Simulation of gas diffusion in highly porous nanostructures by directsimulation Monte Carlo

Jochen A.H. Dreyer a,b, Norbert Riefler b, Georg R. Pesch b, Mirza Karamehmedović b,Udo Fritsching b, Wey Yang Teoh a, Lutz Mädler b,n

a Clean Energy and Nanotechnology (CLEAN) Laboratory, School of Energy and Environment, City University of Hong Kong, Hong Kong SARb Foundation Institute of Materials Science (IWT), Department of Production Engineering, University of Bremen, Germany

H I G H L I G H T S

� Direct Simulation Monte Carlo of gasdiffusion with the open source sol-ver dsmcFOAM.

� Improved accuracy by implementa-tion of the variable soft sphere model.

� Correction of the particle flux accu-mulator to avoid pulsed gas initializa-tion.

� CO diffusion in porous anisotropicnanostructures as formed with FSP.

� Base for describing adsorption, deso-rption and chemical reactions.

G R A P H I C A L A B S T R A C T

a r t i c l e i n f o

Article history:Received 4 September 2013Received in revised form17 October 2013Accepted 22 October 2013Available online 6 November 2013

Keywords:Nanoparticle aggregate filmsGas diffusionDSMCDusty gas modelHigh porosityGas sensor

a b s t r a c t

A Direct Simulation Monte Carlo (DSMC) method is utilized to simulate gas diffusion in nanoscaledhighly porous layers. An open source solver has been extended with the variable soft sphere (VSS) binarycollision model and the inflow boundary model was adjusted for small numbers of DSMC particleinitialization. Comparison with the analytical diffusion equation illustrate the improvement of the VSSmodel compared to the variable hard sphere model (VHS). Subsequently, several highly porous particlelayers (gas sensors synthesized by flame spray pyrolysis and isotropic layers) build up by 10 nm particleshave been investigated. Results for DSMC gas diffusion in the porous structures are in agreement withthe well established dusty gas model (DGM). However, while DGM requires measurements orestimations of pore sizes, porosity, and tortuosity and furthermore is limited to homogenous layers,the present contribution shows significant advantages of DSMC in describing gas diffusion in non-isotropic porous structures.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Diffusion of gas molecules into porous structures is a phenom-enon which occurs in many environmental or technical systemssuch as gas sensors and catalyst beds. The understanding ofmacroscopic effects in such systems often requires modeling atthe mesoscopic scale. To describe rarefied gases or systems with

small pores (Knudsen number Kn40:1), e.g. mesoporous systems(pore-diameter between 2…50 nm), kinetic theory has to beapplied through solving the Boltzmann equation. Early approx-imations of this equation for pure diffusion without walls led(besides the Boltzmann H-theorem) to the Chapman–Enskogperturbation method, followed by the Grad–Zhdanov momenttechnique. The linearization of the Boltzmann equation wasintroduced later by Bhatnagar, Gross and Krook (Cunninghamand Williams (1980)) and serves as the basis of many latticeBoltzmann model (LBM) simulation methods (Succi, 2001).In contrast, the dusty gas model approach (DGM) uses the

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0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ces.2013.10.038

n Corresponding author. Tel.: þ49 421 218 51200; fax: þ49 421 218 51211.E-mail address: lmaedler@iwt.uni-bremen.de (L. Mädler).

Chemical Engineering Science 105 (2014) 69–76

non-linear Chapman–Enskog or the Grad–Zhdanov constitutiveequations of diffusion, where wall effects are considered bycollisions of the gas molecules with stationary rigid spheres(Cunningham and Williams, 1980).

Another approach to calculate diffusive and other kineticproperties is the direct tracking, i.e. calculation of tracks andcollisions of gas molecules. Molecular dynamics (MD) is an oftenapplied method where each atom or molecule is individuallytracked, however, with the expense of very high computationalcosts if larger problems are considered. A good compromisebetween the required computing power and problem size is theDirect Simulation Monte Carlo method (DSMC) (Bird, 1994). In thismethod, the molecules of a volume cell are united into differentparcels. The translation and the collision process are decoupled inevery time step. In the collision process, only a part of themolecules are chosen according to their collision cross section.These collision partners represent statistically the ensemble of allmolecules within the simulation volume.

This work describes the implementation of the variable-soft-sphere (VSS) collision model (Koura and Matsumoto, 1991) in anexisting DSMC solver of OpenFOAM. The results are comparedwith analytical solutions of the diffusion equation. Beyond that,the diffusion processes into gas sensor layers are investigatedbased on this implementation and compared to the well estab-lished DGM method.

2. Theoretical models

2.1. Gas diffusion models

Besides the conservation of mass for each participating com-ponent i, the general relation to describe mass transport withrespect to the concentrations ci of a component i, includesconvection, diffusion and sinks/sources due to chemical reactions(Bird et al., 2002):

∂ci∂t

¼ �ð∇civÞ�ð∇JiÞþRi ð1Þ

with the velocity v, the flux Ji and the rate of production of molesRi of species i per unit volume. In porous media such as catalytic orgas sensor layers, only a small volume on the top of the porouslayer experiences convection (described using the Brinkmanequation Nield and Bejan, 2006). While only the boundary regionat the top of the porous layer is affected by the convective flow,diffusion is the main transport process within the middle and thebottom region of the layer. In most cases the convective part canbe neglected without loss of generality. In this work chemicalreactions are not considered. With these assumptions the masstransport equation (1) simplifies to

εRT

∂xip∂t

¼ �∇Ni ð2Þ

where the concentration ci is replaced by the mole fraction xi usingthe ideal gas law relation (ci ¼ ni=V ¼ pxi=ðRTÞ), with the porosity εand the molar flux Ni with respect to stationary axes. The fluxexpresses the rate of mass transport and can be calculated byvarious methods, e.g. using the 1st Fickian law:

Ni ¼ �D0ijp

RT∇xi ð3Þ

with the bulk molecular diffusion D0ij.

2.1.1. Extended Fickian modelWithin porous media, the diffusion of gas molecules differs

from bulk diffusion, expressed by D0ij for molecules of type i within

molecules of type j, if the mean free path of a molecule is in the

size range of the pore diameter. This case occurs for Knudsennumbers KnZ0:01 with Kn¼ λ=daver:, where daver: is the averagepore diameter and λ the mean free path. In case of non-adsorbinggas molecules (Petropoulos and Papadokostaki, 2012) and forKn41, the molecules will almost exclusively collide with thewalls of the porous structure. This case is expressed by theKnudsen diffusion number DiK given by

DiK ¼daver:3

ffiffiffiffiffiffiffiffiffiffiffi8kBTπmi

sð4Þ

with the molecular mass mi. Additionally, due to the hinderedmolecule movement, the bulk diffusivity in a porous mediumdecrease, expressed, e.g. in Ho and Webb (2006), by Dij ¼ ε=τD0

ijwith the tortuosity τZ1. To take all these wall collisions intoaccount, the diffusion coefficient is expanded using the Bosanquetformula (Veldsink et al., 1995):

Deff ¼1DiK

þ 1Dij

� ��1

ð5Þ

The effective diffusion coefficient Deff applied to Eq. (3) leads toan extended Fickian model, which is in the following used forvalidations of interfacial fluxes.

2.1.2. Dusty gas modelThe dusty gas model (DGM) describes the porous media as a

stationary phase of uniformly distributed dust particles. The fluxexpression in the DGM for a species i in one dimension is(Veldsink et al. (1995) and Ho and Webb (2006)):

∑j ¼ 1; ja i

xiNj�xjNi

Dij� Ni

DiK¼ 1RT

∂ðxipÞ∂z

: ð6Þ

In this equation, pressure gradients can be omitted, whichcorresponds to thin porous layers. For a two component system,after rearranging Eq. (6) and use of Grahams relation betweenfluxes and mole masses of the components, N2=N1 ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM1=M2

p,

N1 can be expressed by

N1 ¼ � pRT

D12D1K

D12þD1K�ax1

∂x1∂z

ð7Þ

with a ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM1=M2

p.

The insertion of Eq. (7) into Eq. (2) leads to partial differentialequation (PDE) of 2nd kind and can be solved either by a PDEsolver (e.g. pdepe-solver of Matlab) or by an ordinary finitedifference scheme. Average pore diameter and ε required forDGM were taken from an analysis of the generated porous layerswhile the tortuosity was calculated by τ¼ ð ffiffiffi

ε3p Þ�1 (Millington,

1959). The DGM is taken as reference for gas diffusion calculationin a porous layer.

2.2. Direct simulation Monte Carlo method

For very small Knudsen numbers (Kno0:01) the Navier–Stokescontinuum model can be used to describe fluid flows. Otherwise,particle based models based on the Boltzmann equation are used.However, there is neither an analytical solution to the Boltzmannequation, nor a simple numerical solution for a partial differentialequation with an integral collision term (Bird, 1994).

The DSMC method circumvents these difficulties throughdirectly simulating molecule collisions with the help of methodsprovided for the solution of the Boltzmann equation (e.g. theChapman–Enskog theory or the Bhatnagar–Gross–Krook (BGK)model) to link the models to the well accepted Boltzmannequation. The essential feature of DSMC is the decoupling of themolecular motion and the intermolecular collisions over a smalltime step. The molecules are represented as real physical

J.A.H. Dreyer et al. / Chemical Engineering Science 105 (2014) 69–7670

quantities, however, the number of molecules in the calculationcan be reduced through so called DSMC-particles that represent afixed number of gas molecules. The number of real gas moleculeswith respect to DSMC-particles is the scaling factor f.

The DSMC-particles undergo collisions according to a particularcollision model. For instance, the hard-sphere (HS) modeldescribes the collision of two particles as a function of the collisionparameter b (Bird, 1994):

b¼ d12 cosχ2

� �ð8Þ

with the mean diameter of the colliding moleculesd12 ¼ ðd1þd2Þ=2 and the deflection angle χ. However, the scatter-ing law given by the HS model is physically not realistic. A morerealistic model is the variable-hard-sphere (VHS) model whichconsiders an effective scattering cross section that decreaseswith increasing translational energy. This relation is expressed ina variable hard sphere molecule diameter dVHS (Koura andMatsumoto, 1991):

dVHS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic � e�ω

π

rð9Þ

with a cross section constant c¼ 3πA2ðνÞðκ=2Þω and relativecollision energy e. Values for the collision integral A2ðνÞ, theinverse power law force exponent ν, the force constant κ andthe energy exponent ω can be found in Chapman and Cowling(1953) or Koura and Matsumoto (1991).

The VHS model leads – in contrast to the HS model – to aphysically realistic temperature dependent viscosity coefficient μ.However, the diffusion coefficient is not described accurately withthis model (see Section 4). A further modification of the impactparameter is the energy dependent exponent of the cosine, α, ofthe deflection angle which enables a correct reproduction of thediffusion coefficient (Koura and Matsumoto (1991)):

b¼ dVSS cos αχ2

� �: ð10Þ

Due to the discretization of DSMC-particle collisions, simula-tion truncation errors can occur if the iteration step size Δt, meshcell size a, or f are chosen incorrectly. It should be noted that onlyone collision per DSMC-particle and Δt is calculated and re-collisions are neglected (Garcia and Wagner, 2000). Furthermore,binary collisions of DSMC-particles are only considered withineach mesh cell and too low numbers of DSMC-particles per cellresult in wrong quantities of collisions (Sun et al., 2011). Thus, acompromise between minimizing the simulation errors (large a;small f and Δt), minimizing computation resources (large fand Δt), and maximize the spatial resolution of the simulation(small a) must be found. The errors occurring from chosen Δt, a,and f can be estimated by calculating the mean free pathλ¼ ð

ffiffiffi2

pπnd2mÞ�1 (with the number density n and molecule dia-

meter dm) and the most probable thermal speed ν¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kT=m

p. A

Δt dependent truncation error was shown to be negligible ifΔtrλ � ð8νÞ�1 (Garcia and Wagner, 2000; Hadjiconstantinou,2000). For a and f no noticeable error can be expected as long asthe a=λ ratio is kept below 0.5 (Sun et al., 2011; Alexander et al.,1998) and the number of DSMC-particles per cell is above 5(Sun et al., 2011).

3. Simulation method

In gas sensors obtained from Flame Spray Pyrolysis (FSP)(Großmann et al., 2011; Mädler et al., 2006; Sahm et al., 2004)two interdigitated electrodes are covered with a porous layer ofnano-particles, as illustrated in Fig. 1. The probe gas diffuses intothe porous layer where it is oxidized/reduced. Due to the chemical

reaction (rate of reaction proportional to probe gas concentration)the electrical resistance of the porous layer changes. The change inresistance is measured by the two electrodes and a signal propor-tional to the probe gas concentration is obtained. One key aspectof sensor response is how much of the probe gas reacts and howclose to the electrodes the reactions take place (Mädler et al.,2006). Examining such process on the nano-meter scale is the aimof this work. After validating the DSMC diffusion simulationsinside an empty cuboid and with symmetric gas molecules, thesimulation will be extended to CO in N2 diffusion and masstransport in porous nano-structures. As the thickness of thenano-particle layer is usually much smaller than the active sensorarea, the analytical diffusion solutions can be approximated by aone dimensional approach.

3.1. Analytical solution

The diffusion equation in one spatial variable, and with thebulk diffusion coefficient D¼D0

ij independent of the concentrationc, is given by the initial-boundary problem to be solved for theconcentration function c:

∂tc¼D∂2x c; xA �0; l½ ; t40;cð0; tÞ ¼ c1; tZ0;∂xcðl; tÞ ¼ 0; tZ0;cðx;0Þ ¼ 0; xA �0; l½:

8>>>><>>>>:

ð11Þ

Here, l is the distance between the inlet and the end of thecalculation domain, and c1 is the constant concentration held atthe inlet. The system (11) can be solved using the separation ofvariables, i.e. first assuming the concentration c is the productcðx; tÞ ¼ XðxÞTðtÞ of functions X and T that respectively depend onthe space and the time variable only, and then enforcing the initialand boundary conditions in (11). This separated-variables solution is

cDðx; tÞ ¼ c1�4c1π

∑1

p ¼ 0

expð�Dtðð2pþ1Þπ=2lÞ2Þ2pþ1

sinð2pþ1Þπx

2l;

xA �0; l½; t40: ð12ÞThe subscript D indicates the value of the diffusion coefficientassociated with the concentration function.

In order to find a value of the diffusion coefficient D that resultsin a good match between (12) and the numerically simulatedsolution cnum at a fixed time t0, a least-squares optimization andminimization of the L2 distance is used as

distðDÞ ¼ ∑xj AG

jcDðxj; t0Þ�cnumðxj; t0Þj2; DA I; ð13Þ

Fig. 1. Scheme of a gas sensor as synthesized by FSP. The probe gas above the nano-particle layer diffuses into the porous structure and, in case of CO, is oxidized to CO2

(see ☆). Due to the chemical reaction the resistance of the electrode bridgingporous layer changes, giving rise to the sensor signal.

J.A.H. Dreyer et al. / Chemical Engineering Science 105 (2014) 69–76 71

where G� �0; l½ is the discrete set of positions x at which thesimulated solution cnum is available, and I� R is the consideredrange of diffusion coefficients. In the numerical evaluation of thedistance distðDÞ, the series in (12) is appropriately truncated. It isexpected that dist(D) attains a global minimum at a value of thediffusion coefficient D that results in a good representation of thesimulation results.

3.2. DSMC simulation

The Direct Simulation Monte Carlo simulations are executedwith the open source software OpenFoam version 2.01 and theimplemented solver dsmcFoam. This package was extended withthe VSS binary collision model using Eq. (10) and the VSS collisioncross section sVSS based on the viscosity coefficient μ (Bird, 1994):

sVSS ¼ π5ðαþ1Þðαþ2Þðm=πÞ0:5ðkBTref Þω

16αΓð9=2�ωÞμEω�0:5 ð14Þ

with the translational kinetic energy E, molecular mass m, and thegamma function Γ. Simulations were conducted inside a cuboid withheight l and cubic cross section with length z. A reflecting wall withMaxwellian thermal boundary condition was implemented at heightvariable x¼ lwhile an open boundary was set at height x¼0. All otherboundaries were cyclic for a 2D diffusion approach. Pristine gas wasinitialized within the volume at t¼0 while a desired gas concentrationc1 was set at the inlet for t40. A constant gas molecule numberdensity of 2.68666�1025 m�3 was set within the volume as well asthe open boundary (inlet) representing the standard number densityat 273.15 K and 1 atm (Bird, 1994). Values of a¼8 nm, Δt ¼ 5�10�12 s, f¼1, and N2/CO parameters shown in Table 1 were utilizedfor all simulations. Larger awould limit the structure resolution of theporous layer while larger f would result in too low DSMC-particle percell values. Thus, a direct numerical representation of all molecules inthe domain has been donewith respect to the scaling factor f¼1.Withthe chosen parameters the number of DSMC-particles per patch andgas type that are introduced each time step (particle flux accumulator;pFA) is much smaller than 1. This causes a pulsed molecule inflowwith the implemented dsmcFoam inflow model (see Fig. S1 in theSupporting Information for details) as the residual of the integernumber of generated DSMC-particles is added to pFA of the followingiteration step. If pFA is much smaller than 1, too large or negative pFAvalues are calculated by this method. Therefore, the codewas adjustedso that pFA is not influenced by the previous residual. Regardingpossible truncation errors the chosen parameters result inΔt � λ � ð30νÞ�1, a � λ�1o7:5�1, and an average DSMC-particlenumber per cell 413. Hence, truncation errors can be expected tobe neglectable small in all cases. The spatial gas composition withinthe cuboid was estimated from z2 � 10 nm volumes assuming idealgas. All simulations were performed three times and averaged.

To generate porous layers as formed during FSP gas sensorsynthesis, the raw data from Mädler et al. (2006) were utilized.Single mesh cells at the positions of the layer's nano-particles (NP)were removed from the empty cuboid with OpenFoam's toolsnappyHexMesh (sHM). At the gas-NP interface a Maxwellianthermal boundary condition was implemented. Typically,

snappyHexMesh (sHM) refines the removed cubes towardsspheres by decreasing a and deforming the rectangular mesh cells.Even though this procedure would result in more realistic NPshapes, a DSMC-particle number below 5 should be avoided (Sunet al., 2011) to prevent inaccurate binary collision numbers.Therefore, the refinement of sHM was disabled resulting in cubicNPs with length a. The validity of this structure simplification wasverified by applying an image based method developed by Yanget al. (2009) to calculate pore size distributions. Binary images ofslices through the porous layer perpendicular to x were generatedevery 50 nm and with a resolution of 1120�1120 pixel (0.25 nmper pixel). Calculated pore sizes were averaged over all slices. Thelayer's porosity ε was calculated by dividing the volume removedby sHM with the total volume of the cuboid.

4. Results

4.1. Evaluation of VHS vs. VSS

Self-diffusion (ε¼100%) of N2 in N2 was computed with theVHS and VSS model and compared to the analytical solution (Eq.(12)). A cuboid with l¼4000 nm and square cross sectionz¼100 nm was chosen resulting in a Kn¼0.017. Due to the largeinlet-wall distance and investigated case of self-diffusion (i.e. wall-gas phase equilibrium with N2 expected before simulation starts)effects such gas adsorption can be neglected. Fig. 2 shows the N2

Table 1Summary of parameters used for the DSMC simulations. All values are given for 273 K and 1 atm (Bird, 1994; Chapman and Cowling, 1953.)

Parameter Molecular mass Molecular diameter Viscosity index Viscosity coefficient Scattering factor

Symbol m d ω μ α

Unit kg m – Nm s�1 –

Collision model VHS/VSS VHS VHS/VSS VSS VSS

N2 4.65�10�26 4.17�10�10 0.74 1.656�10�5 1.36CO 4.65�10�26 – 0.73 1.635�10�5 1.49

Fig. 2. Self-diffusion of 20 vol.% N2 utilizing two different binary collision modelsfor the DSMC simulation. Four different time steps are shown and compared to theanalytical solution of Fick's law of diffusion with a diffusion coefficient of1.783�10�5 m2 s�1.

J.A.H. Dreyer et al. / Chemical Engineering Science 105 (2014) 69–7672

isotope concentration calculated with the three methods at threedifferent time steps. The VHS model results in too low N2

concentrations even after 40 ns when compared to the analyticalresults. The difference further increases with diffusion timeindicating that the VHS model results in a too low diffusioncoefficient. The concentrations computed with the VSS modelmatch well the analytical solution. As numerous literature con-stants such as μ, α, ω, and D were used for the simulations as wellas for the analytical solution, the obtained agreement isremarkably good.

A quantitative comparison between the VHS and VSS model bymeans of least-squares optimization (Eq. (13)) leads to D¼1.33�10�5 m2 s�1 (not shown) and D¼1.62�10�5 m2s�1 (Fig. 4 inset),respectively. The reported D for N2 self-diffusion is 1.783�10�5 m2 s�1 (Massman, 1998) and shows the significant improve-ment of DSMC diffusion simulations using the VSS model.

The validity of the DSMC method and VSS model was furtherevaluated by comparison with the well established semi-infinitediffusion equation (Cussler, 1997):

cinf : ¼ c1�c1 � erfxffiffiffiffiffiffiffiffi4Dt

p� �

ð15Þ

with the inlet gas concentration c1 and error function erf. AsOpenFOAM's DSMC solver contains only a single inflow boundarymodel (in- and outflow not possible) and the implementation ofan additional boundary model exceeds the scope of this work, cinf :was compared to an empty cuboid with l¼10,000 nm. Only resultsare shown where all regarded DSMC-particles are more than1000 nm away from the reflecting wall at x¼ l. For clarity onlythe x¼2500–4000 nm region is shown and results are comparedto DSMC results of diffusion in a 4000 nm cuboid (i.e. wall atx¼4000 nm). As can be seen in Fig. 3 the simulation and analyticresults are identical for small t due to non-existences of gasmolecule-wall interactions. With increasing t the DSMC-particlesstart to be reflected from the wall in the 4000 nm cuboid. Due tothe absence of a wall in the 10,000 nm cuboid, the gas can diffuse

into the x44000 nm region. Consequently, the gas concentrationclose to 4000 nm is lower in the latter case. Again, the DSMCsolver combined with the VSS model results in good agreement tothe analytic solution for cinf :.

4.2. Evaluation of Kn influence

The influence of the Knudsen number on the diffusion is shownin Fig. 4. With increasing Kn the N2 self-diffusion is slowed downindicating a continuum to slip flow transition. The influence ofparticle-wall interactions increases with decreasing wall-inletdistance and thus larger Kn number. In case of large wall-inletdistances the reflecting wall has negligible influence on thediffusion and the simulation results converge towards the analy-tical solution for continuum regime. The low inlet concentration of18.5 vol.% in case of Kn¼0.068 can be explained by insufficienttime to increase the N2 isotope concentration within the first10 nm to 20 vol.% (see Fig. S2 in Supporting Information fordetails). The inset of Fig. 4 shows the Kn number dependance onthe D. Large Kn numbers result in smaller D (i.e. D¼1.21�10�5 m2 s�1 for Kn¼0.068) than the one reported for the con-tinuum regime (1.783�10�5 m2 s�1 Massman, 1998). Withdecreasing Kn number D increases and approaches its final valueof 1.66�10�5 m2 s�1, which is slightly lower than the D used inthe analytical solution (1.783�10�5 m2 s�1 Massman, 1998).

One common gas of interest in sensor or catalytic applicationsis CO in air. Therefore, N2 was chosen as a first approximation ofair and the diffusion of 1 vol.% CO was simulated. An empty cuboidwith length l¼1000 nm and square cross section with z¼280 nmwas used to evaluate the suitability of dsmcFoam for binary gasmixtures and asymmetric gas molecules such as CO. The simula-tion results are again qualitatively in good agreement with theanalytical solution (Fig. 5). A slightly lower DSMC diffusion in caseof Kn¼0.068 can be expected due to a continuum to slip-flowtransition (fitted D¼1.51�10�5 m2 s�1 to reported D¼1.804

Fig. 3. Self-diffusion of 20 vol.% N2 in a cuboid with l of 4000 nm and 10,000 nmand comparison with Eqs. (12) and (15), respectively. Both cases are practicallyidentical until the DSMC-particles start to interact with the reflecting wall in the4000 nm cuboid. As the wall at 4000 nm is absent in the 10,000 nm cuboid, the gasdiffuses in the x44000 nm region results in lower concentrations close tox¼4000 nm.

Fig. 4. N2 self-diffusion within an empty cuboid with different heights l plottedafter t � l�2 ¼ 7500 s m�2 over the relative distance ~x ¼ x � l�1. The DSMC resultsare compared with the analytical solution of Ficks law of diffusion andD¼1.783�10�5 m2 s�1. Inset: Fitted diffusion coefficient for DSMC results takingthe minimum of the sum of squared residuals between the simulation andanalytical solution with varying D.

J.A.H. Dreyer et al. / Chemical Engineering Science 105 (2014) 69–76 73

�10�5 m2 s�1 Massman, 1998) as discussed previously for thecase of N2 self-diffusion in Fig. 4.

4.3. Diffusion in porous layers

Porous layers were generated with two different structureprecisions, i.e. the original one with spherical NPs with 10 nmdiameters and a layer simplified with 8 nm cubes (Fig. 8). Cubeswith 8 nm length were chosen to gain a similar particle volumecompared to the 10 nm spheres. The porosity was constant forboth layers (spherical ε¼93% and cubic ε¼92.6%). An example ofthe image based pore size calculations is shown in Fig. 6 (particlesin black). A similar pattern in particle distribution and pore sizes isvisible. Calculated pore diameters dpore reach in both cases from8 to 110 nm. To further quantify the difference between theoriginal and simplified layers the pore area A at each dpore, thenormalized differential pore area ΔA � ðlog ½Δdpore� � ΣAÞ�1, andmedian pore diameter based on the Q2 pore size distribution

d50;2 was calculated (Fig. 7). The normalized differential pore areashows that both layers (original and simplified) exhibit a broadpeak between 13–100 nm. The sharp peak at 8 nm in the simpli-fied structure can be explained by the structured gaps between thecubes (see Fig. S3 in Supporting Information for details). As thepore area difference between both layers around 8 nm (see Fig. 7inset) is negligibly small, no significant influence is expected. Theratio of overall pore area to image area results in 92.8% and 92.6%for the original and simplified structure, respectively, which agreeswell with the volume based porosity (93% and 92.6%). Also d50;2 ofboth structures are very similar (58 nm for original and 60 nm forsimplified; see inset Fig. 7) and agree with experimental pore sizedistributions of FSP layers (d50;3 ¼ 70 nm for Q3 pore size distribu-tion of as-synthesized layer Schopf et al., 2013). As the wall-molecule collisions in dsmcFoam are exclusively diffusive (i.e.reflection angle independent from impact angle) and ε as well as

Fig. 5. Diffusion of 1 vol.% CO in N2 within an empty cuboid (l¼1000 nm) plotted atdifferent t. The simulation results are slightly slower than the analytical solutioncalculated with D¼1.804�10�5 m2 s�1 as can be expected for Kn¼0.068 (Fig. 4).

Fig. 6. Color-scaled pore sizes within a slice through the original (a) and simplified (b) layers. Both structures show good agreement in pore sizes and particle (shown inblack) distribution. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

Fig. 7. Normalized differential pore area of the original and simplified porous layer.Overall pore to image area ratios of 92.8% and 92.6% for the original and simplifiedstructures can be calculated as well as d50,2 values of 58 nm and 60 nm (see inset).The peak at 8 nm in the normalized differential pore area for the simplifiedstructure can be explained by the gaps between the 8 nm cubes (see Fig. S3 inSupporting Information for details).

J.A.H. Dreyer et al. / Chemical Engineering Science 105 (2014) 69–7674

d50;2 of the original and simplified layers showed good agreement,the structure simplified with 8 nm cubes was taken for thefollowing diffusion simulations (Fig. 8).

Two additional porous structures where generated for a morecomprehensive inside into layer parameters influencing the diffu-sion of gases: (1) An isotropic layer (randomly distributed parti-cles) with same particle size and porosity as above and (2) aporous layer as formed during FSP but with higher porosity(98.1%). The former one is unrealistic from a mechanical point ofview (floating nano-particles) but can be utilized for betterevaluation and comparison with DGM. The isotropic layer exhibitsa volume and area based porosity of 92.6%, which matches the FSPlayer with 92.6%. The pore distribution on the other hand is shiftedtowards lower values (Fig. 9), which is also reflected by the smallerd50;2 (41 nm compared to 60 nm; inset Fig. 9). The decrease in d50;2can be expected as the larger pores caused by the aggregatestructure of the nano-particles (Fig. 6 and S3 in SupportingInformation; see Ref. Mädler et al., 2006 for more details) areabsent in the isotropic layer. An increase in porosity from 92.6% to98.1% results in lager pores and an d50;2 of 144 nm (Fig. 9).

The diffusion through the three layers characterized in Fig. 9was simulated using DSMC and results are compared to the DGMmodel (Fig. 10). Note that CO adsorption was not considered in the

present study. Extension from not CO adsorbing layers to e.g. SnO2

supported Pt may result in deviations. In this case adsorptionshould be considered in the wall interaction scheme of the model.The DSMC results for all three films agree well to the resultsgenerated with the DGM. A comparison between the two FSPlayers shows that a smaller porosity result in slower diffusion. Thiscan be expected as a larger volume of nano-particles in the layerincreases the number of particle-molecule collisions and herebyhinders its diffusion. This effect is even more pronounced withrespect to the empty cuboid (i.e. ε¼100%; Fig. 5). At t¼5 ns the COconcentration between x¼800–1000 nm is close to 0 vol.% inde-pendent of the porosity (Fig. 10) while it already reached about0.03 vol.% in case of the empty volume (Fig. 5). Comparing the FSPand isotropic layer with ε¼92.6% shows the influence of pore sizeon the CO diffusion. Larger pores result in a larger area for the gasmolecules to go through. However, the average pore size andtortuosity of inhomogeneous layers (i.e. FSP layers) can not bederived simply from the porosity. The same tortuosity was utilizedfor the DGM calculations of the FSP and isotropic layer as theporosity is identical (i.e. τ¼ ðε1=3Þ�1 Millington, 1959). It can beexpected from the different pore size distributions (Fig. 9) that thisassumption is inaccurate in case of the FSP layer, which explainsthe better match of the diffusion in the isotropic layer in Fig. 10.Thus DGM forfeits its accuracy with increasing discrepancies fromthe here considered isotropic layer with physically inadequatefloating particles. Also, DSMC solvers require no pore diameter,tortuosity, or porosity information of the porous structure incontrast to other theoretical methods such as DGM. In order togain such structural information time intensive and laboriousstructure analysis methods are required, e.g. the image based poresize calculations applied in this study. Even though DGM requiresmuch less computational time DSMC is more efficient in terms ofcomputational resources when the structure analysis process isconsidered in the overall workload. Thus, DSMC can be easilyapplied to simulate diffusion in graded or inhomogeneous struc-tures where other theoretical methods reach certain limitations.With the correct mass transport verified e.g. chemical reactionscould be implemented in DSMC (Bird, 1994). Investigations such as

Fig. 8. Example for porous structure with 8 nm cubes taken for DSMC diffusionsimulations. Shown is the FSP layer with a porosity of 92.6% and d50;2 of 60 nm.

Fig. 9. Normalized differential pore area of two layers as formed by FSP (ε of 92.6%and 98.1%) and comparison with a isotropic structure (ε¼92.6%). The overall poreto image area ratios result in 92.6% (FSP ε¼92.6%), 92.6% (isotropic ε¼92.6%), and98% (FSP ε¼98.1%). Calculated d50;2 values are 60 nm, 41 nm, and 144 nm,respectively (see inset).

Fig. 10. CO concentration profile generated with DSCM simulations in layers withtwo different porosities and comparison to a isotropic layer as well as DGMcalculations. Results are shown after 5 ns and 20 ns.

J.A.H. Dreyer et al. / Chemical Engineering Science 105 (2014) 69–76 75

ε, tortuosity, particle size, surface area, or particle layer thicknesson sensor response time, gas selectivity or catalytic activity wouldbe of significant scientific relevance.

5. Conclusions

An open-source DSMC code was extended by the VSS binarycollision model to simulate gas diffusion in nano-scaled porouslayers. A pulsed DSMC-particle initialization in case of a particleflux accumulator smaller than 1 was resolved by modifying theexisting inlet model. The simulation results were verified with theanalytical diffusion equation and DGM. It was shown that theimplementation of the VSS model significantly enhances theaccuracy of DSMC diffusion simulations and a pulsed inlet condi-tion can be avoided. Good agreement was observed between theDGM and DSMC simulations in highly porous isotropic layers.However, no prior knowledge about Kn numbers, porosity, ortortuosity as for the analytical equation or DGM are required forDSMC. Furthermore, DSMC is not limited to homogenous struc-tures. With accurate mass transport on the nano-scale by DSMCsimulations, further extensions such as gas adsorption and che-mical reactions are possible.

Acknowledgments

LM would like to thank the German Research Foundation (DFG)for funding this project under Grant MA 3333/2-1. JD is thankful toSven Schopf for helpful discussions regarding pore size distributions.

Appendix A. Supplementary data

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.ces.2013.10.038.

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