Diffusion Fundamentals (E-Journal)

de Rioja, Victor Lopez

Fort, Joaquim

Isern, Neus

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Spread of virus infections 
 
V. L. de Rioja1*, J. Fort1, N. Isern1 
 
1Universitat de Girona, Girona, Spain. 
victor.lopezd@udg.edu 
 
We propose a new reaction-diffusion set of equations to explain the spread of viruses on bacterial 
populations. Although the process  
, (1)
was first described years ago [1] and several modifications have been proposed since then [2-5], in this 
new model we introduce improvements. The two main features of the model are: (i) the second-order 
correction in time that takes care of the time delay due to the fact that viruses spend a time τ inside the 
cells before the new generation disperses away [4, 5], and (ii) a new mathematical formulation for the 
death of infected cells, also dependent on the time delay τ, and which is biologically more reasonable 
than other approaches using a logistic equation [4, 5]. 
We apply our model to experimental data for the front propagation speed of several T7 virus strains 
[6], calculating the necessary parameters from independent experiments (as opposed to other authors 
[2, 3, 7, 8]). We find good agreement between the model and experimental data. We compare our 
results to a classical model with no time delay by Yin and McCaskill [2], as well as to two models that 
describe the death process similarly to our approach but without the including the diffusive delay [7, 
8]. All the other models overestimate the measured speed, therefore showing that our improvements 
are not only biologically sound, but also necessary to properly describe the processes of virus 
infection. 
 
References 
[1] A.L. Koch: The growth of viral plaques during the enlargement phase. J. Theor. Biol., 6 413 (1964). 
[2] J. Yin, J.S. McCaskill: Replication of viruses in a growing plaque: a reaction-diffusion model. Biophys. J., 
61 1540-1549 (1992). 
[3] L. You, J. Yin: Amplification and spread of viruses in a growing plaque. J. Theor. Biol., 200 365-373 
(1999). 
[4] J. Fort, V. Méndez: Time-delayed spread of viruses in growing plaques. Phys. Rev. Lett., 89 178101 1-4 
(2002). 
[5] D.R. Amor, J. Fort: Virus infection speeds: theory versus experiment. Phys. Rev. E, 82 061905 1-9 (2010). 
[6] J. Yin: Evolution of bacteriophage T7 in a growing plaque. J. Bacteriol., 175(5) 1272-1277 (1993). 
[7] A. Jones, H.L. Smith, H.R. Thieme, G. Röst: On spread of phage infection of bacteria in a petri dish. SIAM 
J. Appl. Math., 72 (2) 670-688 (2012). 
[8] S.A. Gourley, Y. Kuang: A delay reaction-diffusion model of the spread of bacteriophage infection. SIAM J. 
Appl. Math., 65 (2) 550-566 (2005). 
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1


English

Spread of virus infections

Gil, S.

Solanpää, Janne

Hämälainen, Timo

Räsänen, Esa

Klages, Rainer

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Chaotic diffusion in periodic lattices with repulsive potentials 
 
S. Gil1*, J. Solanpää2, T. Hämälainen2,  E. Räsänen2, R. Klages1 
 
1Quen Mary University of London, London, United Kingdom 
2Tampere University of Technology, Tampere, Finland 
*s.gilgallegos@qmul.ac.uk 
 
A simple model that has been widely studied to understand the origin of chaotic diffusion is the 
periodic Lorentz gas. This Hamiltonian system consists of a light particle that scatterers with heavy 
ones modeled as hard disks situated on a periodic lattice in the plane. It has been used to derive basic 
physical concepts such as Ohm's law from first principles. More recently, its diffusive properties have 
been explored with respect to changing the density of scatters by matching the dynamics to simple 
random walk approximations [1]. Not known is how diffusion in this model changes by perturbing 
parameters of this system. Here we study the behavior of the diffusion coefficient when the Lorentz 
gas is perturbed by softening the repulsive potential walls of the scatterers. Computer simulations 
show that the diffusion coefficient in the soft system is a highly irregular function of the lattice 
constant as a control parameter. Most surprisingly, we observe an anomalous enhancement of 
diffusion at specific parameter values. We show that these peaks are generated by special periodic 
trajectories of the moving particle that show up at these parameter values. 
 
References 
[1] R. Klages: Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics. Advanced     
Series in Nonlinear Dynamics Journal Vol. 24 (World Scientific, Singapore, 2007).  
 
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1


Chaotic diffusion in periodic lattices with repulsive potentials

Becker, Thijs

Nelissen, Katherine

Cleuren, Bart

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Current fluctuations in boundary driven diffusive systems 
 
T. Becker1*, K. Nelissen2,1, B. Cleuren1 
1Hasselt University, Hasselt, Belgium 
2Universiteit Antwerpen, Antwerpen, Belgium 
*thijsbecker@gmail.com 
 
We consider diffusive systems whose boundaries are connected to particle reservoirs at different 
densities. The object of interest is the net number of particles that have passed through the system in 
the steady state, during a time interval [0,t] , denoted by Qt . Because the dynamics is stochastic, the 
particle flow is a fluctuating quantity described by a probability distributionP(Qt ) . This distribution is 
studied in the thermodynamic limit, i.e., for large system sizes and for large times t . There has 
recently been a strong interest in the study of current fluctuations in the field of nonequilibrium 
statistical mechanics. A proper understanding of these fluctuations could allow one to define 
nonequilibrium thermodynamic potentials, in a conceptually similar way as for equilibrium potentials 
[1]. 
If one knows the transport diffusionD(ρ)and the mobilityσ (ρ) , one can calculate the full current 
distribution in a one-dimensional system. The transport diffusionD(ρ)is defined as 
Qt 
t
= −D(ρ) ∂ρ
∂x
, (1)
with ⋅ the average over P(Qt ) , and the concentration gradient small enough so that linear response is 
valid. The mobility quantifies equilibrium fluctuations of the current 
Qt
2 
t
= 1
L
σ (ρ),    ρA = ρB = ρ , (2)
with ρAand ρB the densities of the two reservoirs, and L the length of the system. All the moments of 
the current distribution can be calculated from the integral [2] 
Im = D(ρ)σ (ρ)
m−1dρ
ρB
ρA
 . (3)
For example, the variance of the particle flow is equal to 
Qt
2 − Qt 
2
t
= 1
L
I
2
I
1
. (4)
Akkermans and co-workers performed a theoretical calculation that showed that the above result can 
also be used to predict current fluctuations in higher-dimensional systems [3]. This result is valid for 
arbitrary shapes of the system and of the contacts between the reservoirs and the system, as long as 
both are of macroscopic size. 
We present numerical results, obtained from kinetic Monte Carlo (kMC) simulations, for the current 
fluctuations in a stochastic lattice gas model that was introduced in [4]. Our results [5] are in full 
agreement with the analytical predictions from [3]. Converge of the Fano factor, 
F = (Qt
2 − Qt 
2 ) / Qt  , to the analytical prediction for a two-dimensional lattice gas with a non-
trivial coupling to the reservoirs is shown in Figure 1. 
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1
 
Figure 1: (left) A square lattice with sides of length L  connected on its left and right to particle reservoirs A and 
B (black dots). Each site can contain maximally one particle. The reservoirs have densities ρA =1 and ρB = 0 . 
The lower left half and the upper right half of the system is connected to the reservoirs. (right) The Fano factor 
as a function of the length L , with one-sigma error bars. Convergence to the analytical prediction F =1/ 3 is 
found for L ≥ 40. 
For a general diffusive system, the transport diffusion depends on the dimension, and the current 
statistics changes for different dimensions. However, the one-dimensional theory can still be used to 
predict the full current distribution. This is shown in Figure 2 for a stochastic lattice gas where each 
site can contain maximally two particles.  
 
Figure 2: (left) Numerically obtained transport diffusion in one, two, and three dimensions. The limiting case of 
infinite dimensions is shown as a line. Error bars are smaller than the symbol sizes. (right) The variance of the 
current L(Qt
2 − Qt 
2 ) / t  in different dimensions, obtained from kMC simulations (black diamonds) and the 
prediction from equations (3) and (4) (blue error bars). The limiting case of infinite dimensions is shown as a 
line. All symbols have one-sigma error bars.  
 
References 
[1] K. Mallick: The exclusion process: a paradigm for non-equilibrium behaviour. Physica A 418, 17-48, 
(2015) 
[2] T. Bodineau, B. Derrida: Current fluctuations in nonequilibrium diffusive systems: an additivity principle. 
Phys. Rev. Lett. 92, 180601 (2004).  
[3] E. Akkermans, T. Bodineau, B. Derrida, O. Shpielberg: Universal current fluctuations in the symmetric 
simple exclusion process and other diffusive systems. EPL 103, 20001 (2013). 
[4] T. Becker, K. Nelissen, B. Cleuren, B. Partoens, C. Van den Broeck: Diffusion of interacting particles in 
discrete geometries. Phys. Rev. Lett. 111, 110601 (2013) 
[5] T. Becker, K. Nelissen, B. Cleuren: Current fluctuations in boundary driven diffusive systems in different 
dimensions: a numerical study. arXiv:1502.07559, (2015) (to be published in New. J. Phys.) 
2


Current fluctuations in boundary driven diffusive systems

Boyer, Denis

Pineda, Inti

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Ultra-slow diffusion in processes with preferential relocations to places 
visited in the past  
 
D. Boyer, I. Pineda* 
 
1Instituto de Física, Universidad Nacional Autónoma de México, Distrito Federal 04510, Mexico 
*intip@fisica.unam.mx 
 
An important class of random walks includes those in which the random increment at time step 
depends on the complete history of the process [1, 2]. These strongly non-Markovian random walks 
offer a promising modeling framework in order to understand the mobility and spreading of living 
organisms [3-5], yet, few analytical results are available for these processes [6, 7]. In this work we 
model a system where a walker can either move randomly (explore locally) or stochastically relocates 
at a site occupied at some earlier time (via long distance steps according to a reinforced rule and 
characterized by a memory parameter). The emergence of frequently visited locations generates ultra-
slow diffusion, logarithmic in time, whereas the walker probability density tends to a Gaussian. In this 
system we also study the response of the non-Markovian walks to the presence of a constant bias, 
where a higher probability is assigned in a prescribed direction. The predictions of the analytical 
expressions obtained in the asymptotic limit are in excellent agreement with numerical simulations. 
 
 
References 
[1] G. M. Schütz, S. Trimper:  Elephants can always remember: Exact long-range memory effects in a non-
Markovian random walk. Phys. Rev. E 70, 045101(R) (2004).   
[2] J. C. Cressoni, M. A. Alves da Silva, G. M. Viswanathan: Amnestically induced persistence in random 
walks. Phys. Rev. Lett. ß98ß, 070603 (2007). 
[3] J. M. Morales, D. T. Haydon, J. Frair, K. E. Holsinger, J. M. Fryxell: Extracting more out of relocation 
data: building movement models as mixtures of random walks. Ecology 85, 2436-2445 (2004). 
[4] A. O. Gautestad, I. Mysterud: Complex animal distribution and abundance from memory-dependent 
kinetics. Ecol. Complex. 3, 44-55 (2006).  
[5] G. M. Viswanathan, M. G. E. da Luz, E. P. Raposo, H. E. Stanley: The Physics of Foraging (Cambridge 
University Press, Cambridge, England, 2011). 
[6] D. Boyer, C. Solis-Salas: Random walks with preferential relocations to places visited in the past and their 
application to biology. Phys. Rew. Lett. 112, 240601 (2014). 
[7] D. Boyer, J. C. R. Romo-Cruz: Solvable random-walk model with memory and its relations with Markovian 
models of anomalous diffusion. Phys. Rev. E 90, 042136 (2014). 
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1


Ultra-slow diffusion in processes with preferential relocations to places visited in the past

Dvoyashkina, Nina

Freude, Dieter

Seidler, Christopher F.

Wark, Michael

Haase, Jürgen

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Composite fuel cell materials studied by 
MAS PFG NMR diffusometry and MAS NMR spectroscopy 
 
N. Dvoyashkina1*, D. Freude1, C.F. Seidler2, M. Wark2, J. Haase1 
 
1Universität Leipzig, Institute für Experimentelle Physik, Leipzig, Germany, 
2Carl von Ossietzky Universität Oldenburg, Institut für Chemie, Oldenburg, Germany 
*nina.dvoyashkina@uni-leipzig.de 
 
Proton exchange membrane (PEM) fuel cells are considered to be one of the most promising energy 
conversion systems for replacement of conventional energy sources based on fossil fuels for 
automobiles and mobile devices [1]. An often used membrane material facilitating the proton transport 
is a Nafion polymer membrane, which has been an industrial standard for many years. However, 
amongst disadvantages for the practical use of Nafion are its stability at temperatures >90 °C and 
maintaining high relative humidity at these temperatures. Incorporation of inorganic–organic additives 
into the polymer membrane is a promising approach to overcome the stability issues at elevated 
temperatures for future generations of membranes for PEM fuel cells. As a host system, mesoporous 
silicas can be utilized for creating hybrid organic–inorganic composites by functionalization them with 
organic moieties.  
In our contribution, we prepared mesoporous Si-MCM-41 
and KIT-6 materials [2], which were subsequently 
functionalized by co-condensation from tetraethyl-
orthosilicate and mercaptopropylmethoxysilane with a 
following thiol-oxidation to SO3H by H2O2. In our earlier 
work [1], we showed that low proton conductivity of pristine 
Si-MCM-41 (2 × 10−6 S cm−1 at 373 K and 100% of 
relative humidity) can be drastically improved (up to 
2 × 10−2 S cm−1) by functionalizing the pore surface with 
sulfonic groups by ~40%. The conductivity of the sample 
KIT-6 reaches a value of the same order.  
Further, magic-angle spinning (MAS) pulsed field gradient 
(PFG) NMR diffusometry has been applied for investigating 
the nature of the conductivity in these materials. In addition, 
MAS NMR spectroscopy provided structural analysis of the 
synthesized materials. The use of high-resolution 1H MAS 
PFG NMR was necessary, since the 1H NMR spectrum 
consists of the signal of mobile hydroxy species and the 
non-decaying signals of the fixed methylene groups (Fig.1). Self-diffusion coefficients in the order of 
magnitude of 4 × 10−9 m2s−1 were obtained and compared with diffusion coefficients derived from 
conductivity data. The deconvolution of the 29Si MAS NMR spectra showed that Si atoms of the 
functionalizing silane are linked to the MCM-41 framework via three or two ≡Si-O-Si≡ bonds. 13C 
cross-polarization MAS spectroscopic experiments confirmed that the majority of the organic 
functional groups remained intact after the oxidation [3]. 
 
References 
[1] M. Phonyiem, S. Chaiwongwattana, C. Laongam, K. Sagarik: Proton transfer reactions and dynamics 
of sulfonic acid group in Nafion. PCCP, 13, 10923-10939 (2011).  
[2] Tae-Wan Kim, Freddy Kleitz, Blain Paul, Ryong Ryoo: MCM-48-like Large Mesoporous Silicas with 
Tailored Pore Structure:  Facile Synthesis Domain in a Ternary Triblock Copolymer−Butanol−Water 
System. J. Am. Chem. Soc., 127, 7601–7610 (2005). 
[3] M. Sharifi, M. Wark, D. Freude, J. Haase: Highly proton conducting sulfonic acid functionalized 
mesoporous materials studied by impedance spectroscopy, MAS NMR spectroscopy and MAS PFG 
NMR diffusometry. Micropor. Mesopor. Mater., 156, 80-89 (2012).  
 
Figure 1:  NMR signal decays with 
linearly increasing strength of the gradient 
pulses for KIT-6 with 20 % 
functionalization by sulfonic groups 
loaded with water vapour at 373 K.  
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1


Composite fuel cell materials studied by MAS PFG NMR diffusometry and MAS NMR spectroscopy

Itckovich, Aleksei A.

Bokstein, Boris

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Computer modeling of atomic clusters formation in grain boundaries 
 
A. Itckovich1*, B. Bokstein1 
 
1The National University of Science and Technology MISiS, Moscow, Russia 
*alexey84.box@gmail.com 
  
 Atomic interaction in grain boundaries (GB) leads to the complexes formation. Up to now two 
types of the complexes were analyzed for binary systems with restricted solubility: AmBn for systems 
with intermediate phases and B2 – without them [1]. It was shown that effect leads to the increase of 
segregation and decrease of diffusivity.  
 There are no direct experimental data on the complexes in GB, only indirect results, obtained 
by molecular dynamics modeling [2]. The author used model of GB containing 18000 Al atoms. The 
part of them was replaced by Fe atoms. With the use of the half-empiric potential Al-Fe it was shown 
that in the Al-based alloy with 2.5 at.% Fe, in symmetric GB (100) the complexes of Fe2-type are 
formed and the coordination number Fe-Fe is close to 1,that is much more than in random distribution. 
It was also shown that the complexes formation leads to decrease of mean-square atomic 
displacements and therefore of diffusion coefficient. 
 At the present study we investigated the mobility of the copper atoms. Model of 269640 atoms 
contained two symmetrical GB's sigma 5 (001) (012). For a few pairs of accidental atoms near the 
grain boundary, placed on 2.6 nm from each other, the energy of interaction (dipole energy, ΔEdipole ) 
between the atoms was introduced, as well as the energy of segregation at GB (ΔEseg).  The mean 
square displacement of atoms within 100 ns at 1000 and 1200 K was measured (Fig.1)  
The data on displacment were recalculated to diffusion coefficients. (Table 1) 
 
 
Fig.1 -  Typical dependence of the mean square displacement of 
pairs of atoms vs time.     
  
 
Table 1 - Data of dipole energy, energy of seregation and diffusion coefficients 
V ΔEdipole eV/atom 
ΔEseg 
eV/atom 
D cm2/s 
1000 1200 
v.1 -0,059 0,008 3,94E-07 2,19E-06 
v.2 -0,199 0,006 2,08E-07 1,46E-06 
v.3 -0,497 0,014 4,01E-08 1,03E-07 
v.3 -0,063 0,478 2,06E-07 9,47E-07 
v.5 -0,285 0,463 5,94E-08 4,87E-07 
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1
 It is seen from the table the diffusion coefficients decrease with increasing the dipole 
interaction energy and segregation energy. 
 
Conclusions. 
A formation of the atomic complexes in GB is possible in systems with restricted solubility. 
The complexes formation leads to decrease of mean-square atomic displacements. 
  1 Esin V., Bokstein B.S., Rodin A.,O., Grain Boundary Diffusion Model in B-regime with regard  to the 
formation of atomic complexes in grain boundaries. Def. and Dif. Forum, vol 309-310 (2011) pp. 29-38 
  2 Mendelev M.I., Rodin A.O., Bokstein B.S. Computer simulation of Fe diffusion in liquid Al and along Al 
grain boundaries. Def. and Dif. Forum, vol 309-310 (2011) pp. 223-230 
 
2


Computer modeling of atomic clusters formation in grain boundaries

Itto, Yuichi

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Anomalous diffusion with heterogeneity in view of superstatistics 
 
Y. Itto 
 
Science Division, Center for General Education, Aichi Institute of Technology, 
Aichi 470-0392, Japan 
 
itto@aitech.ac.jp 
 
There exists an exotic phenomenon from the traditional perspective of anomalous diffusion. 
Such a phenomenon has experimentally been observed in the infection pathway of 
adeno-associated viruses in living HeLa cells [1]. Remarkably, the exponent of anomalous 
diffusion of the virus, which exhibits stochastic motion in the free form as well as the form 
being contained in the endosome, fluctuates depending on localized areas of cytoplasm of the 
cell, in marked contrast to traditional anomalous diffusion. Here, from the viewpoint of 
superstatistics [2], which is a “statistics of statistics” with two largely-separated time scales and 
aims to describe nonequilibrium complex systems with two such time scales, a theoretical 
framework [3] is developed to establish a generalized fractional kinetics [4] for describing the 
infection pathway of the virus in the cytoplasm. In this theory, the existence of a large 
separation of two time scales in the infection pathway is explicitly taken into account. It is also 
shown within the present framework that the motion of the virus may obey a scaling law. 
 
 
References 
[1] G. Seisenberger, M.U. Ried, T. Endreß, H. Büning, M. Hallek, C. Bräuchle: Real-time single-molecule 
   imaging of the infection pathway of an adeno-associated virus. Science 294, 1929-1932 (2001).  
[2] C. Beck, E.G.D. Cohen: Superstatistics. Physica A 322, 267-275 (2003). 
[3] Y. Itto: Heterogeneous anomalous diffusion in view of superstatistics. Phys. Lett. A 378, 3037-3040 
   (2014). 
[4] Y. Itto: Heterogeneous anomalous diffusion of a virus in the cytoplasm of a living cell. J. Biol. Phys. 
   38, 673-679 (2012). 
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1


Anomalous diffusion with heterogeneity in view of superstatistics

Kurzhals, Jan-Magnus

Dierl, Marcel

Maass, Philipp

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Phase transitions in driven single file diffusion of suspended particles 
 
J. Kurzhals*, M. Dierl, P. Maass 
 
Universität Osnabrück, Fachbereich Physik, Barbarastraße 7, 49076 Osnabrück, Germany 
*jkurzhals@uos.de 
 
Driven single-file diffusion through channels occurs in many different areas. Important examples are 
the biased molecular motion through nanotubes or through porous materials with channel-like 
structures, the incoherent electron transport along molecular wires, the ion transport through 
membrane channels, and the directed motion of protein motors along filaments. Much work has been 
devoted in the past to understand anomalous diffusion of tagged particles in single-file transport [1]. 
With respect to collective properties, phase transitions can be expected to occur based on the findings 
for the asymmetric simple exclusion process (ASEP) [2].  
In this work we study the driven motion of suspended particles of radius R through a two-dimensional 
narrow channel of width W<4R that is connected to two particle reservoirs, see Fig. 1(a). To this end 
Brownian dynamics simulations are performed. To cope with the injection of particles into and the 
ejection of particles from the reservoirs, a special simulation method is implemented based on the 
bulk-adapted rate method developed in [3,4]. We find that the bulk density of the suspended particles 
inside the channel exhibits phase transitions as a function of the reservoir densities, reflecting the 
behavior of the ASEP. According to our coupling to the reservoirs, the phase transitions can be 
described by applying the minimum and maximum current principles to the current-density relation in 
a bulk system (channel with periodic boundary conditions), see Fig. 1(b). Possible implications for 
experiments and applications in microfluidics are pointed out.  
       (b) 
      (a) 
 
 
 
 
 
 
Figure 1: (a) Sketch of suspended Brownian particles that are driven by a force f from a left to a right particle 
reservoir through a narrow channel of width W<4R. (b) Relation between current jb and density ρb in a bulk 
system (channel with periodic boundary conditions). 
 
References 
[1] E. Barkai, R. Silbey; Theory of single file diffusion in a force field. Phys. Rev. Lett. 102, 050602 (2009). 
[2] B. Derrida; An exactly soluble non-equilibrium system: The asymmetric simple exclusion process. Phys. 
Rep. 301, 65-83 (1998). 
[3] M. Dierl, P. Maass, M. Einax; Classical driven transport in open systems with particle interactions and 
general couplings to reservoirs. Phys. Rev. Lett. 108, 060603 (2012). 
[4] M. Dierl, M. Einax, P. Maass; One-dimensional transport of interacting particles: Currents, density 
profiles, phase diagrams, and symmetries. Phys. Rev. E 87, 062126 (2013). 
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1


Phase transitions in driven single file diffusion of suspended particles

Saraci, Erisa

Arndt, Robert

Kullmann, Jens

Enke, Dirk

Meier, Tim-Arne

Belder, Detlev

Coppens, Marc-Olivier

Gläser, Roger

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
Diffusion Limitations and Effectiveness Factor of Mesoporous and 
Hierarchically Structured Catalysts for SCR-DeNOx 
 
E. Saraci1*, R. Arndt1, J. Kullmann1, D. Enke1, T.-A. Meier2, D. Belder2,  
M.-O. Coppens3, R. Gläser1 
 
a Institute of Chemical Technology, Universität Leipzig, Leipzig, Germany. 
b Institute of Analytical Chemistry, Universität Leipzig, Leipzig, Germany. 
c Department of Chemical Engineering, University College London, London, United Kingdom. 
*erisa.saraci@uni-leipzig.de 
 
Due to stricter environmental regulations for minimizing NOx emissions, the urge to apply efficient 
catalysts is a worldwide concern. Among the technologies for controlling NOx-emissions, the selective 
catalytic reduction of NOx (SCR-DeNOx) is the main industrial application. Vanadia supported on 
mesoporous titania is the most common catalyst wash-coated on SCR monoliths [1]. Such wash-coats 
lack a well-defined pore architecture, which leads to diffusion limitations and compromised material 
use. Theoretical studies have shown that by introducing macropores in a conventional SCR catalyst, 
its efficiency can be increased by up to 180% [2]. However, a direct proof of the effect of pore 
hierarchy on the catalytic activity of SCR-DeNOx reaction is lacking.  
In order to experimentally investigate the effect of additional macropores in a SCR catalysts, 
mesoporous and hierarchically structured (meso-/macroporous) V2O5/TiO2 mixed oxides with 
different V2O5-contents (1, 3.5 and 5 wt.%) were prepared via a modified sol-gel method [3].  
Characterization results show that the catalysts have similar content and nature of V2O5-species, 
independent of their pore architecture. The diffusion effect on 
the catalytic activity was measured for a mesoporous catalyst by 
varying its grain size. By using the Weisz-Prater criterion to 
estimate the Thiele modulus and the effectiveness factor from 
the observed reaction rates, diffusion limitations were indeed 
found for the DeNOx reaction (ϕ~3 and η~0.7) for ̅cat.grain= 0.25 
mm (Figure 1). The results from SCR of NO with NH3 (mcat.= 
0.2 g, ̅cat.grain = 0.25 mm, T = 423− 773 K, GHSV = 21.300 h−1, 
n(NO)/n(NH3) = 2/1) over mesoporous and hierarchical 
V2O5/TiO2 catalysts show that NO conversion increases with 
V2O5-content. However, the hierarchical catalysts exhibit a 
higher overall reaction 
rate over the whole 
temperature range (TR 
= 423− 773 K) when compared to their mesoporous 
counterparts with same V2O5-content. The NO consumption 
rate (rNO) increases by up to 200%, and the light-off 
temperature (TLight-off) decreases by 50 K when additional 
macropores are present (Figure 2). Considering the surface 
properties of the materials and the fact that DeNOx is limited 
by intraparticle diffusion, the higher activity of the meso-
/macroporous catalysts (for the same V2O5 content) can be 
attributed to diffusion facilitation within the catalyst 
particle. These results provide the first experimental proof 
of the advantage of pore hierarchy in SCR-DeNOx as 
predicted earlier from theoretical calculations [2].  
References 
[1] P. Forzatti: Present status and perspectives in de-NOx SCR catalysis. Appl. Catal. A. 222, 221-236 (2001). 
[2] G. Wang, M.-O. Coppens: Calculation of the optimal macropore size in nanoporous catalysts and its 
application to DeNO(x) catalysis. Ind. Eng. Chem. Res. 47, 3847-3855 (2008). 
[3] G. Hasegawa. K. Kanamori, K. Nakanishi, T. Hanada: Facile Preparation of Hierarchically Porous TiO2 
Monoliths. J. Am. Cer. Soc. 93, 3110-3115 (2010). 
Figure 1: Effectiveness factor and 
Thiele modulus for different grain 
sizes of the 5 wt.% V2O5/TiO2 
mesoporous catalyst. 
Figure 2:  Reaction rate (rNO) at 523 K 
and light-off temperature (TLight-off) 
during SCR-DeNOx over mesoporous 
and hierarchical V2O5/TiO2 catalysts 
with different V2O5-content. 
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1


Diffusion limitations and effectiveness factor of mesoporous and hierarchically structured catalysts for SCR-DeNOx

Wallenstein, Dieter

Fougret, Christoph M.

Brandt, S.

Hartmann, U.

The Open-Access Journal for the Basic Principles of Diffusion Theory, Experiment and Application
 
 
 
 
 
The application of inverse gas chromatography to investigate diffusion 
resistance in FCC catalysts 
 
D. Wallenstein1,*, C.M. Fougret1, S. Brandt1, U. Hartmann1 
 
1 Grace GmbH & Co. KG, Postfach 1445, 67545 Worms, Germany 
* E-mail address: dieter.wallenstein@grace.com 
 
Fluid Catalytic Cracking (FCC) catalysts are used to convert heavy oil fractions (boiling points 
>360°C) into lighter products such as gasoline and liquefied petroleum gas [1]. The diffusion rates of 
feedstock molecules and intermediate products in the FCC catalysts have relevance for the resulting 
conversion and product selectivity levels [2]. Since the cracking reactions take place in the gas phase 
in a riser reactor a corresponding method for diffusion measurements (gas-phase, transport diffusion 
[3]) was developed by means of inverse gas chromatography. In some cases the determined effective 
diffusion coefficients allowed to explain the catalytic properties of FCC catalysts where traditional 
analytical tools such as x-ray diffraction and nitrogen sorption reach their limit. 
Case studies are presented showing diffusion in FCC catalysts and their catalytic performance when 
(1) zeolite crystallite size in an FCC catalyst is varied and when (2) an FCC catalyst is contaminated 
with Fe. 
 
[1] J. W. Wilson, Fluid catalytic cracking, technology and operation, Penn Well Publishing 
Company, ISBN 0-87814-710-1, 1997 
[2] P. Kortunov, S. Vasenkov, J. Kärger, M. Fé Elía, M. Perez, et al, Diffusion in Fluid Catalytic 
Cracking Catalysts on Various Displacement Scales and Its Role in Catalytic Performance, 
Chem. Mater. 17, 2466-2474, 2005 
[3] J. Kärger, Measurement of diffusion in zeolites – a never ending challenge, Adsorption, 9, 29-
35, 2003 
 
This work is licensed under a Creative Commons Attribution 4.0 International License. 
To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
1


The application of inverse gas chromatography to investigate diffusion resistance in FCC catalysts

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