Dominique Barchiesi

B Guizal

Thomas Grosges

Accuracy of local field enhancement models: toward predictive models?

Accuracy of local field enhancement models: toward

predictive models?

Introduction

Since the early developments of near-field optical microscopies, the variety of their applications has not ceased to widen. Recently, a new area of the nanotechnology appeared: the plasmonic. The applications concerned with this field go from telecommunications to biology, passing by the physics of the mesoscopic domain. In plasmonics, the use of electromagnetic field enhancement obtained either by the antenna effect, dependent on the electromagnetic singularities around material tips or by plasmon resonance, opens the way to local measurements of interaction between radiation and matter [1].

However, a direct measurement of this enhancement is not yet possible and the lithography techniques used to build the nanostructure are rather expensive. Therefore, it is necessary to use theoretical forecasts to optimize and functionalize the produced nanostructures. The field enhancement is strongly dependent on physical parameters like wavelength, shape, size, polarization, angle of incidence, and material properties. In the particular case of SNOM (scanning near-field optical microscopy) many papers are dedicated to the computation of the near-field diffracted by a nanometric probe. The computed intensity enhancement varies from 3000 [2] to 10 000 [1], between 10 and 160 000 [3] and it has been shown to be ten ✉ Fax: +33-(0)3-25-71-84-56, E-mail: dominique.barchiesi@utt.fr orders of magnitude in amplifying medium than what would be expected for plasmon resonance in a passive medium [4]. The accurate computation of the field enhancement is necessary to carry out predictions and to design the resonant structures for SERS, and TERS.

In the near-field optical community, several numerical methods have been used for a long time. In principle, all the methods which make possible the computation of the optical near-field around metallic nanostructures, illuminated by an external source, could be used. Nevertheless, only a few of them are able to deal with high local variations of the electric field. This requirement is directly connected with the numerical convergence and the discretization of the problem. Moreover, the discretization required to compute the solution is in the direct space for methods like finite difference time domain (FDTD) or the finite element method (FEM), or in the Fourier space for the spectral methods (like the differential, coupled wave (CWM), volume or surface integral (Green) methods). Let us note that the difference between the methods is sometimes subtle due to the fact that they can be close to each other or even mixed. As an example, the CWM could be considered as a modification of the classical differential methods, or the finite element method can be applied, with some refinements, to any set of differential equations [5].

Various methods (commercial or free of use codes) must be carefully handled especially in the case of high field enhancement computation. To illustrate this important fact, we will present an example of an application in Sect. 2. Section 3 will be devoted to investigating the convergence of three numerical methods, before concluding and proposing possible improvements of the methods.

2

The plasmonic design: the resolution of the inverse problem

An initial problem that has to be solved before designing nanostructures is the measurement of their true permittivity and shape. At present, the bulk permittivities are commonly used in simulations to compute the field enhancement. Moreover, the measurement of the topography of metallic nanostructures with atomic force microscopy (AFM) suffers from the convolution of the AFM probe shape by that of the nanostructure. Therefore, the metrology applications must be performed through an inversion procedure to deduce values from the data recorded experimentally, as it is done in the medical domain. To take into account the important processes of the image formation, an accurate model has to be developed. The principle of an inversion procedure is based on the use of an accurate model to deduce the unknown parameters from experimental data (Fig. 1).

An inversion procedure based on an evolutionary scheme was recently used to perform the recovery of unknown experimental parameters [6]. Also, it was shown theoretically that the measurement of the permittivity of nanostructure is possible [7]. The direct method used in this inversion procedure was FDTD including, an improvement of the model of dispersion [8]. The use of classical FDTD (with Cartesian mesh, commonly used in commercial codes with regular or non-regular Yee's cells) to compute the field enhancement around a metallic nano-wire has been investigated in [9]. It has been shown that to achieve realistic spectroscopic studies, the models must take into account both the geometric and dispersive characteristics of the complex materials. In particular, an appropriate refinement of the grid must be used in order to assure the stabilization and the convergence to the physical solution, especially in the regions where a strong confinement of the light around the nanostructure occurs. In FDTD, artificial plasmons are generated at the vertices of metallic cells, which may prevent computation of the reliable field enhancement [9][10][11]. In plasmonics, the problem of accuracy and the difficulty of obtaining convergence of the models is illustrated in Fig. 2 of [12], where the computed transmission of a square coaxial aperture can vary by more than 10% between FDTD and the Fourier modal method.

In the following section, three popular methods for field computation will be investigated in the case of metallic nanostructures.

Convergence of CWM, FEM and Green methods

To simplify the discussion, the field enhancement is computed in the vicinity of a nano-wire with rectangular shape (200 × 20 nm 2 ) with relative permittivity ε =-10. This permittivity corresponds to gold material at a wavelength FIGURE 1 Principle of the inverse scheme to measure unknown parameters from experimental data. An accurate model of the imaging process and an efficient inversion procedure are needed λ = 632.8 nm. Due to the negative value of the relative permittivity, the excitation of plasmon resonance (which depends on the geometry of the object) can occur. The imaginary part of the permittivity is neglected in order to bench the numerical models with high requirements. Actually, the imaginary part would contribute in widening the resonance peaks and therefore would facilitate the convergence of the methods. Nevertheless, this hypothesis is not restrictive, and all the numerical models used here can deal with complex permittivities.

The nanostructure is illuminated at normal incidence, with a p-polarized plane wave. The computation of the electric field is achieved in reflection, 2 nm above the nanostructure. The intensity enhancement is the square modulus of the electric field and strongly varies at the edges of the nanostructure. Therefore, high order evanescent diffracted waves are required to converge to the solution. Moreover, in order to compare the efficiency of the three investigated methods, the Matlab codes run on the same personal computer.

In the following subsections, the convergence of the Coupled Wave Method, the Volume Integral Method, and the Finite Element Method will be investigated.

3.1

The coupled wave method (CWM)

The coupled wave method (CWM) is widely used due to its implementation simplicity and effectiveness, especially in the case of lamellar structures. This method is a modification of the differential method [13]. In [14] the method was fully described and was applied to the case of a glass aperiodic structure. The code used for the present CWM simulations is the same as in [14], but in this work, the sample is illuminated with a plane wave rather than a gaussian beam. Therefore, the nanostructure is a grating made of gold wires with rectangular shape with a 400 nm period (Fig. 2). The convergence of this spectral method depends only on the number of diffracted waves involved in the computation.

The field enhancement at the edges of the nanostructure is clearly related to the singularity of the electric field in the region where the permittivity abruptly changes. In CWM, the convergence parameter is the truncation order M, which limits the number of diffracted plane waves involved in the computation [14]. Figure 3 shows the convergence of the method with M, by focussing on the maximum of the intensity in the computational window. The regular discretization step in Fourier space is essential to describe resonance in metals and to assure the convergence of the method (which is slower than in the dielectric case [14]). Figure 3a-c show the intensity computed with M = 80, M = 360 and the convergence as a func-FIGURE 2 Schematic of the gold grating with aspect ratio 0.5, height 20 nm, and period 400 nm. The incoming light is a plane wave FIGURE 3 The intensity enhancement computed from CWM for a grating with aspect ratio 0.5, height 20 nm, and period 400 nm. The angle of incidence of the incoming light is 0. The intensity enhancement is computed 2 nm above the grating tracks tion of M, respectively. With the computer used (512 Mo RAM), a memory overflow occurs if M is greater than 360. As shown in Fig. 3c, the convergence is still not reached, even if the energy conservation is satisfied with an accuracy better than 10 -6 . The computation time increases drastically with M (Fig. 3d). This result confirms the study of convergence in [14], where M = 600 is needed to reach convergence, in the case of a dielectric sample. Let us note that the method converges more rapidly and much less harmonics are necessary to reach convergence if either the near-field, a few nanometers away from the sample, or the far-field is required to be diffracted in a given direction (or a scattering section).

In this section, it has been shown that controlling the convergence is essential in the case of computation of the intensity enhancement in the vicinity of the edges of a periodic nanostructure. The investigated case corresponds to a pure reciprocal space method, where the number of plane waves involved in the model of diffraction is the key parameter of the convergence.

Another class of methods uses the Green tensors. In the following section, the classical volume integral method is used to investigate the influence of both the mesh of the nanostructure geometry and the truncation order for the computation of the intensity enhancement.

The volume integral or green volumic method (VIM)

The volume integral method (VIM) is based on the computation of the Green tensors associated with the dis-cretization of a defect above an interface. The convergence of the method depends on the number of cells or dipoles used to mesh the defect and on the discretization in Fourier space used to compute the Green tensor. In this paper, the same numerical method as described in [15], is used to model the diffraction by the nanostructure depicted in Fig. 4.

Like in the CWM case, the goal is to compute the field enhancement 2 nm above the edges of the nanostructures, where a high gradient of electric field is expected. In the following, 2 nm square cells or dipoles are used to mesh the nanostructure. It must be noticed that the size of the dipoles is much smaller than the λ/10 discretization step commonly used in far-field simulations. This small size is necessary to reach convergence. Figure 5c shows the convergence as a function of the discretization used to compute the Green tensor previously defined as the truncation order M. The value of the intensity enhancement cannot be computed with 1 × 1 nm 2 square dipoles, due to the insufficient available memory. The intensity enhancement is equal to 24.6082 with M = 180 FIGURE 4 Schematic of the gold nanostructure with height 20 nm, and period 400 nm. The incoming light is a plane wave FIGURE 5 The intensity enhancement computed with VIM for a nanostructure with height 20 nm, and lateral size 200 nm. The angle of incidence of the incoming light is 0 degrees. The intensity enhancement is computed 2 nm above the nanostructure. The size of the mesh cells is 2 × 2 nm 2 and 26.4890 with M = 360, respectively. These results were obtained close to the maximum of the available memory in the PC.

Figure 5d can be compared to Fig. 3d: the computation time increases rapidly with the order M of truncation. This parameter has to be included in the choice of the models. The performance of the processor influences the computation time. Let us notice that the considered nanostructure is described in a 2 dimensional geometry. Of course, modeling 3D nanostructures would require more computing time and memory resources.

After the study of periodic structure with the CWM, where only the order of truncation was involved in convergence, and those of isolated nanostructures were investigated with the VIM, and where both spatial and truncation influenced the computation of the intensity enhancement, the convergence problem will be illustrated using a pure direct space method: the finite element method applied to the propagation equation.

3.3

The finite element approach to solve the Helmholtz' equation

The last method considered in this study is widely used to solve different kinds of differential equations (such as in mechanical systems, thermic, fluid mechanics etc.) and has been used to compute the near-field around nanostructures in the Green formalism [16]. It is based on the variational formulation of the differential equation and the resolution is achieved by its minimization on an adaptive non-regular non-Cartesian mesh of the computation window, taking into account the boundary conditions on the edges of this window (external boundary conditions). In this paper, the FEM is used to solve the Helmholtz' equation [START_REF] Born | Principles of Optics[END_REF], the boundary conditions are imposed on the incoming plane wave field (Dirichlet's conditions) on the edges of the computational domain [START_REF] Jin | The Finite Element Method in Electromagnetics[END_REF]. The drawback of the finite volume element method is that the whole domain of computation must be meshed, therefore a huge number of cells is needed. Due to the use of an adaptive non-regular non-Cartesian mesh, the refinement of the grid is effective only where high gradients of the solution are located. This may help to spare computer memory. This point has been addressed in [9] by comparison of the FEM with the classical FDTD with regular/non-regular Cartesian Yee's cells.

In this study, the incoming plane wave amplitude is set on the edges of the external boundaries. Consequently, the diffraction of the nanostructure is neglected in this region which assures the unicity of the solution. Nevertheless, to be valid, a sufficiently wide domain of computation must be used (relatively to the scattering cross-section of the nanostructure). In this study, the domain of computation is a 3λ square domain and also the studied nanostructure, situated at the center of the domain is a rectangle of width 200 nm and heigth 20 nm. The intensity in Fig. 6 is computed 2 nm above the nanostructure.

Figure 6 shows the influence of the gridding in the adaptive mesh process. Figure 6a-b shows the confinement of the computed intensity with the initial mesh and after seven re-FIGURE 6 The intensity enhancement computed from FEM for a nanostructure with height 20 nm, and period 400 nm. The angle of incidence of the incoming light is 0 degrees. The intensity enhancement is computed 2 nm above the nanostructure gridings. Even if the problem is symmetric, the height of the peaks is not the same in the plot due to a small decay between the left cell and the right one above the nanostructure edges. As expected, the model is very sensitive to the mesh refinement. The accuracy, obtained for the computation of the field is indicated in Figure 6c and is the maximum of the variation of the field between two adjacent cells. This accuracy is automatically computed in the region of the intensity enhancement, where the gradient of the field is maximum. The advantage of the adaptive non-Cartesian mesh is shown in Figs. 6c-d: the computation time and the number of cells varies slowly with the accuracy p (or the intensity enhancement) and is mainly related to the iterative inversion algorithm process. The smallest cell size reaches 0.1 nm to obtain stabilization of the solution. In this case as for the previous ones, the convergence is probably not completely reached before the memory overflow, but the lateral size of the peaks is smaller, due to the non-regular and non-Cartesian mesh. Let us note that the intensity enhancement is close to those given through the VIM.

Conclusion

In this study, it has been shown that modeling the intensity enhancement near nanostructures edges with classical codes can induce slow convergence (i.e. a great amount of memory, and a large computation time is needed in both the periodic or non periodic cases). The three methods used in this paper use the experimental values of permittivities in contrast to FDTD for which an analytical model of dispersion is required. Therefore, situations where gain or non linearities exist may be described is spite of a probably larger computational time.

Some requirements to accurately compute the intensity enhancement near nanostructures are:

-the use of adaptive non-regular and non-Cartesian mesh.

Indeed, the use of Cartesian grid in FDTD or VIM will result in failure to reach convergence. -the use of a remeshing process and control of the accuracy of the solution. High field gradients are expected near the nanostructures edges and therefore high spatial frequencies or high order diffracted modes are needed to achieve convergence. Moreover, cell sizes much smaller than λ/10 are required (in contrast to those commonly used) where the intensity enhancement is located near the nanostructure. In Fourier space, the contribution of the diffracted plane waves with high spatial frequency cannot be neglected. -the use of the surface methods, for which only the surface of the nanostructure must be discretized. This way may be a memory sparing solution and could be more efficient if a non regular discretization of the surface can be considered. -to smooth the nanostructure shape models in order to avoid the mathematical singularity of the electric field.

Let us note that the CWM, VIM and the FEM have been compared in the same conditions. The language of computation, and the computer are the same, and a similar nanostructure is considered. In this context, the codes have not reached the convergence before a memory overflow. The computation of intensity enhancement requires high performance computers, and a careful control of the convergence. These considerations are critical to tend to accurate computations of the intensity enhancement and predictive plasmonics.


Manouk Abkarian
email: abkarian@lcvn.univ-montp2.fr

Annie Viallat
email: viallat@marseille.inserm.fr

Vesicles and red blood cells in shear flow

We describe the similarities and the specificities of the behaviour of individual soft particles, namely, drops, lipid vesicles and red blood cells subjected to a shear flow. We highlight that their motion depends in a non trivial way on the particle mechanical properties. We detail the effect of the presence of a wall with or without wall-particle attractive interaction from a biological perspective.

-. 45 behaviours that are common to drops, vesicles or RBCs before focussing on specific effects induced by the bending and elastic energy of the membrane and by its incompressibility.

Introduction

Blood circulation, microfluidic devices and emulsion processing bring into play soft micron-size particles like cells, vesicles and drops in creeping flows. We highlight the generic behaviour of these particles in shear flow and its specificities induced by the differences between the various particle structural properties. We focus on three situations relevant for blood circulation: particles in infinite medium, close to a wall (blood cells at a vascular endothelium) or adhered to a wall (leukocytes adhered to the endothelium).

Beyond their common features, like size and fluidity, fundamental differences exist among drops, vesicles and cells: drops and surfactant-covered drops present a viscous extensible interface; while vesicle and cell membranes are essentially incompressible 1 , with no possible transport of 30 molecules between the membrane and the inner volume, ensuring a constant membrane area. Cells exhibit more complicated mechanical properties. Red blood cells' (RBC) membrane, for instance, is underlined by a 2D visco-elastic spectrin network 2 , which confers shear elasticity to their 35 membrane and is responsible for their biconcave discocyte shape. Numerous theoretical works have studied the dynamics of particles with different mechanical properties [3][4][5][6][7] . The approach of the problem is non trivial since the shapes are not given a priori and the equations coupling the the membrane 40 tension and the surrounding flow field are nonlinear. Experimental studies have been developed by several scientific communities coming from fields as diverse as fluid mechanics, soft matter and biorheology, and therefore, were not much compared to one another. Here, we describe

The understanding of individual particles' behaviour in flow is a prerequisite to that of the rheology of suspensions. For instance, the deformability of RBCs plays a crucial role in the rheological properties of the whole blood, both at the largest scale of the arteries 8 , and in the microcirculation through very small capillaries.

Shapes

A first striking feature is that drops 9 , vesicles 10 and RBCs 11 can all exhibit ellipsoidal shapes in shear flow although drop shapes are governed by surface tension, vesicle shapes are determined by the bending rigidity of their membrane (it is easier to bend vesicle and cell membranes than to stretch them) and RBC shapes depend on shear elasticity. These ellipsoidal shapes are, however, reached under different shear rates (small for drops and vesicles, but high for RBCs) since surface tension and bending elasticity involve lower energy than shear elasticity. Moreover, RBCs deform without increasing their surface area whereas, upon increasing the flow strength, the drop stretches and breaks up.

Tanktreading drop-like motion in shear flow

A second striking point is that vesicles and RBCs characterized by a low ratio of their inner fluid viscosity to that of the suspending fluid (c =η i /η o ), exhibit a typical tanktreading motion similar to that observed on drops: their membrane rotates around their centre of mass (Figure 1). This rotating motion transfers the tangential stresses of the flow to the inner fluid, which rotates and dissipates the work done by external flow, allowing the particles to maintain a stationary shape and a steady stationary orientation.

Transition: tumbling solid-like motion

In strong contrast to drops, above a critical value of the viscosity ratio, the motion of vesicles 10,12, and RBCs 13 undergoes a transition towards an unsteady tumbling solidlike motion described, by analogy, with the flipping of a coin (Figure 1).

In order to elucidate this behaviour, Keller and Skalak 3 (KS) treated the particles as ellipsoids of fixed shape with a fluid membrane and used a simplified velocity field. They established the equation of evolution of the inclination angle θ of the particle with respect to the flow direction:

dθ/dt = A + B cos (2θ), (1)
where -2A is equal to the wall shear rate γ, and B is a function of γ, c and ν, the reduced volume defined by the ratio of the volume of the particle to the volume of the sphere with the same surface area. The first term on the right hand of equation ( 1) is a vorticity term equal to -γ/2, while the second term is an extensional term at 45° with the flow direction. Equation (1) predicts tanktreading (-A/B < 1), tumbling (-A/B> 1), and a critical shear rate which depends on c and ν. Theoretical predictions show a rather good qualitative agreement when compared to the observations (Figure 2) of the time evolution of θ measured on tumbling vesicles 12 and RBCs 14 . A good quantitative agreement of the critical shear rate is also found for vesicles 12 (Figure 2c). However, for RBCs, the model does not capture the observed shear-rate dependency of the tanktreading-tumbling transition, and predicts shear rate at the transition in complete disagreement with observations 15 . The model also fails to describe the behaviour of vesicles when they deform in the flow 12 . These discrepancies are due to the 110 shear elasticity of the RBCs' membrane and to vesicles' deformability, we now discuss..

Effect of membrane shear elasticity

We recently showed that RBCs in shear flow present a motion more complex than that described by KS. A typical 115 behaviour 15 is illustrated in Figure 3. For high values of the shear stress (η ο γ), RBCs present a quasi steady tanktreading motion but when η ο γ is decreased their inclination oscillates about a mean angle down to a critical η ο γ c , for which RBCs tumble at least once. This swinging regime is characterized by 120 a quasi-nondeformed cell shape and an oscillation period of the inclination angle equal to half the tanktreading period. The swinging/tumbling transition is induced by tuning down (or up) γ.

Swinging and shear-stress triggered transition are not 125 observed on purely viscous vesicles but swinging has, however, been detected on elastic capsules in shear flow 16 , when they were slightly non spherical at rest. These features are a signature of the membrane shear elasticity. Indeed, if one assumes that the unstrained RBC shape is biconcave, the 130 local elements of the elastic membrane are not equivalent : for instance, during tanktreading, the elements which form the rim at rest rotate about the stationary cell shape to reach the dimples after rotation and vice and versa. They are then locally strained and store elastic energy (Figure 4). After a 135 180° rotation, the elements retrieve their initial shape and are no more strained, thus resulting in a periodic storage of energy. We described this effect by extending the KS model 15 : when writing that the rate of dissipation of energy in the cell equals the rate at which work is done by the external fluid on 140 the cell, we added the elastic power stored in the periodic elastic membrane strain to the cell viscous energy dissipation. The modified model indeed predicts time oscillations (figure 5), and it captures the trends of the γ variations of the amplitude and period of oscillations, as well as the motion 145 transition for decreasing values of the shear stress (Figure 3). We emphasize that the fit by such a model of the characteristics of swinging and tumbling-tanktreading transition on individual flowing RBCs allows the determination of internal viscosity, membrane viscosity and 150 shear modulus of each cell. Therefore, this analytical model, which still has to be refined, or alternatively, a numerical approach 17 hold promises for applications in non-invasive cellular-scale diagnostic in clinical hemorheology..

Effect of vesicle deformability

155 Vesicles are floppy objects whose shape may vary during motion in the flow. In the tumbling regime, they may tremble or successively present ellipsoidal and spherical shapes, which strongly affect the flipping motion 12,18 . This effect, although not relevant for cells which are more rigid, has received recent of the stretching ratio of the deformed vesicle since hydrodynamic pressure induces stretching along the +45°d irection and compression along the -45° direction, where interior flow relaxation can lead to nearly vanishing membrane tension (in case of low internal viscosity) and hence strong shape fluctuations and/or shape relaxation into a sphere.

Presence of a wall: lateral drift 170

The presence of a wall is a situation most relevant to blood physiology, especially for blood cells circulating in arterioles or venules. Indeed, a RBC depleted region near the capillary walls has been long-reported 11,19 originating from the impossibility for RBCs to approach the endothelium walls, while oppositely, white blood cells can approach the endothelium and be captured onto its surface. Migration from the wall is known for deformable drops 20 and RBCs 11,19 since the 1960s and has been recently observed on giant lipid vesicles 21,22 (figure 6). It has been explained in 180 terms of modifications of the flow field induced by the asymmetric shape of the object and the wall 23,24 . The drift velocity away from the surface in absence of external force has been theoretically predicted 24 . No lift force is observed on spherical objects.

The associated lift force Fl is a biological issue since it may counterbalance the adhesion force experienced by leukocytes at their approach to the endothelium. We used giant unilamellar vesicles to determine the variation of the lift force with the aspect ratio, the vesicle radius, the distance to the 190 substrate and the shear rate 21 . These vesicles (closed lipid bilayer membrane) were filled with sucrose solutions of controlled density 25 , which allowed to vary ν by deflating/inflating the vesicles via regulating the outer osmotic pressure. Flowing vesicles were found to unbind from the substrate above a critical value of γ and hover at a distance h from the substrate: h self-adjusts so that the hydrodynamic lift exactly counterbalances the vesicle buoyancy. In our experiments, the lift force Fl ranged from 0.2 to 150pN and wrote as:

200 Fl = f(ν) η ο γ R 3 /h (2)
where f was determined experimentally. By using Eq. ( 2), we estimated the lift force acting blood cells We experimentally showed the existence of a surface flow on giant lipid vesicles 28 . It is divided into two symmetric quadrants with two stagnation points on each vesicle side (Figure 7). The surface streamlines avoid the motionless 230 contact zone thus limiting the friction between the cell and the substrate. As a consequence, the membrane is strongly sheared since the rotational velocity on streamlines close to the contact zone is much smaller than that close to the stagnation points.

235

For cells, the cytoskeleton prevents the surface flow of micron-size defects 29 . It could, however, allow molecular surface flow among the membrane proteins anchoring the underlying cytoskeleton and thus favouring the recruitment in the contact zone of the adhesion receptors of the cell 240 membrane. This point has still to be investigated.

Conclusions

The behaviour of soft particles in flow present specific features which strongly and non-trivially depend on the particles' shape and mechanical properties. These specificities 245 must be accounted for when one considers cells in a flow, since they can be exploited to characterize individual cell mechanics (swinging), and because they involve relevant forces or flows. A lot of work has still to be done in the direction of more complex biomimetic vesicles 30 and cells (for 250 instance tumor cells at the origin of metastasis), towards the effect of interactions between cells (high cell concentration in the blood flow) or to study flows that mimic blood microcirculation through micron-size capillary networks 31 .

Notes and references

160 attention 5 .

 5 It can be understood by considering that the variation of the tumbling angle with time induces a variationThis journal is © The Royal Society of Chemistry[year] [journal], [year], [vol], 00-00 | 3

  205in post-capillary venules, where the wall shear stresses range from 0.2 to 1 Pa. For h = 350 nm (typical size of leukocytes microvilli 26 ), Fl ranges from 31 to 155 pN for RBCs (ν = 0.7). It is equal to 0 for spherical leukocytes while it ranges from 46 to 230 pN if the cell is slightly deformed (ν = 0.95). permitting progressive elimination of the lubrication layer at the vessel surface. When an intimate cell-wall contact occurs, the cell no longer experiences the lift force. It can strongly bind to the endothelium and deform. Particles adhered to the wall 220 Particles adhered to the wall When a soft particle is strongly adhered to the wall in shear flow a new question arises: does the particle present a surface/volume flow? Or, in biological terms: are cell membranar adhesion receptors able to flow towards the cell 225 adhesion line with the substrate and enhance cell adhesion?

Figure 1 :Figure 2 :

 12 Figure 1 : Vesicles and red blood cells in shear flow. A : tanktreading vesicle, viscosity ratio c = η i /η o = 1 ; B : rotation of a bead (diameter 1 µm) stuck on the membrane of a tanktreading RBC with c = 1/47, shear rate : 6 s -1 ; C : tumbling vesicle with c =8 ; D : tumbling RBC with c = 1/47, shear rate : 0.8 s -1 340

Figure 3 :Figure 4 :

 34 Figure 3 : RBC motion in shear flow. A : swinging RBC, c=1/47, shear rate = 1.33 s -1 ; B : Orientation versus the time (c= 1/47, shear rate = 0.8 s -1 ) ; C : Experimental variations of the amplitude and period of oscillations on individual RBCs, (O) , () : c=1/22 ; (), ( ) : c=1/31 ; (∆), (+), (♦), (∇) : c=1/47. Solid lines are obtained by KS modified model.

Figure 5 :Figure 7 :

 57 Figure 5: RBC motion and transition for two shear rates, as computed by using the modified KS model

Hydrodynamically speaking, flowing leukocytes must therefore imperatively preserve their spherical shape to approach vessel walls. Then, selectins binders fast form and break molecular bridges with the cell, preventing its deformation, allowing its capture and slowing down and


G Germani

P Alphonse

M Courty

Y Schuurman

C Mirodatos

Platinum/ceria/alumina catalysts on microstructures for carbon monoxide conversion

Keywords: Water-gas shift, Hydrogen purification, Microreactors, Kinetics, sol-gel, Catalyst coating 1

Platinum/ceria/alumina catalysts have been prepared by a sol-gel method and coated in the microchannels of stainless steel platelets. These catalysts are very active for the water-gas shift reaction between 300 and 400 °C. Moreover, they are non-pyrophoric and thus well suited for the purification of hydrogen for PEM fuel cells. The obtained coatings show good adherence and catalytic activity. The influence of the amount of platinum and ceria as well as the effect of a binder on the catalytic performance has been investigated. The samples have been characterized before reaction by XRD, SEM and by N 2 adsorption measurements. The kinetics, free from internal diffusion limitations, over these thin films have been described by a power law rate equation. An activation energy of 86 kJ/mol has been found and at 260 °C the TOF corresponds to 0.6 ± 0.1 s -1 for all investigated samples. The superior activity of the platelets compared to the powder samples is attributed to the diffusion limitations inside the powder pellets. Thus catalysts deposited on microstructured platelets lead to a better platinum utilization.

Introduction

The generation of high-purity hydrogen is required for efficient operation of the solid polymer electrolyte membrane (PEM) fuel cell. This hydrogen can be produced from hydrocarbon fuels or alcohols by a reforming step. This typically yields a mixture of hydrogen, carbon monoxide, carbon dioxide and steam. The carbon monoxide level must be reduced below 50 ppm to avoid poisoning of the fuel cell electrodes [START_REF] Schuurman | Proceedings of the Seventh World Congress of Chemical Engineering[END_REF]. This is generally accomplished in several steps that might include the high and low temperature water-gas shift (HT-/LT-WGS) as well as a preferential oxidation step. The WGS converts CO into CO 2 by the reaction with water over a suitable catalyst and provides additional hydrogen. The reaction generally attains thermodynamic equilibrium. Two types of catalysts are commonly used in industry: a FeCr based catalyst for HT-WGS and a CuZn catalyst for LT-WGS. The volume of the WGS takes a considerable part of the overall fuel processor due to the thermodynamic constraint to operate at the lowest possible temperatures to achieve high conversions. At these low temperatures most catalysts are not very active. In view of new applications such as on board hydrogen generation for PEM fuel cells, more active catalyst formula are required for down-scaling the reformer, together with fast response, long catalyst lifetime and nonpyrophoric materials [2].

Several new WGS catalysts have been reported in the literature such as Au/Fe 2 O 3 [3] and [4], Au/CeO 2 [4] and [5], Ru/ZrO 2 [6], Rh/CeO 2 [7] and Pt/CeO 2 [4], [8], [9] and [10] as promising catalysts for fuel cell applications as they are highly active in the range of 250-400 °C and they are non-pyrophoric. However these catalysts are expensive and optimum use of the precious metal is required. Metal microstructured platelets which allow a precise temperature control and the use of thin catalyst layers that lead to high efficiency could meet the on board reformer requirements as new compact and efficient systems [11]. This paper reports on the development of platinum/ceria/alumina catalysts on microstructured platelets for the WGS reaction. The focus is on developing methods that lead to strongly adhesive and stable catalyst layers on stainless steel plates that retain their high activity. Sol-gel methods were selected for this purpose, on the basis of literature results [12] and lab experience. It is shown that this strategy leads indeed to a better usage of the precious metal.

Experimental

Experimental set-up

The used microstructured platelets are made of stainless steel and have the following dimensions: 50 mm × 50 mm × 1 mm, containing 49 channels 400 m deep and 600 m wide.

The steel grade was ASTM 316Ti (DIN 1.4571). They are placed together with some filler plates inside a stainless steel housing. Care was taken to assure that no bypassing of the gas occurred. In the initial experiments bypassing was revealed by the fact that the thermodynamic equilibrium was not attained at higher temperatures. By using the proper filler platelets this issue has been solved. The housing is sealed with graphite rectangular seals. This housing allows a proper gas distribution through the microstructured platelets. Moreover, it can be opened up after reaction and the platelets can be taken out for analysis purposes. The heating is provided by means of six 500 W heating cartridges inserted into the housing body.

The reactor temperature was measured with a thermocouple located in the reactor housing close to the exit. Preliminary measurements showed that the temperature deviation between the inlet and outlet is less than 2°.

Mass flow controllers regulate the gas flows. Water is dosed by a syringe pump, vaporized and then mixed into the gaseous reactants stream.

The product gas analysis is carried out at the reactor exit by an Inficon Transpector CIS2 mass spectrometer.

Prior to the activity test, each batch of platelets was reduced in a flow of 10% H 2 in argon (total flow = 200 N ml/min). The temperature was raised from 25 to 450 °C at a rate of 4 °C/min, and then kept at 450 °C for 30 min. The reactor temperature is lowered to 200 °C in a flow of 10% H 2 in argon. The flow is then switched to pure argon and subsequently to the reactive flow with a feed composition of: 32.2% H 2 , 9.6% CO, 8.4% CO 2 , 23.0% H 2 O, 26.8% Ar with a total flow of 200 N ml/min. Such feed composition is typical of the outlet of a gasoline autothermal reformer, after steam addition for favouring the WGS.

Two platelets of each batch were used for the activity tests. The exact amount of catalyst that could be coated on each platelet depended on the batch, but was about 100 mg on average. In one case a powder catalyst was used for comparison. The catalyst was sieved to obtain the particle fraction between 200 and 300 m. This powder catalyst was tested in a 6 mm inner diameter quartz tube under similar conditions as the platelets.

Both carbon and hydrogen balances were established that both amounted to 101 ± 1%.

At temperatures above 350 °C traces of methane were detected, its concentration amounted to maximum 0.05% at 420 °C. In the current paper and for all calculations the methane concentration has been neglected.

Synthesis of sols

The synthesis of the alumina sols, precursor of the catalyst, is based on the method originally proposed by Yoldas [12]. The first step of the process was the hydrolysis of aluminum alkoxide (in this work, aluminum tri-sec-butoxide, Al(OC 4 H 9 ) 3 -Acros Organic) at 85 °C in large excess of water (H 2 O/Al = 100). This gives an aluminium hydroxide slurry that was subsequently peptized, at 85 °C, to a clear sol by addition of nitric acid (HNO 3 /Al = 0.07).

The required amount of the platinum precursor (H 2 PtCl 6 , 6H 2 O-Prolabo Analytical) was dissolved in the water used for the hydrolysis whereas the ceria precursor (Ce(NO 3 ) 3 •6H 2 O-Acros Organic) -dissolved in the minimum amount of water -was added after peptization. Yoldas [12] suggested that the peptization can be done by an acid solution containing the water soluble salt. But it has been observed that, under these conditions, the peptization becomes very slow and several days are needed to obtain a clear sol. Therefore we have preferred to add the solution of the ceria precursor after the complete peptization.

Several sols have been synthesized, in order to prepare catalysts containing about 1 or 2 wt.% of Pt (referred to as P1 or P2) and 10 and 20 wt.% of CeO 2 (referred to as C1 or C2).

To some of the sols a polyvinyl alcohol (Rhodoviol ™ 4/125-Prolabo) has been added (referred to as R). This polymer can adsorb at the surface of the colloïdal particles, thus stabilizing the sol.

The diluted sols prepared according to this procedure are suitable only for the synthesis of thin layers (<1 m). For the synthesis of thicker layers that were needed to attain the required amount of catalyst (about 100 mg of catalyst per microstructured platelet), sols have been prepared loaded with catalyst powder made by firing the same sol at 500 °C.

These loaded sols contained 10 wt.% of catalyst powder mixed with Rhodoviol ™ (5 wt.%). The mixture was ball milled for 30 min. These loaded sols allowed preparing thick layers with a good quality of coating because very little shrinkage occurs during drying. The synthesis parameters are summarized in Table 1 for the various samples.

Coating of the microstructured plates

The first step was the conditioning of the steel in order to ensure the best adhesion between the layer and the substrate. In this operation the plates were immersed into an alkaline cleaner (Turco ™ 4181) at 75 °C for 30 min. Then they were rinsed with water in an ultrasound bath. Finally they were dried and heated in air at 200 °C for 2 h.

The coating was performed by tape casting in such a way that the sol remains only inside the microchannels. After coating, the plates were dried overnight at room temperature. The final step was a calcination in air at 500 °C for 2 h.

Characterization of catalysts

The crystal structure was investigated by powder X-ray diffraction (PXRD). Data were collected on a Seifert 3003TT θ-θ diffractometer in the Bragg-Brentano geometry, using filtered Cu Kα radiation and a graphite secondary-beam monochromator. Diffraction intensities were measured by scanning from 5° to 90° (2θ) with a step size of 0.02° (2θ).

Scanning electron microscopy (SEM) analyses were done on a JEOL JSM-6700 (equipped with a field emission gun). Transmission electron microscopy (TEM) analyses were done on a JEOL 2010. A small amount of sample powder was put in ethanol and dispersed in an ultrasound bath during 1 min. Then the carbon-coated grid was dipped in the suspension and allowed to dry at room temperature.

The nitrogen adsorption-desorption isotherms and the krypton adsorption isotherms were determined at 77 K by volumetric method with a Micromeretics ASAP 2010.

Results

Activity tests

Table 2 gives the characteristics of the different samples that have been tested. The amounts of both ceria and platinum measured on the samples P2C1 and P2C1H are somewhat lower than the amount deduced from the preparation method. The platinum dispersion of the P2C1 sample has been measured by H2 chemisorption according to the protocol outlined in [13].

The dispersion amounted to 58%. The five different samples have been tested for their catalytic activity with the flow composition given above, typical of a reformate gas at the outlet of an autothermal gasoline reformer. b Based on the amount of Pt as determined from the preparation method and assuming a dispersion of 58%.

The carbon monoxide conversions as a function of the temperature for the different microstructures are shown in Fig. 1. From 250 °C onwards a rapid increase of the carbon monoxide conversion is observed for all samples that levels off at higher temperatures to attain the thermodynamic equilibrium conversion. The measured conversions cannot be compared directly to rank the platelets in activity, due to the different amounts of catalyst per platelet, but the use of the same volumetric flow rate. However, a comparison of the initial reaction rates can be performed at low conversions. Therefore the initial reaction rates have been calculated at 260 °C (X CO < 6%) and are reported in Table 2. The data of the P2C1 sample have been modeled according to a power law rate equation. The model consists of a one dimensional plug flow reactor that explicitly takes into account the mass and heat balance in the axial direction as well as over the catalyst layer/particle at each grid point. The least square non-linear regression analysis of the integral experimental carbon monoxide conversions is based on a Marquardt optimization routine. All this is implemented in a FORTRAN code (for more details on the model and the used correlations see [14] and [15]).

(

(2)

(3) Fig. 1. Conversion of carbon monoxide over the different microstructured samples as a function of temperature (note that the lines are just a guide to the eye).

The 95% confidence intervals on the estimated parameters are reported in Eq. ( 1). The fit of the data as a function of temperature is shown in Fig. 2. The reaction orders of carbon monoxide and hydrogen were determined by separate experiments over different samples by varying the partial pressures of carbon monoxide and hydrogen (data not reported here). Only the pre-exponential factor and the activation energy were estimated from the data presented in Fig. 1. An adequate description of the data over the entire temperature range is observed. By only adjusting the relative activity, a similar fit of all the other data can be obtained. Using this rate equation, the activity of the other samples can be estimated by correcting for the different catalyst amounts. It was found that the activity of the different samples did not vary more than ±15%, similar to that of the TOF (turnover frequency) at 260 °C reported in Table 2. for the P2C1 sample.

The sample P2C1 was also tested in the form of a powder, in order to see how the coating procedure influences the activity. Therefore 252 mg of powder (equivalent to two platelets of the stated sample) where loaded into a quartz reactor with an inner diameter of 6 mm, and tested under the same conditions as for the platelets. Fig. 3 compares the carbon monoxide conversion levels for the microstructured platelet and powder catalyst. The graph shows a much lower activity for the powder sample compared to the platelet. Again the relative activity could be calculated by using the rate Eq. ( 1) and amounted to approximately 60% compared to that of the platelet.

X-ray diffraction

Powder X-ray diffraction patterns of platinum/ceria/alumina catalysts are reported in Fig. 4.

To show the effect of the ceria, a pattern of pure γ-alumina has been added. The addition of ceria to the alumina renders the oxide more amorphous (the presence of ceria delays the crystallization of alumina) and the lines of cubic CeO 2 (calculated pattern at the bottom of the graph) are visible. The lines of CeO 2 almost have disappeared after a reduction of the catalyst in hydrogen at 500 °C (Fig. 5). From this pattern it is difficult to know if metallic platinum was formed after reduction, as the line (1 1 1) of metallic platinum can hardly be identified.

Electron microscopy

Measurements by transmission electron microscopy do not allow the proper detection of the platinum particles because of the lack of contrast between the ceria support and the metal crystallites [16] and [13] and instead high-resolution electron microscopy is needed [17]. This has been confirmed for both the 2 and 1 wt.% platinum samples in this study.

Scanning electron micrographs of a catalyst containing 10% of CeO 2 show the very porous texture of catalyst powders prepared from the sol-gel process (Fig. 6). The scanning electron micrographs presented in Fig. 7 illustrates the effect of the addition of Rhodoviol ™ on the texture of the coatings. These micrographs show, at different magnification, the surface of the coatings made consisting of 1% platinum on 20% ceria/alumina. The decomposition of Rhodoviol ™ creates a network with a large porosity. Fig. 8 shows the deposited catalyst layer inside the microstructure. It can be observed that the majority of the catalyst has been deposited on the bottom in a "triangular shape" with an equivalent particle diameter (that is the ratio of the surface area and the wetted perimeter) of 37 m.

Porosity and surface area by gas adsorption

Nitrogen adsorption-desorption isotherms have been measured at 77 K on the powder catalysts. BET surface areas and pore volumes are reported in Table 3. The pore size distributions (PSD) presented in Fig. 9 show that the addition of platinum and/or ceria does not induce strong modifications to the porosity of γ-alumina. Addition of the platinum produces a slight increase in porous volume (≈10%) but no change in the mean pore diameter d h . Addition of ceria induces both a decrease of the pore volume and of the mean pore diameter d h .

Table 3.

The effect of the addition of platinum and ceria on the porosity of alumina The BET and pore volume are calculated from the nitrogen adsorption isotherms (the mean pore diameter d h is calculated assuming a cylindrical pore geometry, S BET being taken as the pore surface area, d h = 4V/S).

Fig. 9. Pore size distributions of alumina, Pt/alumina and Pt/ceria-alumina powder catalysts (calcinated at 500 °C under air) computed from N 2 adsorption isotherm using BJH method.

The surface area of the catalyst coatings on the microstructured platelets has been studied by gas adsorption. Due to the low surface areas involved, krypton has been used as the adsorbate.

The surface enhancement factor (SEF), expressed in m 2 /m 2 , is defined as the ratio of the real surface area (as measured by gas adsorption) over the geometric surface. The values of SEFs, as calculated from surface area measurements by Kr adsorption on platelets coated with various catalysts, have been reported in Table 4. In order to help the interpretation of the data, in the last column the ratio of the SEF to the coating thickness has also been calculated. From these results one can conclude that almost the complete catalyst surface area remains accessible to Kr even when the thickness of the layer increases. On the other hand, loading sols with catalyst powder does not decrease the surface area.

Discussion

The data presented here confirm that platinum/ceria catalysts are indeed very active for the WGS reaction at temperatures above 250 °C. The initial rate data measured for the P2C1 sample corresponds approximately to a turnover frequency of 0.7 s -1 at 260 °C. The initial rates do not vary to a large extent, within approximately a factor of 2, with the different catalyst compositions. However, the turnover frequency for all samples is 0.6 ± 0.1 s -1 , assuming similar Pt dispersions for all samples. Apparently the amount of ceria does not alter the catalyst activity to a great extent. Moreover, no change in activity is observed by increasing the amount of the binder. Similarly, the initial rates obtained over the powder sample compare well with those obtained over the microstructured platelet. However, the data between powder and platelet deviate substantially at higher temperatures and conversion levels (Fig. 3). The conversion over the powder samples is lower than that of the platelets indicating that there might be diffusion limitation inside the pellets, as the pellet size (250 m) is substantially larger than the equivalent particle diameter of the catalyst layer (37 m) inside the microchannel. To verify this hypothesis, a simulation has been performed that takes into account explicitly the diffusion of matter inside the pellets. This simulation was based on a value of the average particle diameter of 250 m, a tortuosity factor of 5 and the mean pore diameter and pellet porosity as calculated from the N 2 adsorption data. Fig. 3 compares the experimental data and the simulation. It can be observed that at 260 °C, the powder and platelet indeed give similar initial rates. Above this temperature the rates are lower due to diffusion limitation inside the pores, until the thermodynamic equilibrium is reached where the rate for the reverse water-gas shift is close to the forward rate and hence the overall rate is low. The model still overestimates the conversions in the medium temperature range. This can be due to an underestimation of the diffusion coefficients or the contribution of the smaller pores has been underestimated. The model evaluated the activity of the powder catalysts to be 60% of that of the thin layer catalysts deposited on the microstructured platelet. Thus the thin layer microstructured catalyst gives an optimum use of the platinum metal. This will be even worse for commercial catalysts in pellet form that usually have sizes in the millimeter range. In that case an eggshell design will be necessary to reduce the quantity of platinum needed. Recently a similar conclusion was reached for monolith-based catalysts [14].

Conclusions

Platinum/ceria/alumina catalysts have been prepared by a sol-gel method and coated in the microchannels of stainless steel platelets. The obtained coatings show good adherence and catalytic activity. Hardly any influence of the amount of ceria as well as the effect of a binder on the catalytic performance has been observed. The kinetics over these thin films has been described by a power law rate equation. An activation energy of 86 kJ/mol has been found and at 260 °C the TOF corresponds to 0.6 ± 0.1 s -1 for all investigated samples. The superior activity of the platelets compared to the powder samples is attributed to the diffusion limitations inside the powder pellets. Thus catalysts deposited on microstructured platelets lead to a better platinum utilization.

Fig. 2 .

 2 Fig. 2. Comparison between the experimental data (symbols) and the model calculations (line)

Fig. 3 .

 3 Fig. 3. Comparison between the samples P2C1 deposited on stainless steel microstructured platelets (grey) and in powder form (black). Symbols: experimental data, solid lines: model calculations.

Fig. 4 .

 4 Fig. 4. Powder XRD patterns of the platinum/ceria/alumina catalysts calcined at 500 °C in air (λ Cu = 0.154184 nm). Calculated lines of cubic CeO 2 have been added at the bottom of the figure.

Fig. 5 .

 5 Fig. 5. Effect of hydrogen reduction on powder XRD patterns of the P2C1 catalyst sample (λ Cu = 0.154184 nm). Calculated lines of cubic Pt(0) have been added at the bottom of the figure.

Fig. 6 .

 6 Fig. 6. SEM of the P2C1 catalyst sample calcined at 500 °C in air.

Fig. 7 .

 7 Fig. 7. SEM of the P1C2R sample coated on SS plates.

Fig. 8 .

 8 Fig. 8. SEM of the P2C1 catalyst sample deposited inside the microchannel.

Table 1 .

 1 Summary of the synthesis parameters

Sample reference Pt (wt.%) CeO 2 (wt.%) Rhodoviol in sol (wt.%) Reduction under H 2

	P1C2	0.8	19.6	5	No
	P1C2R	0.8	19.6	15	No
	P1C1	0.9	9.8	5	No
	P2C1	1.7	10.7	5	No
	P2C1H	1.7	10.7	5	Yes

Table 2 .

 2 Summary of the key parameters of the microstructured samples as well as the powder

	Platelet	Catalyst mass for two platelets (mg)	CeO 2 (%)	Pt (%)	Rate at 260 °C (mmol/s kg cat )	TOF (s -1 ) at 260 °Cb
	P2C1	252	8.3 a	1.4 a	34	0.7
	P2C1H 226	8.3 a	1.4 a	25	0.5
	P1C1	164	9.8	0.9	14	0.5
	P1C2	200	19.6	0.8	15	0.6
	P1C2R 200	19.6	0.8	16	0.7
	Powder -	8.3 a	1.4 a	26	0.5
	a Measured by atomic absorption spectroscopy, all other values have been deduced from the
	preparation method.

S BET (m 2 g -1 ) Pore volume (cm 3 g -1 ) Average pore diameter, d h (nm)

	Alumina	330	0.29	3.5
	1% Pt on alumina	380	0.33	3.5
	1% Pt on 10% ceria-alumina 330	0.26	3.2

Table 4 .

 4 Surface enhancement factors (SEF) of platinum/ceria-alumina coatings

		Coating thickness ( m) SEF (m 2 /m 2 ) SEF ( m)
	1%Pt + 10% Ce(NO 3 ) 3	1.1	1000	909
	1%Pt + 10% Ce(NO 3 ) 3 + powder 8.6	7700	895
	1%Pt + 10% Ce(NO 3 ) 3 + powder 12.3	10300	837

Acknowledgements References

Acknowledgements

Part of this work has been funded by the EU "MINIREF" project (NNE5-2001-00056).

IMM-Institut für Mikrotechnik Mainz GmbH is gratefully acknowledged for providing the microstructured platelets.