From the analytical perspective, we investigate the diffusion processes that arise from a system composed of a surface with a backbone structure coupled to the bulk via the boundary conditions. The problem is formulated in terms of diffusion equations with nonlocal terms, which can be used to model different processes, such as sorption–desorption and reactions on the surface. For the backbone structure, we consider the comb model, which imposes constraints on the diffusion processes in different directions on the surface. The results reveal a broad class of behaviors that can be connected to anomalous diffusion.

1.
A.
Mandelis
,
Diffusion-wave Fields: Mathematical Methods and Green Functions
(
Springer Science & Business Media
,
2013
).
2.
A.
Okubo
and
S. A.
Levin
et al.,
Diffusion and Ecological Problems: Modern Perspectives
(
Springer
,
2001
), Vol. 14.
3.
J.
Crank
,
The Mathematics of Diffusion
(
Oxford University Press
,
1979
).
4.
F. B.
Knight
,
Essentials of Brownian Motion and Diffusion
(
American Mathematical Soc.
,
1981
), Vol. 18.
5.
B.
Wang
,
J.
Kuo
,
S. C.
Bae
, and
S.
Granick
, “
When Brownian diffusion is not Gaussian
,”
Nat. Mater.
11
,
481
485
(
2012
).
6.
M. V.
Chubynsky
and
G. W.
Slater
, “
Diffusing diffusivity: A model for anomalous, yet Brownian, diffusion
,”
Phys. Rev. Lett.
113
,
098302
(
2014
).
7.
B. A.
Grzybowski
,
K. J.
Bishop
,
C. J.
Campbell
,
M.
Fialkowski
, and
S. K.
Smoukov
, “
Micro-and nanotechnology via reaction–diffusion
,”
Soft Matter
1
,
114
128
(
2005
).
8.
I. R.
Epstein
and
B.
Xu
, “
Reaction–diffusion processes at the nano-and microscales
,”
Nat. Nanotechnol.
11
,
312
319
(
2016
).
9.
T. A.
Waigh
and
N.
Korabel
, “
Heterogeneous anomalous transport in cellular and molecular biology
,”
Rep. Prog. Phys.
86
,
126601
(
2023
).
10.
F.
Höfling
and
T.
Franosch
, “
Anomalous transport in the crowded world of biological cells
,”
Rep. Prog. Phys.
76
,
046602
(
2013
).
11.
I. M.
Sokolov
, “
Models of anomalous diffusion in crowded environments
,”
Soft Matter
8
,
9043
9052
(
2012
).
12.
D. W.
Gryczak
,
E. K.
Lenzi
,
M. P.
Rosseto
,
L. R.
Evangelista
, and
R. S.
Zola
, “
Non-Markovian diffusion and adsorption–desorption dynamics: Analytical and numerical results
,”
Entropy
26
,
294
(
2024
).
13.
Q.
Xu
,
L.
Feng
,
R.
Sha
,
N.
Seeman
, and
P.
Chaikin
, “
Subdiffusion of a sticky particle on a surface
,”
Phys. Rev. Lett.
106
,
228102
(
2011
).
14.
P.
Tan
,
Y.
Liang
,
Q.
Xu
,
E.
Mamontov
,
J.
Li
,
X.
Xing
, and
L.
Hong
, “
Gradual crossover from subdiffusion to normal diffusion: A many-body effect in protein surface water
,”
Phys. Rev. Lett.
120
,
248101
(
2018
).
15.
Y.
Liang
,
S.
Wang
,
W.
Chen
,
Z.
Zhou
, and
R. L.
Magin
, “
A survey of models of ultraslow diffusion in heterogeneous materials
,”
Appl. Mech. Rev.
71
,
040802
(
2019
).
16.
J.
Bouchaud
and
A.
Georges
, “The physical mechanisms of anomalous diffusion,” in Disorder and Mixing: Convection, Diffusion and Reaction in Random Materials and Processes (Springer, 1988), pp. 19–30.
17.
D.
Ben-Avraham
and
S.
Havlin
,
Diffusion and Reactions in Fractals and Disordered Systems
(
Cambridge University Press
,
2000
).
18.
A.
Iomin
,
V.
Méndez
, and
W.
Horsthemke
,
Fractional Dynamics in Comb-like Structures
(
World Scientific
,
2018
).
19.
S.
Fedotov
and
A.
Iomin
, “
Migration and proliferation dichotomy in tumor-cell invasion
,”
Phys. Rev. Lett.
98
,
118101
(
2007
).
20.
A.
Iomin
, “
Toy model of fractional transport of cancer cells due to self-entrapping
,”
Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.
73
,
061918
(
2006
).
21.
E.
Agliari
,
D.
Cassi
,
L.
Cattivelli
, and
F.
Sartori
, “
Two-particle problem in comblike structures
,”
Phys. Rev. E
93
,
052111
(
2016
).
22.
E.
Lenzi
,
T.
Sandev
,
H.
Ribeiro
,
P.
Jovanovski
,
A.
Iomin
, and
L.
Kocarev
, “
Anomalous diffusion and random search in xyz-comb: Exact results
,”
J. Stat. Mech.: Theory Exp.
2020
,
053203
(
2020
).
23.
T.
Sandev
,
A.
Iomin
, and
L.
Kocarev
, “
Random search on comb
,”
J. Phys. A: Math. Theor.
52
,
465001
(
2019
).
24.
Y.
Liang
,
T.
Sandev
, and
E. K.
Lenzi
, “
Reaction and ultraslow diffusion on comb structures
,”
Phys. Rev. E
101
,
042119
(
2020
).
25.
L.
Liu
,
L.
Zheng
, and
Y.
Chen
, “
Macroscopic and microscopic anomalous diffusion in comb model with fractional dual-phase-lag model
,”
Appl. Math. Modell.
62
,
629
637
(
2018
).
26.
V.
Arkincheev
and
E.
Baskin
, “
Anomalous diffusion and drift in a comb model of percolation clusters
,”
Soviet physics, JETP
73
,
161
165
(
1991
).
27.
V.
Arkhincheev
, “
Diffusion on random comb structure: Effective medium approximation
,”
Phys. A
307
,
131
141
(
2002
).
28.
V.
Méndez
and
A.
Iomin
, “
Comb-like models for transport along spiny dendrites
,”
Chaos, Solitons Fractals
53
,
46
51
(
2013
).
29.
A.
Iomin
,
V.
Zaburdaev
, and
T.
Pfohl
, “
Reaction front propagation of actin polymerization in a comb-reaction system
,”
Chaos, Solitons Fractals
92
,
115
122
(
2016
).
30.
A.
Iomin
, “
Continuous time random walk and migration–proliferation dichotomy of brain cancer
,”
Biophysical Reviews and Letters
10
,
37
57
(
2015
).
31.
V.
Domazetoski
,
A.
Masó-Puigdellosas
,
T.
Sandev
,
V.
Méndez
,
A.
Iomin
, and
L.
Kocarev
, “
Stochastic resetting on comblike structures
,”
Phys. Rev. Res.
2
,
033027
(
2020
).
32.
R.
Singh
,
T.
Sandev
,
A.
Iomin
, and
R.
Metzler
, “
Backbone diffusion and first-passage dynamics in a comb structure with confining branches under stochastic resetting
,”
J. Phys. A: Math. Theor.
54
,
404006
(
2021
).
33.
E. K.
Lenzi
,
H. V.
Ribeiro
,
M. K.
Lenzi
,
L. R.
Evangelista
, and
R. L.
Magin
, “
Fractional diffusion with geometric constraints: Application to signal decay in magnetic resonance imaging (MRI)
,”
Mathematics
10
,
389
(
2022
).
34.
V.
Méndez
,
A.
Iomin
,
W.
Horsthemke
, and
D.
Campos
, “
Langevin dynamics for ramified structures
,”
J. Stat. Mech.: Theory Exp.
2017
,
063205
(
2017
).
35.
A.
Iomin
, “
Superdiffusive comb: Application to experimental observation of anomalous diffusion in one dimension
,”
Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.
86
,
032101
(
2012
).
36.
E.
Lenzi
,
L.
Da Silva
,
A.
Tateishi
,
M.
Lenzi
, and
H.
Ribeiro
, “
Diffusive process on a backbone structure with drift terms
,”
Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.
87
,
012121
(
2013
).
37.
A.
Iomin
, “
Subdiffusion on a fractal comb
,”
Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys.
83
,
052106
(
2011
).
38.
E.
Baskin
and
A.
Iomin
, “
Superdiffusion on a comb structure
,”
Phys. Rev. Lett.
93
,
120603
(
2004
).
39.
T.
Sandev
,
A.
Schulz
,
H.
Kantz
, and
A.
Iomin
, “
Heterogeneous diffusion in comb and fractal grid structures
,”
Chaos, Solitons Fractals
114
,
551
555
(
2018
).
40.
T.
Gervais
and
K. F.
Jensen
, “
Mass transport and surface reactions in microfluidic systems
,”
Chem. Eng. Sci.
61
,
1102
1121
(
2006
).
41.
R.
Roshandel
and
F.
Ahmadi
, “
Effects of catalyst loading gradient in catalyst layers on performance of polymer electrolyte membrane fuel cells
,”
Renewable Energy
50
,
921
931
(
2013
).
42.
M. K.
Nazeeruddin
,
E.
Baranoff
, and
M.
Grätzel
, “
Dye-sensitized solar cells: A brief overview
,”
Sol. Energy
85
,
1172
1178
(
2011
).
43.
D.
Andreucci
,
D.
Bellaveglia
, and
E. N. M.
Cirillo
, “
A model for enhanced and selective transport through biological membranes with alternating pores
,”
Math. Biosci.
257
,
42
49
(
2014
).
44.
V.
Nikonenko
,
A.
Nebavsky
,
S.
Mareev
,
A.
Kovalenko
,
M.
Urtenov
, and
G.
Pourcelly
, “
Modelling of ion transport in electromembrane systems: Impacts of membrane bulk and surface heterogeneity
,”
Appl. Sci.
9
,
25
(
2018
).
45.
B.
Berkowitz
,
J.
Bear
, and
C.
Braester
, “
Continuum models for contaminant transport in fractured porous formations
,”
Water Resour. Res.
24
,
1225
1236
, https://doi.org/10.1029/WR024i008p01225 (
1988
).
46.
K.
Landman
and
S.
Rosenblat
, “
Diffusion of a contaminant in fractured porous media
,”
Health Phys.
48
,
19
28
(
1985
).
47.
E.
Sudicky
and
E.
Frind
, “
Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures
,”
Water Resour. Res.
18
,
1634
1642
, https://doi.org/10.1029/WR018i006p01634 (
1982
).
48.
S. N.
Shoghl
,
A.
Raisi
, and
A.
Aroujalian
, “
A predictive mass transport model for gas separation using glassy polymer membranes
,”
RSC Adv.
5
,
38223
38234
(
2015
).
49.
E.
Ricci
,
M.
Minelli
, and
M. G.
De Angelis
, “
Modelling sorption and transport of gases in polymeric membranes across different scales: A review
,”
Membranes
12
,
857
(
2022
).
50.
R. S.
Zola
,
E. K.
Lenzi
,
L. R.
Evangelista
, and
G.
Barbero
, “
Memory effect in the adsorption phenomena of neutral particles
,”
Phys. Rev. E
75
,
042601
(
2007
).
51.
H. S.
Fogler
,
Essentials of Chemical Reaction Engineering
(
Pearson Education
,
2010
).
52.
G.
Arfken
,
H.
Weber
, and
F.
Harris
,
Mathematical Methods for Physicists: A Comprehensive Guide
(
Elsevier Science
,
2013
).
53.
H. W.
Wyld
, Mathematical Methods for Physics, 2nd ed., Advanced Book Classics (Advanced Book Program, Perseus Books, 1999).
54.
A. P.
Prudnikov
,
I. A.
Brychkov
, and
O. I.
Marichev
,
Integrals and Series: Direct Laplace Transforms
(
CRC Press
,
1986
), Vol. 4.
55.
T.
Sandev
,
A.
Iomin
, and
H.
Kantz
, “
Fractional diffusion on a fractal grid comb
,”
Phys. Rev. E
91
,
032108
(
2015
).
You do not currently have access to this content.