Charge storage in nanotubes: the case of a 2-1 electrolyte
We consider a 2-1 electrolyte in contact with a narrow nanotube, which only allows one-dimensional storage along the axis. The asymmetry does not allow an a priori definition of the potential of zero charge; instead, the natural reference is the electrode potential at which both ions have the same...
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irk-123456789-1570042019-06-21T01:26:58Z Charge storage in nanotubes: the case of a 2-1 electrolyte Schmickler, W. Henderson, D. We consider a 2-1 electrolyte in contact with a narrow nanotube, which only allows one-dimensional storage along the axis. The asymmetry does not allow an a priori definition of the potential of zero charge; instead, the natural reference is the electrode potential at which both ions have the same electrochemical potential; the value of the latter can serve as a measure of ionophilicity. Near this potential, ionophobic tubes are filled with a dilute gas, ionophilic tubes are filled with a one-dimensional solid containing about the same number of the divalent ions and the monovalent counterions, a structure that is stabilized by a strong screening of the Coulomb interaction by an induced counter charge on the walls of the tube. The filling of the tube by the application of an electrode potential exhibits a complicated pattern of interactions between the two kinds of ions. Ми розглядаємо 2-1 електролiт в контактi з вузькою нанотрубкою, яка дозволяє лише одновимiрне накопичення вздовж осi. Асиметрiя не дозволяє апрiорi означити потенцiал нульового заряду; натомiсть, природною точкою вiдлiку є потенцiал електрода, бiля якого обидва iони мають той же електрохiмiчний потенцiал; значення останнього може служити для вимiрювання iонофiльностi. Поблизу цього потенцiалу iонофобнi трубки наповненi розрiдженим газом, iонофiльнi трубки є наповненi одномiрним твердим тiлом, що мiстить приблизно однакове число двовалентних iонiв i моновалентних контрiонiв, стiйкiсть структури забезпечується сильним екрануванням кулонiвської взаємодiї iндукованим зарядом на стiнках трубки. Наповнення трубки шляхом прикладання електродного потенцiалу демонструє складну картину взаємодiй мiж двома сортами iонiв. 2017 Article Charge storage in nanotubes: the case of a 2-1 electrolyte / W. Schmickler, D. Henderson // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33004: 1–7 . — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 02.70.Lq, 82.20.Wt, 82.45.+z, 83.20 DOI:10.5488/CMP.20.33004 arXiv:1710.01097 http://dspace.nbuv.gov.ua/handle/123456789/157004 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
We consider a 2-1 electrolyte in contact with a narrow nanotube, which only allows one-dimensional storage
along the axis. The asymmetry does not allow an a priori definition of the potential of zero charge; instead,
the natural reference is the electrode potential at which both ions have the same electrochemical potential;
the value of the latter can serve as a measure of ionophilicity. Near this potential, ionophobic tubes are filled
with a dilute gas, ionophilic tubes are filled with a one-dimensional solid containing about the same number
of the divalent ions and the monovalent counterions, a structure that is stabilized by a strong screening of
the Coulomb interaction by an induced counter charge on the walls of the tube. The filling of the tube by the
application of an electrode potential exhibits a complicated pattern of interactions between the two kinds of
ions. |
| format |
Article |
| author |
Schmickler, W. Henderson, D. |
| spellingShingle |
Schmickler, W. Henderson, D. Charge storage in nanotubes: the case of a 2-1 electrolyte Condensed Matter Physics |
| author_facet |
Schmickler, W. Henderson, D. |
| author_sort |
Schmickler, W. |
| title |
Charge storage in nanotubes: the case of a 2-1 electrolyte |
| title_short |
Charge storage in nanotubes: the case of a 2-1 electrolyte |
| title_full |
Charge storage in nanotubes: the case of a 2-1 electrolyte |
| title_fullStr |
Charge storage in nanotubes: the case of a 2-1 electrolyte |
| title_full_unstemmed |
Charge storage in nanotubes: the case of a 2-1 electrolyte |
| title_sort |
charge storage in nanotubes: the case of a 2-1 electrolyte |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157004 |
| citation_txt |
Charge storage in nanotubes: the case of a 2-1 electrolyte / W. Schmickler, D. Henderson // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33004: 1–7
. — Бібліогр.: 21 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT schmicklerw chargestorageinnanotubesthecaseofa21electrolyte AT hendersond chargestorageinnanotubesthecaseofa21electrolyte |
| first_indexed |
2025-07-14T09:21:09Z |
| last_indexed |
2025-07-14T09:21:09Z |
| _version_ |
1837613574680215552 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 3, 33004: 1–7
DOI: 10.5488/CMP.20.33004
http://www.icmp.lviv.ua/journal
Charge storage in nanotubes: the case of a
2-1 electrolyte
W. Schmickler1, D. Henderson2
1 Institute of Theoretical Chemistry, Ulm University, Germany
2 Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602, USA
Received July 25, 2017, in final form August 7, 2017
We consider a 2-1 electrolyte in contact with a narrow nanotube, which only allows one-dimensional storage
along the axis. The asymmetry does not allow an a priori definition of the potential of zero charge; instead,
the natural reference is the electrode potential at which both ions have the same electrochemical potential;
the value of the latter can serve as a measure of ionophilicity. Near this potential, ionophobic tubes are filled
with a dilute gas, ionophilic tubes are filled with a one-dimensional solid containing about the same number
of the divalent ions and the monovalent counterions, a structure that is stabilized by a strong screening of
the Coulomb interaction by an induced counter charge on the walls of the tube. The filling of the tube by the
application of an electrode potential exhibits a complicated pattern of interactions between the two kinds of
ions.
Key words:Monte Carlo simulation, electrical double layer, ions, capacitance, nanocylinder pores
PACS: 02.70.Lq, 82.20.Wt, 82.45.+z, 83.20
1. Introduction
We met Jean-Pierre Badiali first at a workshop on the electrochemical double layer, which was
organized by Roger Parsons at the CNRS Laboratory Bellevue, Paris, in 1982. To our surprise, we found
out that we were working on similar models for the double layer, combining the jellium model for the
metal with the hard-sphere electrolyte model for the solution [1–3]. Subsequently, we worked in parallel
on this topic, exchanging ideas openly, but never writing a joint publication. This only came about much
later, when one of us (WS) worked with him on another topic, on electron transfer reactions [4].
Jean-Pierre worked on a broad spectrum of topics in physics, some of them quite fundamental, and it
was always a challenge, an intellectual and personal pleasure to discuss science with him. But the double
layer is the topic which brought us together, so we thought it fit to dedicate an article on a modern double
layer problem to his memory.
Nowadays there is much interest in double layers in confined spaces, whose extension is smaller than
the Debye length of the solution. An extreme case is a nanotube which is so thin that only one line of
ions can enter. When such a nanotube serves as an electrode and is in contact with an ionic liquid or a
molten salt, ions can enter and form an electric double layer, whose composition can be controlled by the
electrode potential. This case might be thought to be of academic interest only, but there is a practical
side to it: The storage of ions in nanotubes is of great practical importance for supercapacitors and for
batteries, and narrow tubes with diameters of less than 1 nm have an unusually high capacitance per area
[5, 6]. Therefore, much attention has been focused on the storage of a one-dimensional line of ions inside
a narrow tube, e.g., [7–11].
In a very recent work [12] we have investigated a 1-1 electrolyte in contact with a nanotube by
grand-canonical Monte Carlo simulations. In that work, the focus was on the interfacial capacitance.
Strangely, the case of an asymmetrical electrolyte seems to have been neglected so far. So, in this work
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33004-1
https://doi.org/10.5488/CMP.20.33004
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
W. Schmickler, D. Henderson
we consider a 2-1 electrolyte in a nanotube, and focus on the consequences which the composition has
on the structure of the embedded chain of ions, and on the stored charge as a function of potential.
2. Screening of the Coulomb interaction and the formation of a one-
dimensional salt
The total system, i.e., incorporated ions and induced charge on the walls of the tube, is always neutral.
An ion embedded in the tube generates an image charge of equal magnitude and opposite sign, which
spreads in a ring-like manner around the ion.1 When the ion is on the axis of the tube, the electrostatic
problem can be solved for a perfectly conducting, classical tube. In particular, along the axis of the tube,
the electrostatic potential generated by the ion and its image charge is then given by [13, 14]:
Φ(z) =
2
R
∞∑
m=1
exp(−kmz/R)
km |J1(km)|2
, (1)
where R is the radius of the tube, z is the distance from the center of the ion, km denotes the roots of the
Bessel function J0(km) = 0, and J1 is the Bessel function of first order.
A real tube, such as a carbon or gold nanotube, screens the charge even better than a perfect classical
metal, since the electrons spill over the metal surface. Mohammadzadeh et al. [15–18] have shown that
in this case equation (1) can still be used, but the classical radius must be replaced by an effective radius,
which is typically smaller by about 0.5 Å than the physical radius defined by the positions of the atoms.
Figure 1 shows the potential along the axis for a typical case.
0 4 8 12
0.0
0.2
0.4
0.6
0.8
distance / a.u.
p
o
te
n
ti
a
l
/
a
.u
.
Bessel expansion
ring approximation
large distance
Figure 1. (Color online) Potential along the axis of a ring with an effective image radius of 4 a.u. The full
line corresponds to exact solution of equation (1), the ring approximation is given by equation (2), and
the large distance approximation refers to the 1/z3 asymptote.
Formula (1) is of little practical use in Monte-Carlo or molecular dynamics simulations, since its
evaluation requires too much time, and for large distances one needs a large number of terms. A useful
approximation is based on a physical consideration: The image charge forms a ring around the ion. At
large distances, the width of this ring can be neglected; the resulting potential is:
Φ(z) ≈
1
z
−
1
√
z2 + R2
≈
R2
2z3 . (2)
The latter approximation holds when |z | � R; it shows that at large distances the interaction decreases
with 1/z3, while for an unscreened Coulomb potential it decreases with 1/z. Figure 1 compares the exact
1There is a moot discussion in the community whether to call this an image charge or an induced charge. We follow our previous
publications and use both expression as synonymous. In fact, the optical image of a sphere embedded in a reflecting tube is a ring.
33004-2
Charge storage in nanotubes: the case of a 2-1 electrolyte
solution with the approximate expression of equation (2) and the asymptotic form. A tube filled with one
kind of ion, all of the same charge, can be stable, since:
∞∑
n=1
1
n3 = ζ(3) ≈ 1.20205, (3)
where ζ(x) is the Riemann ζ-function. The corresponding sum
∑∞
n=1 1/n for the unscreened Coulomb
potential diverges. One has to keep in mind that the total system, i.e., ions plus nanotube, is always
uncharged. Of course, this holds only for nanotubes with metallic conductivity. In this context it is
noteworthy that semiconducting carbon nanotubes may become conductive by the insertion of ions [16].
The asymptotic behavior allows us to estimate which structure is stable for the case of a 2-1 electrolyte
lined up along the axis of the tube. There are two obvious candidates for a divalent anion, univalent cation:
-2 1 -2 1 -2 1 -21 1 1
structure 1 -2 1 -2 structure 1 1 -2 1 1 -2
In order to find out which of the two is more stable, we calculate the following lattice sums, setting
the lattice constant equal to unity: interaction of one anion with all other ions plus interaction of one
cation with all other ions minus the interaction of a neighboring anion-cation pair, the latter to avoid
double counting. The sums are all related to
∑
1/n3. The corresponding values are: −4.9118 for the
1 -2 1 -2 structure, and −3.3423 for the 1 1 -2 1 1 -2 structure. If we restrict ourselves to nearest neighbor
interactions only, the values are −6 for the 1 -2 1 -2 structure, and −5 for the 1 1 -2 1 1 -2 structure.
Thus, the former is more stable, even though the unit cell carries an excess charge. This estimate will be
confirmed by our Monte Carlo simulations.
3. Grand canonical Monte Carlo simulations
For a 1-1 electrolyte, the potential of zero charge is an obvious reference potential. It coincides with
the potential at which both ions have the same electrochemical potential. For an asymmetrical electrolyte
there is no obvious way to determine the potential of zero charge a priori, but the potential at which both
ions have the same electrochemical potential µ0 can still be defined and can serve as the reference. Thus,
we have for our (1,-2) electrolyte:
µ+ = µ0 + e0∆φ, µ− = µ0 − 2e0∆φ. (4)
An important question is how the the nanotube is filled at the reference potential when the electrochemical
potential µ0 changes.With our sign convention [12] increasing µ0 corresponds to increasing ionophilicity.
The results are shown in figure 2. For low values of µ0, the tube is almost empty, the occupation
corresponds to a dilute gas. At a critical value near µ0 ≈ −3.2 eV, the occupation rises rapidly. Just as
in the case of a 1-1 electrolyte in this region, the fluctuations are large, and the transition region has a
finite width, because there is no phase transition in one-dimensional systems. Above µ0 ≈ −3.0 eV the
tube is filled with a solid structure, i.e., a one-dimensional salt. As predicted above, the number of anions
and cations is almost equal, so that the structure corresponds roughly to 1 -2 1 -2. A typical snapshot
from a tube that is almost filled is seen in figure 3. This implies that the negative excess charge is nearly
equal to the number of anions. With an increasing µ0, the number of cations becomes slightly larger at
the expense of the anions, so that the negative excess charge is somewhat reduced. In accord with the
terminology of Lee et al. [9] we call the tubes that are almost empty at µ0 ionophobic, those that are
filled as ionophilic; the transition region between the two cases is small. In a typical experiment, the
ionophilicity, which depends on the material of the tube and on the electrolyte, is kept constant, while the
electrode potential ∆φ is scanned. The corresponding charging curves depend on µ0. We first consider
the case where the tube is strongly ionophobic, choosing µ0 = −4 eV in accord with figure 2. The results
are shown in figure 4. On a positive scan, the tube starts to fill with anions with a certain delay. At first the
number of particles rises rapidly, but then the Coulomb repulsion between the ions becomes noticeable,
33004-3
W. Schmickler, D. Henderson
–4.0 –3.5 –3.0 –2.5 –2.0
0
100
200
300
µ0 / eV
n
u
m
b
e
r
o
f
p
a
rt
ic
le
s
anions
cations
absolute excess charge
Figure 2. (Color online) Number of particles as a function of the ionophilitity.
cations
anions
Figure 3. (Color online) Snapshot of the particle arrangement for a tube that is almost filled. In this
example there is one vacancy for reasons of entropy.
–10 –5 0 5 10
0
200
400
600
∆ φ / V
n
u
m
b
e
r
o
f
p
a
rt
ic
le
s
anions
cations
absolute excess charge
µ0 = −4 eV
Figure 4. (Color online) Filling of the nanotube as a function of potential for an ionophobic tube with
µ0 = −4 eV.
and the rate of filling slows down. On a negative scan, the filling with cations requires a larger absolute
value of ∆φ since they carry only a single unit of charge. Again, at first the tube is filled rapidly; then, the
rate decreases, but not as strongly as in the positive sweep, since the charges on the ions are smaller and
hence the repulsion is weaker. All in all, the behavior is similar to that of an ionophobic tube in contact
with a 1-1 electrolyte [12].
The filling is most interesting if we start with µ0 near the transition region between the gas-like and
the solid-like region. Figure 5 shows the case with µ0 = −3.2 eV. The negative scan is similar to the
previous case with µ0 = −4 eV: On the application of a negative potential, the tube becomes empty
and stays so, till at about ∆φ = −3 V the cations start to enter, at first rapidly, and then with a smaller
rate. The positive scan is much more interesting: At first anions and cations enter in equal numbers, the
33004-4
Charge storage in nanotubes: the case of a 2-1 electrolyte
–4 –2 0 2 4
–200
0
200
400
∆ φ / V
p
a
rt
ic
le
n
u
m
b
e
r
anions
cations
absolute excess charge
Figure 5. (Color online) Filling of the nanotube as a function of potential for an ionophobic tube with
µ0 = −3.2 eV.
tube is filled almost completely, and a one-dimensional solid is formed. With an increasing potential,
the energetics of the cations becomes less and less favorable. Suddenly, cations start to leave and a few
anions follow them.The excess negative charge also drops, but at higher potentials more anions enter,
and the charge rises accordingly. Nevertheless, the anions that enter drag a few cations with them, but the
negative excess charge keeps on rising. A complicated interplay between the two kinds of ions!
A similar complicated behavior is observed for an even more ionophilic tube with µ0 = −2.5 eV (see
figure 6). For ∆φ = 0, the tube is almost completely filled, the number of cations being slightly larger
than that of the anions — this is the same situation that we observed in figure 5. With an increasing ∆φ,
the cations start to leave, at first slowly, till their number suddenly drops to zero. The number of anions
at first stays roughly constant, till their number starts to rise when all the cations have left.
During a negative sweep of ∆φ, all ions at first leave. Then, the number of cations at first rises rapidly,
and then more slowly as the repulsion becomes noticeable. There is a curious small hump in the number
of cations at the potential where the curve for the cations changes its slope. This effect had also been
observed for the 1-1 electrolyte [12].
Finally, we note that figure 4 illustrates an idea of Kondrat and Kornyshev [19]: Energy storage is most
effective in an ionophobic tube, which can just be completely filled at the limits of the experimentally
accessible potential.
–4 0 4 8
0
200
400
600
∆ φ / V
n
u
m
b
e
r
o
f
p
a
rt
ic
le
s
µ0 = −2.5 eV
anions
cations
absolute excess charge
Figure 6. (Color online) Filling of the nanotube as a function of potential for an ionophobic tube with
µ0 = −2.5 eV.
33004-5
W. Schmickler, D. Henderson
4. Conclusions
An asymmetrical 2-1 electrolyte exhibits a number of intriguing features. Firstly, there is no unique
potential of zero charge. For a symmetrical electrolyte, this is the potential where both kinds of ions
have the same electrochemical potential. As shown in figure 2, only for ionophobic tubes with a low µ0,
the tube is uncharged — in fact it is almost empty. However, in this case it is uncharged over a range of
potentials (see figure 4). By contrast, ionophilic tubes are filled with an almost equal number of cations
and anions at ∆φ = 0, and thus carry an excess charge. Ionophilic tubes have an extended range of zero
charge at negative potentials, where the doubly-charged anions are driven out, and the cations do not yet
enter.
Particularly, at positive potentials the ionophilic curves show a complicated filling pattern: At first,
both types of ions are present, and then the counter-ions start to leave. This expulsion tends to set in
rather abruptly, and in this transition region the total charge may even drop slightly with an increasing
potential, giving rise to a negative capacitance.
As we have stated at the beginning, such one-dimensional storage is difficult to realize experimentally,
so our work should be viewed as a contribution to the study of a model system, which has been discussed
quite extensively in the literature.
Acknowledgements
W.S. thanks the University of Science and Technology of China, Hefei, and particularly his host,
Prof. YanXia Chen, for a visiting professorship. In addition he gratefully acknowledges financial support
by the Deutsche Forschungsgemeinschaft (Schm344/48-1), and thanks CONICET for continued support.
A. Technical details
The grand-canonical Monte Carlo method was pioneered by Torrie and Valleau [20]. Our simulations
were performed by a self-written program based on the book by Allen and Tildesley [21]. Each Monte
Carlo cycle consisted of one attempted insertion and one attempted destruction for each kind of particles,
two attempted particle interchanges, and one ordinaryMonte Carlo attempted displacement for each type.
At each potential, we started the simulations with a random distribution of 50 particles of each type. One
run consisted of between 106 and 107 cycles, depending on the fluctuations observed. Results for each
case considered are based on an average of several runs — since there is always a danger of the system
getting stuck in a metastable state, we preferred to average over several runs rather than over one very
long run.
The simulations were performed for a tube of radius 5 a.u. (2.65 Å) and a radius of 2.0 a.u. (1.06 Å) for
both ions. The radius refers to the radius of the effective position of the image charge. The results reported
are for a tube of length 2000 a.u. (1058 Å); thus, the maximum packing is 500 ions. Periodic boundary
conditions were employed in the z-direction along the axis of the ring. We also performed simulations
for tubes of 4000 a.u. length, and observed no differences in the behavior except larger fluctuations.
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Накопичення заряду в нанотрубках: випадок 2-1 електролiту
В.Шмiклер1, Д. Гендерсон2
1 Iнститут теоретичної хiмiї, Унiверситет Ульма, Нiмеччина
2 Вiддiлення хiмiї i бiохiмiї, Унiверситет Брiгхама Янга, Прово, США
Ми розглядаємо 2-1 електролiт в контактi з вузькою нанотрубкою, яка дозволяє лише одновимiрне на-
копичення вздовж осi. Асиметрiя не дозволяє апрiорi означити потенцiал нульового заряду; натомiсть,
природною точкою вiдлiку є потенцiал електрода, бiля якого обидва iони мають той же електрохiмiчний
потенцiал; значення останнього може служити для вимiрювання iонофiльностi. Поблизу цього потенцiа-
лу iонофобнi трубки наповненi розрiдженим газом, iонофiльнi трубки є наповненi одномiрним твердим
тiлом, що мiстить приблизно однакове число двовалентних iонiв i моновалентних контрiонiв, стiйкiсть
структури забезпечується сильним екрануванням кулонiвської взаємодiї iндукованим зарядом на стiнках
трубки. Наповнення трубки шляхом прикладання електродного потенцiалу демонструє складну картину
взаємодiй мiж двома сортами iонiв.
Ключовi слова:Монте Карло симуляцiї, електричний подвiйний шар, iони, ємнiсть, наноцилiндровi
пори
33004-7
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https://doi.org/10.1039/C5NH00004A
https://doi.org/10.1016/0021-9991(77)90121-8
Introduction
Screening of the Coulomb interaction and the formation of a one-dimensional salt
Grand canonical Monte Carlo simulations
Conclusions
Technical details
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