar (rectangular) ligands lead to an overall lvt/lil topology, as in DUT-10, 153 whereas tetrahedral analogs of the same ligands (e.g. obtained by twisting of the dicarboxylate moieties) lead to the pts/tfk topology, as in Pillaring of SBLs is not the only way to achieve non-default topologies in MOFs assembled from di-isophthalicbased ligands and paddle wheels. As we have discussed above for other MOF families, ligand bend-angles can be chosen to dictate topology. For example, researchers have reported that in SBB assembly with paddle wheels, substituting isophthalate (120º bend) with carbazole dicarboxylate (90º bend) yields PCN-82 (Figure 14d) 156 and DUT-49 with tfb topology. 48,157 In these structures, the dicarboxylate moieties combine with the paddle wheels to form cuboctahedral cages that are bridged by the central core of the ligands to generate the underlying fcu topology.

Square/rectangle ligands with Zr hexanuclear cluster building blocks

In this section, the default topology to assemble square/rectangle ligands with Zr/Hf/RE hexanuclear clusters ("UiO type cluster") is the edge-transitive (4,12)-c ftw. MOFs assemble by linking square ligands with coplanar carboxylate groups. Breaking this coplanarity or deviating from the ideal square shape of the ligand creates geometry mismatch (Figure 15). Note: this section does not cover examples with ligands based on "naturally tetrahedral" ligands (i.e. containing sp 3 carbon, adamantane central core, etc.).

Beyond the expected ftw topology, various other edge-transitive nets have been obtained upon assembly of 4-c ligands with Zr/Hf/RE hexanuclear clusters, including csq, 58 she, 158 scu, 159 shp, 160 flu, 25 ith, 25 sqc, 161 stp 162 and lvt. 163 This is not surprising, given the capacity of such clusters to adapt to lower connectivities (vide supra). Many of these topologies resulted when researchers changed the synthetic conditions or the amount of modulator previously used for ftw MOFs. Although Chen et al. described most of these in their recent review of Zr-MOFs based on edge-transitive nets, 115 we have chosen to highlight a few examples below (Figure 15). Intriguingly, the ftw and scu nets are related. The main cage in ftw-MOFs can be represented as a cube, in which the clusters lying on the vertices are connected through the 4-c ligands that occupy the faces. Accordingly, elongation of the ligand in a single direction (i.e. by using a longer ligand) creates geometry mismatch, as only four of the six faces in the initial cage can accommodate a ligand, leaving the other two free. This scenario favors the scu/tty topology, in which 1D channels exist and the connectivity of the MBBs is reduced to eight (Figure 15). Moreover, if the length of the longer ligand is moderate compared to its width, as in the case of biphenyl-3,3′,5,5′tetracarboxylate, bptet, 163 then an ftw related MOFs with kle topology can still be formed (Figure 15b), as first reported for RE clusters by Luebke et al. 164 Along these lines, Wang et al., in their work on replacing the ligand bptet with the longer ligand 3,3′,5,5′-azobenzene-tetracarboxylate (azotet), obtained a rare, scu-derived net that is not reported in the RCSR database, in which the ligand orientation alternates and the clusters are tilted from their ideal alignment to balance the ligand length/width ratio (Figure 15c). 163 Interestingly, when they evaluated an even longer ligand, [1,1′:4′,1″]terphenyl-3,3″,5,5″-tetracarboxylate (tptet), they obtained the first-ever Zr based lvt/lim-MOF, in which the connectivity of the Zr cluster is reduced to 4 (Figure 15d). In most cases, if a ligand's shape deviates significantly from the ideal square shape required for forming an ftw-MOF, then the resulting topology will be derived from csq (xly or xlz) or from scu (tty or cut). In some cases, the final geometry of the ligand, and ultimately, the resulting topology, can be controlled by introducing steric hindrance or by using ligands with a less rigid core. Indeed, Pang et al. reported selective control among flu/flt (PCN-605, Figure 15g), scu/tty (PCN-606, Figure 15e) and csq/xly (PCN-608, Figure 15f) MOFs, via selective functionalization of a set of 4-c ligands based on 3,3',5,5'-tetra(ethyl-4-carboxyphenyl)-1,1'biphenyl (tpcb). 165 Similarly, Lyu et al., were able to switch from the scu net in CAU-24 166 to either the shp net in NU-904 167 or the csq net in NU-1008, 167 by simply introducing one or two bulky functional groups, respectively, onto the 4,4',4'',4'''benzene-1,2,4,5-tetrayl-tetrabenzoate (tcpb) ligand. These groups directly influence the twisting of the carboxylates and therefore, dictate the resulting topology.

Hexacarboxylate ligands with 4-c square paddle wheel building blocks: not another rht-MOF

In this section, the default topology is ntt, derived from the edge-transitive (3,24)-c net rht. Dicarboxylate bent (120 o ) extremities of trefoil ligands construct externally functionalized MOPs (24-c), that act as SBBs, and are linked together by the central core of the ligand (3-c). Breaking the planarity of the ligand, or preventing the formation of a 24-c MOP by modifying the angles between carboxylates, each introduces geometry mismatch that precludes formation of rht/ntt-MOFs (Figure 16).

In 2008, Nouar et al. described use of externally-functionalized MOPs as SBBs, which they assembled with copper trimers to form rht-MOF-1. [START_REF] Nouar | [END_REF] Interestingly, this trimer can be substituted with 3-connected organic cores 168, 169 to afford overall planar hexacarboxylate ligands, which have been used to assemble various rht-MOFs (also known as ntt-MOFs, Figure 16a) [START_REF] Nouar | [END_REF]47,168,170,171 . In parallel, Guo et al. reported UTSA-20 (Figure 16b), which they built with a similar type of ligand, 3,3',3'',5,5',5''-benzene-1,3,5-triyl-hexabenzoate (bhb), showing that the geometry of this ligand was too intricate and resulted in broken planarity. 172 UTSA-20 exhibits a zyg topology and represents one of the rare MOFs based on paddle wheels and hexacarboxylate ligands that does not exhibit the ubiquitous rht/ntt topology. Another example of a paddle wheel hexacarboxylate MOF that does not exhibit the rht topology is gea-MOF-2 (formally: gwe topology), a representative case to the scope of this review, as the gea topology was discovered by employing a geometry mismatch strategy. 32 Indeed, gea-MOF-2 (Figure 16c) was rationally designed using the SBB approach, to specifically adopt an overall gea topology. Guillerm et al. identified a suitable MOP with 18 vertices 173 matching the geometry of a triangular orthobicupola (eto/ebc), corresponding to the points of extension of a newly discovered RE nonanuclear cluster with 18 points of extension in gea-MOF-1. By incorporating this geometric information to build the eto-MOP in a trefoil hexacarboxylic ligand (i.e. two 120º angles for one 90º angle), they were able to use the ligand 5′,5⁗-((5-((4-(3,6-dicarboxylato-9H-carbazol-9-yl)phenyl)ethynyl)-1,3phenylene)bis(ethyne-2,1-diyl))bis((1,1′:3′,1″-terphenyl]-4,4″-dicarboxylate (lgea2) as a net-coded building unit 4 to replicate the underlying gea net, enabling its use as a blueprint for rational assembly of gea-MOF-2. In this sense, the 90º angle provided geometry mismatch to preclude formation of an rht-MOF (for which the 120º angles are crucial).

Conclusions

We have provided an overview of geometry mismatch-based approaches to assemble MOFs that exhibit nondefault topologies, which in some cases, have enabled discovery of novel clusters that remain to be studied. These approaches obviate classical MOF assembly strategies, in which pre-designed building blocks are used to form specific, edge-transitive or highly regular topologies. Instead, they rely on methods such as transversal reticular chemistry; use of zigzag ligands; introduction of twisting into the ligand carboxylate groups; alteration of ligand bending angles; and changing of the length/width ratio in branched ligands. Thus, by combining classical topology with these techniques to create non-default geometry, researchers are gradually elucidating the pre-requisites for designing MOFs with complex topologies. As researchers have come to understand the basic rules for assembly of some of the topologies previously discovered serendipitously (e.g. gea, sky, agw, and pek), they have been able to exploit these topologies to further rationally design isoreticular materials. In several cases, the MBBs are left unsaturated, such that the aforementioned approaches to non-default topologies can be harnessed to construct multivariate MOFs with precisely-positioned functional groups, with the aim of creating functional sequences that mimicking natural structures such as enzymes or DNA. Finally, given the potential of geometry mismatch to generate countless new MOF topologies, we consider that human efforts to predict MOF topology are about to reach their limit. Accordingly, we believe that if the MOF research community truly wants to attain the next level of structural complexity in rational design, we must now turn to computational approaches to complement our human efforts. Clearly, such computational tools should be openly available and accessible to theoretical and experimental scientists in materials science, chemistry and related fields.

Figure 1 a

 1 Figure 1 a) The three edge-transitive (4,12)-c nets ith, ftw and shp. b) Augmented representation of the three edge-

Figure 2 .

 2 Figure 2. Schematic representation of a) the fcu-a net and how b) twisted, c) bent or d) zigzag ligands would create

Figure 3 .

 3 Figure 3. Assembly of bdc or Br-bdc with paddle wheels leads to a) 2-periodic (MOF-2, sql) or b) 3-periodic

Figure 4 .

 4 Figure 4. Comparison of the ideal orientation and directionality of MBBs with coplanar or 90º-twisted

  and the hbr-MOF DUT-126 [Figure 5e] 85 ), and one 10-c net (the bct-MOF DUT-69 [Figure 5d] 84 ),

Figure 5 .

 5 Figure 5. Varying the angle, and/or the orientation around Zr based clusters, of bent ligands leads to MOFs with a)

Figure 6 .

 6 Figure 6. a) Comparison of the characteristic distances and angle in bdc (linear) and tmuc (zigzag) ligands. b)

Figure 7 .

 7 Figure 7. Detail of the 26ndc ligand disordered in Zr-fcu-26ndc (DUT-52-Zr) and ordered Zr-bcu-26ndc

Figure 8 .

 8 Figure 8. Transversal reticular chemistry enables precise insertion of additional ligands to transform a) Zr-bcu-

Figure 9 .

 9 Figure 9. a) In nht-MOF MIL-142B, substituting linear 44bpdc with b) zigzag azo33 in MOF-909 prevents

6. 1 . 1 .

 11 The rare earths example. In 2014, upon noticing the apparent incompatibility of 3-c ligands with 12-c (cuboctahedra) clusters (which lacks a corresponding edge-transitive net), Guillerm et al. envisioned the possibility

  3,3,12)-c net, flg (PCN-912); the (3,3,18)-c net, ytw (PCN-918); and also isolated a sep-MOF (PCN-909), with the same (3,9)-c net as in the previously reported Zr based MOF, BUT-39 (vide infra). 110

Figure 10 .

 10 Figure 10. a-f) Summary of the RE-based MOFs and g-l) Zr-based MOFs resulting from assembly of RE/Zr with

  3 btb, 6,6′,6″-(2,4,6-trimethylbenzene-1,3,5triyl)tris(2-naphthoate)) (tnna) or 4,4',4''-[benzene-1,3,5-triyl-tris(benzene-4,1-diyl)]tribenzoate (bbc), leads to two other the-MOFs: BUT-13 119 and MOF-1005, respectively. 120 Despite the adaptability of the connectivity of the Zr clusters, which helps to form MOFs of highly regular structure and edge-transitive nets, He et al. found that use of the highly original T-shaped ligand 4,4′,4″-(1H-benzo[d]imidazole-2,4,7-triyl)tribenzoate (btba) led to a novel,

Figure 11 .

 11 Figure 11. Summary of the Cu-based MOFs resulting from assembly of paddle wheels with triangular ligands. The

Figure 12 .

 12 Figure 12. a) Equilateral triangle ligands with coplanar carboxylates assemble with trimers to form supertetrahedra,

Figure 13 .

 13 Figure 13. Combining zinc tetramers with triangular ligands that differ by carboxylate-twisting can yield to a) pyr

Figure 14 .

 14 Figure 14. Assembly of paddle wheels with square/rectangular ligands mainly affords a) pillared kgl layers in

Figure 15 .

 15 Figure 15. Although assembly of square building blocks with the ideally 12-c connected Zr hexanuclear clusters

Figure 16 .

 16 Figure 16. In addition to a) the many rht/ntt-based MOFs (NU-108) based on planar hexacarboxylate ligands, one

AUTHOR INFORMATION

Corresponding Author * vincent.guillerm@icn2.cat, daniel.maspoch@icn2.cat

Author Contributions

All authors have given approval to the final version of the manuscript

Notes

In various cases, the topology used to design specific MOFs, or the reported topology, do not match those obtained by following the recommendations of O'Keeffe and coworkers, on using derived nets. For such cases, we have denoted both topologies in the text (e.g. rht/ntt, ftw/kle, nbo/fof, etc.).

In the case several crystallographically independent ligands are present in a structure, ranges of bend angle or average twist/torsion angle are reported in this review.

ACKNOWLEDGMENT

Dr. Volodymyr Bon and Dr. Feng Liang are acknowledged for their help regarding the signification of some topology acronyms. Borja Ortin-Rubio is acknowledged for his helpful comments on a preliminary version of the review.

ABBREVIATIONS

General acronyms (order of appearance):


Sylvain Lefebvre
email: sylvain.lefebvre@inria.fr

EFFICIENT DIRECT SLICING OF DILATED AND ERODED 3D MODELS FOR ADDITIVE MANUFACTURING TECHNICAL REPORT

In the context of additive manufacturing we present a novel technique for direct slicing of a dilated or eroded volume, where the input volume boundary is a triangle mesh. Rather than computing a 3D model of the boundary of the dilated or eroded volume, our technique directly produces its slices. This leads to a computationally and memory efficient algorithm, which is embarrassingly parallel. Contours can be extracted under an arbitrary chord error, non-uniform dilation or erosion are also possible. Finally, the scheme is simple and robust to implement.

Introduction

Most additive manufacturing (AM) process planning tasks start with the slicing operation [START_REF] Attene | Design, Representations, and Processing for Additive Manufacturing[END_REF], where a model is decomposed into a set of planar slices. Each layer is then transformed into a set of instructions describing how the AM machine will solidify an uncured material (resin, powder, filament) into a solid slab reproducing the slice geometry.

Slicing is typically performed onto a tessellated version of the geometry, often provided as a triangle mesh M under the STL file format. This mesh is expected to properly define the boundary of an inner volume V : it has to unambiguously separate the inner volume from the outside. This is typically defined in the winding number sense [START_REF] Jacobson | Robust inside-outside segmentation using generalized winding numbers[END_REF], using facet orientations to define outside-to-inside and inside-to-outside interfaces. By convention, we consider that faces with counter clock wise (CCW) orientation define an interface from outside to inside (entering), while faces with clock wise orientation define an interface from inside to outside (exiting). This definition is convenient as it naturally captures holes (cavities), and is robust to self-intersections.

Slicing computes the intersection of a set of slicing planes with the geometry, forming a set of slice contours: general 2D polygons representing the intersection area of each slice. Note that other definitions of slicing are possible, where the slice contour is not the result of an intersection [START_REF] Alexa | Optimal discrete slicing[END_REF]. However, in this work we follow the typical point of view of intersecting the model with a plane at each slice height.

When slicing an oriented 3D triangle mesh, the contour orientation is trivially carried from the triangles to the polygon outline segments. Thus, the obtained polygons also obey the winding number rule. Their orientations hence define solid and empty areas (holes). These contours can for instance directly be fed to the OpenGL tessellator to obtain a valid area triangulation through the winding rules, with the added benefit of numerical robustness [START_REF] Mcmains | Geometric algorithms and data representation for solid freeform fabrication[END_REF]. The contours of the triangulated areas form the final, cleaned, slice contours [START_REF] Mcmains | Geometric algorithms and data representation for solid freeform fabrication[END_REF].

While most slicers input triangular meshes, tessellation can be avoided by defining direct slicing procedures [START_REF] Jamieson | Direct slicing of CAD models for rapid prototyping[END_REF], that manipulate the input geometry under its original representation. For instance, slicing a sphere equation leads to disks equations, slicing a cylinder (away from its extremities) leads to ellipses [START_REF] Rosen | Design for additive manufacturing: A method to explore unexplored regions of the design space[END_REF]. In such cases, the produced contours remain under an analytical representation (equations) and thus suffer no loss of precision.

While slicing is a relatively simple process to implement, the process planner often needs to perform more advanced manipulations on the initial 3D geometry. One operation in particular is highly desirable for process planning: the ability to manipulate eroded and dilated versions of the input volumes. Dilated volumes V ↑ r are defined by the Minkowski sum of a ball B r of radius r with the input volume V . While the eroded volume V ↓ r is obtained through the dilation of the complement of V . In the remainder of this document, for the sake of clarity we focus explanations on obtaining the dilated volume V ↑ r . Naturally, this problem has attracted a lot of attention in the research community, both in a general setting and in the specific context of AM. Most techniques attempt to reconstruct a mesh M ↑ r that captures the boundary of V ↑ r . A variety of approaches have been proposed, typically using discrete representations (ray-rep [START_REF] Chiu | Using dexels to make hollow models for rapid prototyping[END_REF][START_REF] Charlie | GPU-based offset surface computation using point samples[END_REF], distance fields [START_REF] Frisken | Adaptively sampled distance fields: A general representation of shape for computer graphics[END_REF], zonotopes [START_REF] Martínez | Chained segment offsetting for ray-based solid representations[END_REF]) or directly manipulating triangle meshes [START_REF] Forsyth | Shelling and offsetting bodies[END_REF][START_REF] Campen | Polygonal boundary evaluation of Minkowski sums and swept volumes[END_REF]. Most often these approaches do not compute the offset with a perfect ball, but rather with a discretized version of B r (tessellation, voxelization). Please refer to Attene et al. [START_REF] Attene | Design, Representations, and Processing for Additive Manufacturing[END_REF] for an overview in the context of AM. Once a mesh is obtained for V ↑ r , it can be sliced alongside the input model. Interestingly, the 3D model of V ↑ r is rarely used directly, only its slices are necessary to define regions of interest within the input model slices. Park [START_REF] Park | Hollowing objects with uniform wall thickness[END_REF] proposed an approximate method based on this observation, where the slices of a dilated model are obtained from the slice contours of the input model. Each original contour is dilated by a ball (or equivalently, a circle is swept along the curve), which results in a 3D torus-like 'inflated' curve. The union of these 3D shapes with the original volume give an approximation of V ↑ r . Rather than performing a 3D union, the union can be performed in the slices. Each inflated curve, when intersected by a slice below or above, produces two contours. The contours may be obtained by dilating the center curve by a radius which depends on the vertical distance between the slice and the curve. This implies that the slices of the approximation of V ↑ r are easily computed using only 2D contouring operations. The main drawback of the technique is that it may miss features which are smaller than the layer spacing used for the dilation. When dilating (or eroding) with a relatively large r, these features may have a significant impact on the result. Another difficult case occurs along near-flat surfaces (tops/bottoms) where the curves may become spaced by more that r leading to artifacts.

In this paper we propose an approach that also operates directly in the slices. We define a direct slicing procedure for the exact offset volume V ↑ r : there is no approximation compared with slicing the boundary mesh M ↑ r . On the contrary our technique affords for a direct control of the chord error with respect to B r , and an exact representation may be extracted. Our approach results in a highly efficient, robust and scalable parallel algorithm.

Method

We seek to slice the dilated volume V ↑ r = V ⊕ B r , where B r is a ball of radius r. V ↓ r is easily obtained by a similar process, which we detail later for the sake of clarity.

We now consider generating a single slice of V ↑ r . The slice j is a plane P j in space. Let us denote by t i the i-th triangle of M. A key observation is that a dilated triangle t i ⊕ B r can be decomposed into three cylinders, three spheres, and a center prism (see Figure 1).

We slice each of these primitive with P j ; that is, for each triangle t i we directly slice the three cylinders, spheres and the prism1 with P j . We thus obtain a set of CCW contours. Direct slicing of simple primitives can be done with high-performance specialized codes: slicing of a sphere produces a circle, direct slicing of a capped cylinder produces ellipses or clipped ellipses (including the rectangle case), direct slicing of a prism produces a triangle or a quad. These contours may be obtained as tessellated contours under a given chord error, or may be retained as exact contour primitives (circular and elliptic arcs, segments). The triangles may be processed in parallel, and only those triangles potentially intersecting the slice P j are considered.

Finding the set of slices intersecting t i ⊕ B r : Given a triangle t i , the set of slices potentially intersecting t i ⊕ B r is easily computed. Assuming the slices are along the z axis, let us denote the z interval covered by the triangle as [z min i , z max i ] (computed from the triangle vertices z coordinates min/max). The interval covered by

t i ⊕ B r is [z min i -r, z max i + r].
Given a slicing height sequence h 0 , ...h n , each specifying the z location of a slicing plane P j , we efficiently retrieve the indices of the first and last slice intersecting t i ⊕ B r . In case of a uniform slicing thickness τ this is a direct computation with a rounding, e.g. j f irst = z min i -r τ . In case of adaptive slicing we perform a logarithmic bisection search in the sequence of slice heights, locating the first slice just above z min i -r (and similarly for the last slice).

When processing in parallel, we launch K threads that visit all triangles simultaneously. Each thread takes a next triangle, determines which slices it impacts, generates the dilated triangle contours for each slice, and stores them in the slice data structure. Once all triangles have been visited by all threads, the final contouring is performed in all slices, again in parallel. As storing the contours may lead to a high memory requirement, this process may be performed for a subset of slices only. For instance, in a first pass, the contours are produced only for the N first slices and discarded for others. Once done with the N first slices, we proceed to the next N slices and start over. This allows to fit computations within a constrained memory budget, with minimal computational redundancy (visiting the triangles being very fast, the cost is in generating and storing the contours).

Producing the final contours of V ↑ r : The CCW contours obtained from the triangles are simply added to the list of contours obtained by slicing M with P j . The winding number contouring directly yields a slice of V ↑ r . Similarly, a slice of V ↓ r is obtained by reversing the contours from the dilated triangles to be CW, before extracting the final contour with the winding number rule. This follows from the definition of the erosion as "the complement of the dilation of the complement" (V ↓ r = M ⊕ B r ): the complement first reverses the contour of M (solid areas become holes, while holes become solid). We then add the CCW contours obtained from the triangles to dilate, and then reverse every contour again. Thus, in the end we only have to revert the contours from the dilated triangles.

Note that on AM processes that take images as input (e.g. DLP SLA), the final contour extraction may be skipped. Rasterizing all polygons using the winding rule in the stencil buffer will yield the final, correct image.

Contour extraction: By default we produce tessellated contours for the spheres and cylinders, enforcing a target chord error. Given these, any polygon contouring algorithm with winding number support may be used (e.g. Vatti's algorithm [START_REF] Bala | A generic solution to polygon clipping[END_REF] or the marching intersections [START_REF] Rocchini | Marching intersections: An efficient resampling algorithm for surface management[END_REF] among many choices in the literature). These can be further sped up, noting that the additional contours produced by the spheres/cylinders/prisms are all convex. This allows to skip some computations, since the number of intersections of a sweep line with convex primitives is always exactly two (excluding exactly aligned vertices).

Exact contours may be obtained by computing proper intersections between circular and elliptic arcs and polygons. Quite remarkably this would produce the contours of the dilation of V by an exact ball B r . We are not aware of any other technique able to achieve this.

When computing contours from dilated triangles in parallel, the contours are sorted by triangle id before contouring, ensuring a deterministic result.

Properties and results

Figure 2 shows slices from a complex 3D model with various amounts of dilation. The model has half a million triangles with intricate geometry, and would be typically very challenging for mesh-based algorithms. We slice the model at 40µm, resulting in 2379 slices (before dilation).

The complexity of computing the contour depends essentially on the number of triangles possibly intersecting a slice. This is a function of triangles per slice (tessellation factor of the input) as well as r (the greater r the more slices are impacted by each dilated triangle). Figure 3 reveals the performance behavior as a function of the radius r and the number of threads. Measurements are conducted on an Intel i7-4790K CPU, four physical cores, with 32GB of memory. As can be seen performance increases linearly with the number of threads -the algorithm is embarrassingly paralleluntil the number of physical cores (four) is exceeded. After this point, smaller improvements are still observed (these occur thanks to hyper-threading and improved scheduling/latency hiding when having many threads). The relationship between r and performance is more subtle. As r increases additional triangles impact each slice, thus increasing the cost. In the case of the Lucy cat model the increase is near linear (this is the case with most smooth models), while it is quadratic with the Bunny model. This depends greatly on the input geometry complexity. An advantage of our technique is that slices may be computed independently, in any order, or in parallel, every time yielding the same result with consistent performance. In particular, consider the case of previewing a single slice of a dilated version of V . With a standard technique, the entire boundary mesh of V ↑ r has to be computed, before a single slice is extracted. With our technique, the computations are performed only for the slice, and most triangles are completely ignored.

The limiting factor in terms of performance is the contouring algorithm, as the number of contours produced from the dilated triangles can become quite large per-slice. We found it best to perform contouring progressively (everytime k contours are stored a new contour is extracted with the winding rules, which accumulates all small contours in a partial result) or equivalently having a divide-and-conquer strategy at the end, recursively performing contouring on half-subsets of all the contours (the subset not being only two contours, but typically a larger number, determined experimentally). Accumulating all the contours before computing the final one with the winding rule usually leads to significantly slower results.

Finally, it is worth noting that our algorithm trivially generalizes to varying dilation/erosion, where the ball radius r varies in the vertices of the input mesh (and is linearly interpolated along edges and triangles). The only change is that dilated triangles produce (truncated) cones instead of cylinders, and the center prism faces become sloped. The radius may even become negative, effectively resulting in a mixed dilation-erosion operator.

However, this algorithm cannot be used to perform opening or closing morphological operations, since the starting point has to be a triangle mesh, and we cannot extract the triangle mesh of the intermediate step.

Figure 1 :

 1 Figure 1: Left: A dilated triangle t i ⊕ B r and its decomposition. Middle: center prism and spheres. Right: cylinders.

Figure 2 :Figure 3 :

 23 Figure 2: Slices of the Voronoi Bunny model for different dilation radii r: original (black), 1 mm (green), 2 mm (orange), 4 mm (blue), 8 mm (purple). The rightmost slice is located through the Bunny's head ; we can see the back of the bunny appearing with large dilation on the right side. Model: Voronoi Stanford Easter Bunny by Virtox https://www.thingiverse.com/thing:303842

as spheres and cylinders are shared by neighboring triangles we can avoid redundant computations