Journal of Membrane Science 278 (2006) 239–250

Optimization of the membrane and poredesign for micro-machined membranes

G. Brans a, R.G.M. van der Sman a, C.G.P.H. Schroen a,∗, A. van der Padt b, R.M. Boom a

a Food and Bioprocess Engineering Group, Wageningen University, P.O. Box 8129, 6700 EV Wageningen, The Netherlandsb Corporate Research Friesland Foods BV, P.O. Box 87, 7400 AB Deventer, The Netherlands

Received 28 July 2005; received in revised form 1 November 2005; accepted 4 November 2005Available online 15 December 2005

Abstract

For micro-machined membranes, it is possible to choose pore size, pore geometry and membrane porosity, within certain limits. Different poregeometries (circular, square, slit shaped and triangular pores), particle size to pore size ratios, pore edges and membrane porosities were evaluatedwith lattice-Boltzmann computer simulations and torque balance considerations for various modes of operation. We focused on hydrodynamicinteractions and assumed uncharged neutral surfaces of the particle and the pore. However, the model can easily be extended with additionalr

so

sa©

K

1

ohtcrmanbaTiu

0d

elations for such interactions in practical systems with defined properties.It was concluded that pore geometry can have a large effect on the flux (up to 60%). Further, the effect of shielding could be quantified. Above a

urface coverage of 0.05, the particles effectively shield each other from the flow field, therewith necessitating either a higher cross flow velocityr a lower transmembrane pressure for particle removal.

Based on the simulations, an extended criterion for the critical flux was developed, which includes the effects of pore geometry, particle to poreize ratio and membrane porosity. Different optimal membrane choices follow for processes aimed at retention of all particles, and for processesimed at fractionation of particles into different fractions.

2005 Elsevier B.V. All rights reserved.

eywords: Microsieve; Membrane; Pore-blocking; CFD; Lattice-Boltzmann

. Introduction

New types of microfiltration membranes, such as siliconr silicon nitride microsieves [1] and metal microfilters [2]ave become available. These are made by photolithographicreatment of a silicon wafer and subsequent etching, or electro-hemical metal deposition on a skeleton in an electrolysis bath,espectively. Compared to conventional ceramic and polymerembranes, these membranes have exemplary properties, such

s a smooth and flat surface, a very low membrane resistance andarrow pore size distribution. The porosity and pore shape cane chosen almost freely, and therewith these membranes openreas for membrane filtration that have not been possible before.he pores are usually placed in a regular pattern and the poros-

ty can be much higher compared to conventional membranes,p to 0.8 for rectangular shapes [3], Fig. 1. By realizing separa-

∗ Corresponding author. Tel.: +31 317 482231; fax: +31 317 482237.E-mail address: karin.schroen@wur.nl (C.G.P.H. Schroen).

tions that were not possible before, these membranes could trulycreate a revolution in microfiltration.

Although the membranes themselves have almost ideal prop-erties, common membrane filtration phenomena, such as con-centration polarization, pore blocking, and cake formation stillaffect membrane performance [4]. In fact, due to the veryhigh permeabilities of these membranes, these phenomena willbe even more important than with conventional membranes.Depending on the degree of “fouling”, three filtration regimescan be identified [5]:

(I) In the sub-critical flux regime, the membrane surface isstill free of particles. Due to the removal of fluid at themembrane, concentration polarization takes place, as inany filtration regime. In this regime, the back-transportof particles away from the membrane can easily keep upwith the convective transport of particles towards the mem-brane. Thus, concentration polarization hardly influencesthe flux (linear relation between transmembrane pressure

376-7388/$ – see front matter © 2005 Elsevier B.V. All rights reserved.

oi:10.1016/j.memsci.2005.11.007

240 G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250

Fig. 1. Microsieve with slit shaped pores. Courtesy Aquamarijn microfiltrationBV.

and flux), because permanent particle deposition or layerformation is absent.

(II) At higher fluxes than the critical flux, an equilibrium isreached between the particle transport towards the mem-brane and the back-transport, after the formation of a steady(cake) layer on the membrane. Steady state fluxes arereached with either Brownian diffusion, shear induced dif-fusion or inertial lift as the relevant back-transport mech-anism, depending on the size of the retained component[6].

(III) At even higher fluxes or particle concentrations, the trans-port of particles towards the membrane is larger than theback-transport. This leads to a continuously growing cakelayer with a continuous flux decline. Regime III can onlybe applied when the cake layer is removed periodically, forexample by pulsed cross flow or backpulsing. Otherwise,the cross flow channel will become blocked.

The type of filtration regime is determined by the hydro-dynamic conditions in the system and is influenced by themembrane resistance, the transmembrane pressure, cross flowvelocity, module geometry and the feed composition. In liter-ature, the transition from regime I to II is often referred to asthe critical flux or critical pressure, whereas the transition fromregime II to III is called the limiting flux or limiting pressure[

amlmahtmith

and pore blocking on the scale of individual pores. This scale,however, is now of utmost important, due to the availability ofmicro-machined membranes.

In this paper, we used a lattice-Boltzmann CFD model to findoptimal pore shape and porosity of microsieves. After validationof the model with analytical solutions for orifice flow and thedrag force on a stationary sphere, we considered the permeabilityof different pore geometries and evaluated the forces acting ona deposited particle for the different pore geometries in crossflow. Focus was on the transition of filtration regime I to II toobtain the maximum flux, in the absence of particle deposition.Furthermore, the effect of membrane porosity and sharpness ofthe pore edge on the drag force was studied on neutral surfaceswith no charge interactions between the particle and the pore.

Finally, the simulation results are compiled into an overallcriterion for the critical flux. The practical relevance and theconsequences for membrane design are discussed for filtrationprocesses aiming at concentration or fractionation of particlesuspensions in the different filtration regimes.

2. Theory

2.1. Single particle release model

Kuiper et al. developed a model that predicts the requiredcbtfaawtnlti

tto(laP

5].

In the sub-critical flux regime (I), particles may deposit onpore, but will be released again. Kuiper et al. developed aodel for the release of a single deposited particle from a circu-

ar pore in regime I [7]. For the actual design and optimization oficro-machined membranes, the effect of different pore shapes

nd membrane porosities should be investigated together withydrodynamic effects for multiple particle situations. Computa-ional fluid dynamics (CFD) have been applied successfully in

embrane technology for the optimization of inserts and spacersn membrane channels [8,9], and in the investigation of concen-ration polarization and cake formation [10,11]. However, CFDas not often been used to study the process of particle deposition

ross flow velocity to release a spherical particle from a mem-rane with circular pores [7]. The model provides guidelines forhe module design and operating conditions for the transitionrom filtration regime I–II. The model is based on a torque bal-nce over the particle under laminar flow conditions and givesfirst impression of the effects that play a role. For particlesith size in the order of 1 �m, the pressure suction force and

he particle drag force are dominant, while lift forces can beeglected. The pressure force and the drag force were calcu-ated with analytical equations. These forces were converted toorques by multiplying with the distance between the point ofmpact of the force and the pivot position.

The particle will be released from the pore if the torque ofhe particle drag force exceeds the torque of the pressure forceowards the pivot position. In that situation, the particle will rollut of the pore and will be taken up by the flow field againFig. 2). Hence, the torque caused by the drag force must bearger than the torque of the pressure force to avoid pore block-ge. When combined with the assumption of fully developedoiseuille flow in the cross flow channel, the following particle

Fig. 2. Forces acting on a deposited particle, after Kuiper et al. [7].

G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250 241

release criterion was obtained for a flat membrane [7]:

�Pinlet

�Pchan< 7.0

hchan

lmem

(d

dp

)3

(1)

With �Pinlet the transmembrane pressure at the inlet, �Pchanthe pressure drop over the cross flow channel and the averagetransmembrane pressure defined as �Pinlet − 0.5�Pchan, hchanis the height of the cross flow channel, lmem the length of themembrane, d the particle diameter and dp the pore diameter. Themodel is only valid for circular pores, low particle concentration,low membrane porosity, and assumes that the particle does notpenetrate into the pore. It is further assumed that a circular poreis completely blocked by a spherical particle and there is nointeraction between particles on neighboring pores.

Hence the model cannot be used when one of these criteriais not met. Since the current micro-machined membranes haveoptions beyond these criteria, extension of the model towardsbroader applicability is required.

2.2. Lattice-Boltzmann CFD model

Lattice-Boltzmann is a relatively new modeling techniquethat has been successful in the simulation of fluid flow in com-plicated geometries, such as porous media and suspension flow[12,13]. It has been proven that the lattice-Boltzmann equationit

wwabsdedokrac

f

it

ν

wsa

f

Fig. 3. Lattice-Boltzmann D3Q19 scheme [15]. The arrows with numbers 1–18indicate the velocities ci in the lattice-Boltzmann model; 0 is the rest parcel, ofwhich the velocity equals zero.

in which ρ is the density of the fluid, ci the velocity in directioni and u is the fluid velocity. For a model with three dimensionsand 19 velocities (D3Q19), the weight factors wi are defined aswi = 1/3 for i = 0, wi = 1/18 for i = 1, . . ., 6 and wi = 1/36 fori = 7, . . ., 18. The set of 19 velocities has been proven to be suf-ficient for simulating fluid flow phenomena with high accuracyin 3D [14].

Simulations were performed in a three-dimensional cubiclattice with dimensions (L) 100 × 100 × 100 and a grid size�x of 0.1 �m. For problems where the effect of the periodicwalls was supposed to be absent, the dimensions of the lat-tice were enlarged. In all cases, the dimensions proved largeenough. To check if the resolution was sufficient, the effect ofgrid refinement was evaluated, but this did not affect the results.The membrane was placed in the middle of the lattice. The flowwas driven by pressure periodic boundary conditions [16], and inthe other dimensions regular periodic boundaries were applied.In cross flow simulations, boundary conditions for velocity andpressure were used according to Zou and He [17]. No-slip bound-ary conditions were applied for rigid walls and the particle [18].

The force acting on the particle was calculated numerically bythe summation of local forces over all the solid–fluid boundariesof the particle. The force on a boundary node can be calculatedfrom the local distribution fi [19]:

F

m 18

wldbn

s equivalent to a discretized version of the Navier–Stokes equa-ion [14].

For a detailed description of the lattice-Boltzmann methode refer to the books of Wolf-Gladrow [14] and Succi [15]; weill outline the method only briefly. In the lattice-Boltzmann

pproach, imaginary fluid parcels move on a regular latticey subsequent collision and propagation steps. During a colli-ion, parcels exchange impulse, and change their velocities andirections. The parcels have the tendency to relax towards anquilibrium distribution. In the collision step, a new equilibriumistribution fi,eq is calculated from the hydrodynamic momentsf the actual local distribution fi. The collision operator ω, alsonown as the inverse of the time relaxation parameter τ, is cor-elated to the viscosity. In the propagation step, the fluid parcelsre propagated into the direction of their corresponding velocity(Fig. 3).

The collision and propagation steps are described by:

i(x + �ci �t, t + �t) = fi(x, t) − ω(fi(x, t) − fi,eq(x, t)) (2)

n which t and x are the discretized time and place, c = �x/�t ishe velocity. Viscosity is defined as:

= c2s

(1

ω− 0.5

)�t (3)

ith ν the kinematic viscosity and cs the (numerical) speed ofound, chosen as c/

√3. The equilibrium distribution function is

s follows:

i,eq = wiρ

[1 + �ci · �u

c2s

+ (�ci · �u)2

2c4s

− �u2

2c2s

](4)

d =∑n=1

∑i=1

Fi, Fi(r + 0.5ci �t, t + 0.5 �t)

= 2(fi(r, t) − fi′ (r + ci �t, t))ci

�x3

�t(5)

here Fi is the force acting on the solid–fluid boundary r of theattice node n in direction i; fi′ is the distribution in the oppositeirection of i. The total force on the object, Fd, is calculatedy summation of the forces over the total amount of boundaryodes m.

242 G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250

Fig. 4. Top view of square, slit (l = 2dp), triangular and circular pore geometry.

2.3. Benchmark studies

The lattice-Boltzmann code was benchmarked with analyt-ical solutions for orifice flow and the drag force coefficient ofa stationary sphere. If the thickness lo of the orifice plate issmaller than the orifice diameter dp, the flow through the orificeis described by Sampson’s equation [3]:

Q = d3p

24η�P if lo <

1

2dp (6)

and for flow through a short channel with thickness lo as [3]:

Q = �P(128loη

πd4p

+ 24η

d3p

) if lo > 2dp (7)

where Q is the volumetric flow rate, �P the pressure difference,η the viscosity of the fluid.

In the orifice benchmark simulations, the maximal porosityconsidered was 0.03. Thus, Eqs. (6) and (7) can be used directly(correction for porosity ∼0.1%) [1]. Simulations of orifice flowwere compared with the analytical solutions for orifices withnegligible length and short channels with lengths of three timesthe orifice diameter. The volumetric flow rate was determinedfor various diameters with a pressure difference of 3.33 kPa.

Another benchmark was performed by simulating the dragf(ta

F

w

Flb

various particle diameters with a constant pressure difference of3.33 kPa. The drag force Fd was calculated with Eq. (5).

2.4. Simulations

After the benchmarks were successfully concluded, we eval-uated various aspects of the design of micro-machined mem-branes, namely the pore geometry, the particle size to pore sizeratio, the sharpness of the pore edge and the membrane surfacecoverage. The different pore geometries that we evaluated werecircular, square, slit shaped (l = 2dp, 3dp), and equilaterally tri-angular (Fig. 4). The characteristic pore size dp was 1.0 �m;the membrane thickness was also 1.0 �m. To compare the samemembrane porosity for all pore geometries, the permeability peropen membrane area k′ (m) of each pore geometry was calcu-lated as:

k′ = ηQ

�PAp(10)

where Q is the volumetric flow rate, �P the pressure difference,η the viscosity of the fluid, and Ap is the open pore area. Thiscorrected permeability can be interpreted as the permeability ofa hypothetical membrane normalized to a porosity of 1, and theindicated pore geometry.

We considered pore blocking by one deposited particle,placed exactly on the pore. The height of the particle h abovetFhdbetpbfmdmp

F branc the pa

orce on a stationary sphere. The drag force coefficient CdN m−1 s) of a periodic packing of spheres was compared withhe asymptotic expression by Hasimoto [20]. This formulaccounts for the periodic boundaries in all dimensions

d = Cd�u = CdQ

L2 (8)

ith

6πη

Cd= 1

r− 2.837

L+ 4.19

L3 r2 − 27.4

L6 r5 (9)

d is the drag force exerted on a sphere with radius r, and L theength of the periodic cubic unit cell in which the sphere haseen placed. Again, the volumetric flow rate was determined for

ig. 5. Position of a particle on a pore. The height of the particle above the memross flow, the particle was located at the end of the slit; for perpendicular slits

he membrane was calculated with Pythagoras’ theorem (Fig. 5).or slits with perpendicular orientation, the particle was placedalfway the largest dimension. For slits oriented in the cross flowirection, the particle was placed at the end of the slit (Fig. 5),ecause the cross flow will transport the particle towards thend of the slit. The transmembrane pressure was 3.33 kPa andhe cross flow velocity 0.16 m s−1. We studied aspect ratios ofarticle diameter to pore size of 1.2, 1.6, 2.0 and 3.0 respectively,ased on the particle size that would just fit inside the pore. Theorces exerted on the particle by the cross flow and the trans-embrane pressure were calculated as described in the section

iscussing the theoretical background of the lattice-Boltzmannodel (Eq. (5)). To study the effect of a deposited particle that

artially penetrates the pore, simulations were performed for the

e (h) was calculated with Pythagoras’ theorem. For slits oriented parallel to therticle was located in the middle.

G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250 243

Fig. 6. Rounded pore throats: geometry of circular pores. Left: Throat withconstant angle (wedge), right: parabolic throat.

drag force on the particle as a function of the three-dimensionalshape of the pore opening; with a perfectly circular pore, with awedge shaped pore and with a pore with rounded edges (Fig. 6).Further, the influence of the membrane porosity and surface cov-erage was studied by decreasing the size of a periodic box, inwhich a particle had been placed on a circular pore with constantcross flow velocity.

3. Results

3.1. Benchmarks with analytical solutions

Both the analytical results and the simulation results for ori-fice flow and flow through a short channel are depicted in Fig. 7.Results are in good agreement with the analytical solution. Smalldeviations were expected due to the fact that a circular orificecannot be exactly defined on the square lattice (for small orifices∼4%). The orifice is constructed by discretization of grid cells,which results in a stair case geometry. For larger diameters of theorifice and the short channels investigated here, the deviationsfrom the analytical solution become around 0.1%.

Another benchmark was the simulation of the drag force coef-ficient on a spherical particle placed in a uniform flow fieldwith periodic boundaries. The drag force coefficient was calcu-lated for different particle sizes and compared with the Hasimotoexpression (Fig. 8). Overall, there is a good agreement betweenodttl

Ft

Fig. 8. Simulation results (squares) and the analytical solution of Hasimoto [20](line) of the drag force coefficient on a periodic array of spheres.

showed that the lattice-Boltzmann method describes this caseaccurately [21].

3.2. Different pore geometries and influence of poreblocking

Subsequently, the effects of different pore geometries (cir-cular, square, slit shaped and triangular) were investigated. Weevaluated the membrane permeability in open (unblocked) sit-uation and the torque balance of a deposited particle in case ofpore blocking.

The permeabilities of the different pore geometries per openmembrane area are depicted in Fig. 9. Compared to circularpores, slits with dimension l = 3dp have almost a double per-meability per area, meaning almost twice the flux at the samemembrane porosity and transmembrane pressure. The orienta-tion of the slits did not influence the permeability; the same istrue for the triangular pores. This was expected, because for openpores the cross flow does not affect the pressure profile aroundthe pore. From a permeability point of view, long slit shapedpores are thus preferred.

The simulations for blocked pores are most relevant in thesub-critical flux regime (I); particles that deposit on the poresby chance can be removed by the force of the cross flow. Theinfluence of the transmembrane pressure on the particle is two-fold: first, there is a direct pressure difference over the particlea

Ffed(t

ur model and the expression, although there were some smalleviations. This can be explained again by the discretization ofhe sphere, similar to the orifice benchmark. The relative devia-ions were 4% for the smallest particle diameter, and 1% for theargest particle diameter. This is in agreement with Ladd, who

ig. 7. Flow simulations of orifices (diamonds) and short channels with lengthhree times the diameter (triangles) and the analytical solutions (lines).

nd second, there is a flow around the particle towards the per-

ig. 9. Permeability of different pore geometries in open condition, correctedor the open pore area (k′). Slits and triangular pores were evaluated in differ-nt orientations: slits parallel (parl) and perpendicular (perp) to the cross flowirection, triangular pores pointing towards the inlet of the feed and the outletexit). The nominal pore size was 1.0 �m, cross flow velocity 0.16 m s−1 andransmembrane pressure 3.33 kPa.

244 G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250

Fig. 10. Pore shape force factor S for different pore geometries and particlediameter to pore size ratios of 1.2, 1.6, 2.0 and 3.0. S is normalized to thepressure force of a particle on round pore (2.0 × 10−9 N). Process parameterssimilar to Fig. 9.

meate side, when the particle does not completely block thepore. This results in an additional drag force. To combine thetwo effects, we may use as a relevant parameter the total (pres-sure and drag) force exerted by the transmembrane pressure andtransmembrane flow on the particle. To compare different poregeometries, we define a pore shape force factor S as the forceexerted on the particle, divided by the force that the particlewould experience on a circular pore of the same size (dp). InFig. 10, the pore shape force factor is depicted for different poregeometries as function of the aspect ratio between the parti-cle and the pore (d/dp). The orientation of the triangular andslit shaped pores toward the feed flow field is important. Thedrag force exerted on a particle positioned on a triangular porepointing to the inlet of the feed is larger than for pores pointingto the feed outlet. This can be explained by the different localpressures around the corners of the triangular pore, dependingon the orientation towards the cross flow. Therewith S is alsoinfluenced.

A similar phenomenon was found for slit-shaped pores. Aslit orientation parallel to the cross flow direction is beneficialbecause of the resulting position of the particle. For slit-shapedpores oriented perpendicular to the cross flow direction, it wasassumed that particles would deposit in the middle of the pore.The velocity of the fluid through the pore is highest there andthe flow field converges near the ends of the slit, leading to aforce directed towards the center of the slit. For slit pores paral-ltobtflth

apitc

effectively increasing the force on the particle. For triangular andsquare pores, this area does not vary too much with increasingparticle diameter, because most of the pore is already coveredby the particle at the lowest aspect ratio of 1.2.

For the smallest particles, the pressure force with a triangularpore pointing to the feed outlet was larger compared to that withcircular pores (S = 1.5). The torque of the pressure force on adeposited particle with these triangular pores, however, is 33%lower compared to circular pores, because of the different loca-tion of the pivot point yielding only half the arm length comparedto other pore geometries (Eq. (14)). Further, the membrane per-meability was 20% larger in open situation (Fig. 9). Thus whenthese effects are combined, the critical flux with triangular porespointing towards the feed outlet could be 60% higher than withcircular pores (assuming a negligible amount of pores is actu-ally blocked). In practice, this can be accomplished by changingthe pore geometry and increasing the transmembrane pressure,while the membrane porosity and cross flow velocity are keptconstant. Since the simulations were performed in the creepingflow regime, the drag force and the pressure force scale linearlywith the cross flow velocity and the transmembrane pressure.The current results can be used for any combination of crossflow velocity and transmembrane pressure by linear scaling, pro-vided that particle size, pore size and membrane porosity do notchange. Based on this, we can find the conditions that corre-spond to the criterion for critical flux, where the torque of theda

3

siflhbstsb

gaaww

ppacrbTed

el to the cross flow direction, the particle is placed at the end ofhe slit. We assumed the particle to be deposited there, becausef the drag force. In this position, the pressure force is lower,ecause most of the fluid flows along one side of the particle. Par-icles in the middle of a slit oriented perpendicularly to the crossow direction had fluid flowing on both sides of the particle,

herewith causing a higher pressure force on the particle, and aigher S.

The shape force factor S for a particle deposited on the squarend triangular pore were very similar to those found for circularores; however for slit shaped pores S increased much faster withncreasing d/dp (Fig. 10). This is caused by the larger pore areahat is covered by the particle. For slit shaped pores, the sphereovers the pores better with increasing particle size, therewith

rag force exceeds the torque of the pressure force and particlesre released.

.3. Pore edge design

The previous simulations were carried out with pores havingharp (90◦) edges. In that case, particles do not deposit too deeplynto the pores and the calculated drag force exerted by the crossow was relatively independent on the pore geometry. In practiceowever, any pore will be at least somewhat rounded, and it maye that production, cleaning and membrane usage cause furthermoothing of the pore edges. Thus, particles deposit deeper intohe membrane and are less exposed to the cross flow than theimulations of the ideal situation, therewith effectively reducingoth the drag force and the arm length.

The effect that small particles penetrate the pore was investi-ated with circular pores with sharp edge, a wedge-shaped pore,nd a pore with rounded edges. The latter was assumed to haveparabolic profile (Fig. 6). The pressure force on the particleas for both rounded pores identical to a regular circular poreith sharp edges (i.e. S = 1).The wedge shaped pore yielded a drag force on the deposited

article that was significantly lower than that of a sharp edgedore (Fig. 11). For the wedge shaped pore, the drag force onparticle varied from 21% for d/dp = 1.2 to 71% for d/dp = 3.0

ompared to the particle on the sharp pore. The effect of theounded pore was slightly smaller than the wedge shaped pore,ecause the flow field can better develop in the pore mouth.he smaller drag force for these geometries was expected, sincespecially particles with small aspect ratio penetrated relativelyeep into the pore.

G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250 245

Fig. 11. Rounded pore edges: comparison of drag force in cross flow directionon particle with circular pore for different ratios of particle diameter to pore size.Pore geometries of Fig. 6 and pore size was 1 �m. The cross flow velocity was0.16 m s−1 and transmembrane pressure 3.33 kPa.

These simulation show that sharp pore edges are essential.Smoother pore geometries, as present in polymer and ceramicmembranes, require either a higher cross flow velocity or lowertransmembrane pressure for a clean membrane surface. A sim-ilar effect is expected for deformable particles. Because theypenetrate deeper in the pore, they become less exposed to thecross flow and experience a lower drag force.

3.4. Membrane porosity and surface coverage

Especially for micro-machined membranes with high poros-ity, it can be expected that the flow field around a particle isinfluenced by the presence of other pores, and of deposited par-ticles nearby. The former is an effect of the overall porosity ofthe membrane; the latter is due to disturbance of the cross flowfield around particles. The fraction of pores on the membranesurface that is blocked by particles is an important parameterhere. This surface coverage θ was defined as the cross sectionalarea of spherical particles divided by the membrane area:

θ =14πd2ε

Ap(11)

with Ap the total open pore area and ε the membrane porosity.For circular pores, this becomes:

θ

( )2

TBwaibfl

siaea

Fig. 12. Influence of surface coverage on drag force in the cross flow directionfor 1.6 �m particle on circular pore in square arrangement according to thesimulations and according to the correlation f1(θ) given by Eq. (16) (line).

only 25% of the drag force at coverage 0.02. For particles onhexagonal arranged pores (surface coverage 0.93), the relativedrag force was 24%, which was very similar to that of squarelyarranged pores. Consequently, the cross flow velocity should befour times higher to create the same torque of drag to removethe particle from the membrane.

Thus a hysteresis effect can be expected: when the transmem-brane pressure is slowly increased, the first particle will depositat the critical transmembrane pressure related to θ = 0. Sincethe critical transmembrane pressure for deposition at higher θ

is smaller, this will trigger fast deposition of other particles,due to the surface coverage effect, and the complete membranesurface will become blocked suddenly (θ high). Resuspensionof particles will only take place at a transmembrane pressurethat is lower than the original pressure at which the first par-ticle deposited (related to θ = 0). Evidently, instead of a lowertransmembrane pressure, one may also read a higher cross flowvelocity. For circular pores, the pressure force is not affected bythe surface coverage, because pores are completely blocked. Forpartially blocked pores, there is still a remaining flux through thepores, which contributes to the pressure force. At higher surfacecoverage, the flux and the pressure force could be affected andwas checked in simulations. Simulations with slit pores (per-pendicular, l = 3d) and surface coverage of 0.25 gave almostidentical pressure force compared to surface coverage of 0.02,and thus we may assume that the effect of surface coverage ont

pbtwt

3

oi

M

= εd

dp(12)

he effect of the surface coverage θ was studied with the lattice-oltzmann model by decreasing the size of a periodic box, inhich a particle was placed on a circular pore. This simulatesdecrease of the interpore (interparticle) distance and hence an

ncrease of the porosity ε. In this system we applied completelocking of pores in square arrangement with a constant crossow velocity.

The drag force on the particle starts to decrease when theurface coverage becomes higher than 0.05 (Fig. 12), whichmplies that once a particle has deposited, the downstream poresre more likely to become blocked too, due to the shieldingffect. At the maximum surface coverage of 0.78 for squarelyrranged pores, the drag force exerted by the cross flow was

he pressure force could be neglected.The simulations show that surface coverage and membrane

orosity do affect particle release. Therefore, these factors muste considered in the design of micro-machined membranes andhe choice of process conditions. In the following section, weill quantify these effects and incorporate them into an extended

orque balance model.

.5. Criterion for critical flux

Based on simulations presented in this article and the modelf Kuiper et al. an extended criterion can be defined for the crit-cal flux with 90◦ pore edges. Starting from the torque balance:

drag > Mpressure (13)

246 G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250

For the torque caused by the drag force we first consider theStokes drag force (Fst) on a spherical particle close to a wall(Fst = 1.7 × 3πηνcd, with 1.7 the wall correction) and multiplywith the height of the point of impact. This height is above thecenter of the particle and equals 0.685 times the particle diameter[7]. The torque of the pressure force is calculated by multiplyingthe pressure force with the arm, given in most pore geometriesby half the diameter of the pore size ((1/2)dp) [7]. The effects ofsurface coverage and the particle to pore size ratio on the dragforce are included in the torque balance

Mdrag > Mpressure

Mdrag = Fdrag · arm − Mdrag (single particle on wall) · f1(θ) · f2

(dp

d

)Mpressure = Fpressure · arm = area · �Pinlet · arm

⎫⎪⎪⎪⎬⎪⎪⎪⎭

→ 3.5πηνcd2,

f1(θ) · f2

(dp

d

)>

{14πd2

pS �Pinlet14d triangular pores pointing to the exit

14πd2

pS �Pinlet12d for all other pore geometries

(14)

S is the pore shape force factor for the pressure force and νc thelocal fluid velocity in the membrane module at the height of thecenter of the particle. �Pinlet is the transmembrane pressure atthe inlet of the cross flow channel, where the particles experiencethe largest pressure force.

f1(θ) is the drag correction factor as function of the surfacecfafgll

trtft

f

F((a

f2

(dp

d

)=

√1 + b0

(dp

d

)2

(16)

with fit parameters a0 = 4.506, a1 = 3.571, a2 = 23.778 andb0 = −0.949.

Further, it is common to use wall shear stress instead of localvelocity, which in the laminar flow regime is defined for thecross flow velocity relevant for the particle νc = (1/2)d τw/η.Together with some rearrangement this leads to:

τw

S �Pinlet

(d

dp

)3

f1(θ) · f2

(dp

d

)> 0.0714 (17)

Note that the wall shear stress is determined by the flow pro-file through the feed channel. Therefore, the module geometryah

be[dsb[nn

fitpslT

Thtifn

overage with spherical particles in square arrangement and2(dp/d) is the torque correction for the height of the particlebove the pore (determined by dp/d). We assume that functions1(θ) and f2(dp/d) act independently. Please note that for trian-ular pores pointing towards the exit of the feed flow, the armength is a quarter of the particle diameter, because of a differentocation of the pivot position.

For circular pores, the pore shape force factor S is definedo be 1 and is independent of the particle diameter to pore sizeatio. For the other pore geometries, S depends also on the par-icle diameter to pore size ratio (Fig. 9). The functions f1(θ) and2(dp/d) were determined by least squares fitting of the simula-ion results (Figs. 12 and 13) and can be expressed as:

1(θ) = 1 + a0θ

1 + a1θ + a2θ2 (15)

ig. 13. Influence of the particle height above the pore, as determined by dp/dFig. 5), on the torque exerted by the cross flow towards the pivot position1.6 �m particle on circular pore) according to the simulations (symbols) andccording to the correlation f2(dp/d) given by Eq. (15) (line).

ffects the wall shear stress via the cross flow velocity and theeight of the cross flow channel.

The dimensionless term τw/�Pinlet is related to an Euler num-er describing the transported energy compared to the dissipatednergy Eu = ρu2/�P and τw = 0.5 ρu2f with f the friction factor22]. In our case, the energy transport and dissipation are notefined in the same direction, but perpendicular. The ratio of wallhear stress to the flux or transmembrane pressure has indeedeen recognized as an important factor in experimental research23]. For the present work, one might define an effective Eulerumber as τw/S �Pinlet which is the determining dimensionlessumber for the system.

The criterion can be simplified for two special situations. Therst situation applies to the filtration of large particles (compared

o the pore size) with a low porosity membrane and circularores. For this process, S equals 1 and because pores are muchmaller than the particle diameter (d/dp large) and porosity isow (surface coverage <0.05), f1(θ) and f2(dp/d) are both unity.hus, the particle release criterion can be simplified to:

τw

�Pinlet

(d

dp

)3

> 0.0714 (18)

he second situation is the filtration of a suspension with aighly porous membrane, where the smallest particle diame-er in the feed is close to the pore size, for example 10% largern case of particle fractionation. Therefore f1(θ = 1) is 0.25 and2(dp/d = 0.9) is 0.48. This can be considered the worst-case sce-ario for the critical flux:

τw

S �Pinlet> 0.45 (19)

G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250 247

Table 1Required cross flow velocity for particle release of 1.2 �m particles from various pore geometries with nominal pore size 1.0 �m at transmembrane pressure of3.33 kPa, assuming linear relation with simulations (cross flow velocity 0.16 m s−1)

Geometrya Fpressure (×10−9 N) Fdrag (×10−9 N) Mpressure (×10−15 N m) Mdrag (×10−15 N m) Required cross flow (m s−1)

Circular 1.94 1.40 0.97 0.90 0.172Square 2.42 1.41 1.21 0.91 0.214Slit l = 2d perpendicular 3.56 1.46 1.78 0.94 0.303Slit l = 2d parallel 3.07 1.45 1.54 0.93 0.263Slit l = 3d perpendicular 3.83 1.47 1.92 0.95 0.323Slit l = 3d parallel 3.11 1.40 1.56 0.90 0.276Triangle inlet 2.44 1.41 1.22 0.91 0.215Triangle outlet 2.81 1.41 7.03 0.91 0.124

a Slits and triangular pores were evaluated in different orientations: slits parallel and perpendicular to the cross flow direction, triangular pores pointing towardsthe feed inlet and the retentate outlet.

This worst-case scenario can be used for membrane mod-ule design and selection of appropriate process conditions.In a rectangular membrane channel, the wall shear can becalculated from the average cross flow velocity (assumingPoiseuille flow) 〈v〉: τw = 6〈v〉η/hchan, with hchan the channelheight. The average cross flow velocity can be calculated with〈v〉 = �Pchanh

2chan/12lmemη, where �Pchan is the pressure drop

over the cross flow channel and lmem the length of the mem-brane. In realistic membrane processes �Pchan/�Pinlet can be 1at maximum. With average transmembrane pressure in the chan-nel equal to 0.5 �Pinlet (=�Pinlet − 0.5�Pchan). Substitution inEq. (19) yields:

hchan

Slmem> 0.225 (20)

This implies that the channel height needs to be at least 0.225times the membrane length. The effect of the criterion on themodule design and appropriate process conditions is more elab-orated in Fig. 14 for circular and triangular pores. The extended

Fhclpd

criterion tells us that depending on the channel dimension, a min-imum pressure gradient over the cross flow channel should beapplied. For very thin channels, for example hchan/2lmem = 0.001,this pressure gradient is very large, even when the pore size ismuch smaller than the particle diameter. This is not realisticbecause the pressure drop over the cross flow channel wouldeasily exceed the transmembrane pressure, causing permeateback flow at the end of the module. For wider channels, suchas hchan/2lmem = 0.1, one can see that as long as the pore sizeis small compared to the particle diameter, the pressure gradi-ent that has to be applied over the feed channel is not too high(Table 1).For fractionation of particles close to the membranepore size (d/dp < 1.1), this implies that the use of the UTP (uni-form transmembrane pressure) concept is essential to overcomeunpractical module designs. With UTP, the permeate is recy-cled along the backside of the membrane to maintain a uniformtransmembrane pressure over the length of the membrane [24].Otherwise, the process will end up in filtration regime II or III,were periodic removal of the cake is required for a stable frac-tionation process.

4. Discussion

4.1. Optimal membrane and process design

Based on the simulation results presented earlier, recom-mmoopmfl

1

ig. 14. Transition from regime I to II for different module designs with

chan/2lmem between 0.001 and 1.0 and membrane porosity ε = 0.5. Above theurves, pore blocking will not occur, of circular pores (solid lines) and triangu-ar pores pointing towards the feed exit (dashed lines). For triangular pores, theore shape force factors S were taken from Fig. 10 (S was only available when

p/d > 0.33).

endations can be given for application of micro-machinedembranes (Table 2). The optimal membrane design and choice

f process parameters depend on the filtration goal (retentionf all particles or fractionation of a poly-disperse particle sus-ension) and the filtration regime. For retention, the objective isaximum flux, while for fractionation it can be either maximumux or maximum selectivity:

. Retention of all particles. For retention of particles in regimeI, a relatively small pore size (d/dp = 3–5) is needed, next tohigh surface porosity. Our simulations indicate that triangu-lar pores pointing to the feed outlet are optimal. A pore sizethat is clearly smaller than the smallest particle size enablesa high enough drag force on particles, because particles can-not penetrate into the pore. The combination of cross flowvelocity and transmembrane pressure must be such that thebacktransport of particles to the cross flow channel is suf-

248 G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250

Table 2Recommendations for membrane choice, depending on the aim of the filtration and the filtration regime

Process/objective Filtration (retention/concentration) high flux, low energy consumption Fractionation high selectivity (full retention and full transmission)

Regime IPore shape Triangular TriangularPore size d/dp = 3–5 Depends on required product sizePorosity Low to average porosity Low porosity (coverage < 0.05)

Regime IIPore shape Not circulara b

Pore size d/dp = 3–5Porosity Average

Regime III (pore blocking by one particlec)Pore shape Slit shapedd CircularPore size d/dp > 1 Depends on required product sizePorosity High porosity High porosity

a The membrane design is not so important, because the cake layer predominantly determines the filtration behavior. However, complete pore blocking may notoccur.

b This process is not applicable. When a steady cake has developed, small particles will be captured in the cake (selectivity is lost). When the cake is removedperiodically, it resembles fractionation in regime III.

c Only possible with periodic removal of the cake.d Selected because of the possibility of highest porosity and lowest membrane permeability. (The particle release criterion is not valid in regime III.)

ficient to keep the membrane clean (single particle releasecriterion, Eq. (17)). In regime II, a steady cake layer is devel-oped, which determines the flux. The membrane propertiesare less important, albeit it is still recommended to use a highaspect ratio to prevent particles getting stuck in the membranepores. For retention in regime III (limiting flux regime), a highporosity and small pore size are recommended (d/dp = 3–5).A high porosity avoids local focusing of the flow around apore, which could effectively increase the resistance of thefirst blocking layer. As in the previous case, the problemof particles getting stuck into the pores is reduced at highd/dp. The pore geometry is not so important. Filtration maycontinue until the flux has become too low due to the cakeformation, and then the cake layer needs to be removed byfor example a backpulsing technique.

2. Fractionation. We assume here that the optimization crite-rion is optimal selectivity (i.e. high retention of large particlesand high transmission of small particles), and not necessar-ily achieving highest flux. In fractionation processes, the poresize is determined by the required selectivity. Large particlesmust be rejected, while smaller particles must be able to passthe membrane. Therefore, regimes II and III are not suit-able, because the cake layer will determine the selectivity,instead of the membrane. For fractionation in the sub-criticalflux regime, the cross flow velocity and transmembrane pres-sure must correspond with the membrane pore size, such

Fractionation in regime III as such is not possible, unless thecake is removed periodically. One might consider two optionshere. When a pore geometry is chosen such that a particlecannot completely block the pore (such as slit shaped or trian-gular pores), blockage will still allow the permeation of fluidthrough the pore. However, the available space will probablybe too small to allow transmission of small particles. Thesesmall particles are therefore retained (for the greater part),and the permeate will contain hardly any of the smaller par-ticles. Alternatively, one might consider using membraneswith circular pores that can be blocked completely withoutaffecting other pores. The flux is reduced more quickly, butthe permeate that is obtained will have a high concentrationof small particles. Combining this with a frequent backpuls-ing strategy to periodically lift off the particles and allowfurther permeation will then result in optimal transmissionof the small particles and retention of the large particles, at areasonable flux level.

Besides an optimal design from hydrodynamic point ofview, micro-machined membranes must meet additional require-ments, such as sufficient mechanical strength [3]. The manufac-turing of robust micro-machined membranes with exact poregeometry, high porosity and a thin active layer is still a chal-lenge. However, given the fast developments in this area we doexpect that progress will be made here.

4

htasot

that rejected particles are removed from the pore (Eq. (17))and transported back to the cross flow channel, while smallparticles can still pass the membrane. As shown by the simu-lations, triangular pores with the side towards the inlet of thefeed may be a good choice. However, fractionation in regimeI can be instable, when particles with diameters close to thepore size are present (d/dp close to 1). These particles thenbecome embedded in the pore (f2(dp/d) low) and very highcross flow velocities are required to remove them. Graduallymore pores will become blocked, leading to a decreasing flux.

.2. Influence of particle–pore interactions

In this paper, the critical flux criterion was developed from aydrodynamic point of view, while neutral interaction betweenhe particle and the pore was assumed. However, in realitylways additional interactions will be present, such as electro-tatic (van der Waals) or double layer interactions. Dependingn the system and distance between the particle and the pore,hese interactions have an attractive or repulsive nature.

G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250 249

When the properties of the system (membrane, particle,medium) are defined, the interaction between the particle and thepore can be quantified with the DLVO theory [25]. The resultingforce can be added to the pressure force on the right hand sideof Eq. (14). Bowen and Sharif showed that electrostatic forcescan indeed play a crucial role in the filtration of 0.1 �m particles[26]. Though hydrodynamic forces become more significant forlarger particles, surface properties could be an important aspectin the design of micro-machined membranes for microfiltrationpurposes as well [27].

5. Conclusion

CFD simulations were used to evaluate the influence of poregeometry and process conditions on deposition or release ofparticles. The simulations indicated that triangular pores point-ing towards the outlet of the feed result in an optimal balancebetween the torque of drag, the torque of pressure force, andthe permeability of the pore. Further, sharp (90◦) pore edgesenhance particle release, while wear and particle deformation(both causing a deeper penetration of the particle into the pore)have a negative influence.

An extended particle release criterion was presented for mem-branes with uniform straight pores, various pore geometries, andmembrane surface coverage. This criterion yields a critical valuefisflm

aat

ppd

A

DStTaMd

cs lattice-Boltzmann speed of sound (m s−1)Cd drag force coefficient (N m−1 s)d particle diameter (m)dp characteristic pore size or diameter (m)Eu Euler numberf friction factorfi density of direction i (kg m−3)fi,eq equilibrium of direction i (kg m−3)fi′ density of direction opposite of i (kg m−3)Fd particle drag force (N)Fi force on boundary in direction i (N)Fst Stokes drag force (N)h height of particle above the membrane (m)hchan height of cross flow channel (m)k′ permeability per open pore area (m)l slit length (m)lmem membrane length (m)lo orifice thickness/small channel length (m)L periodic box size (m)M torque (N m)�P pressure drop (Pa)�Pchan pressure drop over cross flow channel (Pa)�Pinlet transmembrane pressure at inlet (Pa)Q flow rate (mV)

R

or the wall shear stress (and thus the required cross flow veloc-ty) for a given transmembrane pressure. It also shows that themallest particles in the suspension are determining the criticalux. Especially for particle fractionation, this implies a strictembrane and module design.Although the model was developed for hydrodynamic inter-

ctions under neutral conditions, the model can be extended withDLVO term to include interactions between the particle and

he pore for practical systems.Based on our findings, recommendations on membrane and

rocess design were given for processes aimed at retention of allarticles, and processes aimed at fractionation of particles intoifferent fractions.

cknowledgements

The authors would like to acknowledge the support of theutch Ministries of Economic Affairs, Education, Culture andciences, and of Housing, Spatial Planning and Environment

hrough a grant of the Dutch Program Economy, Ecology andechnology (EETK20033). Dr. Rolf Bos of Friesland Foods BV,nd Dr. Cees van Rijn and ir. Wietze Nijdam of Aquamarijnicrofiltration BV are greatly acknowledged for the stimulating

iscussions.

Nomenclature

Ap open pore area (m2)c lattice-Boltzmann velocity (m s−1)ci lattice-Boltzmann velocity in direction i (m s−1)

r particle radius (m)S pore shape force factort discretized time (s)�t integration time step (s)u fluid velocity (m s−1)vc local fluid velocity (m s−1)〈v〉 average cross flow velocity (m s−1)wi lattice-Boltzmann weight factor for direction ix discretized position (m)�x grid size (m)

Greek lettersε membrane porosityη viscosity (Pa s)θ membrane surface coveragev kinematic viscosity (mV)ρ density (kg m−3)τw wall shear stress (Pa)ω lattice-Boltzmann collision operator

eferences

[1] C.J.M. van Rijn, M.C. Elwenspoek, Microfiltration Membrane Sievewith Silicon Micromachining for Industrial and Biomedical Applications,Microelectro Mechanical Systems (MEMS), Amsterdam, The Nether-lands, 1995, p. 83.

[2] A.J. Bromley, R.G. Holdich, I.W. Cumming, Particulate fouling of sur-face microfilters with slotted and circular pore geometry, J. Membr. Sci.196 (2002) 27.

[3] C.J.M. van Rijn (Ed.), Nano and Microengineered Membrane Technol-ogy, Elsevier Science, Amsterdam, The Netherlands, 2004.

250 G. Brans et al. / Journal of Membrane Science 278 (2006) 239–250

[4] G. Brans, J. Kromkamp, N. Pek, J. Gielen, J. Heck, C.G.P.H. Schroen,R.G.M. van der Sman, R.M. Boom, Evolution on microsieve design, J.Membr. Sci. 278 (2006) 344.

[5] R.W. Field, D. Wu, J.A. Howell, B.B. Gupta, Critical flux concept formicrofiltration fouling, J. Membr. Sci. 100 (1995) 259.

[6] G. Belfort, R.H. Davis, A.L. Zydney, The behavior of suspensions andmacromolecular solutions in cross flow microfiltration, J. Membr. Sci.96 (1994) 1.

[7] S. Kuiper, C.J.M. van Rijn, W. Nijdam, G.J.M. Krijnen, M.C. Elwen-spoek, Determination of particle-release conditions in microfiltration: asimple single-particle model tested on a model membrane, J. Membr.Sci. 180 (2000) 15.

[8] S.K. Karode, A. Kumar, Flow visualization through spacer filled chan-nels by computational fluid dynamics. I. Pressure drop and shearrate calculations for flat sheet geometry, J. Membr. Sci. 193 (2001)69.

[9] J. Schwinge, D.E. Wiley, D.F. Fletcher, A CFD study of unsteady flowin narrow spacer-filled channels for spiral-wound membrane modules,Desalination 146 (1–3) (2002) 195.

[10] D.E. Wiley, D.F. Fletcher, Techniques for computational fluid dynamicsmodelling of flow in membrane channels, J. Membr. Sci. 211 (2003)127.

[11] C.J. Richardson, V. Nassehi, Finite element modeling of concentrationprofiles in flow domains with curved porous boundaries, Chem. Eng.Sci. 58 (2003) 2491.

[12] R.J. Hill, D.L. Koch, A.J.C. Ladd, The first effects of fluid inertia onflows in ordered and random arrays of spheres, J. Fluid Mech. 448(2001) 213.

[13] A.J.C. Ladd, R. Verberg, Lattice Boltzmann simulations of particle fluidsuspensions, J. Stat. Phys. 104 (2001) 1191.

[14] D.A. Wolf-Gladrow, Lattice-gas Cellular Automata and Lattice Boltz-

[

[16] T. Inamuro, M. Yoshino, F. Ogino, Lattice Boltzmann simulation offlows in a three-dimensional porous structure, Int. J. Numer. Meth. Fluids29 (1999) 737.

[17] Q. Zou, X. He, On pressure and velocity boundary conditions for thelattice Boltzmann BGK model, Phys. Fluids 9 (6) (1997) 1591.

[18] D. Kandhai, A. Koponen, A. Hoekstra, M. Kataja, J. Timonen, P.M.A.Sloot, Implementation aspects of 3D lattice-BGK: boundaries, accuracyand a new fast relaxation method, J. Comp. Phys. 150 (1999) 482.

[19] A.J.C. Ladd, Numerical simulations of particulate suspensions via adiscretized Boltzmann equation. Part 1. Theoretical foundation, J. FluidMech. 271 (1994) 285.

[20] H. Hasimoto, On the periodic fundamental solutions of the Stokes equa-tions and application to viscous flow past a cubic array of spheres, J.Fluid Mech. 5 (1959) 317.

[21] A.J.C. Ladd, Numerical simulations of particulate suspensions via adiscretized Boltzmann equation. Part 2. Numerical results, J. Fluid Mech.271 (1994) 311.

[22] S. Elmaleh, L. Vera, R. Villarroel-Lopez, L. Abdelmoumni, N. Ghaffor,S. Delgado, Dimensional analysis of steady state flux for microfiltrationand ultrafiltration membranes, J. Membr. Sci. 139 (1998) 37.

[23] O. LeBerre, G. Daufin, Skimmilk cross flow microfiltration performanceversus permeation flux to wall shear stress ratio, J. Membr. Sci. 117(1996) 261.

[24] P.K. Vadi, S.S.H. Rizvi, Experimental evaluation of a uniform trans-membrane pressure crossflow microfiltration unit for the concentrationof micellar casein from skim milk, J. Membr. Sci. 189 (2001) 69.

[25] P.C. Hiemenz (Ed.), Principles of Colloid and Surface Chemistry, 2nded., Marcel Dekker, New York, USA, 1986.

[26] W.R. Bowen, A.O. Sharif, Hydrodynamic and colloidal interactionseffects on the rejection of a particle larger than a pore in microfiltrationand ultrafiltration membranes, Chem. Eng. Sci. 53 (1998) 879.

[

mann Models, Springer Verlag, Berlin, Germany, 2000.15] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and

Beyond, Oxford University Press, Oxford, UK, 2001.

27] A. Arafat, K. Schroen, L.C.P.M. de Smet, E.J.R. Sudholter, H. Zuil-hof, Tailor-made functionalization of silicon nitride surfaces, JACS 126(2004) 8600.