# Spatial characterization of the magnetic field profile of a probe tip used
in magnetic resonance force microscopy

E. Nazaretski  Los Alamos National Laboratory, Los Alamos, NM 87545  E. A.
Akhadov  Los Alamos National Laboratory, Los Alamos, NM 87545  I. Martin  Los
Alamos National Laboratory, Los Alamos, NM 87545  D. V. Pelekhov  Department
of Physics, Ohio State University, Columbus OH 43210  P. C. Hammel  Department
of Physics, Ohio State University, Columbus OH 43210  R. Movshovich  Los
Alamos National Laboratory, Los Alamos, NM 87545

###### Abstract

We have developed the experimental approach to characterize spatial
distribution of the magnetic field produced by cantilever tips used in
magnetic resonance force microscopy (MRFM). We performed MRFM measurements on
a well characterized diphenyl-picrylhydrazyl (DPPH) film and mapped the 3D
field profile produced by a $Nd_{2}Fe_{14}B$ probe tip. Using our technique
field profiles of arbitrarily shaped probe magnets can be imaged.

Magnetic resonance force microscopy attracted a lot of interest in the last
few years due to its high force sensitivity and excellent spatial resolution
of magnetic properties. MRFM has been used in studies of electron and nuclear
spin systems culminating in the detection of the force signal originating from
a single electron spin Rugar 2004 . Recent experiments on nuclear spins of
${}^{19}F$ in $CaF_{2}$ samples demonstrated the spatial resolution of 90 nm
Mamin 2007 , orders of magnitude better than conventional magnetic resonance
imaging technique. In the long term, MRFM is envisioned as a possible route to
achieve imaging of individual molecules. Experiments on ferromagnetic systems
showed the potential for spatially resolved ferromagnetic resonance in
continuous and microfabricated samples Nazaretski 2007 ; Mewes 2006 . In MRFM
experiments, force F exerted on a cantilever, is a convolution of the sample’s
magnetization and the gradient of the magnetic field produced by the probe
tip. To perform correct imaging, quantitative knowledge of the spatial
distribution of the tip field is required. At present, the most common way to
characterize magnetic tips is to use the cantilever magnetometry Rossel 1996 ;
Stipe 2001 . It provides information about the magnetic moment of the tip m,
however, it is also sensitive to the relative orientation of m with respect to
the external magnetic field and the direction of cantilever’s oscillations.
Moreover, the detailed spatial field profile of the magnetic tip can not be
inferred. Alternative approach utilizes the spectroscopic nature of MRFM and
has been demonstrated in previous studies Mamin 2007 ; Chao 2004 ; Wago 1998 ;
Bruland 1998 ; Hammel 2003 . In these experiments the strength of the probe
field has been determined from the position of the onset in the MRFM spectra
as a function of the probe-sample separation $z$. Based on this information,
the point dipole approximation has been used to model the magnetic tip. The
situation becomes more complicated if the shape of the tip is irregular or m
is tilted with respect to the $\hat{z}$ direction. Under these circumstances
the one-dimensional approach is insufficient, and does not reveal the spatial
field profile of the probe tip. In this letter we propose a method for
detailed mapping of the tip magnetic field, free of any assumptions about the
tip shape, size, or composition.
In MRFM experiments the magnetic tip of a cantilever is used to generate the
inhomogeneous magnetic field causing local excitation of the spin resonance in
a small volume of the sample known as sensitive slice. The resonance condition
is written as follows

$|{\bf H}_{tot}(r)|=\frac{\omega_{RF}}{\gamma},$ (1)

where $\gamma$ is the gyromagnetic ratio. The total field ${\bf H}_{tot}(r)$
can be expressed as

${\bf H}_{tot}(r)={\bf H}_{ext}+{\bf H}_{tip}(r),$ (2)

where ${\bf H}_{ext}$ is the externally applied magnetic field and ${\bf
H}_{tip}(r)$ is the field of the probe tip. Width $\Delta z$ of the sensitive
slice is determined by the ratio of the resonance linewidth $\Delta H_{res}$
and the strength of the gradient field $\nabla H_{tip}$ produced by the probe
tip, $\Delta z$ = $\frac{\Delta H}{|\nabla H_{tip}|}$ Suter 2002 . Three
dimensional images of electron spin densities can be reconstructed by
performing lateral and vertical scanning of the sensitive slice across the
sampleWago 1998 ; Chao 2004 .

The concept behind our method for detailed characterization of the tip field
profile is illustrated in Fig. 1. It requires a thin-film sample with sharp
edges. When the sensitive slice touches the sample edge, a leading edge signal
is detected. At this location, the sample edge is a tangent line to the
sensitive slice for a reasonable magnetic tip. Thus, scanning in 3D and
recording the locations corresponding to the leading edge enables full
reconstruction of the sensitive slice. If desired, it can be then
parameterized using dipolar, quadrupolar, etc moments.

To illustrate this procedure, we report on MRFM measurements on a well
characterized DPPH film, while laterally scanning the cantilever over its
edge. We used a commercially available Veeco $Si_{3}N_{4}$ cantilever with the
resonance frequency of $\approx$ 8 kHz and the spring constant $k$ of
$\approx$ 0.01 N/m Veeco . The original tip was removed by focused ion milling
and a small magnetic particle of $Nd_{2}Fe_{14}B$ available from Magnequench
Inc. Magnequench  has been glued to the end of a cantilever with Stycast 1266
epoxy in the presence of an aligning magnetic field. Consequently, the tip has
been magnetized in the field of 80 kOe. The MRFM tip has a spherical shape
with the diameter of $\approx$ 2.4 $\mu$m and its SEM images are shown in
panels (1) and (2) in Fig. 2. The saturation magnetization of $Nd_{2}Fe_{14}B$
particles has been measured in a SQUID magnetometer, and is equal to $4\pi
M_{s}$ = 13 kG Nazaretski 2006a . Based on the SEM image we estimate the probe
moment to be (7.5$\pm$0.4)$\times$10-9 emu, in agreement with the value of
(6.9$\pm$0.5)$\times$10-9 emu measured by the cantilever magnetometry. The
cantilever is mounted on top of a double scanning stage of a low temperature
MRFM system Nazaretski 2006 ; Attocube  . For data acquisition, the
temperature was stabilized at 10 K and the amplitude modulation scheme has
been implemented to couple to the in-resonance spins. The DPPH powder DPPH
was dissolved in acetone and deposited on a 100 $\mu$m thick silicon wafer in
a spin-coater at 3000 rpm. To protect the film, 20 nm of Ti was deposited on
top of DPPH. Approximately 2$\times$1.6 mm2 piece was cleaved from a wafer and
glued to the strip-line resonator of the microscope. The structure of the film
and sharpness of edges were inspected in SEM and are shown in Fig. 2. The film
was found to be continuous, and its thickness varied between 400 and 600 nm.

Fig. 3 shows the typical MRFM spectrum recorded in a DPPH film. When the tip
is located above the film, the strongest tip field experienced by the sample
is situated directly under the probe magnet (assuming ${\bf m}$ $\parallel$
${\bf H}_{ext}$). The field value in the MRFM spectrum where the sensitive
slice just touches the DPPH film is called the leading edge Suter 2002 , and
is indicated by arrows in Fig. 3.

The large positive peak at $\approx$ 3.34 kOe corresponds to the bulk-like
resonance. It originates from the large region of the sample where the tip
field is small, but due to the large number of spins the MRFM signal is
significant. The field difference between the bulk-like resonance and the
position of the leading edge provides the direct measure of the probe field
strength.

Fig. 1 shows the schematic of the characterization experiment. We fixed the
probe-sample separation $z$, and approached different edges of the DPPH film
while tracking the leading edge. The left panel of Fig. 4 shows the field
evolution of the leading edge for two values of $z$ and three different
directions of lateral scanning over the film edge. The almost identical shape
of the curves indicates that m is approximately parallel to the direction of
${\bf H}_{ext}$. In the first approximation, our tip can be modeled as a
magnetic dipole. The field profile produced on the surface of the sample can
be written as follows Jackson 1975 :

$\displaystyle H(R,\theta,\varphi)$ $\displaystyle=\frac{4\pi
M_{s}r_{0}^{3}}{3}\times\\{\frac{-3z(\sin\theta(x\sin\varphi+y\cos\varphi))}{R^{5}}+$
(3)
$\displaystyle+\frac{3z^{2}\cos\theta}{R^{5}}-\frac{\cos\theta}{R^{3}}\\},$

where $4\pi M_{s}$ is the saturation magnetization of $Nd_{2}Fe_{14}B$,
$r_{0}$ is the radius of the tip, $R$ is the vector to the point where the
field is determined, $\theta$ and $\varphi$ are the angles which describe the
spatial orientation of m (see Fig. 1).
The right panel of Fig. 4 shows the z-component of the probe field on the
sample’s surface as a function of $z$. Solid line is the fit using Eq. 3 and
assuming parallel orientation of m and ${\bf H}_{ext}$. Fig. 5(a) shows the
comparison between the lateral field profile of the tip simulated according to
Eq. 3, and the actual data points taken from the left panel of Fig. 4. Good
agreement between the observed and expected behavior suggests that, indeed,
our probe tip can be approximated as a dipole, and its magnetization is
aligned along the direction of ${\bf H}_{ext}$. In case of any significant
misalignment the tip field profile would change substantially, as shown in
Fig. 5(a). For both simulations shown in Fig. 4 and 5, we had to offset the
probe-sample separation by 1.42 $\pm$ 0.03 $\mu$m ($z$ is the only free
parameter in the fit) which suggests that due to the short range probe-sample
interaction the cantilever snaps to the sample at distances smaller than 1.42
$\mu$m Berger 1999 ; Dorofeyev 1999 . The presence of an offset may indicate
the reduced magnetic moment of the tip. However, our cantilever magnetometry
measurements of the tip moment agree well with the expected value, as
mentioned earlier in the paper. Moreover, in Fig. 5(b) we show the calculated
spatial field profile of 2 $\mu$m, 2.2 $\mu$m and 2.4 $\mu$m diameter tips.
The fit for the 2.4 $\mu$m diameter tip provides the best agreement with the
data points. Another argument in support of our tip model pertains to the
magnitude of the MRFM force exerted on a cantilever in a particular sensitive
slice. In Fig. 3 we take the measured MRFM force at $H_{ext}$ = 3.038 kOe and
compare it to our estimates. The calculations yield the force value of
$\approx$ 6.9$\times$10-13 N in good agreement with the measured value of
5.7$\times$10-13 N. Thus, dipolar approximation and our assumptions for the
tip moment were adequate for the present experiment. Importantly, the same
technique could be applied to map field profile from a more irregular tip.

In summary, we have studied the evolution of locally excited electron-spin
resonance in a DPPH film. By tracking the position of the leading edge in MRFM
spectra for different hight and direction of the approach to the sample, we
have determined the spatial field profile of the cantilever tip. Measuring the
MRFM signal onset over the large range of positions with adequate sensitivity
allows to deconvolve the spatial field profile produced by arbitrarily shaped
magnetic tips used in the magnetic resonance force microscopy.

This work was supported by the US Department of Energy and was performed, in
part, at the Center for Integrated Nanotechnologies at Los Alamos and Sandia
National Laboratories. Personnel at Ohio State University was supported by the
US Department of Energy through grant DE-FG02-03ER46054.

## References

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Figure Caption

FIG.1 Schematic of the tip characterization technique. Detection of the
leading edge signal indicates that the sample edge is tangent to the sensitive
slice. 3D scanning can thus be used to fully reconstruct the shape of the
sensitive slice.

FIG.2 Panel (1)and (2): SEM images of the probe magnet. Panel (3) shows the
edge of the DPPH film and panel (4) is the top view showing fine structures on
the surface of the film.

FIG.3 Amplitude and phase of the MRFM signal recorded at $T$ = 10 K,
$\omega_{RF}$ = 9.35 GHz, $z$ = 0.73 $\mu$m. The position of the leading edge
is indicated by arrows.

FIG.4 Left panel: field evolution of the leading edge as a function of lateral
position over the DPPH film edge. The upper and lower set of curves correspond
to $z$ = 2.35 $\mu$m and $z$ = 0.53 $\mu$m respectively. Circles represent the
approach of the sample from side ’1’, squares from side ’2’ and triangles form
side ’3’ of the sample as shown in Fig. 1. Right panel: the $z$-component of
the tip field as a function of the probe-sample separation (left Y-axis) and
the corresponding field gradient (right Y-axis). Solid curve is the fit to Eq.
3.

FIG.5 (a) Lateral field profile of the tip for approaches of sides ’1’ and ’3’
of the sample, as shown in Fig. 1. Data points are taken from the left panel
in Fig. 4. ’0’ on the X-axis corresponds to the edge of the film. Upper and
lower data points correspond to $z$ = 0.53 $\mu$m and $z$ = 2.35 $\mu$m
respectively. Solid curve is fitted to the data using Eq. 3. Dotted and dashed
lines show the expected field profile of the tip where $\theta$ = $\varphi$ =
20∘ and $\theta$ = -20∘, $\varphi$ = 20∘ respectively. (b) expected field
profile for the tip with $r_{0}$=1.2 $\mu$m, z-offset=1.4 $\mu$m (solid line),
$r_{0}$=1.1 $\mu$m, z-offset=1.12 $\mu$m (dotted line) and $r_{0}$=1.0 $\mu$m,
z-offset=0.85 $\mu$m (dashed line).

Figure 1:  Figure 2:  Figure 3:

Figure 4:  Figure 5:


# An Approximation Algorithm for Two-Edge-Connected Subgraph Problem via
Triangle-free Two-Edge-Cover††thanks: This work was partially supported by the
joint project of Kyoto University and Toyota Motor Corporation, titled
“Advanced Mathematical Science for Mobility Society”, and by JSPS KAKENHI
Grant Numbers JP20K11692 and JP22H05001.

Yusuke Kobayashi Research Institute for Mathematical Sciences, Kyoto
University. E-mail: {yusuke<EMAIL_ADDRESS>Takashi
Noguchi22footnotemark: 2

###### Abstract

The $2$-Edge-Connected Spanning Subgraph problem (2-ECSS) is one of the most
fundamental and well-studied problems in the context of network design. In the
problem, we are given an undirected graph $G$, and the objective is to find a
$2$-edge-connected spanning subgraph $H$ of $G$ with the minimum number of
edges. For this problem, a lot of approximation algorithms have been proposed
in the literature. In particular, very recently, Garg, Grandoni, and Ameli
gave an approximation algorithm for 2-ECSS with factor $1.326$, which was the
best approximation ratio. In this paper, we give a
$(1.3+\varepsilon)$-approximation algorithm for 2-ECSS, where $\varepsilon$ is
an arbitrary positive fixed constant, which improves the previously known best
approximation ratio. In our algorithm, we compute a minimum triangle-free
$2$-edge-cover in $G$ with the aid of the algorithm for finding a maximum
triangle-free $2$-matching given by Hartvigsen. Then, with the obtained
triangle-free $2$-edge-cover, we apply the arguments by Garg, Grandoni, and
Ameli.

##  1 Introduction

In the field of survivable network design, a basic problem is to construct a
network with minimum cost that satisfies a certain connectivity constraint. A
seminal result by Jain [13] provides a $2$-approximation algorithm for a wide
class of survivable network design problems. For specific problems among them,
a lot of better approximation algorithms have been investigated in the
literature.

In this paper, we study the $2$-Edge-Connected Spanning Subgraph problem
(2-ECSS), which is one of the most fundamental and well-studied problems in
this context. In 2-ECSS, we are given an undirected graph $G=(V,E)$, and the
objective is to find a $2$-edge-connected spanning subgraph $H$ of $G$ with
the minimum number of edges. It was shown in [4, 5] that 2-ECSS does not admit
a PTAS unless ${\rm P}={\rm NP}$. Khuller and Vishkin [14] gave a
$3/2$-approximation algorithm for this problem, which was the starting point
of the study of approximation algorithms for 2-ECSS. Cheriyan, Sebő, and
Szigeti [1] improved this ratio to $17/12$, and later Hunkenschröder, Vempala,
and Vetta [20, 12] gave a $4/3$-approximation algorithm. By a completely
different approach, Sebő and Vygen [19] achieved the same approximation ratio.
Very recently, Garg, Grandoni, and Ameli [8] improved this ratio to $1.326$ by
introducing powerful reduction steps and developing the techniques in [12].

The contribution of this paper is to present a
$(1.3+\varepsilon)$-approximation algorithm for 2-ECSS for any
$\varepsilon>0$, which improves the previously best approximation ratio.

######  Theorem 1.

For any constant $\varepsilon>0$, there is a deterministic polynomial-time
$(1.3+\varepsilon)$-approximation algorithm for 2-ECSS.

Our algorithm and its analysis are heavily dependent on the well-developed
arguments by Garg, Grandoni, and Ameli [8]. In our algorithm, we first apply
the reduction steps given in [8]. Then, instead of a minimum $2$-edge-cover,
we compute a minimum _triangle-free_ $2$-edge-cover in the graph, which is the
key ingredient in our algorithm. We show that this can be done in polynomial
time with the aid of the algorithm for finding a maximum triangle-free
$2$-matching given by Hartvigsen [10] (see Theorem 4). Finally, we convert the
obtained triangle-free $2$-edge-cover into a spanning $2$-edge-connected
subgraph by using the arguments in [8].

Our main technical contribution is to point out the utility of Hartvigsen’s
algorithm [10] in the arguments by Garg, Grandoni, and Ameli [8]. It should be
noted that Hartvigsen’s algorithm has not received much attention in this
context.

#### Related Work

A natural extension of 2-ECSS is the $k$-Edge-Connected Spanning Subgraph
problem ($k$-ECSS), which is to find a $k$-edge-connected spanning subgraph of
the input graph with the minimum number of edges. For $k$-ECSS, several
approximation algorithms have been proposed, in which approximation factors
depend on $k$ [2, 7, 6]. We can also consider the weigthed variant of 2-ECSS,
in which the objective is to find a $2$-edge-connected spanning subgraph with
the minimum total weight in a given edge-weighted graph. The result of Jain
[13] leads to a $2$-approximation algorithm for the weighted 2-ECSS, and it is
still the best known approximation ratio. For the case when all the edge
weights are $0$ or $1$, which is called the _forest augmentation problem_ ,
Grandoni, Ameli, and Traub [9] recently gave a $1.9973$-approximation
algorithm. See references in [8, 9] for more related work on survivable
network design problems.

It is well-known that a $2$-matching of maximum size can be found in
polynomial-time by using a matching algorithm; see e.g., [18, Section 30]. As
a variant of this problem, the problem of finding a maximum $2$-matching that
contains no cycle of length at most $k$, which is called the _$C_{\leq k}$
-free $2$-matching problem_, has been actively studied. Hartvigsen [10] gave a
polynomial-time algorithm for the $C_{\leq 3}$-free $2$-matching problem (also
called the _triangle-free $2$-matching problem_), and Papadimitriou showed the
NP-hardness for $k\geq 5$ (see [3]). The polynomial solvability of the
$C_{\leq 4}$-free $2$-matching problem has been open for more than 40 years.
The edge weighted variant of the $C_{\leq 3}$-free $2$-matching problem is
also a big open problem in this area, and some positive results are known for
special cases [11, 15, 17, 16]. See references in [16] for more related work
on the $C_{\leq k}$-free $2$-matching problem.

##  2 Preliminary

Throughout the paper, we only consider simple undirected graphs, i.e., every
graph has neither self-loops nor parallel edges.111It is shown in [12] that
this assumption is not essential when we consider $2$-ECSS. A graph $G=(V,E)$
is said to be _$2$ -edge-connected_ if $G\setminus\\{e\\}$ is connected for
any $e\in E$, and it is called _$2$ -vertex-connected_ if $G\setminus\\{v\\}$
is connected for any $v\in V$ and $|V|\geq 3$. For a subgraph $H$ of $G$, its
vertex set and edge set are denoted by $V(H)$ and $E(H)$, respectively. A
subgraph $H$ of $G=(V,E)$ is _spanning_ if $V(H)=V(G)$. In the $2$-Edge-
Connected Spanning Subgraph problem ($2$-ECSS), we are given a graph $G=(V,E)$
and the objective is to find a $2$-edge-connected spanning subgraph $H$ of $G$
with the minimum number of edges (if one exists).

In this paper, a spanning subgraph $H$ is often identified with its edge set
$E(H)$. Let $H$ be a spanning subgraph (or an edge set) of $G$. A connected
component of $H$ which is 2-edge-connected is called a _2EC component of $H$_.
A 2EC component of $H$ is called an _$i$ -cycle 2EC component_ if it is a
cycle of length $i$. In particular, a $3$-cycle 2EC component is called a
_triangle 2EC component_. A maximal $2$-edge-connected subgraph $B$ of $H$ is
called a _block_ of $H$ if $|V(B)|\geq 3$ and $B$ is not a 2EC component. An
edge $e\in E(H)$ is called a _bridge_ of $H$ if $H\setminus\\{e\\}$ has more
connected components than $H$. A block $B$ of $H$ is called a _leaf block_ if
$H$ has exactly one bridge incident to $B$, and an _inner block_ otherwise.

Let $G=(V,E)$ be a graph. For an edge set $F\subseteq E$ and a vertex $v\in
V$, let $d_{F}(v)$ denote the number of edges in $F$ that are incident to $v$.
An edge set $F\subseteq E$ is called a _$2$ -matching_ if $d_{F}(v)\leq 2$ for
any $v\in V$, and it is called a _$2$ -edge-cover_ if $d_{F}(v)\geq 2$ for any
$v\in V$.222Such edge sets are sometimes called _simple_ $2$-matchings and
_simple_ $2$-edge-covers in the literature.

##  3 Algorithm in Previous Work

Since our algorithm is based on the well-developed $1.326$-approximation
algorithm given by Garg, Grandoni, and Ameli [8], we describe some of their
results in this section.

###  3.1 Reduction to Structured Graphs

In the algorithm by Garg, Grandoni, and Ameli [8], they first reduce the
problem to the case when the input graph has some additional conditions, where
such a graph is called a $(5/4,\varepsilon)$-structured graph. In what follows
in this paper, let $\varepsilon>0$ be a sufficiently small positive fixed
constant, which will appear in the approximation factor. In particular, we
suppose that $0\leq\varepsilon\leq 1/24$, which is used in the argument in
[8]. We say that a graph $G=(V,E)$ is _$(5/4,\varepsilon)$ -structured_ if it
is $2$-vertex-connected, it contains at least ${2}/{\varepsilon}$ vertices,
and it does not contain the following structures:

  * •

($5/4$-contractible subgraph) a $2$-edge-connected subgraph $C$ of $G$ such
that every $2$-edge-connected spanning subgraph of $G$ contains at least
$\frac{4}{5}|E(C)|$ edges with both endpoints in $V(C)$;

  * •

(irrelevant edge) an edge $uv\in E$ such that $G\setminus\\{u,v\\}$ is not
connected;

  * •

(non-isolating $2$-vertex-cut) a vertex set $\\{u,v\\}\subseteq V$ of $G$ such
that $G\setminus\\{u,v\\}$ has at least three connected components or has
exactly two connected components, both of which contains at least two
vertices.

The following lemma shows that it suffices to consider
$(5/4,\varepsilon)$-structured graphs when we design approximation algorithms.

######  Lemma 2 (Garg, Grandoni, and Ameli [8, Lemma 2.2]).

For $\alpha\geq\frac{5}{4}$, if there exists a deterministic polynomial-time
$\alpha$-approximation algorithm for 2-ECSS on $(5/4,\varepsilon)$-structured
graphs, then there exists a deterministic polynomial-time
$(\alpha+2\varepsilon)$-approximation algorithm for 2-ECSS.

###  3.2 Semi-Canonical Two-Edge-Cover

A $2$-edge-cover $H$ of $G$ (which is identified with a spanning subgraph) is
called _semi-canonical_ if it satisfies the following conditions.

  1. (1)

Each 2EC component of $H$ is a cycle or contains at least $7$ edges.

  2. (2)

Each leaf block contains at least $6$ edges and each inner block contains at
least $4$ edges.

  3. (3)

There is no pair of edge sets $F\subseteq H$ and $F^{\prime}\subseteq
E\setminus H$ such that $|F|=|F^{\prime}|\leq 3$, $(H\setminus F)\cup
F^{\prime}$ is a $2$-edge-cover with fewer connected components than $H$, and
$F$ contains an edge in some triangle 2EC component of $H$.

  4. (4)

There is no pair of edge sets $F\subseteq H$ and $F^{\prime}\subseteq
E\setminus H$ such that $|F|=|F^{\prime}|=2$, $(H\setminus F)\cup F^{\prime}$
is a $2$-edge-cover with fewer connected components than $H$, both edges in
$F^{\prime}$ connect two $4$-cycle 2EC components, say $C_{1}$ and $C_{2}$,
and $F$ is contained in $C_{1}\cup C_{2}$. In other words, by removing $2$
edges and adding $2$ edges, we cannot merge two $4$-cycle 2EC components into
a cycle of length $8$.

######  Lemma 3 (Garg, Grandoni, and Ameli [8, Lemma 2.6]).

Suppose we are given a semi-canonical $2$-edge-cover $H$ of a
$(5/4,\varepsilon)$-structured graph $G$ with $b|H|$ bridges and $t|H|$ edges
belonging to triangle 2EC components of $H$. Then, in polynomial time, we can
compute a $2$-edge-connected spanning subgraph $S$ of size at most
$(\frac{13}{10}+\frac{1}{30}t-\frac{1}{20}b)|H|$.

######  Remark 1.

In the original statement of [8, Lemma 2.6], $H$ is assumed to satisfy a
stronger condition than semi-canonical, called canonical. A $2$-edge-cover $H$
is said to be _canonical_ if it satisfies (1) and (2) in the definition of
semi-canonical $2$-edge-covers, and also the following condition: there is no
pair of edge sets $F\subseteq H$ and $F^{\prime}\subseteq E\setminus H$ such
that $|F|=|F^{\prime}|\leq 3$ and $(H\setminus F)\cup F^{\prime}$ is a
$2$-edge-cover with fewer connected components than $H$. However, one can see
that the condition “canonical” can be relaxed to “semi-canonical” by following
the proof of [8, Lemma 2.6]; see the proofs of Lemmas D.3, D.4, and D.11 in
[8].

##  4 Algorithm via Triangle-Free Two-Edge-Cover

The idea of our algorithm is quite simple: we construct a semi-canonical
$2$-edge-cover $H$ with no triangle 2EC components and then apply Lemma 3. We
say that an edge set $F\subseteq E$ is _triangle-free_ if there is no triangle
2EC components of $F$. Note that a triangle-free edge set $F$ may contain a
cycle of length three that is contained in a larger connected component. In
order to construct a semi-canonical triangle-free $2$-edge-cover, we use a
polynomial-time algorithm for finding a triangle-free $2$-matching given by
Hartvigsen [10].

######  Theorem 4 (Hartvigsen [10, Theorem 3.2 and Proposition 3.4]).

For a graph $G$, we can find a triangle-free $2$-matching in $G$ with maximum
cardinality in polynomial time.

In Section 4.1, we give an algorithm for finding a minimum triangle-free
$2$-edge-cover with the aid of Theorem 4. Then, we transform it into a semi-
canonical triangle-free $2$-edge-cover in Section 4.2. Using the obtained
$2$-edge-cover, we give a proof of Theorem 1 in Section 4.3.

###  4.1 Minimum Triangle-Free Two-Edge-Cover

As with the relationship between $2$-matchings and $2$-edge-covers (see e.g.
[18, Section 30.14]), triangle-free $2$-matchings and triangle-free $2$-edge-
covers are closely related to each other, which can be stated as the following
two lemmas.

######  Lemma 5.

Let $G=(V,E)$ be a connected graph such that the minimum degree is at least
two and $|V|\geq 4$. Given a triangle-free $2$-matching $M$ in $G$, in
polynomial time, we can compute a triangle-free $2$-edge-cover $C$ of $G$ with
size at most $2|V|-|M|$.

###### Proof.

Starting with $F=M$, we perform the following update repeatedly while $F$ is
not a $2$-edge-cover:

> Choose a vertex $v\in V$ with $d_{F}(v)<2$ and an edge $vw\in E\setminus F$
> incident to $v$.
>
>   1. (i)
>
> If $F\cup\\{vw\\}$ is triangle-free, then add $vw$ to $F$.
>
>   2. (ii)
>
> Otherwise, $F\cup\\{vw\\}$ contains a triangle 2EC component with vertex set
> $\\{u,v,w\\}$ for some $u\in V$. In this case, choose an edge $e$ connecting
> $\\{u,v,w\\}$ and $V\setminus\\{u,v,w\\}$, and add both $vw$ and $e$ to $F$.
>
>

If $F$ becomes a $2$-edge-cover, then the procedure terminates by returning
$C=F$. It is obvious that this procedure terminates in polynomial steps and
returns a triangle-free $2$-edge-cover.

We now analyze the size of the output $C$. For an edge set $F\subseteq E$,
define $g(F)=\sum_{v\in V}\max\\{2-d_{F}(v),0\\}$. Then, in each iteration of
the procedure, we observe the following: in case (i), one edge is added to $F$
and $g(F)$ decreases by at least one; in case (ii), two edges are added to $F$
and $g(F)$ decreases by at least two, because $d_{F}(v)=d_{F}(w)=1$ before the
update. With this observation, we see that $|C|-|M|\leq g(M)-g(C)=\sum_{v\in
V}(2-d_{M}(v))$, where we note that $M$ is a $2$-matching and $C$ is a
$2$-edge-cover. Therefore, it holds that

$|C|\leq|M|+\sum_{v\in V}(2-d_{M}(v))=|M|+(2|V|-2|M|)=2|V|-|M|,$

which completes the proof. ∎

######  Lemma 6.

Given a triangle-free $2$-edge-cover $C$ in a graph $G=(V,E)$, in polynomial
time, we can compute a triangle-free $2$-matching $M$ of $G$ with size at
least $2|V|-|C|$.

###### Proof.

Starting with $F=C$, we perform the following update repeatedly while $F$ is
not a $2$-matching:

> Choose a vertex $v\in V$ with $d_{F}(v)>2$ and an edge $vw\in F$ incident to
> $v$.
>
>   1. (i)
>
> If $F\setminus\\{vw\\}$ is triangle-free, then remove $vw$ from $F$.
>
>   2. (ii)
>
> If $F\setminus\\{vw\\}$ contains a triangle 2EC component whose vertex set
> is $\\{v,v_{1},v_{2}\\}$ for some $v_{1},v_{2}\in V$, then remove $vv_{1}$
> from $F$.
>
>   3. (iii)
>
> If neither of the above holds, then $F\setminus\\{vw\\}$ contains a triangle
> 2EC component whose vertex set is $\\{w,w_{1},w_{2}\\}$ for some
> $w_{1},w_{2}\in V$. In this case, remove $ww_{1}$ from $F$.
>
>

If $F$ becomes a $2$-matching, then the procedure terminates by returning
$M=F$. It is obvious that this procedure terminates in polynomial steps and
returns a triangle-free $2$-matching.

We now analyze the size of the output $M$. For an edge set $F\subseteq E$,
define $g(F)=\sum_{v\in V}\max\\{d_{F}(v)-2,0\\}$. Then, in each iteration of
the procedure, we observe that one edge is removed from $F$ and $g(F)$
decreases by at least one, where we note that $d_{F}(w)=3$ before the update
in case (iii). With this observation, we see that $|C|-|M|\leq
g(C)-g(M)=\sum_{v\in V}(d_{C}(v)-2)$, where we note that $C$ is a $2$-edge-
cover and $M$ is a $2$-matching. Therefore, it holds that

$|M|\geq|C|-\sum_{v\in V}(d_{C}(v)-2)=|C|-(2|C|-2|V|)=2|V|-|C|,$

which completes the proof. ∎

By using these lemmas and Theorem 4, we can compute a triangle-free $2$-edge-
cover with minimum cardinality in polynomial time.

######  Proposition 7.

For a graph $G=(V,E)$, we can compute a triangle-free $2$-edge-cover of $G$
with minimum cardinality in polynomial time (if one exists).

###### Proof.

It suffices to consider the case when $G$ is a connected graph such that the
minimum degree is at least two and $|V|\geq 4$. Let $M$ be a triangle-free
$2$-matching in $G$ with maximum cardinality, which can be computed in
polynomial time by Theorem 4. Then, by Lemma 5, we can construct a triangle-
free $2$-edge-cover $C$ of $G$ with size at most $2|V|-|M|$.

We now show that $G$ has no triangle-free $2$-edge-cover $C^{\prime}$ with
$|C^{\prime}|<2|V|-|M|$. Assume to the contrary that there exists a triangle-
free $2$-edge-cover $C^{\prime}$ of size smaller than $2|V|-|M|$. Then, by
Lemma 6, we can construct a triangle-free $2$-matching $M^{\prime}$ of $G$
with size at least $2|V|-|C^{\prime}|$. Since $|M^{\prime}|\geq
2|V|-|C^{\prime}|>2|V|-(2|V|-|M|)=|M|$, this contradicts that $M$ is a
triangle-free $2$-matching with maximum cardinality. Therefore, $G$ has no
triangle-free $2$-edge-cover of size smaller than $2|V|-|M|$, which implies
that $C$ is a triangle-free $2$-edge-cover with minimum cardinality. ∎

###  4.2 Semi-Canonical Triangle-Free Two-Edge-Cover

We show the following lemma saying that a triangle-free $2$-edge-cover can be
transformed into a semi-canonical triangle-free $2$-edge-cover without
increasing the size. Although the proof is almost the same as that of [8,
Lemma 2.4], we describe it for completeness.

######  Lemma 8.

Given a triangle-free $2$-edge-cover $H$ of a $(5/4,\varepsilon)$-structured
graph $G=(V,E)$, in polynomial time, we can compute a triangle-free $2$-edge-
cover $H^{\prime}$ of no larger size which is semi-canonical.

###### Proof.

Recall that an edge set is identified with the corresponding spanning subgraph
of $G$. Starting with $H^{\prime}=H$, while $H^{\prime}$ is not semi-canonical
we apply one of the following operations in this order of priority. We note
that $H^{\prime}$ is always triangle-free during the procedure, and hence it
always satisfies condition (3) in the definition of semi-canonical $2$-edge-
cover.

  1. (a)

If there exists an edge $e\in H^{\prime}$ such that
$H^{\prime}\setminus\\{e\\}$ is a triangle-free $2$-edge-cover, then remove
$e$ from $H^{\prime}$.

  2. (b)

If $H^{\prime}$ does not satisfy condition (4), then we merge two $4$-cycle
2EC components into a cycle of length $8$ by removing $2$ edges and adding $2$
edges. Note that the obtained edge set is a triangle-free $2$-edge-cover that
has fewer connected components.

  3. (c)

Suppose that condition (1) does not hold, i.e., there exists a 2EC component
$C$ of $H^{\prime}$ with fewer than $7$ edges that is not a cycle. Since $C$
is $2$-edge-connected and not a cycle, we obtain $|E(C)|\geq|V(C)|+1$. If
$|V(C)|=4$, then $C$ contains at least $5$ edges and contains a cycle of
length $4$, which contradicts that (a) is not applied. Therefore, $|V(C)|=5$
and $|E(C)|=6$. Since operation (a) is not applied, $C$ is either a bowtie
(i.e., two triangles that share a commmon vertex) or a $K_{2,3}$; see figures
in the proof of [8, Lemma 2.4].

    1. (c1)

Suppose that $C$ is a bowtie that has two triangles $\\{v_{1},v_{2},u\\}$ and
$\\{v_{3},v_{4},u\\}$. If $G$ contains an edge between $\\{v_{1},v_{2}\\}$ and
$\\{v_{3},v_{4}\\}$, then we can replace $C$ with a cycle of length $5$, which
decreases the size of $H^{\prime}$. Otherwise, by the $2$-vertex-connectivity
of $G$, there exists an edge $zw\in E\setminus H^{\prime}$ such that $z\in
V\setminus V(C)$ and $w\in\\{v_{1},v_{2},v_{3},v_{4}\\}$. In this case, we
replace $H^{\prime}$ with $(H^{\prime}\setminus\\{uw\\})\cup\\{zw\\}$. Then,
the obtained edge set is a triangle-free $2$-edge-cover with the same size,
which has fewer connected components.

    2. (c2)

Suppose that $C$ is a $K_{2,3}$ with two sides $\\{v_{1},v_{2}\\}$ and
$\\{w_{1},w_{2},w_{3}\\}$. If every $w_{i}$ has degree exactly $2$, then every
feasible $2$-edge-connected spanning subgraph contains all the edges of $C$,
and hence $C$ is a $\frac{5}{4}$-contractible subgraph, which contradicts the
assumption that $G$ is $(5/4,\varepsilon)$-structured. If $G$ contains an edge
$w_{i}w_{j}$ for distinct $i,j\in\\{1,2,3\\}$, then we can replace $C$ with a
cycle of length $5$, which decreases the size of $H^{\prime}$. Otherwise,
since some $w_{i}$ has degree at least $3$, there exists an edge $w_{i}u\in
E\setminus H^{\prime}$ such that $i\in\\{1,2,3\\}$ and $u\in V\setminus V(C)$.
In this case, we replace $H^{\prime}$ with
$(H^{\prime}\setminus\\{v_{1}w_{i}\\})\cup\\{w_{i}u\\}$. Then, the obtained
edge set is a triangle-free $2$-edge-cover with the same size, which has fewer
connected components.

  4. (d)

Suppose that the first half of condition (2) does not hold, i.e., there exists
a leaf block $B$ that has at most $5$ edges. Let $v_{1}$ be the only vertex in
$B$ such that all the edges connecting $V(B)$ and $V\setminus V(B)$ are
incident to $v_{1}$. Since operation (a) is not applied, we see that $B$ is a
cycle of length at most $5$. Let $v_{1},\dots,v_{\ell}$ be the vertices of $B$
that appear along the cycle in this order. We consider the following cases
separately; see figures in the proof of [8, Lemma 2.4].

    1. (d1)

Suppose that there exists an edge $zw\in E\setminus H^{\prime}$ such that
$z\in V\setminus V(B)$ and $w\in\\{v_{2},v_{\ell}\\}$. In this case, we
replace $H^{\prime}$ with $(H^{\prime}\setminus\\{v_{1}w\\})\cup\\{zw\\}$.

    2. (d2)

Suppose that $v_{2}$ and $v_{\ell}$ are adjacent only to vertices in $V(B)$ in
$G$, which implies that $\ell\in\\{4,5\\}$. If $v_{2}v_{\ell}\not\in E$, then
every feasible 2EC spanning subgraph contains four edges (incident to $v_{2}$
and $v_{\ell}$) with both endpoints in $V(B)$, and hence $B$ is a
$\frac{5}{4}$-contractible subgraph, which contradicts the assumption that $G$
is $(5/4,\varepsilon)$-structured. Thus, $v_{2}v_{\ell}\in E$. Since there
exists an edge connecting $V\setminus V(B)$ and $V(B)\setminus\\{v_{1}\\}$ by
the $2$-vertex-connectivity of $G$, without loss of generality, we may assume
that $G$ has an edge $v_{3}z$ with $z\in V\setminus V(B)$. In this case, we
replace $H^{\prime}$ with
$(H^{\prime}\setminus\\{v_{1}v_{\ell},v_{2}v_{3}\\})\cup\\{v_{3}z,v_{2}v_{\ell}\\}$.

In both cases, the obtained edge set is a triangle-free $2$-edge-cover with
the same size. Furthermore, we see that either (i) the obtained edge set has
fewer connected components or (ii) it has the same number of connected
components and fewer bridges.

  5. (e)

Suppose that the latter half of condition (2) does not hold, i.e., there
exists an inner block $B$ that has at most $3$ edges. Then, $B$ is a triangle.
Let $\\{v_{1},v_{2},v_{3}\\}$ be the vertex set of $B$. If there are at least
two bridge edges incident to distinct vertices in $V(B)$, say $wv_{1}$ and
$zv_{2}$, then edge $v_{1}v_{2}$ has to be removed by operation (a), which is
a contradiction. Therefore, all the bridge edges in $H^{\prime}$ incident to
$B$ are incident to the same vertex $v\in V(B)$. In this case, we apply the
same operation as (d).

We can easily see that each operation above can be done in polynomial time. We
also see that each operation decreases the lexicographical ordering of
$(|H^{\prime}|,{\rm cc}(H^{\prime}),{\rm br}(H^{\prime}))$, where ${\rm
cc}(H^{\prime})$ is the number of connected components in $H^{\prime}$ and
${\rm br}(H^{\prime})$ is the number of bridges in $H^{\prime}$. This shows
that the procedure terminates in polynomial steps. After the procedure,
$H^{\prime}$ is a semi-canonical triangle-free $2$-edge-cover with
$|H^{\prime}|\leq|H|$, which completes the proof. ∎

###  4.3 Proof of Theorem 1

By Lemma 2, in order to prove Theorem 1, it suffices to give a
$\frac{13}{10}$-approximation algorithm for 2-ECSS in
$(5/4,\varepsilon)$-structured graphs for a sufficiently small fixed
$\varepsilon>0$. Let $G=(V,E)$ be a $(5/4,\varepsilon)$-structured graph. By
Proposition 7, we can compute a minimum-size triangle-free $2$-edge-cover $H$
of $G$ in polynomial-time. Note that the optimal value ${\sf OPT}$ of 2-ECSS
in $G$ is at least $|H|$, because every feasible solution for 2-ECSS is a
triangle-free $2$-edge-cover. By Lemma 8, $H$ can be transformed into a semi-
canonical triangle-free $2$-edge-cover $H^{\prime}$ with
$|H^{\prime}|\leq|H|$. Since $H^{\prime}$ is triangle-free, by applying Lemma
3 with $H^{\prime}$, we obtain a $2$-edge-connected spanning subgraph $S$ of
size at most $(\frac{13}{10}-\frac{1}{20}b)|H^{\prime}|$, where $H^{\prime}$
has $b|H^{\prime}|$ bridges. Therefore, we obtain

$|S|\leq\left(\frac{13}{10}-\frac{1}{20}b\right)|H^{\prime}|\leq\frac{13}{10}|H|\leq\frac{13}{10}{\sf
OPT},$

which shows that $S$ is a $\frac{13}{10}$-approximate solution for 2-ECSS in
$G$. This completes the proof of Theorem 1. ∎

##  5 Concluding Remarks

In this paper, we have presented a $(1.3+\varepsilon)$-approximation algorithm
for 2-ECSS, which achieves the currently best approximation ratio. We give a
remark that our algorithm is complicated and far from practical, because we
utilize Hartvigsen’s algorithm [10], which is quite complicated. Therefore, it
will be interesting to design a simple and easy-to-understand approximation
algorithm with (almost) the same approximation ratio as ours. Another possible
direction of future research is to further improve the approximation ratio by
improving Lemma 3.

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# Closed formula for the transport of micro- nano-particle across model porous
media

Paolo Malgaretti<EMAIL_ADDRESS>Helmholtz Institute Erlangen-
Nürnberg for Renewable Energy (IEK-11), Forschungszentrum Jülich, Erlangen,
Germany  Jens Harting  Helmholtz Institute Erlangen-Nürnberg for Renewable
Energy (IEK-11), Forschungszentrum Jülich, Erlangen, Germany  Department of
Applied Physics, Eindhoven University of Technology, Eindhoven, The
Netherlands

###### Abstract

In the last decade the Fick-Jacobs approximation has been exploited to capture
the transport across constrictions. Here, we review the derivation of the
Fick-Jacobs equation with particular emphasis on its linear response regime.
We show that for fore-aft symmetric channels the flux of non-interacting
systems is fully captured by its linear response regime. For this case we
derive a very simple formula that captures the correct trends and that can be
exploited as a simple tool to design experiments or simulations. Finally, we
show that higher order corrections in the flux may appear for non-symmetric
channels.

##  I Introduction

It is common to experience long queues form when a constriction occurs on a
highway Lighthill and Whitham (1955); Wang et al. (2013). Such an (unlucky)
phenomenon is clearly the result of the “local” confinement: due to the
constriction, vehicles slow down hence reducing the local “mass” flux as
compared to the clear part of the highway. Such a local reduction of the mass
flow causes the onset of the annoying queues that every now and then we
experience. This phenomenon does not occur only on highways. It becomes a
major issue close to emergency exits in the case of panic Vermuyten et al.
(2016). The very same dynamics occurs also at smaller scales and for simpler
systems. For example, it is common experience that it is difficult to extract
pills from a container if the opening is too small. Here, pills tend to “clog”
i.e., to form stable structures close to the opening of the container that
prevent pills from going out. The very similar dynamics occurs in silos
containing crops Jeong et al. (2018), in erosion Jäger et al. (2018), in
suspensions of hard and soft particles Marin et al. (2018); Kusters et al.
(2014); Bielinski et al. (2021), in herds of sheep Garcimartín et al. (2015),
in the onset of panic in ants Altshuler et al. (2005), and even humans
Zuriguel et al. (2020).

The effect of confinement does not have to be unpleasant, as it is for traffic
jams, or inconvenient, as it is for clogging of silos. Vice versa, tuning the
shape of the confining media can be an intriguing and novel way to control the
dynamics of the confined system. For example, microfluidic devices exploit
variations of the section of the micro channels they are made of to control
the dynamics of fluid and to induce the formation of droplets Squires and
Quake (2005); Dressaire and Sauret (2017); Douféne et al. (2019); Convery and
Gadegaard (2019). Similarly, Tunable Resistive Pulse Sensing (TRPS) techniques
exploit micro- nano-pores to analyze small particles ranging from a few tens
of nanometers up to micrometric scale Weatherall and Willmott (2015). In
particular, TRPS has been used to direct detect antibody-antigen binding Saleh
and Sohn (2003), to measure elecrophoretic mobility of colloidal particles Ito
et al. (2004), to perform single-molecule detection Heins et al. (2005) and to
measure the zeta-potential of nanometric particles Arjmandi et al. (2012).
Alternatively, Chromatography techniques have been developed to separate
micro- or nano-particles depending on both their size as well as their surface
properties Robards and Ryan (2022); Reithinger and Arlt (2011); Michaud et al.
(2021); Seidel-Morgenstern et al. (2008). Finally, at even smaller scales,
nanopores have been designed to sequence DNA molecules Soni et al. (2010).

Transport in confinement is not relevant only for particle detection/analysis.
Indeed, the flow of fluids across a porous medium is crucial in diverse
scenarios. For example, oil recovery industries have put much effort into
developing techniques to maximize the extraction of oil from the rock matrix
it is embedded in Carvalho (2015); Foroozesh and Kumar (2020). Similarly,
understanding the dependence of the flow of water on the porosity of the soil
is crucial in environmental sciences Farhadian and Nikvar-Hassani (2019).
Moreover, diverse techonlogies related to the energy transition such as blue-
energy Boon and Roij (2011), hydrogen technology P. Preuster (2017); Solymosi
et al. (2022), electrolyzers and fuel cells Suter et al. (2021); Du et al.
(2022), or $CO_{2}$ segregation Hepburn et al. (2019) rely on the transport of
(charged) chemical species across nanoporous materials.

Finally, several biological systems are controlled by the transport of
confined complex fluids. For example, neuronal transmission relies on the
transport of neuro-receptors among neurons and to their specific binding sites
Alberts et al. (2007). Moreover, cell regulation relies on the proper tuning
of the concentrations of electrolytes inside the cell. Such a regulation
occurs via dedicated pores and channels whose shape makes them very sensitive
to specific ions Pethig (1986); Dubyak (2004); Calero et al. (2011); Peyser et
al. (2014); Lee et al. (2017) and RNA is transported across the nuclear
membrane Melnikov et al. (2017); Bacchin (2018); Berezhkovskii et al. (2019).
Moreover, the lymphatic and circulatory systems in mammals rely on the
transport of quite heterogeneous suspensions composed of a variety of
components, spanning from the nanometric size of ions up to the micrometric
size of red blood cells, across varying-section elastic pipes Kusters et al.
(2014); Nipper and Dixon (2011); Wiig and Swartz (2012); Yoganathan et al.
(1988). Finally, the survival of plants relies, at large scales, on the proper
circulation of liquid (sap) along the trunk Jensen et al. (2016) and at short
scales on the cytoplasmic streaming within the cells Shimmen and Yokota
(2004).

All the above mentioned systems rely or depend on the dynamics under
confinement. Therefore, understanding the dynamics and transport properties of
confined complex systems such as ions, molecules, polymers, colloidal
particles, and suspensions is of primary importance for the understanding of a
wide spectrum of phenomena and for the development of technological
applications. Even more, identifying the relevant parameters controlling key
features, like transport or phase transitions, will open a new route for
controlling the dynamics of confined systems upon tuning the geometry of the
confining media.

Up to now, there has been no systematic study of the dependence of the
dynamics of confined systems upon changing the shape of the confining walls.
The main reason is the large effort that such a study requires. Indeed,
experimentally tuning the shape of a pore is a tremendous task since, if
possible at all, it requires to synthesize every time a new item from scratch.
On the theoretical side, studying the dynamics and the transport of confined
systems is a tremendous task since it requires to capture several length, time
and energy scales. In fact, the length scales range from the nanometric scale,
typical for ions and for van der Waals interactions to the micrometric scale
of colloids, polymers and macromolecules up to the millimeters/centimeters
scale of microfluidic devices. Concerning time scales, the spectrum spans the
diffusion time of small particles and ions over their size $\sim\mu\text{sec}$
up to the long time scales typical of transport $\sim\text{sec}$. Concerning
energy scales, they range from thermal energy $k_{B}T$ ($\sim
10^{-21}\text{J}$) up to van der Waals and electrostatic interactions whose
magnitude can be of several $k_{B}T$. On the top of these “direct”
interactions also the effective interactions induced by the confinement should
be accounted for. For example, squeezing a deformable object, like a polymer
or a vesicle, through a constriction can require quite an amount of energy
that can easily reach the order of $100-1000\,k_{B}T$. Given such a
complexity, one typically would rely on numerical techniques such as molecular
dynamics. However, the wide range of interactions (Van der Walls,
electrostatic..) jointly with the wide range of time and length scales imposes
to put forward numerical approaches capable of properly resolving the smallest
length, time and energy scales. At the same time, such an approach should also
resolve the large length, time and energy scales. Accordingly, the numerical
route becomes quite demanding from the perspective of the computational time.

Since both experimental and numerical routes are quite expensive, an
approximated analytical route based on some controllable expansions may become
appealing. Intriguingly, it is possible to obtain simple analytical models
that capture some features of the dynamics of confined systems. The key idea,
is to “project” the dynamics of the system onto some relevant coordinate (in
chemistry sometimes called “reaction coordinate”) and then to study the
dynamics of these few (typically one) degrees of freedom. For example, in the
case of polymer translocation across pores, the most important observable is
the time the polymer takes to cross from one side to the other of the pore.
Therefore, the relevant degree of freedom is the position of the center of
mass of the polymer whereas the degrees of freedom associated with the
position of the monomers can be integrated out.

In this contribution, we briefly review the derivation of the Fick-Jacobs
approximation Zwanzig (1992); Reguera and Rubi (2001); Kalinay and Percus
(2005a); Kalinay and Percus (2005b); Kalinay and Percus (2008); Martens et al.
(2011); Chacón-Acosta et al. (2013); Malgaretti et al. (2013) and its use in
studying transport across corrugated pores and channels. The Fick-Jacobs
approximation has been shown to be applicable to the transport of ions
Malgaretti et al. (2014, 2015); Malgaretti et al. (2016a); Chinappi and
Malgaretti (2018); Malgaretti et al. (2019), colloids Reguera et al. (2006,
2012); Marini Bettolo Marconi et al. (2015); Malgaretti et al. (2016b);
Puertas et al. (2018), rods Malgaretti and Harting (2021), polymers Bianco and
Malgaretti (2016); Malgaretti and Oshanin (2019); Bodrenko et al. (2019), and
more recently even active systems Malgaretti and Stark (2017); Kalinay (2022);
Antunes et al. (2022), chemial reactors Ledesma-Durán et al. (2016) and
pattern-forming systems Chacón-Acosta et al. (2020).

In the following we re-derive the Fick-Jacobs approximation with particular
emphasis on the regime in which the current is proportional to the applied
force. In such a regime, it is possible to derive a closed formula that
accounts for the dependence of the flux on the geometry of the channel.
Interestingly, our derivation naturally highlights a few relations between the
underling Smoluchowski equation and the linear response theory. Even though
this work is motivated by the transport in confined pores and channels, the
results we derive are valid for all $1D$ systems (independently of the
physical origin of the effective potential) in the dilute regime (for which
mutual interactions can be neglected) and whose dynamicsis governed by the
Smoluchowski equation (i.e. in the overdamped regime).

Figure 1: Cartoon of the varying-section channel.

##  II Model

In the following we are interested in the transport of a single colloidal
particle confined in an axially symmetric channel characterized by its half
section (see Fig. 1 for a scketch of the system)

$\displaystyle h(x)=h_{0}+h_{1}\cos\left(2\pi\frac{x}{L}\right)\,.$ (1)

The time evolution of the probability density is governed by the Smoluchowski
equation

$\displaystyle\dot{\rho}(\mathbf{r},t)=\nabla\cdot\left[D\nabla\rho(\mathbf{r},t)+D\beta\rho(\mathbf{r},t)\nabla
W(\mathbf{r})\right]\,,$ (2)

where $D$ is the diffusion coefficient, $\beta^{-1}=k_{B}T$ is the inverse
thermal energy, $k_{B}$ the Boltzmann constant, $T$ the absolute temperature
and

$\displaystyle W(\mathbf{r})=\begin{cases}\phi(\mathbf{r})&|r|<h(x)\\\
\infty&\text{else}\end{cases}$ (3)

is the effective potential responsible for both confining the particle within
the channel and for additional soft interactions, $\phi(\mathbf{r})$ with the
channel walls. For smoothly-varying channel cross-sections,
$\partial_{x}h(x)\ll 1$, it is possible to factorize the probability density
Zwanzig (1992); Reguera and Rubi (2001); Kalinay and Percus (2008); Martens et
al. (2011); Chacón-Acosta et al. (2013); Malgaretti et al. (2013)

$\displaystyle\rho(\mathbf{r},t)=p(x,t)\dfrac{e^{-\beta
W(\mathbf{r})}}{e^{-\beta A(x)}}\,,$ (4)

where

$\displaystyle A(x)=-k_{B}T\ln\left[\frac{1}{\pi
h_{0}^{2}}\int_{-\infty}^{\infty}e^{-\beta W(\mathbf{r})}rdr\right]$ (5)

is the local free energy Malgaretti et al. (2016c). Moreover, integrating
along the radial direction leads to

$\displaystyle\dot{p}(x,t)=\partial_{x}\left[D\partial_{x}p(x,t)+D\beta
p(x,t)\partial_{x}A(x)\right]\,.$ (6)

Such a procedure is called Fick-Jacobs approximation Zwanzig (1992); Reguera
and Rubi (2001); Malgaretti et al. (2013). Its regime of validity has been
assessed by several groups Reguera et al. (2006); Berezhkovskii et al. (2007);
Burada et al. (2007); Berezhkovskii et al. (2015); Kalinay and Percus (2005a);
Kalinay and Percus (2005b); Kalinay and Percus (2006); Martens et al. (2011);
Pineda et al. (2012); García-Chung et al. (2015). In particular, it has been
shown that the quantitative reliability of the Fick-Jacobs approximation can
be enhanced by introducing a position dependent diffusion coefficient Reguera
et al. (2006); Berezhkovskii et al. (2007); Burada et al. (2007);
Berezhkovskii et al. (2015); Kalinay and Percus (2005a); Kalinay and Percus
(2005b); Kalinay and Percus (2006); Martens et al. (2011); Pineda et al.
(2012); García-Chung et al. (2015), $D(x)$, hence leading to the set of
equations

$\displaystyle\dot{p}(x,t)$ $\displaystyle=-\partial_{x}J(x,t)$ (7)
$\displaystyle\frac{J}{D(x)}$ $\displaystyle=-\partial_{x}p(x)-\beta
p(x)\partial_{x}A(x)\,.$ (8)

Eq. (8) is completed with the boundary conditions

$\displaystyle p(-L)$ $\displaystyle=p(L)$ (9)
$\displaystyle\int_{-L}^{L}p(x)dx$ $\displaystyle=1.$ (10)

We decompose the effective force $-\partial_{x}A(x)$ as the net force

$\displaystyle f=-\frac{1}{2L}\int_{-L}^{L}\partial_{x}A(x)dx=-\frac{\Delta
A}{2L}$ (11)

and

$\displaystyle A_{eq}(x)=A(x)+fx.$ (12)

$f$ accounts for the net force responsible of the flux and $A_{eq}(x)$
accounts for all the other conservative forces that will not give rise to any
flux. In the following, we expand both the flux, $J$, and the density, $p$,
about the equilibrium case:

$\displaystyle J=$ $\displaystyle J_{0}+J_{1}+J_{2}+...$ (13) $\displaystyle
p(x)=$ $\displaystyle p_{0}(x)+p_{1}(x)+p_{2}(x)+...$ (14)

Note that due to Eq. (10) at zeroth order we have

$\displaystyle\int_{-L}^{L}p_{0}(x)dx$ $\displaystyle=1\,.$ (15)

This implies

$\displaystyle\int_{-L}^{L}p_{n}(x)dx$ $\displaystyle=0\,\,\,\,\forall n\neq
0$ (16)

Accordingly, at order zero we have

$\displaystyle p_{0}(x)$ $\displaystyle=\tilde{p}e^{-\beta A_{eq}(x)}$ (17)
$\displaystyle J_{0}$ $\displaystyle=0$ (18) $\displaystyle\tilde{p}$
$\displaystyle=\frac{1}{\int_{-L}^{L}e^{-\beta A_{eq}(x)}dx}\,.$ (19)

At the generic $n$-th order we have

$\displaystyle\frac{J_{n}}{D(x)}=-\partial_{x}p_{n}(x)-\beta
p_{n}(x)\partial_{x}A_{eq}(x)+\beta p_{n-1}(x)f\,,$ (20)

the solution of which reads

$\displaystyle p_{n}(x)=e^{-\beta
A_{eq}(x)}\left[\int\limits_{-L}^{x}\left[\beta
p_{n-1}(y)f-\frac{J_{n}}{D(y)}\right]e^{\beta A_{eq}(y)}dy+\Pi_{n}\right]\,.$
(21)

Here, $J_{n}$ and $\Pi_{n}$ are integration constants. Imposing the periodic
boundary conditions, $p_{n}(-L)=p_{n}(L)$, and recalling that
$A_{eq}(-L)=A_{eq}(L)$ leads to

$\displaystyle\int_{-L}^{L}\left(\frac{J_{n}}{D(y)}-\beta
p_{n-1}(y)f\right)e^{\beta A_{eq}(y)}dy=0\,,$ (22)

with

$\displaystyle J_{n}=\beta f\dfrac{\int_{-L}^{L}p_{n-1}(y)e^{\beta
A_{eq}(y)}dy}{\int_{-L}^{L}\dfrac{e^{\beta A_{eq}(y)}}{D(y)}dy}=\beta
f\tilde{p}\dfrac{\int_{-L}^{L}\frac{p_{n-1}(y)}{p_{0}(y)}dy}{\int_{-L}^{L}\dfrac{e^{\beta
A_{eq}(y)}}{D(y)}dy}\,.$ (23)

In the last step we used Eq. (17). Finally, $\Pi_{n}$ is determined by
imposing Eqs. (15), (16)

$\displaystyle\Pi_{n}=-\tilde{p}\int_{-L}^{L}e^{-\beta
A_{eq}(x)}\int\limits_{-L}^{x}\left[\beta
p_{n-1}(y)f-\frac{J_{n}}{D(y)}\right]e^{\beta A_{eq}(y)}dydx\,.$ (24)

At leading order in the force, Eqs. (21) (23) read

$\displaystyle p_{1}(x)=$ $\displaystyle\,e^{-\beta A_{eq}(x)}\left[\beta
f\tilde{p}(x+L)-J_{1}\int_{-L}^{x}\frac{e^{\beta
A_{eq}(y)}}{D(y)}dy\right]\,,$ (25) $\displaystyle J_{1}=$
$\displaystyle\,\dfrac{2\beta fL}{\int_{-L}^{L}e^{-\beta
A_{eq}(x)}dx\int_{-L}^{L}\frac{e^{\beta A_{eq}(x)}}{D(x)}dx}\,.$ (26)

Interestingly, from Eq. (26), it is possible to identify a force-independent
channel permeability

$\displaystyle\chi=\dfrac{2\beta L}{\int_{-L}^{L}e^{-\beta
A_{eq}(x)}dx\int_{-L}^{L}\frac{e^{\beta A_{eq}(x)}}{D(x)}dx}\,.$ (27)

As expected, Eq. (27) agrees with the derivation of the effective diffusion
coefficient for a particle at equilibrium and in the presence of entropic
barriers Lifson and Jackson (1962); Reimann et al. (2001). This is in
agreement with the linear response theory within which the transport
coefficients that determine the flux under external forces can be determined
from equilibrium properties.

Some general remarks can be derived in the case of fore-aft symmetric
channels, for which $A_{eq}(x)=A_{eq}(-x)$, and diffusivities, $D(x)=D(-x)$.
For such cases, the magnitude of the flux should depend solely on the
magnitude of the force and not on its sign. This implies that

$\displaystyle J_{2n}=0,\quad\forall n>0\,.$ (28)

In order to proceed, we recall that, for fore-aft symmetric $f(x)$ and $g(x)$,
the following equality holds:

$\displaystyle\int_{-L}^{L}g(x)\int_{-L}^{x}f(y)dydx=\frac{1}{2}\int_{-L}^{L}f(x)dx\int_{-L}^{L}g(x)dx\,$
(29)

Enforcing the condition in Eq. (28) into Eq. (23) and using the last
expression leads to

$\displaystyle\Pi_{n}=0,\quad\forall n>0\,.$ (30)

and, substituting again into Eq. (23) eventually leads to

$\displaystyle J_{n}=0,\quad\forall n\geq 1\,.$ (31)

Interestingly, we note that even though $\Pi_{n>0}=0$ and $J_{n>1}=0$ the
density profile is still sensitive to higher order corrections in the force,
i.e. in general $p_{n}\neq 0$. According to this analysis, Eq. (26) is not
just the linear contributions to the flux rather it provides the exact
expressions at every order in the external force. The outcome of this analysis
is indeed intuitive since it states that for non-interacting systems confined
within fore-aft symmetric channels non-linear effects are absent. The same
results are indeed valid for any $1D$ problem with such a symmetry.

In contrast, if neither the potential, $A(x)$, nor the diffusion profile,
$D(x)$, have a defined parity, then the left-right symmetry is broken, Eq.
(28) does not hold anymore, and indeed a diode effect may set for sufficiently
large external forces. We can assess the dependence of the diode effect on the
geometry of the channel by calculating

$\displaystyle J_{2}=\beta
f\dfrac{\int\limits_{-L}^{L}\int\limits_{-L}^{x}\\!\\!\\!\beta\tilde{p}f-\frac{J_{1}}{D(y)}e^{\beta
A_{eq}(y)}dy+\Pi_{1}dx}{\int_{-L}^{L}\dfrac{e^{\beta A_{eq}(y)}}{D(y)}dy}\,.$
(32)

Using

$\displaystyle\Gamma(x)=\int_{-L}^{x}\dfrac{e^{\beta A_{eq}(y)}}{D(y)}dy$ (33)

and the definition of $J_{1}$ we obtain

$\displaystyle J_{2}=\frac{\beta
f}{\Gamma(L)}\int\limits_{-L}^{L}\beta\tilde{p}f(x+L)-2\beta\tilde{p}fL\dfrac{\Gamma(x)}{\Gamma(L)}+\Pi_{1}dx\,.$
(34)

Finally, using the definition of $\Pi_{1}$ we obtain

$\displaystyle J_{2}=$ $\displaystyle\frac{(\beta
fL)^{2}\tilde{p}}{\Gamma(L)}\frac{1}{L}\int\limits_{-L}^{L}\left[\left(\frac{x}{L}+1\right)-2\dfrac{\Gamma(x)}{\Gamma(L)}\right]\left[1-e^{-\beta
A_{eq}(x)}\right]dx\,.$ (35)

###  II.1 Transport across free energy barriers

In the case of transport of point-like particles across $3D$ varying-section
channels with axial symmetry the effective potential reads

$\displaystyle A^{(id)}_{eq}(x)=-2k_{B}T\ln\left[\frac{h(x)}{h_{0}}\right]\,,$
(36)

where $h(x)$ is the local half-section of the channel and $h_{0}$ its average
value (see Fig.1). Accordingly, Eq. (26) reads

$\displaystyle J_{id}=\dfrac{2\beta
fL}{\int_{-L}^{L}\frac{h^{2}(x)}{h_{0}^{2}}dx\int_{-L}^{L}\frac{h_{0}^{2}}{h^{2}(x)D(x)}dx}\,.$
(37)

In the case of micro- or nano-particles that undergo solely excluded volume
interactions with the channel walls, the effective channel half-section
becomes $h(x)-R$ where $R$ is the particle size and we obtain

$\displaystyle
A^{(pcl)}_{eq}(x)=-2k_{B}T\ln\left[\frac{h(x)-R}{h_{0}}\right]\,,$ (38)

which leads to

$\displaystyle J_{pcl}=\dfrac{2\beta
fL}{\int_{-L}^{L}\frac{(h(x)-R)^{2}}{h_{0}^{2}}dx\int_{-L}^{L}\frac{h_{0}^{2}}{(h(x)-R)^{2}D(x)}dx}\,.$
(39)

We recall that $R<h_{0}-h_{1}$ for the particle to be able to cross the
channel. Finally, several groups have shown that the Fick-Jacobs approximation
can be improved by assuming a position-dependent diffusion coefficient Zwanzig
(1992); Reguera and Rubi (2001); Kalinay and Percus (2006, 2008); Martens et
al. (2011); Pineda et al. (2012); Berezhkovskii et al. (2015); García-Chung et
al. (2015). Nowadays, there is general agreement that the approximated formula
for the diffusion coefficient reads Reguera and Rubi (2001) (or is in practice
equivalent to)

$\displaystyle D(x)=\dfrac{D_{0}}{\sqrt{1+(\partial_{x}h(x))^{2}}}\,.$ (40)

Figure 2: Transport across porous media. Upper left: permeability, $\chi$, as
obtained form Eq. (46) (solid lines), Eq. (26) with constant diffusion
coefficient (dashed lines) and Eq. (26) with a diffusion coefficient as given
by Eq. (40) (dashed-dotted lines) normalized by the one across a constant-
section channel $\chi_{o}=D\beta/4L$, as function of the geometry of the
channel $\Delta
S=\ln\frac{h_{0}+h_{1}}{h_{0}-h_{1}}=\ln\frac{h_{max}}{h_{min}}$ for different
values of the particle radius. Upper right: ratio of $\tilde{\chi}$ over
$\chi$ normalized by $\chi$ for the data sets shown in the left panel. Bottom:
permeability, $\chi$, normalized by the one across a constant-section channel
$\chi_{o}=D\beta/4L$, as function of the radius of the particle, $R$,
normalized by the average channel width, $h_{0}$, for different channel
geometries captured by $\Delta S$. Bottom right: ratio of $\tilde{\chi}$ over
$\chi$ normalized by $\chi$ for the data sets shown in the left panel.

###  II.2 Piece-wise linear potential and homogeneous diffusion coefficient

In order to get analytical insight it can be useful to approximate the
effective potential $A(x)$ by

$\displaystyle A_{eq}(x)=-\frac{\Delta A_{eq}}{L}|x|\,,$ (41)

where

$\displaystyle\Delta A_{eq}=A^{max}_{eq}-A^{min}_{eq}$ (42)

is the piece-wise linear difference between the maximum and minimum values of
$A_{eq}$. Moreover, if we assume that the diffusion coefficient is homogeneous

$\displaystyle D(x)=D_{0}$ (43)

we get

$\displaystyle\int_{-L}^{L}e^{\beta A_{eq}(x)}dx=$
$\displaystyle\frac{2L}{\beta\Delta A_{eq}}\left(1-e^{-\beta\Delta
A_{eq}}\right)$ (44) $\displaystyle\int_{-L}^{L}e^{-\beta A_{eq}(x)}dx=$
$\displaystyle\frac{2L}{\beta\Delta A_{eq}}\left(e^{\beta\Delta
A_{eq}}-1\right)$ (45)

and finally by substituting the last expressions into Eq. (27) we obtain an
approximated expression for the permeability

$\displaystyle\tilde{\chi}=\frac{D\beta}{4L}\dfrac{\left(\beta\Delta
A_{eq}\right)^{2}}{\cosh(\beta\Delta A_{eq})-1}\,.$ (46)

Interestingly, Eq. (46) shows that $\chi$ is an even function of $\Delta
A_{eq}$. This implies that the transport is insensitive upon flipping the sign
of the free energy barrier $\Delta A$. Finally, Eq.(46) shows that $\chi$
decays exponentially with $\beta\Delta A_{eq}$.

##  III Discussion

The reliability of the Fick-Jacobs approximation, namely Eq.(26), has been
addressed for point-like particles and it has shown good quantitative
agreement for forces up to $\beta fL\simeq 10$ Burada et al. (2007). However,
Eq. (26) still requires to numerically compute integrals, whereas Eq. (46)
provides a direct (yet approximated) dependence of $\tilde{\chi}$ on $\Delta
A$. Therefore, it is important to address the reliability of Eq. (46) as
compared to the full solution Eq. (26). Indeed, all the panels of Fig. 2 show
that the permeability calculated with the piece-wise linear model, Eq. (46),
shows some discrepancies as compared to the full expression give in Eq. (26).
In particular, as shown in Fig. 2 for the case under consideration
($h_{0}/L=0.1$) the corrections due to the inhomogeneous diffusion (dashed-
dotted lines) are indistinguishable from those with constant diffusion
coefficient (dashed lies) and hence they do not improve the approximation. On
the other hand Fig. 2 shows that the simple formula in Eq. (46) is sufficient
to properly capture the trends and indeed can be used to estimate the
transport of colloidal particle across porous media. Interestingly, concerning
the magnitude of $\chi$, the bottom panels of Fig. 2 show that the channel
permeability decreases upon increasing the particle size. Interestingly, the
decrease is almost linear for larger corrugations of the channel (larger
values of $\Delta S$) whereas for smaller values of the corrugation it
plateaus at smaller values of $R$.

Figure 3: Dependence of the approximated channel permeability, $\tilde{\chi}$,
(as defined in Eq. (46)) normalized by that of a constant section channel,
$\chi_{o}$ as function of the amplitude of the dimensionless free energy
barrier $\beta\Delta A$ which encodes the physical properties of the confined
system.

Finally, we discuss the dependence of $\tilde{\chi}$ in $\beta\Delta A$ as per
Eq. (46). As shown in Fig. (3), $\tilde{\chi}$ has a maximum for $\beta\Delta
A=0$ and then it decays exponentially for larger values of $\beta\Delta A$.
Interestingly, $\tilde{\chi}$ attains values close to unity up to $\beta\Delta
A\simeq 5$, i.e. for a free energy barrier much larger than the thermal
energy.

The fact that Eq. (46) depends solely on $\Delta A$ allows one to estimate the
transport also in situations in which the particles may have some soft
interactions with the walls, like electrostatic interactions. In that case the
free energy barrier will depend not only on the size of the particle and on
the geometry of the channel but also on the charge of both the particle and
the walls of the channels Malgaretti et al. (2015); Malgaretti et al. (2016c).
Moreover, Eq. (46) allows also to predict the transport of soft or deformable
objects, like proteins or polymers Bianco and Malgaretti (2016); Malgaretti
and Oshanin (2019); Carusela et al. (2021).

##  IV Conclusions

We have derived closed formulas for the transport within linear response
theory as well as for higher order corrections. In particular, we have shown
that for the case of non-interacting systems confined in fore-aft symmetric
channels the higher order corrections in both the flux and in the density are
identically zero. Hence, for fore-aft symmetric channels, the full expression
for the flux is indeed the one obtained within the linear response regime.
Accordingly, the channel permeability derived within linear response, Eq.
(27), is related to the well known expression of the effective diffusion
coefficient reported in the literature Lifson and Jackson (1962); Reimann et
al. (2001). Moreover, we have shown that, within the linear response, the
formula for the permeability $\chi$, Eq. (27), can be further simplified by
approximating the local free energy by piece-wise linear potential (Eq. (41))
to obtain Eq. (46), whose overall drop is determined by the values of the free
energy at the bottleneck and at the waist of the channel. We have shown that
such an approximation provides the correct trends and it is reliable within
$\simeq\pm 50\%$ as shown in the right panels of Fig. 2. This feature is
crucial since Eq. (46) can be easily computed and it is valid for all soft-
interactions between the particle and the channel walls.

## Acknowledgments

We thank I. Pagonabarraga and J. Schmid for insightful discussions and
acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project-ID 416229255 – SFB 1411 and Project-ID
431791331 – SFB 1452.

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