Diffusion Fundamentals (E-Journal)

Sikorski, Andrzej

Polanowski, Piotr

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   diffusion-fundamentals.org, Vol. 35 (2022) 29, URL: https://diffusion-fundamentals.org/journal/35/2022/29.pdf 
Diffusion in a crowded environment 
Andrzej Sikorski1*, Piotr Polanowski2 
1Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland 
2Department of Molecular Physics, Łódź University of Technology, 90-924 Łódź, Poland 
*sikorski@chem.uw.edu.pl
We present results of extensive and systematic simulation studies of two-dimensional fluid motion in a 
complex crowded environment. In contrast to other studies we focused on cooperative phenomena that 
occurred if the motion of particles takes place in a dense crowded system, which can be considered as 
a crude model of a cellular membrane. Our main goal was to answer a fundamental question: how do 
the fluid molecules move in a disordered environment with a complex structure. The influence of 
crowding is especially interesting in two-dimensional cases, e.g. in membranes where the presence of 
macromolecules, proteins and cytoskeleton often changed the mobility, and anomalous diffusion 
appeared [1]. The dynamic lattice liquid (DLL) model, which can work at the highest fluid density, 
was employed for this purpose [2]. This model became a basis for a parallel algorithm which took into 
account coincidences of elementary molecular motion attempts resulting in local cooperative structural 
transformations. Within the frame of the DLL model we considered systems that contained mobile and 
immobile objects of various kinds [3-4]. The dynamic properties of the system were studied. The 
subdiffusive motion of fluid molecules was found in the crowded system and described. The 
simulation and the analysis emphasize the influence of the movement correlation between moving 
particles and obstacles. 
Figure 1: A scheme of the DLL model: collisions (yellow), vacant creations (red), open paths (pink), and closed 
motion paths (green). 
References 
[1] F. Höfling, T. Franosch, Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys. 
76, 046602 (2012). 
[2] T. Pakula, Simulations on the completely occupied lattice, in: Simulation methods for polymers, ed. by M. 
Kotelyanskii and D. N. Theodorou, Marcel Dekker, New York-Basel, 2004. 
[3] P. Polanowski, A. Sikorski, Diffusion of small particles in polymer films, J. Chem. Phys. 147, 14902 
(2017). 
[4] P. Polanowski i A. Sikorski, Molecular transport in systems containing binding obstacles, Soft Matter 15, 
10045 (2019). 


English

Diffusion in a crowded environment

Beckwée, Emile

Baron, Gino V.

Ciocarlan, Radu-George

Cool, Maarten

Houlleberghs, Eric

Breynaert, Johan Martens

Denayer, Joeri F.M.

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   diffusion-fundamentals.org, Vol. 35 (2022) 58, URL: https://diffusion-fundamentals.org/journal/35/2022/58.pdf 
Kinetics & thermodynamics of CH4 storage in porous solid aided clathrates 
E.J. Beckwée1*, G.V. Baron1, R.-G. Ciocarlan2, P. Cool2, M. Houlleberghs3, E. Breynaert3, J. 
Martens3, J.F.M. Denayer1 
1 Vrije Universiteit Brussel, Brussels, Belgium 
2 University of Antwerp, Antwerp, Belgium 
3 KU Leuven, Leuven, Belgium 
*emile.jules.beckwee@vub.be
Clathrate hydrates are inclusion compounds consisting of a host lattice, formed by hydrogen bonded 
water molecules, in which gas molecules, such as CH4, CO2 and H2 can reside by attractive van der 
Waal’s forces with the host. Typically, these structures are formed at elevated pressures and 
temperatures around or below the freezing point of water. In the past, clathrates were merely regarded 
an obstacle in gas and oil transportation due to spontaneous formation in pipelines leading to flow 
assurance problems. Recently, however, they have received increased attention as medium to store 
gases, given their non-explosivity, environmentally benign nature, moderate storage conditions, low 
production cost, low risk of handling and high volumetric storage capacity, i.e. up to 160 m3 CH4 (at 0 
°C and 1 bar) is stored in 1 m3 CH4 clathrate [1]. 
Clathrate formation in bulk water is hampered by low water to hydrate conversions, high pressures to 
initiate formation and slow nucleation and growth kinetics. A promising solution is the use of 
nanoporous materials to confine the water phase (Figure 1). Under nanoconfinement the gas-liquid 
interface is increased compared to bulk water, allowing clathrate formation to occur at milder conditions, 
with improved kinetics and with increased gas storage [2]. However, while plenty of research has been 
devoted to identifying moderately hydrophobic mesoporous materials as optimal clathrate formation 
promotors in terms of formation pressure and gas uptake, quantitative kinetic data of confined clathrate 
nucleation, growth and reformation is greatly underrepresented. Yet this is crucial for the potential of 
clathrates as a future energy storage medium. 
Two high-pressure experimental setups were developed in-house to quantitively monitor the kinetics 
and thermodynamics of solid aided clathrate formation. A gravimetric system was constructed by 
changing the electronics and software of a magnetic suspension balance, allowing the highly accurate 
and direct monitoring of gas uptake/release kinetics during clathrate formation/dissociation, 
respectively. A robust volumetric system was built to mimic the future gas uptake/release process in a 
storage vessel. Hence, the poster will cover results of nucleation, growth and reformation of methane 
clathrates aided by mesoporous silicas (hydrophobized SBA-15) showing the effect of temperature, 
pressure, water loading and multiple reformations on the kinetics and thermodynamics of methane 
clathrate formation. 
Figure 1: Graphical representation of porous solid aided (confined) CH4 clathrate formation, highlighting the 
need for quantitative kinetic data. Crystal cages of CH4 clathrate are depicted in partially transparent red & blue. 
References 
[1] C. Koh, E. Sloan: Natural gas hydrates: Recent advances and challenges in energy and environmental 
applications. AIChE Journal 53, 1636-1643 (2007). 
[2] M. Casco, J. Silvestre-Albero, A. Ramírez-Cuesta, Rey F., J. Jordá, A. Bansode, A. Urakawa, I. Peral, M. 
Martínez-Escandell, K. Kaneko, F. Rodríguez-Reinoso: Methane hydrate formation in confined nanospace 
can surpass nature. Nature Communications 6, 1-8 (2015). 


Kinetics & thermodynamics of CH4 storage in porous solid aided clathrates

Barkai, Eli

This work is licensed under a Creative Commons Attribution 4.0 International License. 
Packets of Diffusing Particles Exhibit Universal Exponential Tails 
Eli Barkai1 
1Physics Department, Bar Ilan University, Ramat Gan, Israel 
*barkaie@biu.ac.il
Brownian motion is a Gaussian process described by the central limit theorem. However, exponential 
decays of the positional probability density function 𝑃(𝑥, 𝑡) of packets of spreading random walkers, 
were observed in numerous situations that include glasses, live cells, and bacteria suspensions. We show 
that such exponential behavior, historically related to the work of Laplace, is generally valid in a large 
class of problems of transportin random media. By extending the large deviations approach for 
a continuous time random walk, we uncover a general universal behavior for the decay of the density 
[1]. It is found that fluctuations in the number of steps of the random walker, performed at finite time, 
lead to exponential decay (with logarithmic corrections) of 𝑃(𝑥, 𝑡). This universal behavior also holds 
for short times, a fact that makes experimental observations readily achievable. Time permitting, we 
will discuss the Hitchhiker model [2] which gives a microscopical description of the observed behavior, 
in terms of a model describing aggregation processes, that lead to a distribution of molecule sizes and 
to Laplace diffusion. 
[1] E. Barkai and S. Burov: Packets of diffusing particles exhibit universal exponential tails. Phys. Rev. Lett. 124, 
[2] M. Hidalgo-Soria and E. Barkai: The Hitchhiker model for Laplace diffusion processes in the cell 
   diffusion-fundamentals.org, Vol. 35 (2022) 2, URL: https://diffusion-fundamentals.org/journal/35/2022/2.pdf 
References 
060603 (2020). 
environment. Phys. Rev. E 102, 012109 (2020). 


Packets of Diffusing Particles Exhibit Universal Exponential Tails

Hołyst, Janusz

This work is licensed under a Creative Commons Attribution 4.0 International License. 
   diffusion-fundamentals.org, Vol. 35 (2022) 7, URL: https://diffusion-fundamentals.org/journal/35/2022/7.pdf 
Discontinuous Transitions in Stochastic Models 
 of Collective Opinion Dynamics  
Janusz Hołyst 
Faculty of Physics, Warsaw University of Technology 
jholyst.fens@gmail.com
Changes of our opinions can be caused by many reasons and sometimes it is useful to describe a part of 
them as a stochastic process or an effect of a  social temperature. It follows   the collective dynamics of  
social groups depends not only on deterministic interactions between group members but also  on a 
social noise coming for example from communication disturbances  or complexity of our beliefs.   
The lecture will show that a combination of stochastic and deterministic social interactions can lead to 
discontinuous  transitions in collective group opinions. The effect will be demonstrated with several 
examples: (i) a social impact  model  of  a strong leader,  (ii) a model of  interactions between competing 
social groups, (iii) a  temporal bilayer echo-chamber model and  (iv) a model of   structural  balance 
dynamics. For all these systems critical values of the social noise as calculated from analytical 
approaches are in a good agreement  with agent based numerical simulations.  
Figure 1: Social structures emerging in the coevolving voter model in which an agent can take the opinion of his 
neighbour or stick to his own view and seek a connection to an agent of the same state. Left:  the case when the 
probability of social link rewiring is less than a critical value and society is differentiated  (green and blue 
agents) but opposite  groups are connected (red links).  Right: the probability of social link rewiring is larger 
than a critical value and society has split into two disjointed subgroups with opposing views. The evolution will 
be illustrated by on-line computer animations. 
References 
[1] JA Hołyst, K Kacperski, F Schweitzer: Phase transitions in social impact models of opinion formation. 
Physica A: Statistical Mechanics and its Applications 285,199-210 (2000). 
[2] A Jędrzejewski, J Toruniewska, K Suchecki, O Zaikin, JA Hołyst: Spontaneous symmetry breaking of 
active phase in coevolving nonlinear voter model. Physical Review E 102, 042313 (2020). 
[3] K Suchecki, JA Hołyst: Bistable-monostable transition in the Ising model on two connected complex 
networks. Physical Review E 80, 031110 (2009). 
[4] ŁG Gajewski, J Sienkiewicz, JA Hołyst: Transitions between polarization and radicalization in a 
temporal bilayer echo-chamber model. Physical Review E 105, 024125 (2022). 
[5] PJ Górski, K Bochenina, JA Hołyst, RM D’Souza: Homophily based on few attributes can impede 
structural balance, Physical Review Letters 125, 078302 (2020). 


Discontinuous Transitions in Stochastic Models of Collective Opinion Dynamics

Mandelis, Andreas

Photothermal Coherence Tomography: Three-Dimensional Images based on Diffusion Localization

Selhuber-Unkel, Christine

This work is licensed under a Creative Commons Attribution 4.0 International License. 
   diffusion-fundamentals.org, Vol. 35 (2022) 14, URL: https://diffusion-fundamentals.org/journal/35/2022/14.pdf 
Intracellular motion is an essential mechanism to ensure the function of biological cells. It is not only 
important for the transport of stored molecules, e.g., lipids and enzymes, but can also be a decisive factor 
for the development of diseases. This is particularly interesting in investigations of the highly motile 
pathogenic amoeba Acanthamoeba castellanii (A. castellanii), which has a supercrowded intracellular 
space.  Using high-speed live cell imaging in combination with single-particle tracking analysis, we 
have shown that the motion of endogenous intracellular particles in the size range from a few hundred 
nanometers to several micrometers in A. castellanii is strongly superdiffusive and influenced by cell 
locomotion, cytoskeletal elements, and myosin II. We demonstrate that cell locomotion significantly 
contributes to intracellular particle motion, but is clearly not the only origin of superdiffusivity. By 
analyzing the contribution of microtubules, actin, and myosin II motors we show that myosin II is a 
major driving force of intracellular motion in A. castellanii. The cytoplasm of A. castellanii is 
supercrowded with intracellular vesicles and granules, such that significant intracellular motion can only 
be achieved by actively driven motion, while purely thermally driven diffusion is negligible. 
References 
[1] M. Timmermann, N. Lukat, L. Schneider, IV WC Shields, G. Lopez, C. Selhuber-Unkel: Migration of 
microparticle-containing amoebae through constricted environments. ACS Biomater. Sci. Eng. 6, 889-897 
(2020).  
[2] S. Thapa, N. Lukat, C. Selhuber-Unkel, A. G. Cherstvy, R. Metzler: Transient superdiffusion of polydisperse 
vacuoles inside highly-motile amoeboid cells. J. Chem. Phys. 150, 144901 (2019). 
[3] S. Gutekunst, K. Siemsen, S. Huth, A. Möhring, B. Hessler, M. Timmermann, I. Paulowicz, Y. Mishra, L. 
Siebert, R. Adelung, C. Selhuber-Unkel: 3D Hydrogels Containing Interconnected Microchannels of 
Subcellular Size for Capturing Human Pathogenic Acanthamoeba Castellanii. ACS Biomater. Sci. Eng. 5, 
1784-1792 (2019). 
[4] J. F. Reverey, J. H. Jeon, H. Bao, M. Leippe, R. Metzler, C. Selhuber-Unkel: Superdiffusion dominates 
intracellular particle motion in the supercrowded cytoplasm of pathogenic  Acanthamoeba castellanii. Sci. 
Rep. 5, 11690 (2015). 
Intracellular Diffusion: From Crowding to Migration 
Christine Selhuber-Unkel 
Heidelberg University, Institute for Molecular Systems Engineering and Advanced Materials 
(IMSEAM), Heidelberg, Germany 
selhuber@uni-heidelberg.de


Intracellular Diffusion: From Crowding to Migration

Sepiol, Bogdan

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   diffusion-fundamentals.org, Vol. 35 (2022) 15, URL: https://diffusion-fundamentals.org/journal/35/2022/15.pdf 
Studies of diffusion in ionic glasses on atomic scale 
K. Holzweber, C Tietz and B. Sepiol* 
University of Vienna, Faculty of Physics, Austria 
*
bogdan.sepiol@univie.ac.at 
Ionic glasses are of great interest due to their potential for energy storage solutions, sensor technology 
or as a way to immobilize nuclear waste over a very long time.  
Direct experimental access to processes of atomic diffusion in amorphous solids on the relevant time-
scales are very important for the understanding and the quantification of dynamical processes. Atomic 
scale X-ray Photon Correlation Spectroscopy (aXPCS) provides the necessary experimental access to 
follow atomic motion in space and time in a model systems of borate glasses with various alkali 
contents.  
We used aXPCS to study diffusion in alkali borate glasses serving as a good example for amorphous 
ionic conductors [1]. We found for instance [2], that ionic diffusion process as a vacancy mechanism 
on a network of largely occupied cationic sites strongly decoupled from the slower network dynamics, 
where alkali ions perform up to 10
9
 jumps before the network structure changes, see Fig. 1. While no 
beam damage is observed, the structural rearrangements indicate instantaneous response proportional 
to the dose rate. We could associate observed dynamical effects like correlation time length or the 
shape of the correlation function with various macroscopic properties of glasses like a glass-transition 
temperature, alkali type and the sample stoichiometry. 
Unique possibility of following atomic motion even at room temperature could be seen as a great 
opportunity to probe various dynamical properties of materials and to study an impact of radiation 
field on diffusion. 
Figure 1: Visualization of the diffusion model by a series of snapshots: 
large magenta spheres are Rb+ ions, white spheres vacancies, small red 
circles oxygen and blue are boron atoms. From top to bottom: vacancy 
makes successive jumps on the three-dimensional arrangement of Rb+ 
ions. 
References 
[1] K. Holzweber, C. Tietz, M. Leitner, T.M. Fritz and B. Sepiol, Beam-induced motion in alkali borate glasses. 
Phys. Rev. B 100, 214305 (2019). 
[2] C. Tietz, T.M. Fritz, K. Holzweber, M. Legenstein, B. Sepiol and M. Leitner, Diffusive dynamics in an 
amorphous superionic conductor. Phys. Rev. Research 2, 043141 (2020). 


Studies of Diffusion in Ionic Glasses on Atomic Scale

Savtchenko, Leonid

Zheng, Kaiyu

Rusakov, Dmitri

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   diffusion-fundamentals.org, Vol. 35 (2022) 23, URL: https://diffusion-fundamentals.org/journal/35/2022/23.pdf 
Electro-porosity of media depends on external electric fields 
Leonid P. Savtchenko*, Kaiyu Zheng, Dmitri A. Rusakov 
UCL Queen Square Institute of Neurology, University College London, UK 
*leonid.savtchenko@ucl.ac.uk
From nanomaterials to living tissues, the electro-porosity and thus electrodiffusion of media have been 
attributed entirely to their structure. However, brain tissue suggests prominent non-ohmic properties, 
the origins of which continue to be poorly understood. 
To address the question [1], we explore Monte Carlo simulations and the Nernst-Planck mathematical 
formalism of ion electrodiffusion in a space filled with overlapping spheres. The stochastic 
computations indicate that the permeability of such media is saturated non-ohmically with stronger 
external electric fields, with this dependence growing stronger with lower medium porosity.  
We tested the simulation results empirically using voltammetry measurements in electrolytes filled with 
non-polarizing dielectric (glass) microscopic spheres and found good correspondence with our 
predictions.  
A first-approximation theoretical insight relates this non-ohmic conductance to a disproportionately 
increased dwell time of diffusing ions at potential barriers (or traps), represented by geometric obstacles 
when the field strength increases. 
These discoveries have some potentially fundamental implications for understanding electricity in 
porous media. For instance, it suggests that routinely recorded brain field potentials may not necessarily 
scale linearly with the strength of current sources inside the brain tissue. At the microscopic scale, this 
predicts retarded diffusion of charged molecules, heterogeneous charge accumulation, and possibly 
supra-linear heat dissipation - with increased electrical activity. The dependence of medium 
permeability on the electric field could help accurately interpret electric activity recordings in brain 
tissue.  
This work was supported by the Wellcome Trust Principal Fellowship Award (212251_Z_18_Z), ERC 
Advanced Grant (323113) and European Commission NEUROTWIN grant (857562), to D.A.R. 
Reference 
[1] L.P. Savtchenko, K. Zheng, D.A. Rusakov: Conductance of porous media depends on external electric 
fields. Biophysical Journal 120 (8), 1431-1442 (2021). 


Electro-porosity of media depends on external electric fields

Mailhiot, Sarah

Molaiyan, Palanivel

Kantola, Anu M.

Hu, Tao

Lassi, Ulla

Telkki, Ville-Veikko

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   diffusion-fundamentals.org, Vol. 35 (2022) 43, URL: https://diffusion-fundamentals.org/journal/35/2022/43.pdf 
Li+ Ion Diffusion in Solid State Electrolyte Li3InCl6 measured by 7Li Liquid 
State NMR 
Sarah E. Mailhiot1*, Palanivel Molaiyan1, Anu M. Kantola1, Tao Hu1, Ulla Lassi1, Ville-Veikko 
Telkki1 
1University of Oulu, Oulu, Finland 
*sarah.mailhiot@oulu.fi
Conventional lithium-ion batteries have changed modern life. However, concerns remain about their 
energy density limit and safety risks introduced by flammable liquid electrolytes. One alternative is all-
solid-state Li batteries (ASSBs) which are not only safer but also can achieve higher energy densities. 
One of the most vital components of ASSBs is the solid-state electrolyte (SSEs). The optimization and 
design of SSEs is still underway, of which ion conductivity, Li+ diffusion mechanisms, and their stability 
are key. For the study of ion conduction mechanisms and Li+ diffusion, pulsed gradient spin echo 
(PGSE) NMR methods are ideal. The PGSE method provides diffusion time dependent 7Li apparent 
diffusion coefficients (D) and distances on the scale of 0.2-10 µm. In the short diffusion time limit, the 
electrical conductivity is directly proportional to D via the Nernst-Einstein equation,  
𝜎 = !"#
!
$%
, (1) 
in which s is the electrical conductivity, N is the number of carrier ions, e is the elementary charge, k is 
the Boltzmann constant, and T is the temperature. In the long diffusion time limit, D scales with 
tortuosity and restricted motion, such as the effect of grain boundaries, and the mean squared 
displacement can be calculated as 
< 𝑧& >	= 6𝐷Δ (2) 
in which D is the diffusion time. In this study, the Li+ diffusion mechanisms in Li3InCl6 are studied as a 
function of treatment condition. Li3InCl6 is a halide-based SSEs with a stable monoclinic crystal 
structure and good ionic conductivity. It is known that the ionic conductivity increases after annealing; 
however, the mechanism by which conductivity increases is unknown. The results indicate that the SSEs 
is stable when exposed to air and that the Li+ D is enhanced by annealing while the presence of 
intraparticle restrictions increase. The Li3InCl6 samples are also characterized by D-T2, T2, T2-T2, and T1 
measurements as well as FESEM, TEM/STEM and XRD. 
Figure 1: a) D is plotted as a function of diffusion time (D). b) The mean squared displacement, <z2> is plotted as 
a function of D. When D is short (D= 10 ms), D reflects conductivity. In comparison to the control sample, 7Li 
D(D= 10) ms increases after annealing and shows no change when the sample is exposed to air. When D is long, 
(D= 150 ms), D reflects long range mobility and tortuosity. In comparison to the control sample, 7Li D(D= 150 
ms) decreases after annealing and shows no change when the sample is exposed to air.  


Li+ Ion Diffusion in Solid State Electrolyte Li3InCl6 measured by 7Li Liquid State NMR

Kristinaityte, Kristina

Mames, Adam

Pietrzak, Mariusz

Ratajczyk, Tomasz

Urbańczyk, Mateusz

This work is licensed under a Creative Commons Attribution 4.0 International License. 
  diffusion-fundamentals.org, Vol. 35 (2022) 49, URL: https://diffusion-fundamentals.org/journal/35/2022/49.pdf 
Synergy of Time-Resolved NUS and DOSY for the monitoring of 
photopolymerization of anthracene derivatives 
Kristina Kristinaityte1, Adam Mames, Mariusz Pietrzak1,  Tomasz 
Ratajczyk1 and Mateusz Urbańczyk*1 
1Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland 
*murbanczyk@ichf.edu.pl
Currently, photochemistry is experiencing an upswing. This is due to the material sciences and green 
energy industries' interest in the practical applications of photoactive materials. The 
photopolymerization of anthracene derivatives is of pivotal interest as functional photoresponsive 
polymers can be obtained. Unfortunately, the  monitoring of photopolymerization is not an easy task, as 
the composition of the sample changes during the photochemical processes, and the analysis of post-
photoreaction products can only indirectly give information about photoreaction. Therefore, 
comprehensive monitoring has to be carried out in real-time mode.  
Figure 1: (a)-(c) HSQC spectra at different reaction times. Squares indicate the analyzed peaks. Colors indicate the specific 
n-mers as in (d); (e): Integrals calculated for the peaks for different n-mers; (e): The average mass of the mixture calculated 
from Diffusion and HSQC data. 
Here, it is demonstrated that comprehensive monitoring of photopolymerization can be achieved through 
simultaneous time-resolved DOSY [1, 2] interleaved by time‐resolved non‐uniform sampling (TR‐NUS) 
[3]. TR-DOSY enabled us to follow the change of the mass of a polymer as time progressed, which can 
be correlated with the TR-NUS HSQC spectra. Thanks to this, we observed step-by-step the creation 
and consumption of specific n-mers.  The presented approach can significantly improve the 
methodology of the investigation of various photoreaction processes -in particular, 
photopolymerization.  
References 
[1] Urbańczyk, M., Bernin, D., Czuroń, A., & Kazimierczuk, K. Monitoring polydispersity by NMR diffusometry 
with tailored norm regularisation and moving-frame processing. Analyst, 141(5), 1745–1752 (2016) 
[2] MacDonald, T. S. C., Price, W. S., & Beves, J. E. Time-Resolved Diffusion NMR Measurements for Transient 
Processes. ChemPhysChem, 20(7), 926–930 (2019) 
[3] Mayzel, M., Rosenlöw, J., Isaksson, L., & Orekhov, V. Y.  Time-resolved multidimensional NMR with non-
uniform sampling. Journal of Biomolecular NMR, 58(2), 129–139 (2014). 


Synergy of Time-Resolved NUS and DOSY for the monitoring of photopolymerization of anthracene derivatives

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