Diffusion Fundamentals (E-Journal)

Some aspects of grain boundary segregation (GBS) are discussed. This paper adds two new sprouts. The first is connected with formation of the atomic complexes in boundary region and their effect on grain boundary diffusion (GBD). The second – with a nonhomogeneity of energy distribution between boundary sites

Bokstein, Boris

Itckovich, Alexei

Pokhvisnev, Yury

Rodin, Alexei

English

Grain Boundary Segregation: the New Sprouts

Abdank-Kozubski, Rafał

Kärger, Jörg

Tomczy, Wojciech

Foreword: Special Issue "IXth Diffusion Fundamentals Conference"

Schmidt, Johannes

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   diffusion-fundamentals.org, Vol. 35 (2022) 13, URL: https://diffusion-fundamentals.org/journal/35/2022/13.pdf 
Stochastic Models for Interdisciplinary Problems 
Andreas Schadschneider1, Johannes Schmidt2* 
1Institute for Theoretical Physics and Institute for Physics Education, Universität zu Köln, 
50937 Köln, Germany 
2Institute for Theoretical Physics, Technische Universität Berlin, 10623 Berlin, Germany 
and Bonacci GmbH, 50937 Cologne, Germany 
*johannes.schmidt@bonacci.de
Stochastic models based on driven diffusion have been very useful in several interdisciplinary 
applications, especially those related to transport and traffic problems [1]. One of the paradigmatic 
models for traffic flow, the Nagel-Schreckenberg model, can be viewed as a generalization of the 
asymmetric simple exclusion process (ASEP) which is a paradigmatic model for driven diffusion. Other 
generalizations have been used e.g. for crowd dynamics which describes motion in two dimensions, or 
intracellular transport processes where diffusion competes with Langmuir kinetics. 
This presentation gives an overview over such models and their applications. The research on traffic 
systems shows a close interplay between empirical observations, experiments (Fig. 1, left) and 
theoretical investigations that rely strongly on ideas and concepts from physics. From this more 
theoretical perspective, universal properties of the underlying models are discussed in the presentation. 
Generically the models are superdiffusive and belong to the Fibonacci family [2] of dynamical 
universality classes which generalize the KPZ universality class [3]. Specifically, the Nagel-
Schreckenberg model shows KPZ behavior (Fig.1, right) [4]. This behavior is surprisingly robust and 
even persists for 2-lane generalizations which are no longer single-file motion [4]. 
Figure 1: Left: Experiment on jam formation in vehicular traffic: Right: Dynamical structure function of the 
Nagel-Schreckenberg model for various model parameters. 
References 
[1] A. Schadschneider, D. Chowdhury, K. Nishinari: Stochastic Transport in Complex Systems. Elsevier (2010). 
[2] V. Popkov, A. Schadschneider, J. Schmidt, G.M. Schütz: Fibonacci family of dynamical universality classes. 
Proc.Nat. Acad. Sci.112, 12645 (2015). 
[3] M. Kardar, G. Parisi, Y.-C. Zhang: Dynamic Scaling of Growing Interfaces. Phys. Rev. Lett. 56, 889 (1986). 
[4] J. de Gier, A. Schadschneider, J. Schmidt, G.M. Schütz: Kardar-Parisi-Zhang universality of the Nagel-
Schreckenberg model. Phys. Rev. B 100, 052111 (2019). 


Stochastic Models for Interdisciplinary Problems

Caro, Jürgen

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   diffusion-fundamentals.org, Vol. 35 (2022) 20, URL: https://diffusion-fundamentals.org/journal/35/2022/20.pdf 
1st Fickian Law to predict gas selectivity of a nanoporous membrane 
Juergen Caro 
Institute of Physical Chemistry and Electrochemistry, Leibniz University Hannover, Germany 
The separation of short chain olefin/paraffin pairs like ethylene/ethane and propylene/propane is a hot 
topic in chemical industry. The common technology is cryogenic distillation, but adsorption and espe-
cially membrane permeation processes would be much more energy-efficient. However, the existing 
polymers do not allow the fabrication of powerful membranes for the olefin/paraffin separation. 
Since the olefin molecules ethylene and propylene are slightly smaller than the corresponding paraffins, 
molecular sieve membranes with a pore size exact between the olefin and paraffin molecules could be 
applied. At first sight, MOFs are the disruptive material that could allow the construction of such mo-
lecular sieve membranes. However, MOFs show the effect of framework flexibility with no sharp mo-
lecular sieving effect. Therefore, theoretical calculations predicting the membrane selectivity turned out 
as difficult. 
The poster will discuss the applicability of the 1st Fickian Law to predict gas separation through a MOF 
membrane based on independent experimental data. The flux density of component A is described by  
𝑗𝐴 = −𝐷𝑇𝐴  
𝜕𝑐
𝜕𝑥
 (1) 
with jA as the flux density in mol of A per time and area, DTA is the transport diffusivity of A, and ∂c/∂x 
is the concentration gradient of component A across the membrane. As a rough estimate, we substitute 
the concentration gradient for the concentration difference over the membrane divided by the thickness 
of the membrane. Further, due to the presence of vacuum or a sweep gas on the permeate side of the 
membrane, the concentration of A at the desorption side of the membrane is considered zero. 
Thus, the selectivity of the membrane is the ratio of the fluxes A and B acc. to (1), 
𝛼𝐴,𝐵 =  
𝑗𝐴
𝑗𝐵
=  
𝐷𝑇𝐴
𝐷𝑇𝐵
𝑐𝐴
𝑐𝐵
(2) 
To predict the selectivity of a membrane, we need independent data for the transport diffusivities DTA 
and DTB as well as adsorption data cA and cB. Infrared microscopy is an excellent novel method to de-
termine transport diffusivities [1]. The following table shows that the MOF membrane of structure type 
ZIF-8 should be very promising in propylene/propane separation but not suitable for the ethylene/ethane 
separation. Experimental permeation studies at room temperature on supported ZIF-8 membranes have 
exactly found this prediction. 
Ethylene Ethane Propylene Propane 
DT from ref. [2] in m2/s 
c at 1 bar in mmol/g 
410-11 
1.5 from [3] 
110-11 
2.3 from [3] 
210-13 
4.2 from [4] 
410-16 
4.5 from [4] 
Predicted selectivity acc. (2) 2.6 470 
Measured selectivity 2.4 from [5] > 200 from [6] 
References 
[1] J. Kärger, T. Binder, C. Chmelik, et al.: Microimaging of transient guest profiles to monitor mass transfer in 
nanoporous materials. Nature Mater. 13, 333–343 (2014). https://doi.org/10.1038/nmat3917 
[2] C. Chmelik, J. Kärger: The predictive power of classical transition state theory revealed in diffusion studies with MOF ZIF-
8. Micropor. Mesopor. Mater. 225, 128–132 (2016). https://doi.org/10.1016/j.micro meso.2015.11.051
[3] P. Krokidas, M. Castier, I.G. Economou: Computational study of ZIF-8 and ZIF-67 performance for separation of gas 
mixtures. J. Phys. Chem. C 121, 17999–18011 (2017). https://doi.org/10.1021/acs.jpcc.7b05700  
[4] D. Liu, X. Maa, H. Xi, Y.S. Lin: Gas transport properties and propylene/propane separation characteristics of ZIF-8 
membranes. J. Membr. Sci. 451, 85-93 (2014). https://doi.org/10.1016/j.memsci.2013.09.029  
[5] H. Bux, C. Chmelik, R. Krishna, J. Caro: Ethene/ethane separation by the MOF membrane ZIF-8: Molecular correlation 
of permeation, adsorption, diffusion. J. Membr. Sci. 369, 284-289 (2011). https://doi.org/10.1016/j.memsci.2010.12.001  
[6] S. Park, K.Y. Chob, H.K. Jeong: Enhancing the propylene/propane separation performances of ZIF-8 membranes by 
postsynthetic surface polymerization. J. Mater. Chem. A, in press (2022). https://doi.org/10.1039/D1TA09096H  


1st Fickian Law to predict gas selectivity of a nanoporous membrane

Cano, Nerea Alcazar

Buscalioni, Rafael Delgado

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   diffusion-fundamentals.org, Vol. 35 (2022) 38, URL: https://diffusion-fundamentals.org/journal/35/2022/38.pdf 
Complexity of anomalous diffusion processes 
Nerea Alcázar-Cano 1* and Rafael Delgado-Buscalioni 1,2  
1Theoretical Condensed Matter Physics Department, Universidad Autónoma de Madrid, Spain 
2Condensed Matter Physics Center (IFIMAC), Madrid, Spain 
*nerea.alcazar@uam.es
The main motivation for studying anomalous diffusion is the great variety of phenomena that give rise 
to it. The structure of the environment, the interactions with other particles of the system, or the presence 
of an external force field are examples that modify the dynamics and lead to anomalous diffusion. Over 
the last few years we have been working on two very different systems in which anomalous diffusion is 
the main character: the subdiffusion of passive tracers in disordered media and the collective diffusion 
of a population of excitons in two-dimensional perovskite layers.  Both works are nice examples of the 
complexity of the anomalous diffusion processes. 
In disordered or heterogeneous media, a diffusive particle encounters obstacles which slow down its 
dynamics. We observe that, although some dynamic properties as the subdiffusion exponent does not 
depend on the structure of the media, the full dynamics given by displacement probability distribution 
of the particle strongly depends on the heterogeneity of the structure [1,2]. 
Figure 1: (Right) One of the random networks where passive tracers diffuse anomalously. (Left) Pictorial 
representation of our model for the exciton diffusion in perovskite layers. 
Experimentally it has been measured that an exciton population presents anomalous diffusion when it 
moves in two‐dimensional metal‐halide perovskites. This anomalous behaviour is due to the presence 
of defects in the material, where excitons get trapped. We develop a continuum model for the exciton 
concentration field which is able to reproduce the spatial dynamics of the exciton population and allow 
us to estimate important quantities such as the defect density and the diffusion coefficient [3]. 
References 
[1] N. Alcázar‐Cano and R. Delgado‐Buscalioni. Hydrodynamics induce superdiffusive jumps of passive tracers 
along critical paths of random networks and colloidal gels. Soft Matter 18, 1941-1954 (2022). 
[2] N. Alcázar‐Cano and R. Delgado‐Buscalioni. A general phenomenological relation for the subdiffusive 
exponent of anomalous diffusion in disordered media. Soft Matter 4, 9937-9949 (2018). 
[3] M. Seitz, M. Meléndez, N. Alcázar‐Cano, D. N. Congreve, R. Delgado‐Buscalioni, and F. Prins. Mapping the 
trap‐state landscape in 2D metal‐halide perovskites using transient photoluminescence microscopy. Adv. Opt. 
Mater. 9 (18): 2001875 (2021). 


Complexity of anomalous diffusion processes

Binder, Christian

Krondorfer, Johannes

This work is licensed under a Creative Commons Attribution 4.0 International License. 
   diffusion-fundamentals.org, Vol. 35 (2022) 62, URL: https://diffusion-fundamentals.org/journal/35/2022/62.pdf 
ZoRRO – A new software package for ab initio calculation of molecular 
diffusion rates 
Christian Binder, Johannes Krondorfer and Andreas W. Hauser, 
Graz University of Technology, Petersgasse 16, 8010 Graz, Austria 
Molecular diffusion in porous structures is a highly interesting topic, as a variety of applications are 
associated with this process. Prominent examples are seawater desalination, gas separation on an 
industrial scale, the chiral separation of drug molecules via two-dimensional materials such as porous 
graphene or flat metal-organic frameworks, but also gas storage in zeolite structures. Describing these 
processes ab initio is a challenging task, since molecular propagation is a highly disordered, complex 
process with, in most cases, seemingly random trajectories. As a result, molecular dynamics 
simulations using cost effective force fields have become the established standard approach for 
theoretical estimates of permeabilities and diffusion constants. Unfortunately, these force fields often 
exhibit significant errors in energy prediction that can falsify the correct values by orders of magnitude. 
Another popular approach is to identify a given molecular propagation process as a chemical reaction 
and to employ Eyring theory. However, as can be shown, in cases of pronounced anharmonicity and 
particularly low-lying vibrational frequencies, the Eyring based predictions can deviate substantially 
from the correct result. 
We developed an alternative method for determining the relevant quantities for problems of 
molecular diffusion, aiming for the best of both worlds. Achieving the low computational cost of an 
Eyring based calculation, enabling the use of high-level methods, e.g. DFT, while keeping the 
methodological accuracy of MD-simulations. Our protocol is based on an extension of Eyring theory 
but goes beyond the harmonic approximation through a cost-effective evaluation of the partition sum 
factors, relevant for determining diffusion constants (reaction rates) of molecules in larger external 
structures. We treat the molecule the external structure of interest as rigid objects, effectively 
reducing the dimensionality of the problem. In our treatment we fall back on the concept of the ̀ ridge', 
a hypersurface which has a dividing character for the reactant and product volume. In order to provide 
simple access to our ideas and methods we have developed a python-based software package named 
ZoRRO, where the acronym stands for Zustandssummen1 of Rigit Rotor Objects. Our package is an 
extension of the well-known atomic simulation environment (ASE). 
Testing our package for the case of natural gas purification, the separation of N2, CH4 and CO2 through 
a graphdiyene membrane, we find that it provides excellent predictions for the propagation rates, 
thereby justifying the rigid rotor approximation. Moreover, a tremendous advantage can be seen in 
terms of computational effort compared to a molecular dynamics simulation, which, in our case, 
requires 106 to 107 potential energy surface (PES) evaluations. Our software package, on the other 
hand, yields reasonable estimates with about 100 evaluations of the PES, providing enormous savings 
in computing time. Furthermore, we examined the accuracy of different levels of energy predictors, 
ranging from simple force fields to accurate DFT based methods, revealing the necessity for high level 
predictors in order to sufficiently describe generic processes in molecular diffusion.  
Potential future efforts to refine the discussed methods might be the inclusion of various effects. E.g. 
inter molecular interactions, enabling a description of processes in solution, and quantum effects, 
which might be relevant for the description of light molecules such as H2. A further interesting 
improvement would be the generalization of the method to larger (biological) molecules by allowing 
for deformations of the studied molecule. 
1 Zustandssumme = german term for partition sum 


ZoRRO - A new software package for ab initio calculation of molecular diffusion rates

Górska, Anna

Szyszkiewicz-Warzecha, Krzysztof

Filipek, Robert

Lewenstam, Andrzej

Wilczek-Vera, Grazyna

This work is licensed under a Creative Commons Attribution 4.0 International License. 
   diffusion-fundamentals.org, Vol. 35 (2022) 63, URL: https://diffusion-fundamentals.org/journal/35/2022/63.pdf 
The influence of chemical activity on transport of ions in cementitious 
materials of real microstructure 
Anna Górska1, Krzysztof Szyszkiewicz-Warzecha1,  
Grażyna Wilczek-Vera2, Andrzej Lewenstam1 and Robert Filipek1,* 
1 AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland 
2 McGill University, Montreal, Quebec, Canada 
*anna.chyba@agh.edu.pl
Thermodynamic characteristics of pore solution influence behavior of porous materials. In case of 
cement-based materials electrolytic solutions present in pores can be highly concentrated. The activity 
of ions contained in the pores affects their diffusion inside the material and, consequently, may influence 
the kinetics of degradation of reinforcement concrete. Many models have been proposed to calculate the 
activity of ions. In this paper we investigated extended Debye-Hückel, Davies, Truesdell-Jones, Pitzer 
and MacInnes activity models. We show how the activity obtained from different models influences ion 
transport in porous concrete material.  
A 3D model has been considered to calculate distribution of ions concentration and electric potential 
in concrete. Four ions, major components of pore solution were considered: Na+, Cl–, K+ and OH–. The 
Nernst−Planck−Poisson (NPP) system of equations was used to compute ion fluxes in porous materials 
in space and time. COMSOL Multiphysics software was used to solve the NPP problem in 3D geometry 
of a real concrete sample, which has been obtained from X-ray computed tomography. Exemplary 
results of simulations are shown in Figures 1 and 2. Figures suggest that impact on concentration may 
be non-negligible when activity is taken into account.  
0 5 10 15 20 25
0
400
800
1200
1600
2000
2400
2800
3200
3600
4000
c C
l-  (
m
ol
/m
3 )
Distance (mm)
 Pitzer
 MacInnes
 f=1
5 days
1 day
10 days
3 4 5
1400
1600
1800
2000
Fig. 1. Calculated concentration distribution of Cl- in 
the cement sample after 1200 h of immersion in 20% 
NaCl water solution assuming Pitzer and MacInnes 
models compared with ideal solution approximation 
(f = 1). 
Fig. 2. Difference of chloride concentration (mol/m3) 
obtained by Pitzer activity model and ideal solution 
approximation (f = 1) in the concrete sample (3D 
model) after 1200 h of immersion in 20% NaCl water 
solution (2D cross-section). White areas correspond to 
basalt grains.
Acknowledgments: This research was supported by the National Science Centre, Poland, grant No. 
2020/37/B/ST8/02805. 
References 
[1] J. J. Jasielec, J. Stec, K. Szyszkiewicz-Warzecha, A. Łagosz, J. Deja, A. Lewenstam, R. Filipek, Effective 
and Apparent Diffusion Coefficients of Chloride Ions and Chloride Binding Kinetics Parameters in 
Mortars: Non-Stationary Diffusion–Reaction Model and the Inverse Problem, Materials 13, 5522-5545 (2020). 


The  influence of chemical activity on transport of ions in cementitious materials of real microstructure

This work is licensed under a Creative Commons Attribution 4.0 International License. 
   diffusion-fundamentals.org, Vol. 35 (2022)3, URL: https://diffusion-fundamentals.org/journal/35/2022/3.pdf 
Typical planar ceramic membrane obtained by sequential tape casting with a gas selective layer on top. 
a) A metal-organic 
framework (MOF) type 
ZIF-8 layer on an asym-
metric graded titania 
support [4], b) Principle 
of a supported mem-
brane: The µm-thick 
separation layer is de-
posited on a 
macroporous ceramic or metallic support. To reduce the pressure drop across the support, i.e. to mini-
mize the flow resistance, usually asymmetric (graded) supports with hierarchical cross section are used. 
As a rough estimate, mass transport through a membrane can be described using 1st Fickian Law. The 
flux density of component A through the membrane is described by   𝑗𝐴 = −𝐷𝑇𝐴  
𝜕𝑐
𝜕𝑥
with jA as the flux 
density in mol of A per time and area. DTA is the transport diffusivity of A, and ∂c/∂x is the concentration 
gradient of component A across the membrane. Exact knowledge of the transport diffusivity is thus an 
important prerequisite for a knowledge-based optimization of separation devices [5]. A discussion of 
the challenges and traps of such measurement is the focus of the Conference Workshop on “Diffusion 
in Nanoporous Materials” and part of an IUPAC initiative (https://iupac.org/project/2015-002-2-100).  
Adsorbents for rapid cycle pressure swing adsorption (psa) processes must allow a quick fluid 
flow of the feed into the nanopores of a zeolite or coal, where a 
diffusion-limited adsorption into the micropore system takes place. 
The shaped zeolite or carbon powders should show transport path 
like a human lung. For a short cycle time, however, the pressure 
drop in a packed adsorbent bed becomes a problem. The choice of 
an appropriate model accounting for intra-particle diffusional limi-
tations is essential to simulate accurately the pressurization and 
blowdown steps of a psa processes [6]. 
[1] A. Bunde, J. Caro, J. Kärger, G. Vogel (eds.): Diffusive spreading in nature, technology and society, Springer, (2018).  
[2] J. Caro: Fluid Flow, in: F. Schüth, K.S.W. Sing, J. Weitkamp (eds.), Handbook of Porous Solids, Wiley-VCH, pp 352-370 (2002). 
[3] J. Caro: Diffusion coefficients in nanoporous solids derived from membrane permeation measurements, Adsorption 27, 283-293 (2021).   
[4] J. Caro: Hierarchy in inorganic membranes, Chem. Soc. Rev. 45, 3468-3478 (2016). https://doi.org/10.1039/C5CS00597C 
[5] J. Kärger, D.M. Ruthven, R. Valiullin (Eds.): Diffusion in Nanoporous Solids, Thematic Issue of the Adsorption Journal 27 (2021). 
[6] S. Farooq, D.M. Ruthven: Numerical simulation of a kinetically controlled pressure swing adsorption bulk sepa-ration process based
   on a diffusion model, Chemical Engineering Science 46, 2213-2224 (1991). 
Diffusion in nanoporous materials as rate-limiting step in gas separation: 
Membrane permeation and pressure swing adsorption 
Juergen Caro 
Institute of Physical Chemistry and Electrochemistry, Leibniz University Hannover, Germany 
Nanoporous materials like zeolites are widely used for industrial gas separation through pressure swing 
adsorption (psa), new disruptive materials like metal-organic frameworks (MOFs) and covalent organic 
frameworks (COFs) are under evaluation. First LTA zeolite membranes are used in about 500 plants 
worldwide for the de-watering of solvents, especially bio-ethanol. MOF membranes are tested for their 
unique ability to separate short chain olefin/paraffin mixtures. 
Industrial adsorbents and membranes usually show a hierarchical transport-optimized structure. Such 
hierarchical structure as shown in the figure allows an effective interplay of nanopore diffusion [1] in 
the separation layer with fluid flow through the macroporous support [2] to minimize the overall 
transport resistance [3].  


Diffusion in nanoporous materials as rate-limiting step in gas separation: Membrane permeation and pressure swing adsorption

Danielewski, Marek

This work is licensed under a Creative Commons Attribution 4.0 International License. 
   diffusion-fundamentals.org, Vol. 35 (2022) 5, URL: https://diffusion-fundamentals.org/journal/35/2022/5.pdf 
An Ontological Basis for the Diffusion Theory 
Marek Danielewski 
Faculty of Materials Science & Ceramics, AGH UST, Mickiewicza 30, 30-059 Kraków, Poland 
A question of interpretation of diffusion is answered in the affirmative. Fick’s diffusion 
equation represents physical reality that has been interpreted by Einstein and Smoluchowski. This 
gives rise to new question, how broad is spectrum of physical reality that diffusion could in 
principle give a complete account. Here we will base on the elegant mathematical foundations 
formulated three decades before Fick by French mathematician Augustin Cauchy (~1822). It will 
be shown that the diffusion equation follows from his model of the ideal elastic continuum, 
namely the classical energy and momentum balance equations and is the consequence of the 
stable wave existence.   
This demonstrates that the complete ontological construal of diffusion theory exists. 
Explicitly, the interpretation of the diffusion equation and of the flux constative formulae. The 
two terms in the flux equations; the driving forces defined by the potential gradients and the 
kinetic coefficients in front of the driving forces are discussed in this paper. The fundamental 
consequences of all derived equations and relations for physics, chemistry and the future 
prospects are presented. 


An Ontological Basis for the Diffusion Theory

Zschech, Ehrenfried

This work is licensed under a Creative Commons Attribution 4.0 International License. 
   diffusion-fundamentals.org, Vol. 35 (2022) 19, URL: https://diffusion-fundamentals.org/journal/35/2022/19.pdf 
Tortuosity and Transport in Systems for Electrochemical Energy Storage 
and Conversion 
Ehrenfried Zschech1*, Robert Filipek2 
1deepXscan GmbH, Dresden, Germany 
2AGH University of Science and Technology, Krakow, Poland 
*ehrenfried.zschech@deepxscan.com
The efficiency of systems for electrochemical energy storage and conversion is based on their 
advantageous surface chemistry, i.e., the crystalline structure of the materials and the chemical bonding. 
In addition to the intrinsic properties of the constituent components, the 3D morphology of 
electrocatalysts has to be tailored, as it is closely related to the exposed facets and active sites. The 
design of electrocatalytic systems with an optimized hierarchical morphology and surface structures 
with high activity, resulting in highly performant and durable systems, is a task that is expected to play 
an increasing role in the future [1]. 
The total surface area of an electrocatalytic active material and the fluid-dynamic processes depend on 
the morphology of the formed nanostructures. Parameters such as porosity and tortuosity strongly 
influence the transport properties of the material. Multiphase and porous materials can be effectively 
modelled taking into account their hierarchical structure. Detailed 3D information about the morphology 
of porous materials and quantitative parameters that describe the transport properties (permeability, 
tortuosity) can be derived from tomography studies. Depending on the typical size of the studied objects, 
electron tomography (ET) in the TEM or X-ray computed tomography (XCT), both nano-XCT and 
micro-XCT, are suitable techniques to provide the morphology of the (sub-)structures of the hierarchical 
materials.  
High-resolution allows to distinguish existing sub-structures. Each sub-structure can be modelled as a 
continuum, without explicitely separating it into the pore space and the solid skeleton. Thus, 
macroscopic concentrations, fluxes and other properties describing fluid-dynamic processes are given 
for each point x  in the space occupied by material. However, these quantities are defined as the averages 
over a representative elementary volume. For example, the macroscopic concentration i (x,c t)  is the 
microscopic concentration ( , t)mici  averaged over a specific volume. Using the homogenization method 
[2], it can be shown that the mass balance equation for nonhomogeneous materials must be transformed 
to a form taking into account the material porosity and a so-called effective diffusion coefficient 
2eff
i i=D D /   which is a molecular diffusion coefficent modified by two empirical parameters that 
characterize a porous material: porosity   and tortuosity  [3]. Similarly, the homogenization method 
can be effectively used for energy, charge and momentum balance equations, enabling the description 
of kinetic processes in real electrochemical systems considering their 3D morphology. 
Specially designed operando studies in laboratory XCT tools provide information about the material’s 
morphology and kinetic processes in electrochemical systems. We will demonstrate materials ageing 
and system degradation during charging/discharging cycles of batteries [4]. 
References 
[1] Topal, E.; Liao, Z.; Loeffler, M.; Gluch, J.; Zhang, J; Feng, X.; Zschech, E. Multi–scale X–ray Tomography 
and Machine Learning Algorithms to Study MoNi4 Electrocatalysts Anchored on MoO2 Cuboids Aligned on Ni 
Foam. BMC Materials 2, 5 (2020). 
[2] Hornung, U. Homogenization and Porous Media, Springer-Verlag, New York (1997). 
[3] Stec, J.; Tarasiuk, J.; Nagy, S.; Smulski, R.; Gluch, J.; Filipek, R. Non-destructive investigations of pore 
morphology of micropore carbon materials, Ceramics International 45, 3483 (2019). 
[4] Shapovalov, V.; Kutukova, K.; Maletti, S.; Heubner, C.; Butova, V., Shukaev, I.; Guda, A.; Soldatov, A.; 
Zschech, E. Laboratory X-ray microscopy study of microcrack evolution in a novel sodium iron titanate-based 
cathode material for Li-ion batteries. Crystals 12, 3 (2022) 


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Grain Boundary Segregation: the New Sprouts

Foreword: Special Issue "IXth Diffusion Fundamentals Conference"

Stochastic Models for Interdisciplinary Problems

1st Fickian Law to predict gas selectivity of a nanoporous membrane

Complexity of anomalous diffusion processes

ZoRRO - A new software package for ab initio calculation of molecular diffusion rates

The influence of chemical activity on transport of ions in cementitious materials of real microstructure

Diffusion in nanoporous materials as rate-limiting step in gas separation: Membrane permeation and pressure swing adsorption

An Ontological Basis for the Diffusion Theory

Tortuosity and Transport in Systems for Electrochemical Energy Storage and Conversion

1,189

1,234