Tiziana Stomeo
email: tiziana.stomeo@unile.it

T F Krauss

Frederick Van Laere

Dries Van Thourhout

R Baets

Cyril Cambournac

Melanie Ayre

Henri Benisty

Successful fabrication and integration of multifunctional photonic-crystal devices on bonded InP membrane chip

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Nowadays, low-cost and small-footprint integrated optical devices for metropolitan optical network are actively demanded. One of the requirements for photonic integrated circuits is the need for system-oriented multi-functionality, i.e. the ability to perform different operations desired by system engineers, under the same materials technology simply by adjusting the geometry of the structure itself and by joining several devices together to form a complete integrated sub-system. This multi-functionality -a core concept of the EU project FUNFOX (IST-04582)-is addressed here by implementing two kinds of photonic-crystal (PhC) structures in bonded InP membranes.

The fabrication of these PhCs is typically carried out using electron-beam lithography and dry etching, and the precise fabrication of the 2D structure is always a critical issue for achievement of low-loss and high-quality components in photonic integrated circuits. Well-controlled pattern features are necessary in order to get a PhC device exhibiting the desired functions. There are various kinds of fabrication errors that might cause deviation from perfect photonic-crystal structures, such as the deviation of the periodicity due to the hole misplacement, hole size non-uniformity due to errors in the e-beam writing, proximity-effect-induced non-uniformity in the e-beam lithography, or systematic hole size change due to imperfect sidewall features formed during the dry etching process.

The variation of hole size due to fabrication errors could have a significant impact on the photonic bandgap. In this work, special attention is given to the electron-beam lithography (EBL) process for making the PhC patterns with well-controlled size, in order to integrate PhC DeMUX devices with 1D grating coupler without spectral mismatch. We also report on a highly controlled EBL approach to align different structures, which allows us to successfully fabricate efficient polarization diversity couplers integrated with photodetectors on bonded InP-membrane.

1D Grating Coupler integrated with 2D PhC DeMUX

The integrated device we implemented consists of 1D grating-coupled [1][2] PhC-based demultiplexer on bonded InP membrane, for application in coarse-WDM systems operating at 1500-1560 nm wavelength window. The DeMUX action is based on the anticrossing phenomenon, so-called mini-stopband (MSB) [3][4], supported by PhC multimode waveguides.

The basic heterostructure consists of a 300 nm-thick InP membrane sandwiched between air and benzocyclobutene (BCB, a low-index polymer, n = 1.54 at 1.55 µm) on a host substrate. The 1D coupler and the seven-channel DeMUX consisting of a W 5+x 2D PhC waveguide (|x |< 0.26) were written in a single step using a hybrid LEO Gemini 1530/RAITH ELPHY e-beam lithography tool operating at 30 kV. The ZEP520-A resist thickness of 400 nm was chosen to ensure sufficient durability as a mask for the RIE transfer of the pattern into an underlying SiO 2 layer and at the same time, to ensure good resolution of e-beam writing. The key issue in the fabrication process is to achieve PhC devices with well-controlled patterns size. A proximity error correction (PEC) had to be applied to reach this target. Moreover, in order to achieve smoother and circular holes and faster exposure, the EBL system was used in "circular mode". In this mode, every circular hole is exposed by the deflection of the beam along concentric circles.

High-quality grating coupler (660 nm period and 50 % duty cycle) and PhC DeMUX (362 nm hole diameter and 540 nm period) with well-controlled sizes were achieved after the resist development. The other demux-device's fabrication steps are described in [START_REF] Stomeo | Integration of Grating Couplers with Compact PhC Demux on InP Membrane[END_REF]. In Fig. 1 are shown SEM pictures of the devices prior to bonding. The demultiplexing operation of the PhC device is demonstrated by the experimental data of Fig. 2. Note the precise 10 nm spacing, and the non-next-channel cross-talk of ~ -10 dB.

Polarization diversity grating-coupled 2D PhC DeMUX integrated with photodetectors

Thanks to the progresses in surface couplers on InP membrane [START_REF] Van Laere | [END_REF][7][START_REF] Van Laere | Multifunctional Photonic Crystal Compact Demux-Detector on InP[END_REF], we could fabricate a complete integrated device consisting of (i) a polarization diversity surface coupler, (ii) a MSB-based DeMUX , and (iii) integrated photodiodes.

The basic system is always a BCB-bonded InP membrane. Also in this case a single ebeam process was performed, but it required a highly controlled approach to align the 2D polarization diversity coupler integrated with 2D PhC DeMUX to the photodetectors. The process is based on a two-step procedure: (i) First detector mesas and alignment marks have been defined using optical lithography and etching until the InP-membrane layer is reached; (ii) In a following step, 2D coupler, waveguides, and 2D PhC DeMUX have been aligned to the detector mesas using a controlled EBL process carried out by means of a hybrid LEO Gemini 1530/RAITH ELPHY e-beam lithography system. The approach is reproducible and widely applicable allowing tight alignment accuracy.

A complete layout of the device after e-beam alignment process and mask etch is shown in Figure 3. It involves a 2D polarisation diversity coupler on the input side, tapering down to single mode waveguides and bends, entering into the 2D PhC demultiplexer section. The subsequent detection of each wavelength is achieved by an integrated detector. The overall footprint is small but can be significantly shrunk to 100×150 µm with curved grating couplers and much shorter tapers. Figure 4 shows the spectral response (photo-current, linear scale) of the three rightmost channels and the through detector (Channel 4 was damaged). These results highlight that very high standards of fabrication accuracy need to be met in order to integrate high-quality multifunctional PhC devices.

Figure 1

 1 Figure 1: (a) Complete device layout: 1D Grating Coupler integrated with PhC DeMUX. (b) Input 1D grating coupler (shallow etch). (c) Entrance of the PhC waveguide. (d) Channel1, at DeMUX entrance. (e) PhC holes (deep etch).

Figure 2 :

 2 Figure 2: Experimental data from seven-channel demux with cleaved edge.A good separation of 7 optical channels with a spectral resolution of 10 nm nearly suitable for coarse WDM at 20 nm spacing is achieved.

Figure 3 :

 3 Figure 3: Layout of the polarization diversity PhC DeMUX with integrated photodetectors. The two polarization diversity paths being deviated by the pair of demux 1&2, and recombined in the array of photodiodes, including the 'through' waveguide.

Figure 4 :

 4 Figure 4: Spectral response (photocurrent, linear scale) of the three rightmost channels and the through detector. Channel 4 was damaged.

Acknowledgements

Authors acknowledge the FP6 European Project IST-04582 "FUNFOX" (2004-2007).


Hugo Reynes
email: hugo.reynes@supergrid-institute.com

Eric Vagnon

Cyril Buttay

Martin Guillet

Enhancing voltage rating of standard power modules for harsh environment applications

Recent advances in wide band-gap semiconductors technologies enable an increase in the power density of converters. The advantages of silicon carbide (SiC) devices are well documented and packaging solutions for high voltage components (6.5 -10 kV) are available. However, when insulation coordination constraints are considered, isolation distances become a major issue, as they result in larger packages. In this paper, we describe and demonstrate a simple solution which allows to extend the capability of a standard power module package to higher voltages.

Introduction

Wide band-gap semiconductors are widely used for power conversion and are a core-technology for medium voltage power conversion and high voltage direct current (HVDC) power transmission. For these latter applications, power modules must offer high power density, while withstanding high voltage stress. Current research tends to get the best compromise between electrical insulation performance and compactness, by adjusing the geometry [START_REF] Reynes | Protruding ceramic substrates for high voltage packaging of wide bandgap semiconductors[END_REF], [START_REF] Hourdequin | Metallized ceramic substrate with mesa structure for voltage ramp-up of power modules[END_REF], the design [START_REF] Dimarino | Design and development of a high-density, high-speed 10 kV SiC MOSFET module[END_REF] or by material processing [START_REF] Diaham | An original insitu way to build field grading materials (fgm) with permittivity gradient using electrophoresis[END_REF], [START_REF] Donzel | Nonlinear resistive electric field control for power electronic modules[END_REF].

For higher voltages (6.5 kV and more), the external surface of the power modules must be considered carefully: the voltage ratings of a package are dictated by clearance and creepage distances, as defined in IEC standards [START_REF]Insulation coordination for equipment within low-voltage supply systems -part 1: Principles, requirements and tests, en/fr, es[END_REF]. Creepage distances (the distance between two conductors along a solid insulating surface) are set by the geometry of the package, the class of its material, and the level of environmental pollution.

Regarding the geometry of the package, an open standard has been introduced for power modules [START_REF] Paques | A new HiPak module platform with improved reliability[END_REF], [START_REF] Buchholz | Electrical performance of a low inductive 3.3 kV half bridge IGBT module[END_REF], with a maximum voltage rating of 6.5 kV. An example of its implementation is Mitsubishi's IGBT Dual type X-series HV100 power module (Fig. 1). With this open standard module, the connector pattern and the footprint remain the same from one implementation to another, while manufacturers are free to design the internal layout or the external creepage features (Infineon, Wolfspeed, Hitachi with the XHP-3, XHV-7 and nHPD²-HV power modules respectively).

Because the location of the terminals is set with this open standard, there is little headroom for dramatic changes, e.g. to increase the creepage distances enough to operate at 10 kV or more. The material class also offers little possibility for improvements, as the materials used for power modules usually belong to material group I with a Comparative Tracking Index (CTI) ≥ 600, the best class defined in the IEC standards. As a consequence, limiting the environmental pollution may be the only possibility to use a power package at a higher voltage than what it was designed for (packages are typically designed considering a pollution degree 2 according to the IEC standards). As a result, it can be interesting to extend the voltage rating of power modules, or, on the other hand, to limit the voltage de-rating due to harsh environment, by adding a protective element. In this paper, a focus is made on open standard power modules and more specifically their use ranging from pollution degree 1 (PD1) to pollution degree 3 (PD3).

Protective device concept

A protective device is designed to fit between the power module and the busbar connection (Fig. 2). The design is adjusted to the external surface of the power module for a perfect fit. Double lips are formed around the connections for a good sealing of the normally exposed conductive surfaces of the busbar. By doing so, the protection breaks the clearance and creepage lines between both DC terminals of the power module. In addition, the sealing can offer a strong resistance to creepage and provide a pollution-proof protection. As a result, the new shortest creepage path is now on the top surface of the busbar (between the heads of both connection screws in Fig. 2). To overcome this issue in our tests, the connection screws are encapsulated with some silicone sealant. Other solutions could be to use insulating screws or to add rims around any connections, using the remaining space without degradation of the switching performances.

A similar approach can be found in [START_REF] Dimarino | Design and experimental validation of a wire-bond-less 10 kV SiC MOSFET power module[END_REF] where the high voltage connections are isolated from each other with no creepage path in-between; in addition, some shielding of the terminals is achieved using field plates on the connection board to prevent field reinforcements.

Experimental setup

All the shortest creepage path lengths from any pair of terminals of an XHP-3 power module are reported in Tab. 1. Some short paths can be ignored since they are not undergoing high differential voltage (i.e.: pin n o 1 and pin n o 5 and 6 are source, source kelvin and gate terminals of the same switch, respectively). It appears that the minimum relevant creepage distance is observed between terminals n o 3 and 4 (connected together inside the module) and the base-plate (BP). However, for handling and manufacturing reasons, this study focuses on the 60 mm creepage path between pin n o 1 and pin n o 2.

Maximum voltages applicable on this specific power module can then be determined with the creepage distance Tab. 2 derived from the IEC standard [START_REF]Insulation coordination for equipment within low-voltage supply systems -part 1: Principles, requirements and tests, en/fr, es[END_REF].

Considering the casing material is class I with a CTI ≥ 600, the maximum allowed voltage for this specific casing design should not exceed 5 kV in PD3 conditions.

Tab. 2: Maximum rated rms voltage for a 60 mm creepage path between two terminals according to the material group of the considered insulation and the pollution degree [START_REF]Insulation coordination for equipment within low-voltage supply systems -part 1: Principles, requirements and tests, en/fr, es[END_REF].

Mat I. Mat. II Mat III PD1 12.5 kV 12.5 kV 12.5 kV PD2 10 kV 7 kV 5 kV PD3 5 kV 4 kV 4 kV

Test vehicles

The protective device is made using injection molding of silicone rubber (Elastosil ® LR3003, Wacker Chemie AG). The choice of this rubber is driven by its mechanical behavior (tear strength and hardness), as well as for the good contact quality and the insulation properties of silicone interfaces [START_REF] Finis | On the electric breakdown behavior of silicone gel at interfaces[END_REF].

The protection device can be seen in Fig. 3(a).

Because the study focuses on the isolation between terminals 1 and 2, only a small part of the power module is needed (corresponding to the highlighted area in Fig. 1). Solid plastic parts (referred to as "dummy halp-power modules" in the remaining of the article) are designed and fabricated. The halfpower module is injection molded with a polybuthylene terephtalate (PBT) plastic with 25% glass fiber for mechanical resistance. This choice of plastic (PBT ULTRADUR ® B 4450 G5 from BASF) is also driven by fire resistance but above all for electrical properties such as resistivity and high CTI. Brass inserts with nickel finish are used to mimic the terminals of an actual power module. This is important, as these terminals could provide short paths for tracking failure. The power module terminals sheetmetal pads are reproduced with copper plates fixed flush to the corresponding pads with the brass insert underneath. A sample without protection and connector is depicted on Fig. 3(b) where the sheetmetal power terminals and brass inserts for signal connection are visible.

Flat connectors are used to reproduce the mechanical and electrostatic behavior of a typical busbar connection. All the exposed mechanical parts are nickel-plated to avoid unwanted galvanic corrosion in the presence of electrolytes. Stacks of washers are used to act as spacers between the connector and the power terminals, with a total height shorter than that of the protective device by about 500 µm, resulting in the compression of the silicone part to ensure a good sealing . This assembly is mounted with a 3.5 N m torque, as specified in the power module data-sheets. The screw heads are encapsulated with a salt water resistant silicone putty (SI 5366, Loctite ® ) to minimize the risk of arcing on the top of the assembly.

A total of 120 samples is mounted: 60 samples have a protection device mounted between the connectors and the module, while the 60 others have no protection. A protected sample can be seen in Fig. 3(c).

Artificial pollution pre-conditioning

Pre-conditioning is performed on samples to simulate harsh environmental conditions. Indeed, the hydrophobic properties of polymeric insulators can be affected by pollution. The pollution degrees are simulated as follows:

• PD1: laboratory conditions, no particular preconditioning is performed, the power modules are air-blown to remove any dust before mounting. Theses samples are considered PD2 when sprayed with water.

• PD2: non-conductive dust is applied. This non-conductive dust can change the surface tension of the power module casing, allowing a thin continuous film of water to form if high humidity or condensation is present. The pollution layer is simulated using kaolin (white clay). A suspension of 40 g L -1 according to IEC 60507 is sprayed on the mounted power module. The sample is then dried at 50 °C for 30 min. This cycle is repeated 5 times until a continuous layer of kaolin is formed on the power module. This method, inspired from the dry powder method presented in [START_REF] Gutman | Pollution tests for polymeric insulators made of hydrophobicity transfer materials[END_REF], ensures a pre-conditioning on both the silicone rubber and PBT. The hydrophilic effect of the pollution pre-conditioning is visible for wet testing.

• PD3: this case corresponds to a nonconductive pollution which can become conductive in the presence of humidity. A method to simulate seashore pollution is described in [START_REF] Dernfalk | Pollution test methods for polymeric insulators: Simulation of coastal environment and recovery of hydrophobicity[END_REF]. Here, the power modules are mounted and placed in a salt-mist chamber (Fig. 4) for a total of 168 h according to the IEC 60068-2-11 standard.

The pollution degree reached according to the artificial pre-conditioning of the samples is summarized

Breakdown voltage and partial discharge tests

To perform the breakdown voltage tests, an ac 50 Hz voltage waveform is applied on the sample connectors through a 230 V / 50 kV transformer (Fig. 5). A 22 Ω resistor limits the current in the primary coil of the high voltage transformer. An isolation transformer with a 1 to 1 ratio is used for safety reasons. The voltage is controlled manually with an auto-transformer, following a 1 kV s -1 ramp as close as possible. Since there is no arc detection in this setup, the circuit is manually deenergized when a first flash-over breakdown is observed. Breakdown voltage tests are destructive but they must be performed first to define the voltage range at which partial discharge measurement can be performed safely. Each breakdown voltage measurement series consist of 5 samples for all pre-conditioning referenced in Tab. 3.

The partial discharge tests are performed in a high voltage cabinet with an ac 50 Hz voltage source of maximum 24 kV and a background noise of 0.2 pC. The detection system (Omicron) is connected in accordance with IEC 60270. The sample is placed in parallel of a coupling capacitor through which high frequency signals generated by partial discharges are measured according to IEC 60270 guidelines. To avoid introducing parasitic partial discharges and corona discharges at the sharp edges of a screw or crocodile clip connector, a custom part (an oblong piece of brass with a female banana socket) is used to connect the samples to the test setup.

To measure the partial discharge inception voltage (PDIV) a ramp of 1 kV s -1 is applied until the sustained partial discharges are generated; the voltage is then reduced to 0 and this cycle is repeated at least 5 times (Fig. 6). The PDIV threshold is 10 pC as it is a standard value for electronic devices. A set of 7 to 10 samples are used for each partial discharge test series as mentioned in Tab. 4. Partial discharge tests are considered non-destructive, so to minimize the use of samples, the bare PD3 dry samples are reused for bare PD3 wet samples.

Test Results and Discussion

Breakdown voltage tests results

The results of breakdown voltage tests are shown in Fig. 7(a). Protected samples have a higher breakdown voltage than non protected samples. For non protected samples, artificial pollution lowers the breakdown voltage when activated with humidity. The breakdown voltage for protected samples in PD1 and PD3 are within the same statistical range. The effect of humidity is no longer visible except for salt polluted samples. However, the breakdown mechanism differ from bare samples: the flashover takes place along the connectors (busbars) for protected samples (Fig. 8(a)), while on the bare samples, it occurs between the power terminals of the module (Fig. 8(b)). This means that with the protection in place, the breakdown voltage of the module and its terminals exceeds that of the busbars.

Partial discharge tests results

Following the test procedure, a minimum of 5 PDIV measurement per sample is recorded. The mean value for each sample is calculated. for actual power modules. When the samples are protected, even salt-conditioned and wet samples have a PDIV comparable to that of dry PD1 protected samples. This result demonstrates the effect of pollution on PDIV values of devices, but it also demonstrates the efficiency of the protection.

Conclusion

A protection has been designed to address the voltage de-rating of power modules under harsh environment up to PD3. At first, breakdown voltages are performed to test the capability of such a device to withstand electrical stress. It shows + 47 % increase of breakdown voltage for PD1 samples and + 120 % for wet PD3 samples over non-protected samples. The protection is found to be efficient to prevent arcing since no breakdown is observed any more at the protection -module interface.

Partial discharge tests are also performed to ensure that the protection does not degrade the dielectric performance of the interconnects. The PDIV levels of protected samples remain similar to those of un-protected clean samples, while un-protected samples exhibit a strong drop in PDIV for wet PD3 conditions. Here again, the protection is found to be efficient to maintain PDIV constant despite harsh environmental conditions.

The dielectric characterizations validate the concept of the protection for high voltage power modules, allowing to use compact power module outlines for higher voltages. On the other hand, the protection can be a solution to avoid the voltage derating when the power module is used under harsh environment. The breakdown voltages on protected samples highlighted a new weakness point on the power module and busbar assembly as shown in Fig. 8. However, this weak point could easily be managed with a proper busbar design considering standard creepage, clearance, coating or potting solution.

Fig. 1 :

 1 Fig. 1: Mitsubishi's IGBT Dual type X-series HV100 power module. The clear part of the picture represents the tested zone of this experimental study. The pins n°1 and n°2 are designed to fulfill the dielectric requirement from the IEC guidelines.

Fig. 3 :

 3 Fig. 3: (a) Protective device made from silicone rubber; (b) Sample before mounting, with sheet-metal terminals and brass insert connections; (c) Test vehicles assembly with the protective device.

Fig. 4 :Tab. 3 :

 43 Fig. 4: A total of 40 samples are placed in a salt mist chamber for PD3 pre-conditioning. Half of the samples are protected while the other half are not.

Tab. 4 :Fig. 5 :

 45 Fig. 5: Breakdown Voltage test setup.

Fig. 6 :

 6 Fig. 6: Partial discharge test procedure consist of repeated voltage application on the test sample.The PDIV is measured in the voltage zone 1, the partial discharge regime is observed in the voltage zone 2.

Figure 7

 7 (b) represents a boxplot of average values for each sample sorted by type of series. A first result is the degradation of the PDIV for non-protected wet samples with a salt pollution, which could be expected

Fig. 7 :Fig. 8 :

 78 Fig. 7: (a) Boxplot of the breakdown voltage tests results versus the conditioned type samples; (b) Boxplot of the Partial Discharge Inception Voltage tests results versus the conditioned type samples.

Acknowledgement

This work was supported by a grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" Program (ANE-ITE-002-01).


B G Sébastien

Blanquer

Maike Werner

Markus Hannula

Shahriar Sharifi

Guillaume P R Lajoinie

David Eglin

Jari Hyttinen

André A Poot

Dirk W Grijpma

Keywords: triply-periodic minimal surface (TPMS), surface curvature, stereolithography, poly(trimethylene carbonate) (PTMC), tissue engineering scaffold

Reproduction of the anatomical structures and functions of tissues using cells and designed 3D scaffolds is an ongoing challenge. For this, scaffolds with appropriate biomorphic surfaces promoting cell attachment, proliferation and differentiation are needed. In this study, eight triply-periodic minimal surface (TPMS)-based scaffolds were designed using specific trigonometric equations, providing the same porosity and the same number of unit cells, while presenting different surface curvatures. The scaffolds were fabricated by stereolithography using a photocurable resin based on the biocompatible, biodegradable and rubber-like material, poly(trimethylene carbonate) (PTMC). A numerical approach was developed to calculate the surface curvature distributions of the TPMS architectures. Moreover, the scaffolds were characterized by scanning electron microscopy, microcomputed tomography and water permeability measurements. These original scaffold architectures will be helpful to decipher the biofunctional role of the surface curvature of scaffolds intended for tissue engineering applications.

Introduction

Scaffold design plays a major role in cell and tissue organization in tissue-engineered constructs. There is an increasing body of evidence that cells can sense and respond to surface curvature, thereby making it an intriguing scaffold functionality for tissue engineering purposes. Local tissue growth rate was shown to depend on the pore geometry. Tissue growth was investigated in channels of different geometries (with perimeters in the millimeter range) and it was shown that tissue growth increased with increasing concave curvatures while tissue growth was minimal on convex areas [START_REF] Bidan | Geometry as a factor for tissue growth: towards shape optimization of tissue engineering scaffolds[END_REF][START_REF] Rumpler | The effect of geometry on three-dimensional tissue growth[END_REF]. Furthermore, other approaches demonstrated the effect of curved surfaces on cell migration.

It was observed that cells actively migrated out of concave pits while they adhered and proliferated on convex structures [START_REF] Park | Study of cellular behaviors on concave and convex microstructures fabricated from elastic PDMS membranes[END_REF]. Recently, it was shown that single cells show different attachment modes to concave and convex spherical substrates with diameters ranging from 250 μm (≈2-3 times mesenchymal stem cell size) to 750 μm (much larger than cell size). Cells in concave spherical substrates showed an upward lift of the cell body and migrated faster compared to cells on flat or convex surfaces, while convex substrates led to a flattened cell morphology, nuclear deformation and promoted osteogenic differentiation [START_REF] Werner | Surface curvature differentially regulates stem cell migration and differentiation via altered attachment morphology and nuclear deformation[END_REF]. In addition, it was recently shown that mouse T lymphocytes cultured on sinusoidal wavy structures preferentially migrated along grooves with the most concave surface curvature [START_REF] Song | Sinusoidal wavy surfaces for curvature-guided migration of T lymphocytes[END_REF]. In view of the remarkable effect of curved surfaces on cell behavior, the availability of scaffolds containing highly controllable surface curvatures might contribute to scaffold design optimization for tissue reconstruction.

Triply-periodic minimal surfaces (TPMS) are mathematically defined surface curvatures, infinite and periodic in the 3D Euclidean space, making them interesting surfaces for highly controllable and homogeneous scaffold designs. Minimal surfaces are frequently encountered in nature and play an essential role in guiding chemical, biochemical and cellular processes. Minimal surfaces are present in the atomic organization of molecules [START_REF] Cook | Coherent dynamics of N-Level atoms and molecules: III. Analytically soluble periodic case[END_REF] and the lipid bilayer of cellular membranes [START_REF] Shearman | Inverse lyotropic phases of lipids and membrane curvature[END_REF][START_REF] Tenchov | Cubic phases in membrane lipids[END_REF]. They are also reported in the description of the domain morphology of selfassembled block copolymers and macromolecules [START_REF] Angelova | Self-assembled multicompartment liquid crystalline lipid carriers for protein, peptide, and nucleic acid drug delivery Acc[END_REF][START_REF] Matsen | Block copolymer microstructures in the intermediate-segregation regime[END_REF][START_REF] Thomas | Periodic area-minimizing surfaces in block copolymers[END_REF][START_REF] Urbas | Bicontinuous cubic block copolymer photonic crystals[END_REF]. The specific conformations of minimal surfaces allow understanding of phase behavior and structurefunction relations.

The natural organization of minimal surface structures responds to the physical principle which governs the forms and motions of objects, the principle of free energy minimization. In nature and man-made environments, systems normally try to arrange themselves to reduce their potential energy in order to consume less energy and increase their stability [START_REF] Eriksson | The mechanical surfacetension and stability of minimal surface-structures[END_REF]. The term 'minimal surface' is directly linked to the surface energy and represents the lowest possible potential energy that a surface can have if its energy is proportional to the surface area. In consequence, structures that respect this configuration will be thermodynamically stable. The minimization of the surface energy leads automatically to typical curved structures with respect to the initial fixed boundary [START_REF] Gandy | Nodal surface approximations to the P, G, D and I-WP triply periodic minimal surfaces[END_REF][START_REF] Mackay | Periodic minimal-surfaces from finiteelement methods[END_REF][START_REF] Hyde | The Language of Shape the Role of Curvature in Condensed Matter: Physics[END_REF].

As TPMS structures adopt a periodic curved conformation, it seems relevant to investigate a new generation of tissue engineering scaffolds based on TPMS to emphasize the potential of surface curvature in tissue engineering. Surprisingly, besides the typical scaffolds with Gyroid [START_REF] Melchels | Mathematically defined tissue engineering scaffold architectures prepared by stereolithography[END_REF] and Schwartz P architecture [START_REF] Rajagopalan | Schwarz meets Schwann: design and fabrication of biomorphic and durataxic tissue engineering scaffolds[END_REF][START_REF] Ronca | Preparation of designed poly(D,L-lactide)/nanosized hydroxyapatite composite structures by stereolithography[END_REF], TPMS structures for tissue engineering applications have only been investigated as computational models [START_REF] Kadkhodapour | Investigating internal architecture effect in plastic deformation and failure for TPMS-based scaffolds using simulation methods and experimental procedure[END_REF][START_REF] Kapfer | Minimal surface scaffold designs for tissue engineering[END_REF][START_REF] Olivares | Finite element study of scaffold architecture design and culture conditions for tissue engineering[END_REF][START_REF] Shin | Finite element analysis of Schwarz P surface pore geometries for tissue-engineered scaffolds Math[END_REF][START_REF] Yoo | Porous scaffold design using the distance field and triply periodic minimal surface models[END_REF]. Therefore, the aim of our work was to prepare a new library of mathematically designed tissue engineering scaffolds with sophisticated pore architecture based on TPMS. These complex scaffolds were built by stereolithography (SL), the most versatile rapid prototyping technique with the highest accuracy and precision to produce 3D structures at high resolution [START_REF] Melchels | Mathematically defined tissue engineering scaffold architectures prepared by stereolithography[END_REF][START_REF] Cooke | Use of stereolithography to manufacture critical-sized 3D biodegradable scaffolds for bone ingrowth[END_REF][START_REF] Jacobs | Rapid Prototyping & Manufacturing: Fundamentals of StereoLithography[END_REF][START_REF] Jacobs | Stereolithography and other RP&M Technologies: From Rapid Prototyping to Rapid Tooling[END_REF]. Flexible and elastic scaffolds were prepared from a photo-curable resin based on reactive poly(trimethylene carbonate) (PTMC) oligomers which is recognized as biocompatible and bioerodible material [START_REF] Chapanian | The role of oxidation and enzymatic hydrolysis on the in vivo degradation of trimethylene carbonate based photocrosslinkable elastomers[END_REF]. The pore characteristics of the built scaffolds were investigated a.o. by micro-computed tomography (μCT) analysis as well as by determination of their water permeability. In addition, we developed a mathematical method to determine the surface curvature distribution for any type of TPMS structure which would be a more precise approach to define the porous features.

Materials and methods

Materials

Trimethylene carbonate (1,3-dioxan-2-one, TMC) was obtained from Foryou Medical, China. Stannous octoate (tin 2-ethylhexanoate, SnOct 2 ), trimethylolpropane (TMP), trimethylamine, methacrylic anhydride, hydroquinone, methanol absolute, anhydrous dichloromethane and propylene carbonate were purchased from Sigma-Aldrich, Germany. Lucirin TPO-L (ethyl 2,4,6-trimethylbenzoylphenyl phosphinate) was provided by BASF, Germany. Orasol Orange G was obtained from Ciba Specialty Chemicals, Switzerland. Reagents and solvents were of analytical grade and used as received.

Synthesis and characterization of macromer

Three-armed PTMC of 5000 g mol -1 was typically obtained by ring-opening polymerization. PTMC oligomer was prepared by polymerization of TMC (0.98 mol; 100 g) initiated by TMP (0.0196 mol, 2.62 g) and catalyzed by Sn(Oct) 2 (0.05 wt%), at 130 °C for 48 h under an argon atmosphere. The monomer conversion was determined by 1 H-NMR spectroscopy (Varian Innova 300 MHz, USA), using CDCl 3 as solvent.

End-functionalization of the PTMC oligomer was achieved by reacting hydroxyl end-groups with an excess of methacrylic anhydride (0.15 mol; 22 ml) in the presence of triethylamine (0.15 mol; 20 ml) in 100 ml of dry dichloromethane. The reaction was performed at room temperature under stirring and argon atmosphere for 5 d. The macromer was then isolated by precipitation in cold methanol. The degree of functionalization was determined by 1 H-NMR analysis in CDCl 3 .

Stereolithography

A SL apparatus (EnvisionTec Perfactory Mini Multilens, Germany) equipped with a digital mirror device allowing projections of 1280×1024 pixels, each measuring 16×16 μm 2 , was used to fabricate eight different TPMS scaffolds. The wavelength of the blue light irradiation ranged from 400 to 550 nm, with a peak at 440 nm. Mathematical formulas were developed and used to generate 3D structures using K3dSurf v0.6.2 software (freeware from http:// k3dsurf.sourceforge.net). The generated structures were then modified to a volume of 10×10×10 mm 3 and converted into .stl files using computer-aided design software (Rhinoceros 3D, McNeel, Europe).

Resin formulation

Resin was prepared by diluting PTMC macromer in propylene carbonate (30 wt%), followed by mixing with Lucirin TPO-L initiator (5 wt% relative to the macromer) and Orasol Orange dye (0.15 wt% relative to the macromer). Using this resin, layers with a thickness of 25 μm were sequentially polymerized upon illumination for 50 s in the SL apparatus. After building, the scaffolds were carefully washed in acetone and ethanol.

Surface curvature 3D rendering of the structures and mapping of the curvatures were computed using Matlab (The Math-Works, Inc., Natick, MA, USA), according to the formula given in equation (1) (see below). Symbolic derivation was used to solve the equations and differentiate the expressions, except for two architectures (S and DG see figure 1 and table 1) that required a completely discretized numerical method.

The calculation of the Gaussian curvature distribution was implemented using the method described by Mackay et al [START_REF] Mackay | Periodic minimal-surfaces from finiteelement methods[END_REF]. The mathematical equations that approximate the typical surface of TPMS structures are expressed in the trigonometric function of x, y and z [START_REF] Gandy | Nodal surface approximations to the zero equipotential surfaces for cubic lattices[END_REF]. The trigonometric equations can be solved with regard to z, leading to a simplified expression [START_REF] Mackay | Periodic minimal-surfaces from finiteelement methods[END_REF][START_REF] Rajagopalan | Schwarz meets Schwann: design and fabrication of biomorphic and durataxic tissue engineering scaffolds[END_REF] for the surfaces in the form = å ( ) F F x y , .

i i

Hence, the Gaussian curvature K for each TPMS design can be expressed by the Laplace-Young equation (equation ( 1)), involving the first and second derivatives of the function

= - + + ( ) ( ) ( ) K x y , , 1
F F F F F F rad 1 xx yy xy yx x y 2 2 2
where the first and second derivatives are abbreviated as follows:

= = = = ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ F F F F ; ; ; ; x F x y F y xy F x y xx F x 2 2 2 = ¶ ¶ F . yy F y 2 2
Each nodal equation presented in table 1, descriptive of a minimal surface, was correlated to its curvature computed using equation [START_REF] Bidan | Geometry as a factor for tissue growth: towards shape optimization of tissue engineering scaffolds[END_REF]. The Gaussian curvature given by this equation can also be written as a function of the two non-dimensional local radii of curvature r 1 and r 2 as expressed in equation (2). Practically, the size and the repetition of the unit cells within the investigated scaffold are bound to vary in view of the specific tissue reconstruction targeted. Therefore, it is convenient to consider the bare equation given in table 1 and determine how the curvature will scale when changing the size of the unit cells, which allows to generalize the characterization of the surface curvature for any structure based on TPMS architecture.

F(x) is considered as a direct solution of an equation from table 1, and F 2 (x) will be the scaled version of the TPMS architecture investigated. The structures being triply periodic, the unit cell is covered for

Î p w ⎡ ⎣ ⎤ ⎦ x 0, , 2 i.e. w p = Î [ ] x x 0, 2 2
(and identically for y).

Here, we defined w as w = = and maintaining the aspect ratio results in

w w w = = .
x y z Therefore the rescaled surface F 2 is now written as w w w

= ( ) ( ) F x y F x y , , 2 leading to w w w w = = ¶ ¶ w ¶ ¶ ¶ ¶ ( ) ( ( )) ( ) x y F x y F x x y , , , , F x x 1 2 and similarly, w w w = = w w ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ( ) ( ) ( ) ( )
x y x y , , .

F x x F x y x F x , 2 2 2 2 2
Finally, from equation (1) we can extract equation (3):

w = ( ) ( ) ( ) K x y K x y , , , 3
2 2 2 2
where K is the local Gaussian surface curvature. The surface curvature distribution has been developed from only 90% of the surface area limit for each structure, as standard methods to describe distributions. This, is justified on the one hand by the fact the distributions tend to be mathematically infinite (or starting from 0), which renders errors bars useless in terms of statistical data interpretation. Furthermore, one needs to bear in mind that the extreme values of statistical distributions based on discrete data is intrinsically dependent on the chosen discretization step, and therefore do not carry any actual information on the system.

Scanning electron microscopy

Visualization of the built microsctructures was carried out by scanning electron microscopy (SEM, Philips XL 30 ESEM-FEG, The Netherlands) at an operating voltage of 5 kV, using pristine specimens sputtercoated with gold.

Micro-computed tomography

Structural analysis of the prepared scaffolds was performed by μCT scanning (MicroXCT-400 Carl Zeiss x-ray Microscopy, Pleasanton, USA) at an operating voltage of 60 kV and a tube current of 166 μA. Pixel size was 19.4 μm. Visualizations were done with Avizo (FEI Visualization Sciences Group). Porosities and pore size distributions were assessed with Fiji [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] program using BoneJ [START_REF] Doube | BoneJ free and extensible bone image analysis in imageJ[END_REF] plugin. Interconnectivity analysis for different pore sizes was also done with Matlab.

Scaffold permeability

Permeability was measured using a water flow set-up in the vertical direction [START_REF] Chor | A permeability measurement system for tissue engineering scaffolds[END_REF][START_REF] Pennella | A survey of methods for the evaluation of tissue engineering scaffold permeability[END_REF]. The principle was to let water flow through a scaffold and to collect the volume every 10 s for 10 min. A liquid tank was used to deliver the water at low pressures (25, 50 and 75 mbar). The permeability was determined by Darcy's law expressed in equation (4)

= m D ( ) Q , 4 A k P L . . .
where Q is the volumetric flow rate, A the surface area, k the intrinsic permeability, ΔP the pressure difference, μ the dynamic viscosity of water and L the thickness. The intrinsic permeability was determined from the slope c of the water flow (Q, measured in triplicate) versus ΔP using equation (5)

= m ( ) k . 5 c L A . .

Results

TPMS designs

Many 3D geometric architectures have been described as periodic minimal surfaces [START_REF] Schoen | Infinite periodic minimal surfaces without selfintersections[END_REF]. Among them, bicontinuous or biphasic TPMSs form a solid/void interface where the void space represents the pores of the 3D scaffold and the solid space is materialized by the polymer. The bicontinuous surfaces of TPMS structures are represented by complex harmonic equations. To allow analytical evaluation and fabrication, the harmonic equations were approximated by the periodic nodal surface [START_REF] Gandy | Nodal surface approximations to the P, G, D and I-WP triply periodic minimal surfaces[END_REF]. Table 1 lists the approximated periodic nodal equations for eight TPMS structures [START_REF] Yoo | Porous scaffold design using the distance field and triply periodic minimal surface models[END_REF][START_REF] Schoen | Reflections concerning triply-periodic minimal surfaces Interface[END_REF]. These equations allowed generating a CAD file, and extracting an STL file for use in the manufacturing by SL. By modulating a linear term (C) to the nodal equation, it is possible to define the pore characteristics and the surface curvature. Therefore, the role of the constant C can be defined as offsetting value increasing the volume fraction of the void space [START_REF] Yoo | Porous scaffold design using the distance field and triply periodic minimal surface models[END_REF].

The investigated TPMS structures (table 1 and figure 1) have the capacity to yield stable 3D scaffolds, having a common porosity of 65% and a common number of 64 unit cells per scaffold, with a fixed cubic size of = m 10 mm. As shown in figure 1, TPMS structures are periodic in three independent directions and can be infinitely reproduced by repetition of a cubic translational unit cell and thus might be used to construct a 3D porous scaffold. The bicontinuous geometry of TPMS structures gives a with total pore interconnectivity.

Schwarz P (P), Diamond (D) and Gyroid (G) geometries have been widely investigated from a mathematical perspective and are designed from catenoid and helicoid surfaces [START_REF] Colding | Shapes of embedded minimal surfaces[END_REF]. These structures were already proposed for tissue engineering scaffolds [START_REF] Melchels | Mathematically defined tissue engineering scaffold architectures prepared by stereolithography[END_REF][START_REF] Ronca | Preparation of designed poly(D,L-lactide)/nanosized hydroxyapatite composite structures by stereolithography[END_REF]. Double Schwarz P (DP), Double Gyroid (DG), Double Diamond (DD), Fischer-Koch S (S) and F-RD are new geometries for tissue engineering applications, with varying surface curvature distributions through intertwined or interpenetrated surfaces. These structures present a high intrinsic complexity, moreover most of these new structures exhibit a high specific tortuosity which induce a relatively large surface area and would significantly increase the contact and the exchanges with the surfaces, which is required for cells adhesion. In addition, these 3D structures display specific and periodic surface curvatures.

Calculation of surface curvature

Although the definition of curvature is basically defined as the inverse of the radius of the curvature, in 3D the description of curvature is more complex. In 3D, at each point of the surface two different principal curvatures (k 1 and k 2 ) can be defined. Principal curvatures are used to calculate the mean curvature ( ) H and the Gaussian curvature ( ) K (table 2). For any type of minimal surface structure, at each point of the surface the principal curvatures have an equal magnitude but opposite sign, illustrating convex and concave shapes [START_REF] Gandy | Nodal surface approximations to the zero equipotential surfaces for cubic lattices[END_REF]. Therefore, the mean curvature ( ) H , which is defined as the average of the principal curvatures (equation ( 6)), is equal to zero at any point of a TPMS structure. Gaussian curvature ( ) K is an intrinsic measure defined as the product of the principal curvatures (equation ( 7)). In contrast with the mean curvature, Gaussian curvature represents a local curvature within the structure.

As expressed in table 2, the mean curvature is at every point at the surface equal to zero = ( ) H 0 for any type of minimal surface. Consequently, the Gaussian curvature varies in a periodic fashion with the curved trajectory along the surface, from a negative minimum value to a maximum value equal to zero on a flat surface.

In order to study the typical Gaussian curvature distribution of a specific TPMS scaffold, it is essential to fully define the surface curvature.

From equation (3), established in the Materials and methods section, the Gaussian curvature | | K distributions were calculated for each TPMS function based on structures made with 4 unit cells in all three directions ( ) n and 10 mm length ( ) m . The curvature mapping of each TPMS architecture was computed using equation (3) and presented in figure 2.

Scaffold designs based on TPMS display a defined periodic porous geometry with variation of the Gaussian curvatures. Therefore, several distinct variations of the Gaussian curvature can be obtained according to the type of TPMS architecture. Interestingly, besides the calculation of the curvature distribution, this method enables the determination of the number of unit cells ( ) n necessary to design a cubic or rectan- gular prism scaffold with a predefined size ( ) m , required for a TPMS scaffold with desired Gaussian curvature values. This would allow for example, to investigate the influence of surface curvature on the behavior of seeded cells. In practice, it is essential to be able to vary the size of the scaffold according to the target application, which confirms the need to develop a method like this that allows a systematic determination of the curvature for any given TPMS scaffold.

Figure 3 reports the Gaussian curvature distribution numerically determined for the eight different TPMS scaffold designs calculated in mm -2 . The heterogeneity of the surface curvature distribution is clearly distributed on figure 3(a) which thus allows to

Gaussian curvature = ⋅ K k k 1 2 (7) When = H 0, = - k k 1 2
emphasize the % of surface area dedicated to a specific surface curvature value. It is therefore clear that the TPMS structures can present several maxima in the % surface area at some values of the surface curvature. This aspect may be crucial for the creation of cell-supporting scaffolds since cell behavior is affected by the local curvature. To simplify these distributions, the figure 3(b) gives an idea of the mean curvature with the interval of surface curvature distributions for each structure.

The eight structures can be classified in two groups: G, DG, DP, P and D with relatively low surface curvatures and narrow curvature distributions, and DD, S and F-RD with substantially higher maximum surface curvatures and wide curvature distributions.

Fabrication of scaffolds with TPMS

Biodegradable PTMC oligomer was synthesized as previously described [START_REF] Schuller-Ravoo | Preparation of flexible and elastic poly(trimethylene carbonate) structures by stereolithography[END_REF]. The monomer-to-initiator ratio was adjusted to obtain a molecular weight of 5000 g mol -1 . A high degree of acrylate functionalization of more than 95%, contributed to efficient photocrosslinking by SL. A solution of 30 wt% of this low molecular weight macromer in propylene carbonate has a viscosity between 5 and 10 Pa s, which is well suited for manufacturing by SL. An optimized Orasol Orange dye concentration of 0.15 wt% relative to the macromer was used to control the penetration depth of the blue light and thus to avoid over-crosslinking between the printed layers. SEM images in supplementary information is available online at stacks.iop.org/ BF/0/000000/mmedia show the typical layer-bylayer fabrication for the S structures.

Characterization of built scaffolds with TPMS structures

All designed TPMS structures were successfully built by SL using PTMC macromer. Using a stiffer material such as PDLLA macromer (see supplementary information), the TPMS scaffolds were built with the same efficiency (data not shown), indicating the versatility of this fabrication technology. Representative SEM images of the TPMS structures made of PTMC show the remarkable resolution of the manufacturing process, with a precise reproduction of the curved surfaces (figure 4).

The complex internal 3D structure of the scaffolds was investigated high spatial resolution by μCT analysis (figure 5). Moreover, this technique offers insight in the accuracy of SL fabrication, which was found to be around 94%-98% for the PTMC built scaffolds with respect to the CAD design (table 3).

The μCT reconstructions of the fabricated scaffolds show homogeneous and highly interconnected pores.

As expected, the scaffolds with DD, DG and DP architectures show perfect interpenetration of each respective simple D, G and P geometry. The table 4 reports the pore characteristics of the scaffolds measured by μCT, in terms of pore size and strut thickness which defines the material thickness from edge to edge. These measurements correspond to the diameter of a simulated permeating sphere inside the pore or inside the material [START_REF] Melchels | Mathematically defined tissue engineering scaffold architectures prepared by stereolithography[END_REF][START_REF] Hildebrand | A new method for the model-independent assessment ofthickness in threedimensional images[END_REF]. The same approach was used to measure the pore characteristics of the designed scaffolds. These data confirm the high precision of the fabrication method, which is around 92% as compared to the designed scaffolds.

It is important to notice that this method of calculation take only into account the accessible volume in the void or material interface and therefore neglect the surface topography. Indeed, TPMS architectures are defined from their surface topography and it seems relevant to also calculate a radius of curvature which will concretely consider the surface architecture. Thus, we propose in this article to change the vision of the pore features by introducing a new approach and method to calculate the radius of surface curvature from the calculated curvature and the equation (2). The two approaches are undeniably distinct and give two complementary indications on the pore features.

Pore interconnectivity of the built TPMS scaffold μCT analysis was also used to determine the interconnectivity of the pores in the TPMS scaffolds using an algorithm that mimics mercury porosimetry. Figure 6(A) shows a μCT reconstruction of a scaffold with Fischer-Koch S architecture, as well as three different cross-sections through the scaffold. It clearly shows the reproduction of the S architecture with a completely open internal structure of the scaffold. Figure 6(B) describes the interconnectivity of the void space in the Fischer-Koch S scaffold, based on the accessible pore volume for simulated spheres of different diameters permeating through the scaffold [START_REF] Melchels | Mathematically defined tissue engineering scaffold architectures prepared by stereolithography[END_REF]. The graph shows that the interconnectivity of the pores in this scaffold is 100%. This specific interconnectivity of the pores was found for all TPMS structures investigated.

Water permeability measured with built TPMS scaffolds

The intrinsic water permeabilities of the TPMS scaffolds as determined according to Darcy's law are reported in table 5. As the porosity of each scaffold is constant (65% of porosity with 94%-98% of reproducibility, table 3), the evolution of the permeability for each scaffold cannot be modulated by the porosity. The highest intrinsic permeabilities were found for the scaffolds with the pore channel architectures (D, G and P). Moreover, the data show that the intrinsic water permeability is inversely related to the specific surface area, and thus the complexity of the inner structure of the scaffold, ranging from 7.34 mm -1 for the DD to 3.69 mm -1 for the P scaffold (see also table 4).

Discussion

To advance the field of regenerative medicine, tissue engineering scaffolds with precise and optimized properties are needed to induce effective tissue regeneration. With respect to scaffold architecture, TPMS scaffolds might be very useful as they present geometries based on the principle of minimal surfaces. Our approach combining SL, degradable polymers and TPMS-based scaffold designs, allows building scaffolds with specific optimized geometry and therefore targeting the regeneration of a wide variety of tissues.

It was shown before that cell morphology, migration behavior, stem cell fate and local tissue formation can be affected by surface curvature in the range of curvatures larger than cell size [START_REF] Bidan | Geometry as a factor for tissue growth: towards shape optimization of tissue engineering scaffolds[END_REF][START_REF] Rumpler | The effect of geometry on three-dimensional tissue growth[END_REF][START_REF] Park | Study of cellular behaviors on concave and convex microstructures fabricated from elastic PDMS membranes[END_REF][START_REF] Werner | Surface curvature differentially regulates stem cell migration and differentiation via altered attachment morphology and nuclear deformation[END_REF][START_REF] Lo | Three-dimensional spherical spatial boundary conditions differentially regulate osteogenic differentiation of Q4 mesenchymal stromal cells[END_REF]. Therefore, in addition to conventional properties like surface chemistry and stiffness, which affect cell behavior, also surface curvature must be taken into account as design parameter for advanced tissue engineering scaffolds.

TPMS architectures can be described by trigonometric equations, which allow drawing up a library of different architectures all based on minimal surfaces. TPMS scaffolds display specific functionalities such as pore interconnectivity and a periodic surface curvature distribution that may be of considerable benefit in tissue engineering. Each TPMS architecture presents a specific and periodic surface curvature distribution that can be tuned to the target tissue. The surface curvature of TPMS scaffolds has never been described in the literature, and we especially focused our study on this original functional parameter for 3D scaffolds. The surface curvature, and therefore the pore characteristics of a specific TPMS scaffold, can be easily modulated by varying the number of unit cells (n) and/or the length of the cubic side (m) of the scaffold. The maximum Gaussian curvatures calculated using equation (3) allow predicting the surface curvature of TPMS scaffolds and to potentially tailor cell behavior. Indeed, surface curvature gives a much more relevant indication on pore characteristics and surface topography, especially required in tissue engineering as cell function and behavior depend principally on the scaffold surface features. By our approach, we proposed to orient the focus on surface curvature supported by the TPMS family architecture, and we proposed a method to calculate the surface curvature distribution which can be easily modulated for specific applications.

The Gaussian curvature distributions of the eight investigated TPMS structures are not uniform (figure 2). The intrinsic complexity of the DD, S and F-RD structures leads to a broader curvature distribution with the highest surface curvature values, while D, G, P, DG and DP structures have a more narrow distribution with the lowest values for the surface curvature. Surface curvature distribution could be a useful variable in the development of co-culture systems, in which specific cell types differentially respond to low or high surface curvatures. The different equations given in table 1 require the use of a constant (C), which was chosen to keep the porosity of the scaffolds constant (65%). Changing this constant will have different effects on each structure. More precisely, this constant defines the amount of material placed around the structure skeleton. Changing of the constant leads to differences in shape, density and consequently surface curvature of each structure.

The mathematical determination of the surface curvatures revealed that the structures have not only different mean curvatures but also different curvature distributions, which vary from narrow to relatively wide. This opens the possibility to change the properties of the structures, which may be crucial for the creation of cell-supporting scaffolds since cells appear to be affected by the local surface curvature. An interesting option would be to study the effect of scaffolds with narrow curvature distributions compared with scaffold architectures with wider curvature distributions on cell behavior and tissue formation. It can be hypothesized that surface curvature with the smoother distribution might favor a homogeneous tissue reconstruction, whereas the more inhomogeneous surfaces in terms of curvature might favor cell differentiation and may be used for the replacement of more complex tissues.

The permeability of a scaffold for a fluid describes the scaffold structure independent of the sample size and fluid used, and is determined by a number of parameters including porosity, pore size, pore orientation and pore interconnectivity [START_REF] O'brien | The effect of pore size on permeability and cell attachment in collagen scaffolds for tissue engineering[END_REF]. Permeability is also related to the fluid flow in the scaffold. Fluid flow mediated shear stress is recognized to strongly influence parameters such as cell function [START_REF] Bancroft | Fluid flow increases mineralized matrix deposition in 3D perfusion culture of marrow stromal osteloblasts in a dose-dependent manner[END_REF][START_REF] Raimondi | Mechanobiology of engineered cartilage cultured under a quantified fluid-dynamic environment[END_REF] and cell distribution after perfusion seeding [START_REF] Cartmell | Effects of medium perfusion rate on cell-seeded threedimensional bone constructs in vitro[END_REF], and may therefore be important as bioreactor design parameter [START_REF] Maes | Modeling fluid flow through irregular scaffolds for perfusion bioreactors[END_REF][START_REF] Porter | 3D computational modeling of media flow through scaffolds in a perfusion bioreactor[END_REF]. Relationships between permeability and porosity and permeability and pore size have been described in the literature [START_REF] Dias | Permeability analysis of scaffolds for bone tissue engineering[END_REF][START_REF] Li | The effect of scaffold architecture on properties of direct 3D fiber deposition of porous Ti6Al4V for orthopedic implants[END_REF]. The complete pore interconnectivity of the TPMS scaffolds contributed to relatively high water permeabilities in the order of 10 -9 m 2 , providing an environment through which nutrients and metabolic wastes are able to diffuse, which is a prerequisite for homogeneous tissue growth [START_REF] Hou | Preparation of interconnected highly porous polymeric structures by a replication and freeze-drying process[END_REF][START_REF] Lu | Role of interconnections in porous bioceramics on bone recolonization in vitro and in vivo[END_REF][START_REF] Moore | Quantitative analysis of interconnectivity of porous biodegradable scaffolds with micro-computed tomography[END_REF]. As expected, the highest water permeabilities were found for the scaffolds with the most open pore architecture (D, G and P). Moreover, the permeability was inversely related to the specific surface area and thus the total surface area of the scaffolds, in agreement with data reported by Jung and Torquato [START_REF] Jung | Fluid permeabilities of triply periodic minimal surfaces[END_REF]. This can be explained by increasing friction, impeding fluid flow with increasing surface area. Determination of the scaffold permeability may allow to predict mass transport under dynamic conditions in a bioreactor [START_REF] Coletti | Mathematical modeling of three-dimensional cell cultures in perfusion bioreactors[END_REF], cell differentiation [START_REF] Pennella | A survey of methods for the evaluation of tissue engineering scaffold permeability[END_REF][START_REF] Dias | Permeability analysis of scaffolds for bone tissue engineering[END_REF] as well as degradation kinetics of the material [START_REF] Agrawal | Effects of fluid flow on the in vitro degradation kinetics of biodegradable scaffolds for tissue engineering[END_REF].

In summary, the fabrication of 3D scaffolds based on TPMS architectures with specific and controlled surface curvatures offers exciting new functionalities in the field of tissue engineering. This approach allows to create a library of mathematically designed scaffolds with heterogeneous zonal features by combining different TPMS architectures. This provides great potential for regeneration of anisotropic tissues such as osteochondral bone and the intervertebral disc. The next step is to investigate the effects of different TPMS architectures on cell behavior in 3D.

Conclusions

A series of eight scaffolds with mathematically defined TPMS architectures were produced by SL using the biocompatible, degradable and rubber-like material PTMC. For the first time, we report on the potential of surface curvature as distinct design parameter for advanced tissue engineering scaffolds. An equation method to determine the surface curvature of TPMS scaffolds has been developed, which allows tailoring of the scaffolds to a desired surface curvature, which potentially has an effect on cell behavior. By means of this new approach, the pore characteristics of tissue engineering scaffolds can be precisely defined. In addition to the surface curvature, the different TPMS architectures exhibit distinct specific surface areas and permeabilities, which may also influence tissue formation. Finally, we highlighted the potential benefits of TPMS architectures and their specific surface curvatures on tissue engineering applications, but they may also be of importance in many other applications such as the science of fluid adsorption or for heterogeneous catalysis devices.

Figure 1 .

 1 Figure 1. TPMS structures: (a) Diamond (D); (b) Gyroid (G); (c) Schwarz P (P); (d) Fischer-Koch S (S); (e) Double Diamond (DD); (f) Double Gyroid (DG); (g) Double Schwarz P (DP); (h) F-RD, for each structure the top design is one single unit cell and the bottom structure represents 4×4×4 unit cells.

  to the spatial period of the structure, n the number of unit cells along one side of the structure (which is fixed at 4 unit cells for the structures tested in this work) and m the size of the cubic structure in mm. Considering the new set of variables w

Figure 2 .

 2 Figure 2. Gaussian curvature | | K distribution along the surface of each TPMS structure. Color bars represent the Gaussian curvature values in mm -2 . (a) Diamond (D); (b) Gyroid (G); (c) Schwarz P (P); (d) Fischer-Koch S (S); (e) Double Diamond (DD); (f) Double Gyroid (DG); (g) Double Schwarz P (DP); (h) F-RD.

Figure 3 .

 3 Figure 3. Complete distribution of the curvature in terms of surface area for each TPMS structures (a), and representation of the surface curvature distribution in histogram with the mean surface curvature distribution depicted by a square symbol and the full curvature distribution on 90% of the surface area limit is given by the interval bars.

  The tortuous and intertwined architectures of the scaffolds present in figures 4(d)-(h) and 5(d)-(h) are clearly visible. By

Figure 4 .

 4 Figure 4. SEM visualization of the built TPMS scaffolds based on PTMC: (a) Diamond (D); (b) Gyroid (G); (c) Schwarz P (P); (d) Fischer-Koch S (S); (e) Double Diamond (DD); (f) Double Gyroid (DG); (g) Double Schwarz P (DP); (h) F-RD, (scale bars represent 1 mm).

Figure 5 .

 5 Figure 5. μCT reconstruction of the eight TPMS scaffolds based on PTMC: (a) Diamond (D); (b) Gyroid (G); (c) Schwarz P (P); (d) Fischer-Koch S (S); (e) Double Diamond (DD); (f) Double Gyroid (DG); (g) Double Schwarz P (DP); (h) F-RD.

Figure 6 .

 6 Figure 6. μCT imaging and pore interconnectivity of a built PTMC scaffold with Fischer-Koch S architecture. (A) μCT reconstruction in 3D and 2D cross-sections through different parts of the scaffold. (B) The red curve represents the pore volume that is accessible for simulated spheres of different diameters permeating through the scaffold. The histogram shows the accessible fraction of the total pore volume related to the diameter of the simulated permeating sphere. Scale bars represent 1 mm.

Table 1 .

 1 Approximated periodic nodal equations for eight TPMS structures[START_REF] Yoo | Porous scaffold design using the distance field and triply periodic minimal surface models[END_REF][START_REF] Schoen | Reflections concerning triply-periodic minimal surfaces Interface[END_REF].

	TPMS structures	Nodal equations for TPMS: f (x,y,z)=C a

Table 2 .

 2 Expression of curvatures for minimal surfaces.

	Mean curvature	H	=	k 1	+ 2	k	2 (6)

Table 3 .

 3 Reproducibility of the fabrication calculated from the porosity (in %) determined by μCT for the designed and built TPMS scaffolds based on PTMC.

	TPMS structures	DD	S	F-RD	DG	DP	D	G	P
	Porosity of designed scaffolds (%)	65.3	64.6	65.3	65.2	64.7	65.4	65	65.3
	Porosity of built scaffolds (%)	61.2	62.1	61.7	66.4	60.9	64.5	66.4	66.7
	Reproducibility	93.7%	96.1%	94.5%	98.1%	94.1%	98.6%	97.9%	98%

contrast, the pore geometries of D, G and P scaffolds are notably straightly open from one side of the scaffold to the other, and therefore affect substantially the specific surface area determined by μCT and reported in table 4, compared to the other geometries much more intertwined.

Table 4 .

 4 Specific surface area (SSA), mean pore size and strut thickness for the designed and built TPMS scaffolds based on PTMC, determined by μCT.

		SSA of built	SSA of	Mean pore size of	Mean pore size of	Mean strut thick-	Mean strut thick-
	TPMS	scaffolds	designed scaf-	built scaf-	designed scaf-	ness of built scaf-	ness of designed
	structures	(mm -1 )	folds (mm -1 )	fold (μm)	fold (μm)	fold (μm)	scaffold (μm)
	DD	7.40	8.08	545±256	578±63	678±138	616±133
	S	5.90	7.40	720±132	750±105	619±131	529±129
	F-RD	5.96	6.34	1001±328	1117±256	964±318	850±306
	DG	5.45	6.55	878±333	700±110	589±76	541±57
	DP	4.53	5.56	818±333	830±125	1240±440	1103±420
	D	4.55	5.30	1205±180	1185±147	854±144	797±158
	G	3.94	4.42	1445±240	1467±189	881±104	827±132
	P	3.69	4.02	2069±481	2069±457	1384±498	1260±547

Table 5 .

 5 Intrinsic permeabilities of the eight TPMS scaffolds based on PTMC.

									Biofabrication 00 (2017) 000000
	10
	TPMS scaffolds	DD	S	F-RD	DG	DP	D	G	P
	Permeability×10 -9 (m 2 )	2.81±0.24	3.31±0.18	4.11±0.41	4.20±0.11	4.59±0.16	4.72±0.35	5.59±0.65	6.44±0.37
									S B G Blanquer et al

Biofabrication 00 (2017) 000000 S B G Blanquer et al

Acknowledgments

We thank Frédéric Fernandez from the platform of electronic microscopy at the University of Montpellier for the SEM analyses.

PAPER

Surface curvature in triply-periodic minimal surface architectures as a distinct design parameter in preparing advanced tissue engineering scaffolds