Upkar Kumar

Sviatlana Viarbitskaya

Aurelien Cuche

Christian Girard

Sreenath Bolisetty

Raffaele Mezzenga

Gérard Colas

Des Francs

Alexandre Bouhelier

Erik Dujardin

Aurélien Cuche

Gérard Colas Des Francs

Designing plasmonic eigenstates for optical signal transmission in planar channel devices

Keywords: . 2D crystalline gold, surface plasmon, transmittance and routing devices, plasmonic eigenmodes, non-linear luminescence, LDOS

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Introduction

Signal transfer at faster rates, longer distances, along narrower pathways, with reduced losses and a higher bandwidth are some crucial impediments to the improvement of any information processing technologies. Optical transfer in free space, dielectric waveguides or fibers offers the best prospects for several of these criteria except for sub-wavelength spatial confinement.

Yet, this particular point is of paramount importance to interface optical signal with electronic information processing and has been partially addressed by the photon-plasmon conversion using noble metal circuitry. [1][2][3][4][5][6] Surface plasmon (SP) propagated in films, [7][8] waveguided in grooves [9][10][START_REF] Gramotnev | Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface[END_REF] or stripes [START_REF] Weeber | Near-field observation of surface plasmon polariton propagation on thin metal stripes[END_REF][START_REF] Dionne | Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model[END_REF][START_REF] Kalavrouziotis | 0.48Tb/s (12x40Gb/s) WDM transmission and high-quality thermo-optic switching in dielectric loaded plasmonics[END_REF] has proven efficient in transmitting or even processing information to and from optical and optoelectronic planar devices. [START_REF] Haffner | All-plasmonic Mach-Zehnder modulator enabling optical high-speed communication at the microscale[END_REF][START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF][START_REF] Muehlbrandt | Siliconplasmonic internal-photoemission detector for 40 Gbit/s data reception[END_REF] The optoelectronic conversion has become more challenging as the size of electronic devices is reaching the nanoscale though recent advanced functionalities based upon particle chains [START_REF] Solis | Electromagnetic Energy Transport in Nanoparticle Chains via Dark Plasmon Modes[END_REF][START_REF] Teulle | Multimodal plasmonics in fused colloidal networks[END_REF][START_REF] Volpe | Deterministic Subwavelength Control of Light Confinement in Nanostructures[END_REF] and atomic-size gaps [START_REF] Ward | Optical rectification and field enhancement in a plasmonic nanogap[END_REF][START_REF] Stolz | Nonlinear Photon-Assisted Tunneling Transport in Optical Gap Antennas[END_REF][START_REF] Emboras | Atomic Scale Plasmonic Switch[END_REF][START_REF] Kern | Electrically driven optical antennas[END_REF][START_REF] Buret | Spontaneous Hot-Electron Light Emission from Electron-Fed Optical Antennas[END_REF][START_REF] Parzefall | Antenna-coupled photon emission from hexagonal boron nitride tunnel junctions[END_REF] could allow the next level of downsizing. Notably, even at the nanoscale, the dominant approach to signal transmission consists in propagating along a selected wave vector or inside a onedimensional (1D) waveguide, but lossy in-and out-coupling usually limit the overall transfer efficiency. [START_REF] Kinsey | Examining nanophotonics for integrated hybrid systems: a review of plasmonic interconnects and modulators using traditional and alternative materials Invited[END_REF][START_REF] Wei | Cascaded logic gates in nanophotonic plasmon networks[END_REF] A further step is to design the spatial distribution of the in-plane plasmon modes to implement the in-, out-coupling as well as the directional transmission and routing functions. [START_REF] Volpe | Deterministic Subwavelength Control of Light Confinement in Nanostructures[END_REF] The use of spatial and spectral properties of higher order plasmon modes in 2D crystalline Au cavities have recently been proposed as a generic approach to integrated optical logic gates and, further, information processing. [START_REF] Viarbitskaya | Tailoring and imaging the plasmonic local density of states in crystalline nanoprisms[END_REF] The implementation of plasmonic devices from the properties of 2D cavity modes pertain to the ability to tailor their size and shape. [START_REF] Gu | Resonant wedge-plasmon modes in single-crystalline gold nanoplatelets[END_REF][START_REF] Viarbitskaya | Morphology-induced redistribution of surface plasmon modes in two-dimensional crystalline gold platelets[END_REF][START_REF] Bellido | Electron Energy-Loss Spectroscopy of Multipolar Edge and Cavity Modes in Silver Nanosquares[END_REF] While simple and usually symmetrical shapes can be obtained directly by chemical synthesis, the full potential of arbitrarily shaped crystalline cavities can, so far, only be attained by the physical or chemical etching of Au platelets which has already led, for example, to antennas with improved performances. [START_REF] Huang | Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry[END_REF][START_REF] Mejard | Advanced engineering of single-crystal gold nanoantennas[END_REF] Using this approach, we have recently demonstrated that a small defect, such as a hole, could significantly alter the modal behavior of 2D cavities pointing out the potential for more complex modal design. [START_REF] Cuche | Modal engineering of Surface Plasmons in apertured Au Nanoprisms[END_REF] Here, we report the directional signal transmittance and routing mediated by the rich and complex spatial and spectral variation of the plasmonic eigenmodes supported by 2D crystalline structures. The potential of multiple input-output devices with a deterministic transmission between one entry and one exit ports is demonstrated in a diabolo-like structures composed of a rectangular channel flanked by two mesoscale triangular pads that sustain higher order modes in the visible spectral range, [START_REF] Viarbitskaya | Tailoring and imaging the plasmonic local density of states in crystalline nanoprisms[END_REF][START_REF] Cuche | Beyond dipolar regime in high-order plasmon mode bowtie antennas[END_REF] as schematized in Figure 1. The size of the structure is intermediate between the supra-and sub-wavelength size ranges so as to combine long range delocalization and sharp field enhancement at the input and output ports that are therefore precisely defined. In Figure 1c, an in-plane near-field intensity map is calculated over the entire diabolo for an optical excitation consisting of a Gaussian laser beam impinging on the structure at normal incidence. The beam is kept fixed at one corner of the triangular input cavity. A clear remote response is observed at the output cavity as far as the diagonally opposed corner. The full map suggests that the remote response is mediated by the excitation of a mode bridging the input and output locations via the channel. Interestingly, the output signal is almost annihilated when the incident polarization is rotated by 90° (Fig. 1d). The same near-field transmittance simulations provide the spectral distribution of the delocalized plasmon modes (Fig. 1b). The simulated transmittance spectra are monitored in (O) when the diabolo is excited with horizontal or vertical polarizations in (I). Upon horizontal polarization excitation (black curve), the symmetrical diabolo exhibits a 100-nm wide transmittance peak culminating at 810 nm.

When the polarization is flipped (red curve), the spectrum is completely attenuated in this spectral region. These simulations strongly suggest that the transmittance is obtained by populating polarization-dependent 2D delocalized modes. Indeed, by simply moving the excitation to the lower corner of the input cavity, the signal is redirected to the upper corner of the diabolo output cavity through a symmetrical and energetically degenerated set of delocalized eigenstates. We experimentally explore this concept and demonstrate the effective spectral and spatial engineering of higher order plasmon modes in channel devices comprising input and output cavities. These information transfer structures may comply with the upcoming demand of compact processing devices, such as logic gates, [START_REF] Wei | Cascaded logic gates in nanophotonic plasmon networks[END_REF][START_REF] Xie | Ultracompact all-optical full-adder and half-adder based on nonlinear plasmonic nanocavities[END_REF] coupled to quantum emitters in integrated architectures aimed at transferring 6,[START_REF] Rousseaux | Adiabatic passage mediated by plasmons: A route towards a decoherence-free quantum plasmonic platform[END_REF][START_REF] Roller | Hotspot-mediated non-dissipative and ultrafast plasmon passage[END_REF] and manipulating [START_REF] Kewes | A realistic fabrication and design concept for quantum gates based on single emitters integrated in plasmonic-dielectric waveguide structures[END_REF][START_REF] Wang | A 14 x 14 mu m(2) footprint polarization-encoded quantum controlled-NOT gate based on hybrid waveguide[END_REF] quantum information.

Results and Discussion

Focused ion beam (FIB) milling of crystalline gold microplatelets is used to design a structure supporting cavity modes that can be spatially and spectrally optimized to perform the desired transmission and routing functions from a chosen input port to a specific output location (Fig. 2a). This nanofabrication method provides plasmonic nanocircuits and devices with superior structural and optical quality. [START_REF] Huang | Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry[END_REF] The microplatelets are synthesized by a simple green reducing process reported in reference [START_REF] Bolisetty | Amyloid-mediated synthesis of giant, fluorescent, gold single crystals and their hybrid sandwiched composites driven by liquid crystalline interactions[END_REF] . The colloidal suspension is drop-casted onto cross-marked glass coverslip and thoroughly washed. The microplatelets are natively stabilized by strongly adsorbed β-lactoglobulin amyloid fibrils, which are removed by three successive O2-plasma treatments. We conducted scanning electron microscopy (SEM) and atomic force microscopy (AFM) to assess the scarce density of colloidal objects on the surface in order to address single objects. The in-plane dimensions (~ 2-3 µm) and thickness (20-40 nm) of the targeted native hexagonal microplatelets are recorded individually (Fig. 2b). Gallium ion FIB milling is performed, first, to etch out the diabolo shape (Fig. 2c) and then to remove the peripheral fragments of the mother microplatelet. The milling protocol is optimized using a pattern generator and nanolithography software to reduce edge amorphisation, metal re-deposition and gallium contamination. The typical side length of the equilateral triangular pads ranges from 600 to 1000 nm, which is comparable to chemically synthesized prismatic cavities, and the connecting channel is 500 nm long and 200 nm wide (See Experimental Section). [START_REF]Complementary experimental and simulated data are provided in Supporting Information[END_REF] Non-linear photoluminescence (nPL) microscopy is a convenient technique to map out the intensity of the local field inside plasmonic devices without resorting to slow and invasive nearfield scanning probes. [START_REF] Viarbitskaya | Tailoring and imaging the plasmonic local density of states in crystalline nanoprisms[END_REF][START_REF] Teulle | Scanning optical microscopy modeling in nanoplasmonics[END_REF] nPL imaging acts as a local probe that benefits from a higher spatial resolution and sensitivity conferred by the non-linear mechanism. In this work, the nPL of gold is excited with a 180-fs pulsed Ti:Sapphire laser focused in a diffraction-limited spot by a high numerical aperture objective (N.A. 1.49). The excitation spot diameter is about 300 nm and the mean power density at the sample is 20 mW.µm -2 . The power dependency of the non-linear emission signal collected in the 375-700 nm spectral range is quadratic and assimilated as such in corresponding simulations. Two different nonlinear optical mapping modes are implemented to obtain the modal landscape and the transmittance of the device. The first mode provides confocal maps reconstructed by raster scanning the device through the focal spot (Fig. S2). [START_REF]Complementary experimental and simulated data are provided in Supporting Information[END_REF] The intensity of the detected nPL is proportional to the squared in-plane surface plasmon local density of states (SP-LDOS) and the maps reveal the spatial distribution of the eigenmodes of the 2D confined plasmons (Figs. S3 andS4). In the second imaging mode which is schematized in Figure 2d, the nPL response of the entire device is recorded for a fixed excitation position to visualize the plasmon-mediated signal transfer and distant nPL generation within the device. [START_REF] Viarbitskaya | Delocalization of Nonlinear Optical Responses in Plasmonic Nanoantennas[END_REF] Such wide-field images, displayed in Figures 2e and2f, are obtained in leakage radiation (LR) imaging when the diabolo shown in Fig. 2a is excited in position (I) with a linearly polarized beam. In Figure 2e which is obtained for a 0° polarized excitation, nPL is not only observed as a local response coinciding with the excitation spot but also from several remote regions of the input triangle, the channel and even from the distant corner of the right triangle, where a localized bright spot can be observed in position (O). The nPL signal emitted in the substrate along the transfer path is collected in LR imaging mode with the high N. A. objective thus producing an image that can be qualitatively compared to the near-field response of the diabolo. [START_REF] Hohenau | Surface plasmon leakage radiation microscopy at the diffraction limit[END_REF][START_REF] Huang | Farfield imaging of the electromagnetic local density of optical states[END_REF] Indeed, the nPL distribution recorded in Figure 2e faithfully matches the transmittance map calculated in Figure 1c for the same excitation polarization direction, thus providing a compelling correlation between the transmission of the near-field from a fixed excitation and the remote production of nPL signal.

The nPL images suggest that the pulsed excitation in (I) is indeed carried through the channel, as was recently observed in straight micrometer-long rods. [START_REF] Viarbitskaya | Delocalization of Nonlinear Optical Responses in Plasmonic Nanoantennas[END_REF] Yet, in contrast to long rods, which sustain a continuum of accessible resonances, the non-linear emission from the diabolo is not uniformly scattered from the remote (right) edge but it emerges from a very localized and specific location close to the corner, which coincides with an antinode of the SP-LDOS distribution (see Fig. 3a). This remote output spot could be used to further excite a collecting waveguide, forming an all-optical switch in an integrated nanophotonic platform. [START_REF] Xie | Ultracompact all-optical full-adder and half-adder based on nonlinear plasmonic nanocavities[END_REF][START_REF] Weeber | Colloidal Quantum Dot Integrated Light Sources for Plasmon Mediated Photonic Waveguide Excitation[END_REF] A comparison of the wide-field and confocal nPL maps confirms that the transmittance takes place between two locations of high SP-LDOS intensity, i.e. showing intense confocal nPL signal (Figs. S3 andS4). Indeed, the discrete modal structure of the diabolo is revealed in the computed maps of the projected in-plane SP-LDOS (Fig. 3 a-b) and the simulated nPL maps for an incident linearly polarized beam with a waist of 300 nm (Fig. 3 c-d). In particular, the four extremal hotspots in the corners of the triangular pads (Fig. 3c) are reminiscent of the ones observed in crystalline triangular prisms. [START_REF] Viarbitskaya | Tailoring and imaging the plasmonic local density of states in crystalline nanoprisms[END_REF] The input (I) and output (O) should be chosen in registry with these hotspots but the confocal nPL maps do not provide the transmittance connectivity between them. The simulated SP-LDOS map in Fig. 3a suggests a connecting modal path enabling a transfer of plasmon information through the channel, when the SP-LDOS is projected along the horizontal linear polarization. The transmission configurations with horizontally polarized excitation (Figs. 1c, 2e and S5) reveal that the four corners of the diabolo are connected in pairs by two energetically degenerated and diagonally delocalized modes that mediate the effective transmittance between (I) and (O) only or, similarly, between (I') and (O') only as schematized in Figure 3e (See also Fig. S5). In the confocal nPL maps, the relative intensity of the spots is adjusted by rotating the polarization direction away from horizontal. [START_REF]Complementary experimental and simulated data are provided in Supporting Information[END_REF] Hence, an excitation with a vertical polarization does not result in a bright nPL emission from the distal tip of the triangular cavity as previously observed in the case of isolated prisms of similar size, [START_REF] Viarbitskaya | Tailoring and imaging the plasmonic local density of states in crystalline nanoprisms[END_REF][START_REF] Viarbitskaya | Morphology-induced redistribution of surface plasmon modes in two-dimensional crystalline gold platelets[END_REF] but in a strongly reduced emission from the proximal corner and channel (Fig. 3d) associated with a minimal SP-LDOS distribution within the channel (Fig. 3b). In Figures 2 and3 indicate that the transmittance is enabled by a spatial and spectral excitation of plasmonic eigen-modes of the diabolo. To further confirm the mode-mediated signal transfer, we considered exciting the structure in other locations while monitoring the non-linear emission at the output port (O). For example, in Figure 4, the excitation beam is parked at the left entrance of the channel (channel input point Ci, in Fig. 2a) which results in a striking change of behavior compared to the excitation in (I). A horizontally polarized excitation does not produce any delocalized nPL emission from the right pad (Fig. 4a) while a vertical polarization generates both an intense emission at the excitation point and a remote nPL spot in (O) as seen in the inset of Figure 4b. The corresponding simulated near-field SP transmittance maps are presented in Figures 4c and4d (~ 1.9 µm). Significant dissipation effects that would limit the plasmon propagation can thus be excluded. In light of Figure 3, Ci appears as a location with low density of states, for a 0° polarized excitation, which makes the energy in-coupling less effective and so results in a reduced transmittance efficiency in Fig. 4a. For a 90° polarized excitation in Ci, the in-coupling is slightly higher as indicated by a larger confocal response in Ci seen both in Fig. 3d and in Fig. 4b, which coincides a significant nPL response in output port O in Fig. 4b. Interestingly, the polarization dependence of the transmittance when directly exciting the entrance of the channel at (Ci) is orthogonal to the case of a beam impinging in (I). This counter-intuitive behavior indicates that the channel cannot be considered as a rod nor the diabolo as the simple juxtaposition of two triangular and a rod-shaped sub-structures. The diabolo sustains specific resonances, the spatial distribution of which specifically enables the transport of plasmon information from a high SP-LDOS area to another via paths of finite SPLDOS present in the channel. This is precisely modelled in our plasmon transmittance simulation tool, in which the effective transmittance depends on the existence or not of a phase matching throughout the device (See Supporting Information, SI). [START_REF] Teulle | Multimodal plasmonics in fused colloidal networks[END_REF] In Figure 4f, the transmittance of the device calculated for inputs (I) or (Ci) and output (O) is compared to the corresponding experimental nPL intensity measured in image plane maps for the two polarization configurations. The transmittance calculations faithfully follow the relative intensity of the output nPL intensity, which is maximized for the horizontal polarization when exciting in (I) and by the vertical polarization when exciting in (Ci). Moreover, for a given input location, the output transmittance in (O) in Figure 4f match the relative variation of the confocal nPL intensities in Figures 3 andS4 suggesting that the major contribution to the transmittance variation comes from the in-coupling efficiency. Indeed, a lower confocal nPL intensity in (Ci) is measured for the horizontal polarization (Figs. 3c andS4i) than for the vertical polarization (Figs. 3d andS4j), while the opposite is observed in (I). The efficiency to generate a non-linear output varies in other locations and is provided by the confocal nPL maps in Figures 3e-f and S4e-l. Some minor relative differences between the simulated and experimental transmittances are ascribed to the fact that our transmission model computes the linear near-field rather than the non-linear emission and does not account for the full complexity of the out-coupling to the far field and nPL collection, which may vary from one excited modes and for a particular polarization to another.

The operation of the transfer devices at a given energy relies on the spatial and spectral tailoring of a supporting delocalized mode that simultaneously enables an effective transmission and localized in-and out-coupling. This spatial and spectral matching of the cavities can be altered by varying the global geometry of the device. For instance, the symmetry breaking created in bowtie-like antennas comprising two mesoscale triangular prisms with different sizes was shown to have a marked effect on the local field distribution and spectral response. [START_REF] Cuche | Beyond dipolar regime in high-order plasmon mode bowtie antennas[END_REF] We have fabricated non-symmetrical diabolos by changing the relative sizes of the triangular pads while keeping the same channel geometry. In Figure 5, we examine the case of a structure made of a 640-nm left side triangular pad connected to a 730-nm right side triangular pad (Fig. 5a). The total in-plane SPLDOS plotted in Figure 5b highlights the non-symmetrical modal structure with a predominant m=3 mode on the left side and a predominant m=5 mode on the right side. When the diabolo is excited at input (I) with a linearly polarized beam along the horizontal or vertical direction, no output is observed in (O) as shown in Figures 5c and5d.

The series of experimental nPL profiles in Figure 5g demonstrates that the transmittance along the I-Ci-O path remains inhibited for all polarization directions. The incident excitation propagates as far as the mid-channel area but is completely suppressed in the right side of the device. As the incident polarization direction is rotated, one can notice that the intensity of the nPL is uniformly affected over the entire structure, in striking contrast to the periodic intensity redistribution observed with the symmetrical diabolo for a similar sequence (See details in SI and Figure S6). The reshaping of the diabolo keeps the system off-resonance and does not allow for a signal transmission through the channel. We confirm the transmittance suppression by calculating the near-field SP transmittance maps for horizontally (Fig. 5e) and vertically (Fig. 5f) polarized excitation in (I). Once again, a very good match with the corresponding experimental maps is obtained. In the excitation regions, the confined hotspot and the extension along the edges observed in Figure 5c are matched by the two intense lobes in Figure 5e, while the weak and diffuse signal around (I) in Figure 5d is also obtained in the simulated Figure 5f.

Furthermore, in the output regions, both polarization configurations yield a weak signal, which is marginally more intense in Figure 5e (horizontal polarization) than in Figure 5f (vertical polarization), in agreement with the weak transmittance measured in Figures 5g andS6. The spatially modulated, long-range transmittance monitored in the nPL images merely reveal in space the spectral match or mismatch of the device with the excitation conditions. In Figure 5h, the smaller non-symmetrical diabolo exhibits a sharper and blue-shifted transmittance peak at 775 nm (black curve) compared to the 820-nm resonance of the larger symmetrical device upon horizontally polarized excitation (Fig. 1b). Additionally, a low energy peak beyond 850 nm is observed in the non-symmetrical diabolo for the orthogonal excitation (red curve). Both excitation configurations though show a vanishing transmittance intensity at 810 nm, which confirms the off-resonance response of the non-symmetrical geometry at this wavelength.

Similarly, the off-resonance excitation of the symmetrical diabolo at 750 nm results in no detectable nPL and in marginal transmittance in simulated maps in the output port, as shown in SI. The spectral differences between the two diabolos are both related to their shape differences and relative sizes, the non-symmetrical one comprising smaller triangular pads features resonances that are blue-shifted compared to the ones of the symmetrical device. Yet the smaller size itself does not warrant a higher transmittance which can be suppressed when operated out of resonance.

Conclusion

In this work, we illustrate how the modal engineering in two dimensions can be harnessed to design and optimize the optical signal transmission in compact and single crystalline devices that could be docked to other information processing modules. We have shown that the structural design of 2D mesoscale resonators from crystalline gold platelets allows to spatially and spectrally engineer SP modes exhibiting both delocalized extension and strong spatial modulation. Diabolo structures composed of a transmission channel flanked by two triangular resonators are investigated in detail. The excitation of the diabolo structures at specific input sites with suitable wavelength and polarization results in a remote but confined response, therefore implementing an efficient multi input/output signal transmission and routing through a two-dimensional device. The transmission mechanism is related to the existence of a complex plasmonic modal landscape. The transmission and routing functions are demonstrated experimentally using a fixed excitation and a detection of the spatial extension of the response through the channel. Our experimental results are confirmed numerically using a dedicated near-field SP transmittance code with a realistic polarized Gaussian excitation. The performances of the transmission modulation upon polarization flipping at 810 nm show a large ON/OFF ratio exceeding 20dB power attenuation for the spectrally-matched symmetrical diabolo. Our modal design approach to engineer and modulate SP transmission in compact wavelength-size devices contribute to the recent strategies to embed active information processing functions into pure or hybrid plasmonic structures such as the insertion of phase changing material into gold transmission lines [START_REF] Rude | Active Control of Surface Plasmon Waveguides with a Phase Change Material[END_REF] or the ultrafast electrical switching of SP in indium tin oxide waveguides. [START_REF] Abb | All-Optical Control of a Single Plasmonic Nanoantenna-ITO Hybrid[END_REF][START_REF] Lee | Nanoscale Conducting Oxide PlasMOStor[END_REF][START_REF] Guo | Ultrafast switching of tunable infrared plasmons in indium tin oxide nanorod arrays with large absolute amplitude[END_REF] The diabolo structures examined in this work are merely one first example of the potential of modal engineering in pure plasmonic systems toward information processing and transfer in complex devices, which could be used to create new computing architectures for classical and quantum optical technology. [START_REF] Rousseaux | Adiabatic passage mediated by plasmons: A route towards a decoherence-free quantum plasmonic platform[END_REF][START_REF] Roller | Hotspot-mediated non-dissipative and ultrafast plasmon passage[END_REF][START_REF] Tame | Quantum plasmonics[END_REF] Experimental Section Sample preparation. The crystalline gold microplatelets are synthesized using βlactoglobulin protein fibrils according to the methods reported in references [START_REF] Bolisetty | Amyloid-mediated synthesis of giant, fluorescent, gold single crystals and their hybrid sandwiched composites driven by liquid crystalline interactions[END_REF][START_REF] Li | Hybrid Nanocomposites of Gold Single-Crystal Platelets and Amyloid Fibrils with Tunable Fluorescence, Conductivity, and Sensing Properties[END_REF] Wide-field non-linear optical microscopy. Wide-field non-linear optical microscopy is performed on an inverted microscope. A 180 fs pulsed Ti:Sapphire laser tuned at λ = 810 nm is focused in a diffraction-limited spot by a high numerical aperture objective (oil immersion, 100x, NA = 1.49). The full width half maximum (FWHM) spot diameter of the excitation beam is about 300 nm. The laser average power density at the sample is 20 mW.µm -2 . The linear polarization of the excitation beam is rotated by a half-wavelength plate inserted at the laser output. Non-linear photoluminescence (nPL) is collected through the same objective after filtering in the 375-700 nm spectral range from the backscattered fundamental beam by a dichroic beam splitter. Wide-field images of the signal emitted from the entire diabolo structure for a fixed excitation position are collected on a CCD camera placed in the conjugated object plane. Since the non-linear photoluminescence is quadratic with the excitation power, the experimental maps are normalized with the square of the excitation power measured at the back aperture of the objective.

Simulations. Numerical simulations of the local electromagnetic field, SP-LDOS and nPL signal were performed using home-made codes based on the 3D Green Dyadic Method as described in earlier reports. [START_REF] Teulle | Scanning optical microscopy modeling in nanoplasmonics[END_REF] The generalized propagator K(r,r ' ,ω) is computed first that gives the total electromagnetic response of the complex metallic nanostructure under any arbitrary illumination field E 0 (R 0 , r, ω). The diabolo nanostructure is discretized into hexagonal lattice of cells. The local electromagnetic field inside the metallic nanostructure is then given by 𝐸𝐸(𝑹𝑹 𝟎𝟎 , 𝒓𝒓, 𝜔𝜔) = � 𝐾𝐾(𝒓𝒓, 𝒓𝒓 ′ , 𝜔𝜔) 𝑉𝑉 . 𝐸𝐸 0 (𝑹𝑹 𝟎𝟎 , 𝒓𝒓′, 𝜔𝜔)𝑑𝑑𝒓𝒓′ where R 0 represents the center of the light beam. Simulations of confocal nPL intensity generated by a Gaussian beam excitation are obtained by integrating the square of local electric near-field intensity on the whole structure. The computed nPL maps are constructed by raster scanning the position of the sample through the focused Gaussian beam. [START_REF] Viarbitskaya | Morphology-induced redistribution of surface plasmon modes in two-dimensional crystalline gold platelets[END_REF][START_REF] Teulle | Scanning optical microscopy modeling in nanoplasmonics[END_REF] The same formalism is used to calculate SPLDOS spectra and maps inside a metallic nanostructure.

The SP-LDOS transmittance calculations are performed with an excitation located at a fixed input co-ordinate (position vector r1). The transmitted near-field intensity generated by the excited plasmonic field at a distal output co-ordinate (position vector r2) is calculated. The energy transferred from input to output is obtained by solving the Dyson sequence equation which gives the Dyadic tensor S (r 2 , r 1 , ω). Supporting Information. Supporting information is available in the online version. Diabolo nanofabrication from single crystalline Au microplatelets (Fig. S1). Confocal nPL mapping (Fig. S2). SPLDOS, simulated and experimental confocal TPL maps (Figs. S3,S4). Signal routing through symmetrical degenerate plasmon modes (Fig. S5). Polarization dependency of the transmitted nPL intensity (Fig. S6). Calculation of near-field transmittance maps and spectra (Figs. S7, S8, S9). Resonant and non-resonant near-field transmittance maps and spectra in symmetrical diabolos (Fig. S10). References.

Corresponding Authors

* Email: Erik.Dujardin@cemes.fr * Email : Alexandre.Bouhelier@u-bourgogne.fr

For table of contents only

Designing plasmonic eigenstates for optical signal transmission in planar channel devices.

Upkar Kumar, Sviatlana Viarbitskaya, Aurélien Cuche, Christian Girard, Sreenath Bolisetty, Raffaele Mezzenga, Gérard Colas des Francs, Alexandre Bouhelier, Erik Dujardin. Plasmonic modes in 2D gold cavities are designed to route non-linear signal from one input port to one output port and to modulate the transmitted power with the exciting polarization.

S3 SP-LDOS, simulated and experimental confocal nPL maps

Confocal non-linear photoluminescence (nPL) mapping visualizes the spatial distribution of the surface plasmon local density of states (SP-LDOS) as demonstrated in refs [1][2][3] . Here we apply confocal nPL mapping to diabolo structures and we evidence modal behavior reminiscent to the one displayed by triangular nanoprisms. Experimental maps are compared to simulated nPL and SP-LDOS maps using the tools developed and described in our earlier work. In particular, the SP-LDOS and nPL patterns generated by the symmetrical diabolo structure shown in Fig. S1e are displayed in Fig S3 for four different incident linear polarizations. Fig. S3a to S3d are the maps of the partial SP-LDOS at 810 nm projected on the polarization direction 0°, 90°, 120° and 150° with respect to horizontal. Our Green Dyadic Method code is used to simulate the realistic confocal nPL response (beam waist 300 nm) at 810 nm excitation for these four linear polarizations (Fig. S3e-S3h). These simulated maps can be compared to the experimental ones (Fig. S3i-S3l) acquired by raster scanning the 300-nm focused beam waist of the pulsed Ti : Sapphire laser operated at 810 nm and by collecting the non-linear luminescence signal (cut-off filter at 500 nm). Both experimental and simulated nPL maps patterns match and relative intensities on the apexes and in the channel recorded experimentally are well accounted for in the simulations. The observation of intense and sharply localized luminescence spots at the outer apexes of the triangular pads, the intensity of which is modulated upon rotating the excitation polarization direction, is reminiscent of our earlier reports on the non-linear patterns produced by triangular nanoprisms. In particular, the longitudinal excitation (0° polarization; Figs. S3a, S3e and S3i) is associated with a apex and the output remain close to their minimal value with a very small modulation also in phase. When excited at 810 nm, the asymmetrical produces a transmitted luminescence that is spatially attenuated (See Fig 5) and modulated uniformly by the incident polarization.

The polarization dependency reported here reinforce the attribution of the effective transmittance to the excitation of a delocalized SP mode that display SP-LDOS intensity extrema at opposite apexes used as input (I) and output (O). The transmittance though mesoscopic symmetrical diabolo structures presents resonances in the visible and near-IR region as shown in Figure 6h. A small variation of the lateral size of the triangular pads has limited effect on the spectral characteristics of the transmittance modes. Indeed, in Figure S10a, the transmittance of the diabolo structures shown in Fig. 2a and Fig. S1e, which have the same channel size but a pad size difference of ca. 70 nm, is virtually identical. In particular, the transmittance curves coincide at 810 nm (vertical dash-dot line in Fig. S10a). Experimentally, the image plane nPL maps recorded on the larger diabolo show significant transmittance when the excitation is polarized along the longitudinal direction (Fig. S10b) but none when the polarization is orthogonal to the main axis (Fig. S10c) as already described for the smaller diabolo in the main text (See Fig. 2d,e). The spectra suggest that the transmittance can be spectrally suppressed for both polarizations by exciting the structures out of the resonances peaks, for example at the wavelength of 750 nm (dotted vertical line in Fig. S10a) where the spectral intensity is incidentally the same for both structures and low for both polarization. Accordingly, simulated near-field transmission maps for an excitation in the upper left corner at a wavelength of 750 nm yields no or very small transmittance in the output (O) region for either 0° (Fig. S10d) or 90° (Fig. S10e) excitation polarization. Experimental image plane nPL upon excitation at 750 nm are displayed in panels (f) and (g) of Figure S10. The patterns in the excitation pad match closely the simulated near-field and none of the two polarization conditions yields measurable transmittance as observed in the output region plotted with a 10x magnified color scale. This further confirms the delocalized mode-mediated transmittance in the resonant diabolo structures.

Figure 1 .

 1 Figure 1. Modal transmittance in mesoscopic crystalline gold, diabolo-shaped devices. (a)

29

 29

Figure 2 .

 2 Figure 2. Experimental demonstration of the modal transmittance in diabolo-shaped devices (a) Scanning electron microscopy (SEM) image of a symmetrical diabolo-shaped structure. The sample is composed of two identical triangles (side length 925 nm) connected by a 500-nm long, 200-nm wide channel. (I), (Ci) and (O) indicated the specific input, channel input and output locations considered in this work. (b) SEM image of the pristine crystalline Au microplatelet deposited on ITO-coated glass coverslip prior to ion milling the sample shown in (a). (c) SEM image of the same microplatelet after nanopatterning the diabolo structure and prior to ion etching the unwanted peripheral parts of the starting platelet. (d) Schematics of the experimental set-up for wide-field non-linear photoluminescence (nPL) mapping. (e, f) Nonlinear photoluminescence (nPL) wide-field CCD image of the diabolo structure shown in (a) when the 810-nm excitation is placed in (I) for an incident beam polarized at (d) 0°and (e) 90°. The insets on the right display the signal from the output right triangular area with a 10x

Figure 2f ,

 2f Figure 2f, this polarization configuration for the beam parked in input (I) markedly alters the

Figure 3 .

 3 Figure 3. Plasmon modal properties of the mesoscale diabolo structure. (a-b) Partial in-plane

Figure 4 .

 4 Figure 4. Direct channel excitation. (a, b) Non-linear photoluminescence wide-field CCD

  Figures 4c and 4d for horizontally and vertically polarized excitation in Ci, with insets showing

Figure 5 .

 5 Figure 5. Transmission blockade by spectral mismatch in a non-symmetrical diabolo. (a) SEM

  . The microplatelets studied here are prepared by initial mixing of 0.01M HAuCl4 salt with 0.6 wt% of β-lactoglobulin fibrils and then heated at 55° C for 24 hours. The microplatelets are 20-40 nm in thickness and the typical in-plane dimensions are in the range of 1-15 µm. The colloidal suspension is drop-casted onto glass coverslips (22 × 22 × 0.15 mm) coated with a nominal 10nm indium-tin oxide (ITO) layer. Cross-mark arrays are designed by photolithography and Au metallization. Organic residues are thoroughly washed with 10% aqueous ethanol solution and deionized water and the gold surface is decontaminated by three successive 5-min O2 plasma treatments. The diabolo structures are fabricated by focused ion milling of hexagonal microplatelets with diameter of 3-5 µm using a gallium ion beam on a Zeiss 1540 XB dual beam microscope interfaced with a Raith Elphy Multibeam pattern generator. To ensure minimal substrate milling while thoroughly etching the gold, typical ion current of 1.0 pA and dose of 8000 µC.cm -2 are used. The milling protocol consists in etching out the diabolo pattern first with minimal beam rastering along the diabolo edges. The peripheral portions of the starting hexagonal colloid are milled in a second step.

Figure S3 .

 S3 Figure S3. (a-d) Partial SP-LDOS maps at 810 nm, obtained for projection along the (a) 0°, (b) 90°, (c) 120°, (d) 150° polarization directions. (e-h) Simulated confocal nPL maps for an excitation at 810 nm, with a realistic beam waist of the incident Gaussian beam of 300 nm diameter. (i-l) Experimental confocal maps recorded for the same corresponding polarization directions. The sample is described in Fig. S1e.

Figure S9 .

 S9 Figure S9. (a) Schematic view of the Gaussian excitation spot position (red dot) with respect to the 1.5 µm long gold wire. (b-c) Corresponding 2 x 2 µm² simulated images showing the intensity distribution of the total field propagating in the wire once illuminated by the Gaussian spot. (d-e) Similar to (b-c) with a logarithmic color scale. The incident polarization is indicated by the white arrows. The four maps are computed at a distance of 30 nm above the metallic wire.

authors acknowledge the support of the massively parallel computing center CALMIP (Toulouse, Fr).

Author Contributions

ED, CG, AB designed the experiments. RM and SB synthesized the Au microplatelets. UK and ED conducted the sample preparation and nanofabrication steps. AB, SV implemented the experimental optical set-up. UK, SV, AC performed the optical experiments. CG, AC, UK, GCdF developed the simulations tools and performed all calculations. ED and UK processed the data. All authors contributed to the data analysis and manuscript writing. All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT

This work was funded by the French Agence Nationale de la Recherche (Grant ANR-13-BS10-0007-ANR-PLACORE), the region of Burgundy under the PARI II Photcom and the European Research Council under the FP7/ 2007-2013 Grant Agreement No. 306772. The

Designing plasmonic eigenstates for optical signal transmission in planar channel devices

Upkar Kumar, 1 Sviatlana Viarbitskaya, 2 Aurélien Cuche, 1 Christian Girard, 1 Sreenath Bolisetty, 3 Raffaele Mezzenga, 3 Gérard Colas des Francs, 2 Alexandre Bouhelier, 2 * Erik Dujardin 1 * 1 CEMES CNRS UPR 8011, 29 rue J. Marvig, 31055 Toulouse, France. 2 Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS UMR 6303, Université de Bourgogne Franche-Comté, 9 Av. A. Savary, Dijon, France. 3 ETH Zurich, Department of Health Sciences and Technology, Schmelzberg-strasse 9, CH-8092 Zurich, Switzerland * Corresponding author: dujardin@cemes.fr

Supporting information

Table of contents

S1. Diabolo nanofabrication from single crystalline Au microplatelets

The diabolo-shaped transmission devices were produced from crystalline gold microplates deposited on ITO-coated glass coverslip as described in the Methods section.

The large polydispersity of the colloidal suspension allowed to easily select hexagonal plates of adequate sizes (typically 3-5 micrometer diameter) that could be located optically or in the electron microscope with respect to labelled crossmarks (Fig. S1a andS1b). The Ga ion milling is performed in two stages. First, the structure is defined and isolated by a sequence of irradiation boxes optimized to reduce the edge amorphisation and metal re-deposition (Fig. S1c).

Then the peripheral areas of the starting platelet are removed leaving an isolated structure on the ITO/glass substrate (Fig. S1d). Figures S1e to h show a series of several diabolo structures fabricated and studied in this work.

The fabrication protocol is also illustrated by the movie provided as a separate supplementary material. Confocal non-linear optical microscopy is performed on an inverted microscope. A 180 fs pulsed Ti:Sapphire laser tuned at λ = 810 nm is focused in a diffraction-limited spot by a high numerical aperture objective (oil immersion, 100x, NA = 1.49). The full width half maximum (FWHM) spot diameter of the excitation beam is about 300 nm. The laser average power density at the sample is 20 mW.µm -2 . The linear polarization of the excitation beam is rotated by a half-wavelength plate inserted at the laser output. Non-linear photoluminescence (nPL) is collected through the same objective, as the sample is raster scanned with a XY piezo stage (step size 25-50 nm), and filtered in the 375-700 nm spectral range from the backscattered fundamental beam by a dichroic beam splitter and a collection filter. Confocal maps are recorded on an avalanche photodiode. Since the non-linear photoluminescence is quadratic with the excitation power, the experimental maps are normalized with the square of the excitation power measured at the back aperture of the objective. minimal confocal signal in the channel and a maximal intensity emitted from the outer apexes. On the contrary, the transverse excitation (90° polarization; Figs. S3b, S3f and S3j) exhibits a reinforcement of the confocal emission localized in the channel. Intermediate projections, such as the one shown in Figs. S3c, S3g and S3k (polarization angle 120°) indicate that high nPL (i.e. high SP-LDOS) is distributed along a path connecting one apex on the left triangular pad to the opposite apex on the right pad, through the channel.

The exact same observations are made for the diabolo presented in Figure 2 of the main text. The SP-LDOS (Fig. S4a-d) and simulated nPL (Fig. S4e-h) match the corresponding experimental confocal nPL displayed in Fig. S4i-l. However, one can notice one bright spot on the lower edge of the right triangular pad is always visible in the confocal maps, irrespective of the excitation polarization direction, and is attributed to a small defect in the substrate near the diabolo. Such a defect does not affect transmittance when the diabolo is excited far from the defect (in position (I) for example) and the signal collected in leakage image plane mapping. This indicates that the defect might not be directly connected to the diabolo structure but within a spot radius when the confocal detection is recorded close to the channel entrance. Incidentally, this defect close but not on the diabolo may explain the small but real asymmetry observed in image plane maps recorded when exciting in Ci (Figs. 4a and4b) that are not accounted for by the simulation on the realistic model (Figs. 4c and4d). Indeed, when exciting in Ci, the defect may be excited alongside and generate a non-symmetrical input configuration.

S4 Signal routing through symmetrical degenerate plasmon modes

The four SP-LDOS hot spots can be exploited to input and output optical signal as shown both numerically and experimentally in Figure S5 in complement to Figures 1 and2. When the diabolo device is excited at position (I) (Fig. S5a) and (I') (Fig. S5a), the simulated transmission of the plasmonic signal to the right cavity of the diabolo device is seen at the diagonally opposed (O) and (O') respectively, for an excitation polarization along the channel. These identical but symmetrical transmission patterns establish the mirror symmetry of the optical signal transmission between paired SP-LDOS hotspots.

The experimental excitation in (O') (Fig. S5c) and (O) (Fig. S5d) results in localized outputs in (I') and (I) respectively and shows the same symmetrical coupling of the SP-LDOS hotspots (I)-(O) and (I')-(O') through degenerated modes linking these diagonally opposed apexes.

Incidentally, Fig. S5d demonstrates the reversible excitation at the position (O) with a horizontally polarized beam and output in (I) compared to Fig. 2d, where the excitation in (I) and the output in (O). The experimental wide field images and computed transmission images presented in Figure S10, thus confirm the reversibility and full symmetry of the optical transmission functionality of the diabolo device. The transmittance is mediated by a set of energetically degenerated plasmonic eigenstates that are all accounted for together in the corresponding confocal SP-LDOS maps (Fig. S4).

These results demonstrate that the diabolo structure acts as a routing transmission devices in which the choice of the excitation location, (I) or (I'), results in the non-linear emission from either (O) or (O') and vice-versa. The diabolo used here is described in Fig. 2a. The left side of the cavity is color enhanced by a factor 10x.

S5 Polarization dependency of the nPL intensity in image plane maps

The transmittance through the diabolo structure is highly modulated by the excitation polarization direction (See Figs. 1,2,4). In Figure S6, we show the full polarization dependency of the signal recorded on image plane nPL maps at the location of the excitation beam (Input I), at the channel entrance (Ci) and at the readout output (O) for the symmetrical (Fig. S5a) and non-symmetrical (Fig. S5b) diabolos studied in the main text.

In Figure S6a, one observes that the nPL signal is alternatively maximum above each successive apex (the first being (I), the second one being the channel input Ci and the third one being the lower apex). Strikingly, the output (O) polarization dependency strictly matches the one in (Ci). The maximum intensity on (I), (Ci) and the lower apex are shifted by about 80°-90°. This graph suggest that the transmittance remains low for most polarization configurations until the channel is adequately excitedfor angles close to 0° -and then the funneling as far as the the output (O) is set without further polarization shift (red and blue curves overlap). This behavior indicates the excitation of a resonance mode delocalized over the 2D diabolo structure that is characterized by a sharp polarization-dependent condition. On the contrary, the transmittance intensity along the non-symmetrical diabolo shows a perfectly overlapping polarization evolution in (I) and (Ci) suggesting a global non-resonant modulation. The third

S6 Calculation of near-field transmittance maps and spectra

Here, we describe the numerical tool based on the Green dyadic formalism that provides the computation of the electric field transfer between two remote locations in an arbitrary shaped metallic nanostructure as illustrated in Figure S7.

Figure S7

. Schematic geometry of a plasmonic 1D channel, supported by a transparent dielectric substrate, and the associated observation window above the structure used for simulations. The size of this window can be adjusted to the dimensions of the structure. In a metallic system where surface plasmons can be excited by a focused illumination E0, of angular frequency ω and positioned at r0, the expression of the electric field Eout at a given output location r' can be expressed as follows: 4,5 𝑬𝑬 𝒐𝒐𝒐𝒐𝒐𝒐 (𝒓𝒓′, 𝒓𝒓 𝟎𝟎 , 𝜔𝜔) = ∫ 𝑲𝑲(𝒓𝒓′, 𝒓𝒓, 𝜔𝜔). 𝑬𝑬 𝟎𝟎 (𝒓𝒓, 𝒓𝒓 𝟎𝟎 , 𝜔𝜔)

where V is the volume of the metallic object, and K(r', r, ω) is the generalized field propagator that contains the entire response of the sample under any incident illumination: 4

𝑲𝑲(𝒓𝒓′, 𝒓𝒓, 𝜔𝜔) = 𝛿𝛿(𝒓𝒓 -𝒓𝒓 ′ ) + 𝜒𝜒(𝒓𝒓, 𝜔𝜔). 𝑺𝑺(𝒓𝒓′, 𝒓𝒓, 𝜔𝜔)

S(r', r, ω) is the Green Dyadic tensor describing the whole system that includes the plasmonic structure and the dielectric substrate. In addition, the frequency-dependent optical response of the metallic structure is given by:

and 𝜒𝜒(𝒓𝒓, 𝜔𝜔) = 0 outside of the metallic object.

In this work, we focused on the plasmon-mediated field transfer in the metallic structures. We therefore excluded the direct contribution of the incident illumination at the output location. 6 This can be done by removing the delta Dirac distribution in equation (2):

𝑲𝑲(𝒓𝒓′, 𝒓𝒓, 𝜔𝜔) = 𝜒𝜒(𝒓𝒓, 𝜔𝜔). 𝑺𝑺(𝒓𝒓′, 𝒓𝒓, 𝜔𝜔)

The whole set of experimental data presented in this study has been acquired using a focused Gaussian illumination. Consequently, the illumination field E0 has been modelled as a Gaussian spot through an expansion in a plane waves in the wave vector domain: [7][8][9][10] 𝑬𝑬 𝟎𝟎 (𝒓𝒓, 𝒓𝒓 𝟎𝟎 , 𝜔𝜔) = ∫ 𝑑𝑑𝑑𝑑

𝑘𝑘 0 is the vacuum wave vector of the incident light and 𝒓𝒓 𝟎𝟎 = (𝑒𝑒 0 , 𝑦𝑦 0 , 𝑧𝑧 0 ) defines the center of the excitation Gaussian spot in the Cartesian system of coordinates shown in Fig. S7. 𝜀𝜀 𝑠𝑠𝑠𝑠𝑠𝑠 is the dielectric function of the substrate and the beam waist 𝑤𝑤 0 describes the lateral extension of this incident Gaussian beam. The integration in equation ( 5) is performed in the 2D reciprocal space defined by the vector k∥ = (α, β).

In order to take into account the realistic geometry of the system with a transparent substrate, the tangential components of the field vector ζ are expressed as follows:

The transmission matrix T reads:

𝜏𝜏 ∥ and 𝜏𝜏 ⊥ are the Fresnel coefficients for the interface. Here, δ is an angle in the xy plane between the x-axis in Cartesian coordinates and the orientation of the planar component of the wave vector k∥ . In equation ( 5), the normal component 𝜁𝜁 𝑧𝑧 of 𝜻𝜻 is obtained using the following expression:

Finally, the electric intensity distributions generated in the simulated maps of the main manuscript and the figures S8 and S9 below take the following form:


A Michrafy
email: michrafy@enstimac.fr

M Michrafy

M.S Kadiri

J A Dodds

Predictions of tensile strength of binary tablets using linear and power law mixing rules

Keywords: Compaction, Tensile strength, Binary mixtures, Tablets, Mixing rules

There has recently been an increased interest in predicting the tensile strength of binary tablets from the properties of the individual components. In this paper, measurements are reported for tensile strength of tablets compressed from single-component and binary powder mixtures of lactose with microcrystalline cellulose (MCC), and lactose with two types of silicified microcrystalline cellulose (SMCC and SMCC-HD), which are different in compressibility. Measurements show the tensile strength increases with the relative density for single powders, and both with the relative density and the mass fraction of cellulose in the mixtures. It was also observed, for binary mixtures compacted at 50 and 150 MPa, that there was a slight variation in porosity with the mass fraction of celluloses.

The predictions of the tensile strength of binary tablets from the characteristics of the single-components was analysed with the extended Ryshkewitch-Duckworth model by assuming both linear and power law mixing rules for the determination of the parameters "tensile strength at zero porosity and bonding capacity constant". As consequence, four models were analysed and compared with measurements using criteria based on the standard deviation from the mean values. Results showed a good prediction using a linear mixing rule combined with the power law. However, as the predictions of these models depend on the powders and the porosity range for the characterization of single-components, none of them can be systematically considered as being the best to predict binary behaviour from data for individual powders.

Introduction

Studies of tablet formation by direct compression are often based on a single free powder, a granulated powder or a powder with an addition of a small percentage of binder. As the majority of formulated pharmaceutical tablets consist of mixtures of more than one powder, an important problem is to predict the properties of tablets obtained by compression of powder mixtures from the properties of the individual components. In most published work, tensile strength of tablets is implicitly assumed to be a reflection of the bonding contact between particles following the compaction. However, factors such as particle size, particle shape, porosity distribution may also be involved. In binary tablets, several studies have been carried out on the correlation between tensile strength and composition of the two components, particularly in binary mixtures where one of powders is ductile and the other is fragmentary [START_REF] Sheikh-Salem | Compaction characteristics of mixtures of materials with dissimilar compaction mechanisms[END_REF][START_REF] Leuenberger | Compression of binary powder mixtures and solubility parameters of solids[END_REF][START_REF] Vromans | Densification properties and compactability of mixtures of pharmaceutical excipients with and without magnesium stearate[END_REF][START_REF] Riepma | Consolidation and compaction of powder mixtures. I. Binary mixtures of same particle size fractions of different types of crystalline lactose[END_REF][START_REF] Kuentz | A new theoretical approach to tablet strength of binary mixture consisting of a well and poorly compactable substance[END_REF][START_REF] Hutin | Formulation of solid products: two component tablets with a disintegrating agent[END_REF][START_REF] Ramirez | Comparison of different mathematical models for tensile strength-relative density profiles of binary tablets[END_REF][START_REF] Van Veen | Predicting mechanical properties of compacts containing two components[END_REF][START_REF] Wu | A simple predictive model for the tensile strength of binary tablets[END_REF]. Two main methods are used in such studies. The first involves the characterization of compaction properties of mixtures and the influence of mixture composition on pressure-density relationships. The second is oriented toward predictions of the properties of binary tablets from the properties of the individual components [START_REF] Chan | The tensile strengths of single powders and binary mixtures[END_REF][START_REF] Kuentz | A new theoretical approach to tablet strength of binary mixture consisting of a well and poorly compactable substance[END_REF][START_REF] Ramirez | Comparison of different mathematical models for tensile strength-relative density profiles of binary tablets[END_REF][START_REF] Wu | A simple predictive model for the tensile strength of binary tablets[END_REF]Busignies et al., 2006a,b). Literature results for the compaction of binary mixtures show the complexity of the phenomena involved and the problems inherent in defining a single theory to explain the tensile strength of binary tablet from the behaviour of individual components. The work of [START_REF] Sheikh-Salem | Compaction characteristics of mixtures of materials with dissimilar compaction mechanisms[END_REF] showed that the addition of small quantity of lactose to sodium chloride considerably reduces the tensile strength. The study of [START_REF] Vromans | Densification properties and compactability of mixtures of pharmaceutical excipients with and without magnesium stearate[END_REF] showed that the addition of small quantities of roller-dried ␤-lactose to cellulose does not produce a significant decrease of tensile strength even if cellulose and sodium chloride seem to have similar mechanical behaviour in compaction. The physicochemical properties (adhesive or cohesive forces) may play a significant role in bonding particles and could explain the results mentioned above. In particular [START_REF] Sheikh-Salem | Compaction characteristics of mixtures of materials with dissimilar compaction mechanisms[END_REF] assumed that lactose and sodium chloride have no bonding affinity. In addition to these complexities, factors such as powder flowability and compressibility are also pertinent parameters that could induce effects on values of the tensile strength of tablets formed from mixed powders. Indeed, good flowability would favour the homogeneity of the mixing, whereas the good compressibility gives the possibility of increasing tensile strength at low pressures. This latter behaviour is important for selecting the appropriate excipient for mixing with powders such as nutraceutical and functional foods where the compaction load has to be moderate at the same time as the tablet must have a sufficient resistance for handling.

The objective of this study is first to experimentally determine tensile strength of tablets compressed from binary mixtures of lactose and one of three different microcrystalline celluloses. Then the prediction of tensile strength of binary tablets from individual properties of the components is analysed by assuming both linear and power law mixing rules for calculating the parameters of the modified Ryshewitch-Duckworth model. Predictions of the different proposed combinations are compared with the experimental data by a statistical criterion and the possibility of finding better combinations is discussed.

Materials and methods

Materials

Excipients differing in mechanical properties such as flowability, compressibility or compactability are often used to optimize the user properties of tablets and confer functionalities to a formulation. Excipients containing microcrystalline cellulose have been developed by JRS Pharma to give powders such as Prosolv ® 90 and Prosolv ® 90 HD for pharmacy and foods formulation. Compared with microcrystalline cellulose Vivapur ® 102 (MCC), these silicified microcrystalline cellulose excipients (noted SMCC) contain 98% of MCC and 2% of colloidal silicon dioxide co-processed to give intimate mixtures [START_REF] Sherwood | A new class of high functionality excipients: silicified microcrystalline cellulose[END_REF]. They present a rougher surface, a better flowability and compressibility, this is particularly the case for Prosolv ® HD 90 (noted SMCC-HD). The basic characteristics of these powders (particle size, bulk density and true density) were presented in Table 1.

To analyse properties of tablets compacted from binary mixtures, MCC, SMCC and SMCC-HD excipients were added to the lactose Granulac ® 140 (Meggle Gmbh) in different compositions. Scanning electron microscopy images of these excipients are shown in Fig. 1.

Preparation of the binary mixtures

The lactose powder was blended with MCC (respectively, SMCC and SMCC-HD) in various mass fractions in 100 g lots that were thoroughly mixed in a Turbula mixer (from Turbula ® , System Schatz, Basel Switzerland) for 15 min at 32 rpm. The true density of single-components and binary mixtures was measured using a Micrometrics AccuPyc 1330 helium gas displacement pycnometer. Results of the true densities of powders and mixtures are reported in Table 1 for single powders and in Table 2 for mixtures.

Tablets preparation

Cylindrical flat tablets have been made from these powders by the following procedure. A mass of 0.4 ± 0.003 g of powder (respectively, single-components or binary mixtures of powders) was manually poured into a cylindrical die 11.28 mm in diameter (die section 1-cm 2 ) and compacted at different pressures using an Instron universal testing machine with a 30 kN load cell. The compaction was made at the speed of 10 mm/min at the ambient temperature of 25 ± 1 • C. Once the tablet was ejected, its mass was measured with an electronic balance (CP 224S, Sartorius, Germany) and its dimensions (diameter and thickness) were measured with digital micrometer (Mitutoyo). The apparent and relative densities were recorded. The diametrical crushing load was then measured with the Erweka TBH 30 apparatus (from Erweka ® Gmbh Heussenstamm, Germany). For each pressure level, three compacts were made using this procedure and each measurement presented in this study is the mean of the characteristics of three tablets.

Characterization of tablets

The relative density of a tablet is defined as the ratio of the apparent density ρ a of tablet to the true density ρ t of powder. It is equal to 1 minus the porosity. For each composition, the true density of powders was measured with a helium pycnometer and the apparent density was calculated from the powder mass and the tablet dimensions (diameter and thickness) as:

ρ a = m πhD 2 /4 (1)
where m is the mass of the tablet, and h and D its thickness and diameter.

The tensile strength (σ t ) of tablet is calculated as follows [START_REF] Fell | Determination of tablet strength by the diametrical compression test[END_REF]:

σ t = 2F πDh (2)
where F is the load required to break the tablet diametrically.

Approaches to predict tensile strength of binary mixtures tablets

In the literature, two approaches have been investigated for predicting the tensile strength σ t,m of binary tablets from the relative density ρ r,m (or the porosity ε = 1ρ r,m ) of the mixture. One is based on Percolation theory [START_REF] Leuenberger | Compression of binary powder mixtures and solubility parameters of solids[END_REF][START_REF] Kuentz | A new theoretical approach to tablet strength of binary mixture consisting of a well and poorly compactable substance[END_REF][START_REF] Ramirez | Comparison of different mathematical models for tensile strength-relative density profiles of binary tablets[END_REF] and the second on the Ryshkewitch-Duckworth model [START_REF] Ryshkewitch | Compression strength of porous sintered alumina and zirconia[END_REF][START_REF] Duckworth | Discussion of Ryshkewitch paper[END_REF][START_REF] Wu | A simple predictive model for the tensile strength of binary tablets[END_REF].

Percolation theory applied to the tensile strength of binary mixtures results in a power law expression involving the relative density and a critical value corresponding to the percolation threshold ρ c,m (minimum solid fraction needed to build a network of relevant contact points). It is assumed that one component dominates the overall strength of the tablet. For tablets with a relative density ρ r,m > ρ c,m , the tensile strength σ t,m is given as:

σ t,m = k m (ρ r,m -ρ c,m ) 2.7 (3)
where k m is a material constant and the 2.7 is a universal constant related to the structure of the particle contact network. The critical relative density of the mixtures ρ c,m is related to those of the components ρ c,1 and ρ c,2 as: The constant α is the mass fraction of the well compactable component in the binary mixtures. However, in our case it was difficult to determine the critical relative density of lactose powder. Between 50 and 70 MPa of pressure, the tensile strength of tablets was zero. At 100 MPa of pressure, the tensile strength was non-null. The tensile strength versus relative density curve of the lactose is plotted in Fig. 2. The change of the slope that corresponds to the percolation threshold could occur between 75 and 95 MPa. Unfortunately, in this range, no data was recorded. Our study was then oriented towards the application of the Ryshkewitch-Duckworth approach. The Ryshkewitch-Duckworth model was first applied to the tensile strength of porous sintered alumina and zirconia [START_REF] Ryshkewitch | Compression strength of porous sintered alumina and zirconia[END_REF] and discussed in [START_REF] Duckworth | Discussion of Ryshkewitch paper[END_REF]. The model was recently adapted and applied to predict the tensile strength of binary tablets of MCC and HPMC or MCC and Starch [START_REF] Wu | A simple predictive model for the tensile strength of binary tablets[END_REF]. The predictive equation of tensile strength for binary tablets is as follows:

ρ c,m = αρ c,1 + (1 -α)ρ c,2 (4)
ln σ t,m σ ∞,m = -κ m ε m (5)
where σ ∞,m is the tensile strength of binary tablet at zero porosity and κ m is a material constant representing the bonding capacity of the material. These two constants depend on the composition of the powder mixture. The tensile strength of binary tablets at zero porosity σ ∞,m is here assumed to be determined from those of components σ ∞,1 and σ ∞,2 using two different combination rules (i and ii):

(i) The linear mixing rule which assumes that in a fully densified material the volumes of the components do not undergo notable changes:

σ ∞,m = ησ m,1 + (1 -η)σ m,2 . (6)
(ii) A power law relation as in [START_REF] Leuenberger | Compression of binary powder mixtures and solubility parameters of solids[END_REF][START_REF] Kuentz | A new theoretical approach to tablet strength of binary mixture consisting of a well and poorly compactable substance[END_REF]:

σ ∞,m = (σ m,1 ) η (σ m,2 ) 1-η . ( 7
)
Here η is the volume fraction of SMCC that is related to the weight fraction α, the true density ρ t,1 and the true density of the binary mixtures ρ t,m as:

η = αρ t,m ρ t,1 (8)
The true density ρ t,m of the binary mixtures can be expressed by linear combination as:

ρ t,m = α ρ t,1 + 1 -α ρ t,2 -1 (9)
It follows that predictions of tensile strength of a binary tablet can be determined from the properties of the individual components if the material constant of the binary mixtures κ m can be estimated from the material constants κ 1 and κ 2 of the two components. Here also there are two possibilities: the linear mixing and the power law rules:

κ m = ηκ 1 + (1 -η)κ 2 ( 10
)
κ m = κ η 1 κ 1-η 2 (11)
Finally, by determining material parameters of components from experimental data such α, ρ t,i , κ i and σ m,i (i = 1, 2) and assuming a linear or power law mixing rule (Eqs. ( 6), ( 7), ( 10) and ( 11)), the material parameters of the binary mixtures such as true density ρ t,m , tensile strength at zero porosity σ ∞,m and the constant κ m can be predicted. It follows from Eq. ( 5) and expressions of σ ∞,m and κ m that there are four possibilities for predicting the tensile strength of binary tablets from the relative density:

σ t,m = σ ∞,m exp(-κ m (1 -ρ r,m )) ( 12
)
where κ m and σ ∞,m are calculated from the following equations:

Linear-linear (LL) Eqs. ( 6) and (10) Power-power (PP) Eqs. ( 7) and (11) Linear-power (LP) Eqs. ( 6) and (11) Power-linear (PL) Eqs. ( 7) and (10)

Predictions from the above models will be compared with the experimental data of the powder mixtures "lactose/MCC", "lactose/SMCC", "lactose/SMCC-HD" using a statistical criterion (mean values and standard deviation) over the range of porosity studied.

Criteria for comparisons

One way to compare and discriminate between the predictions of the four models presented below (Eq. ( 12) and (LL), (PP), (LP) and (PL) combinations) is to calculate, for each mixture, the mean of the absolute difference between the measured σ t,meas and the predicted σ t,pred ( σ = |σ t,measσ t,pred |) tensile strength over the studied range of porosity.

mean = Σ σ N ( 13
)
and standard deviation, which measures the dispersion of values about the mean

disp = NΣ σ 2 -(Σ σ) 2 N 2 1/2 ( 14
)
where N is the number of data in the range of porosity and |•| represents the absolute value. Results of this comparison are presented in Tables 4b, 5b and 6b for each mixture.

Results and discussion

Tensile strength of tablets made from single-component powders

Tensile strength of tablets prepared from single-component powders (lactose, MCC, SMCC and SMCC-HD) is plotted in Fig. 2a as a function of relative density. It can be seen that for all powders the tensile strength increases with the relative density. Errors of measurements of tensile strength (error bar was plotted in Fig. 2a were less than 0.15 MPa for SMCC (respectively, 0.15 for SMCC-HD, 0.15 for MCC and 0.12 for lactose). As it is shown in Fig. 2a, tensile strength of single-components is in the order "lactose < SMCC-HD < MCC < SMCC".

The compressibility of powders MCC, SMCC, and SMCC-HD (the compressibility is the ability of powder to reduce in volume under pressure) is represented by the "porosity versus pressure" curves plotted in Fig. 2b. It can be seen that SMCC-HD is more compressible than MCC and SMCC. This means that, tablets of SMCC-HD need less pressure than do those made from MCC and SMCC, to reach the same porosity and hence the same tensile strength (porosity is the primary factor influencing tensile strength). As a consequence, SMCC-HD seems to be more appropriate than MCC and SMCC for the optimization of nutraceutical and enzymes formulations where the required cohesion has to be sufficient at moderate pressures.

Properties of binary powder mixtures and tablets

The true density of binary mixtures "MCC/lactose", "SMCC/lactose" and "SMCC-HD/lactose" were measured for the proportions 25-75, 50-50 and 75-25% in mass. Results are reported in Table 2. Predictions of the true densities of binary mixtures based on Eqs. ( 8) and ( 9) and the true densities of the single-components (Table 1), are given in Table 2. The predicted and measured true densities are in agreement. This result is con- the studied mixtures. The values of these constants are reported in Table 3. The correlation coefficients R 2 (R 2 > 0.98) are also reported and indicate the good fit of data with the model.

The variation of the tensile strength of binary tablets (MCC, SMCC or SMCC-HD) with the mass fraction of the more compactable powder (the compatibility is the ability of powder to produce a tablet with a specific strength) is plotted in Figs. 6 and7 for pressures of 50 and 150 MPa. Changes of porosity as a function of the composition are also reported in the same figures. At a pressure of 50 MPa the porosity increases slightly and linearly with the mass fraction of the cellulose, whereas the strength increases. At higher pressure (150 MPa), the porosity shows a small variation as a function of mixture composition, whereas the tensile strength increases. This is a consequence of the plastic behaviour of the cellulose where at high pressure, the greater is the composition of MCC (respectively, SMCC) the greater the increase in strength with small changes of volume. Measurements of pore diameters by mercury porosimetry may help to understand this increase in strength of tablets with the increase of cellulose composition.

Predictions of tensile strength for binary mixtures

The combinations (LL), (PP), (LP) and (PL) of the linear mixing rule and power law predicting κ m and σ ∞,m and hence tensile strength of a binary mixture from Eq. ( 12), were applied to the powder mixtures studied (SMCC/lactose, SMCC-HD/lactose and MCC/lactose). The determination of κ m and σ ∞,m for each composition requires values of bonding constant and tensile strength at zero porosity of single-components (reported in Table 1) and the volume fraction η. Then the tensile strength was evaluated from Eq. ( 12) for the same porosities as measured. In Table 4a (respectively, Tables 5a and6a) are given predictions of tensile strength of mixtures SMCC/lactose (respectively, SMCC-HD/lactose, MCC/lactose). The measured tensile strength is also listed.

In order to compare the predictions of the different combinations with the measurements for all compositions and over the range of porosities, the average and standard deviation (Eqs. ( 13) and ( 14) for each composition and for each range of porosity were used. Results of the evaluation are presented in Table 4b (respectively, Tables 5b and6b) for SMCC/lactose mixtures (respectively, SMCC-HD/lactose and MCC/lactose). It can be seen from results for mixtures SMCC/lactose and SMCC-HD/lactose, that combination (LP) gives the best predictions, whereas combination (PL) was the best for mixtures MCC/lactose. However, this result should not be systematically generalised for others powders. The best combination depends on powders, the number of measurements of single-components, the precision on the measurements . . ..

Conclusion

Experiments have been performed to determine the tensile strength of tablets compressed from single powders of lactose, microcrystalline cellulose (MCC) and two silicified microcrystalline cellulose having different levels of compressibility (SMCC and SMCC-HD). Binary tablets have also been made from mixtures of lactose and these components and their tensile strength determined as a function of mixture composition. It is found that the tensile strength increases with the relative density, for single and binary powder mixtures. For binary tablets, it is shown that the tensile strength increases with the cellulose composition while the porosity undergoes slight changes. As the porosity is the primary factor influencing tensile strength, estimation of changes of median pore diameter (mercury porosimetry) as a function of the composition of cellulose may be a more pertinent analysis.

Predictions of the tensile strength of binary tablets from the properties of single-components have been analysed with Ryshkewitch-Duckworth model by assuming either a linear mixing rule or a power mixing law rule for the determination of material parameters. The different combinations (four) were analysed and compared with measurements by using statistical criteria based on the mean values and the standard deviation over the range of porosities studied. Good predictions were obtained with the linear mixing rule combined with the power law. However, as the predictions of these models depend on the powders and the range of porosity rmeasurements, the conclusions are only valid for the powders used in this study and should not be systematically generalised to other powders.

Fig. 1 .

 1 Fig. 1. Scanning electron microscopy images of powders lactose (Granulac ® 140) (a), MCC (Vivapur ® 102) (b), SMCC (Prosolv ® 90) (c) and SMCC-HD (Prosolv ® 90 HD) (d).

Fig. 2 .

 2 Fig. 2. (a) Tensile strength of tablets compacted from powder of lactose, MCC, SMCC, SMCC-HD (tablets prepared at different pressures in a 11.28 mm die diameter). (b) Porosity-pressure correlations for MCC, SMCC and SMCC-HD powders.

Fig. 3 .

 3 Fig. 3. Tensile strength against relative density: mixture of SMCC and lactose (tablets prepared in a 11.28 mm die diameter at different pressures.

Fig. 5 .

 5 Fig. 5. Strength of binary tablets of MCC/lactose vs. relative density. Effect of the fraction mass of MCC in mixtures.

Fig. 6 .

 6 Fig. 6. Tensile strength (TS) and porosity vs. composition of celluloses. Binary mixtures: lactose/SMCC, lactose/SMCC-HD and lactose/MCC (pressure = 50 MPa).

Fig. 7 .

 7 Fig. 7. Correlation between tensile strength (TS) and porosity vs. composition of celluloses. Binary mixtures: lactose/SMCC, lactose/SMCC-HD and MCC/lactose (pressure = 150 MPa).

Table 1

 1 Basic characteristics and fitted material parameters of Ryshkewitch-Duckworth model from "porosity-tensile strength" curve of powders: lactose (Granulac ® 140 from Meggle Gmbh), MCC (Vivapur ® 102), SMCC (Prosolv ® 90) and SMCC-HD (Prosolv ® 90 HD) from J. Rettenmaier & Söhn

	Materials	Average particle	Bulk density (g/cm 3 )	True density (g/cm 3 )	Bonding constant	Tensile strength at
		size (m)				zero porosity (MPa)
	SMCC SMCC-HD MCC Lactose	90 90 90 70	0.32 0.32 0.32 0.66	1.5981 ± 0.003 1.5898 ± 0.001 1.599 ± 0.0012 1.5617 ± 0.0022	7.72 7.14 5.95 20.15	37.14 24.19 21.11 12.89

Table 3

 3 Fitted material parameters of Ryshkewitch-Duckworth model from "porosity-tensile strength" measurements of binary mixtures "MCC/lactose", "SMCC-HD/lactose" and "SMCC/lactose"

	Binary mixtures	Mass fraction of lactose	κ m	σ ∞,m	R 2
		0.75	9.93	4.32	0.9824
	MCC/lactose	0.5	10.45	14.33	0.9883
		0.25	7.53	15.79	0.9933
		0.75	13.41	11.53	0.9982
	SMCC-HD/lactose	0.5	10.56	13.37	0.9979
		0.25	10.47	23.28	0.9915
		0.75	12.76	12.85	0.9976
	SMCC/lactose	0.5	9.76	19.66	0.9921
		0.25	8.50	25.94	0.9926

Table 4a

 4a Predictions of tensile strength for binary mixtures tablets (SMCC/lactose) with Ryshkewitch-Duckworth model according to (LL), (PP), (LP) and (PL) combinations for calculating σ ∞,m and κ m from values of components in Table1

	Table 5b
	Corresponding mean and dispersion values for models LL, PP, LP and PL (SMMC-HD/lactose mixtures)
	Mass fraction of mixtures Mixtures 75% SMCC-HD/25% lactose 50% SMCC-HD/50% lactose 25% SMCC-HD/75% lactose	Porosity Porosity range 0.53 0.44 0.16-0.48 0.14-0.4 0.15-0.29	σ t (MPa), measured Mean ± dispersion (LL) σ t (MPa), model (LL) Mean ± dispersion (PP) σ t (MPa), model (PP) Mean ± dispersion (LP) σ t (MPa), model (LP) 0.24 0.10 0.15 0.17 0.66 0.26 0.37 0.41 0.53 ± 0.97 0.18 ± 0.22 0.33 ± 0.27 0.28 ± 0.12 0.11 ± 0.09 0.11 ± 0.15 0.22 ± 0.07 0.10 ± 0.03 0.07 ± 0.03	Mean ± dispersion (PL) σ t (MPa), model (PL) 0.09 0.23 0.11 ± 0.07 0.32 ± 0.13 0.24 ± 0.09
			0.38		1.14	0.50	0.68	0.77	0.46
	75% SMCC/25% lactose Table 6a Predictions of tensile strength of binary mixtures tablets (MCC/lactose) with Ryshkewitch-Duckworth model according to (LL), (PP), (LP) and (PL) combinations 0.34 1.61 0.80 1.04 1.17 0.74 0.30 2.24 1.25 1.54 1.74 1.14 0.22 3.84 2.70 3.11 3.50 2.47 0.20 4.72 3.53 3.97 4.47 3.24 for calculating σ ∞,m and κ m from values of components in Table 1
	Mass fraction of mixtures		0.14 Porosity		6.95 σ t (MPa),	6.64 σ t (MPa),	7.03 σ t (MPa),	7.92 σ t (MPa),	6.08 σ t (MPa),
			0.41		0.29 measured	0.08 model (LL)	0.13 model (PP)	0.15 model (LP)	0.07 model (PL)
	50% SMCC/50% lactose 75% MCC/25% lactose		0.36 0.32 0.29 0.29 0.23 0.22 0.20 0.18 0.16 0.15 0.13 0.11		0.53 0.85 0.14 1.14 0.43 2.06 2.46 0.78 1.23 3.73 1.57 4.09	0.17 0.29 1.14 0.45 2.16 1.08 1.58 3.42 4.18 2.99 5.28 6.50	0.25 0.40 1.71 0.59 2.95 1.32 1.85 4.35 5.15 3.26 6.28 7.49	0.29 0.46 1.75 0.68 3.01 1.51 2.11 4.44 5.25 3.73 6.41 7.65	0.15 1.12 0.25 2.12 0.39 0.95 3.35 4.09 1.38 5.17 2.61 6.37
	25% SMCC/75% lactose 50% MCC/50% lactose		0.33 0.29 0.29 0.26 0.24 0.21 0.18 0.19 0.15 0.15 0.15 0.12		0.18 0.32 0.19 0.44 0.32 0.89 0.55 1.14 0.75 1.83 1.29 1.91	0.07 0.13 0.39 0.20 0.75 0.50 1.54 0.72 2.36 1.48 2.36 3.34	0.09 0.16 0.70 0.25 1.20 0.57 2.20 0.80 3.14 1.56 3.14 4.21	0.10 0.18 0.72 0.28 1.24 0.64 2.26 0.91 3.24 1.76 3.24 4.34	0.06 0.38 0.11 0.73 0.18 1.50 0.44 2.29 0.64 2.29 1.31 3.25
			0.10		2.81	4.36		5.26	5.42	4.23
			0.28		0.24	0.14		0.23	0.23	0.14
	Table 4b		0.21		0.33	0.46		0.64	0.65	0.45
	Mean and dispersion for models LL, PP, LP and PL (SMMC/lactose mixtures) 25% MCC/75% lactose 0.18 0.64	0.71		0.95	0.97	0.69
	Mixtures	Porosity range 0.15 0.14	Mean ± dispersion (LL) 0.96 1.60	Mean ± dispersion (PP) 1.14 1.34	Mean ± dispersion (LP) 1.46 1.49 1.68 1.72	1.12 Mean ± dispersion (PL) 1.31
	75% SMCC-25% lactose 50% SMCC-50% lactose 25% SMCC-75% lactose Table 6b Mean and dispersion values	0.14-0.53 0.15-0.41 0.15-0.33	0.70 ± 0.37 0.63 ± 0.25 0.29 ± 0.11	0.46 ± 0.26 0.47 ± 0.18 0.23 ± 0.09		0.40 ± 0.25 0.30 ± 0.18 0.16 ± 0.07	0.87 ± 0.42 0.75 ± 0.34 0.34 ± 0.015
	Mixtures	Porosity range		Mean ± dispersion (LL)	Mean ± dispersion (PP)	Mean ± dispersion (LP)	Mean ± dispersion (PL)
	Table 5a Predictions of tensile strength of binary mixtures tablets (SMCC-HD/lactose) with Ryshkewitch-Duckworth model according to (LL), (PP), (LP) and (PL) combi-75% MCC/25% lactose 0.11-0.29 2.41 ± 0.86 3.28 ± 1.0 3.38 ± 1.03 2.33 ± 0.83 50% MCC/50% lactose 0.10-0.29 1.04 ± 0.51 1.72 ± 0.71 1.80 ± 0.75 0.98 ± 0.47 25% MCC/75% lactose 0.14-0.28 0.15 ± 0.07 0.24 ± 0.18 0.26 ± 0.18 0.14 ± 0.08 nations for calculating σ ∞,m and κ m from values of components in Table 1 Mass fraction of mixtures Porosity σ t (MPa), σ t (MPa), σ t (MPa), σ t (MPa), t (MPa), Comparison between models LL, PP, LP and PL (MMC/lactose mixtures).
					measured	model (LL)		model (PP)	model (LP)	model (PL)
			0.48		0.14	0.15		0.25	0.26	0.14
			0.39		0.43	0.35		0.54	0.56	0.34
	75% SMCC-HD/25% lactose		0.33 0.28		0.78 1.23	0.67 1.13		0.94 1.51	0.98 1.56	0.64 1.09
			0.26		1.57	1.40		1.82	1.88	1.35
			0.16		4.09	4.18		4.84	5.00	4.04
			0.40		0.19	0.07		0.14	0.15	0.07
			0.35		0.32	0.15		0.26	0.27	0.15
			0.31		0.55	0.27		0.42	0.44	0.25
	50% SMCC-HD/50% lactose		0.27		0.75	0.43		0.64	0.67	0.41
			0.22		1.29	0.89		1.23	1.29	0.85
			0.19		1.91	1.45		1.87	1.97	1.38
			0.14		2.81	2.59		3.13	3.28	2.47
			0.29		0.24	0.11		0.16	0.17	0.11
			0.26		0.33	0.18		0.25	0.26	0.17
	25% SMCC-HD/75% lactose		0.22		0.64	0.39		0.50	0.52	0.37
			0.18		0.96	0.71		0.87	0.91	0.68
			0.15		1.60	1.28		1.50	1.56	1.23

Acknowledgement

We would like to thank Mr. Laurent Ouvrard from the company J. Rettenmaier & Shöne (France) for the given lots of powders Prosolv ® 90 and Prosolv ® 90 HD.