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Takahiro Ichikawa, Takeshi Yamada et al.
Surface proton hopping conduction mechanism dominant
polymer electrolytes created by self-assembly of
bicontinuous cubic liquid crystals

Volume 15
Number 19
21 May 2024
Pages 6981–7360

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Surface proton hopping conduction mechanism
dominant polymer electrolytes created by selfassembly of bicontinuous cubic liquid crystals†
Takahiro Ichikawa, *a Takeshi Yamada,Kaori Sudaa and Tsubasa Kobayashi a*b Nanami Aoki,a Yuki Maehara,a

For the development of the next generation of fuel cells, it is essential to create an innovative designprinciple of polymer electrolytes that is beyond extension of the existing strategy. In the present study,
we focused on the surface hopping proton conduction mechanism where an activation energy for
proton conduction is greatly reduced by decreasing the distance between SO3− groups. Our gyroid
nanostructured polymer ﬁlm (Film-G), with a hydrophilic surface where the SO3− groups are aligned
densely and precisely, shows high proton conductivity of the order of 10−2 S cm−1 when the water
Received 20th February 2024
Accepted 8th April 2024

content is about 15 wt%. We reveal that the high proton conductivity of Film-G is attributed to theexhibition of an extremely-fast surface hopping conduction mechanism due to the reduced activation

DOI: 10.1039/d4sc01211a

energy barrier along the gyroid minimal surface. This ﬁnding should introduce a game-changing novel

rsc.li/chemical-science

opportunity in polymer electrolyte design.

Introduction
Naon, which was developed in the late 1960s by DuPont, issurely the most successful polymer electrolyte because of its
advanced proton conductivity and high mechanical stability.1–3
Signicant eﬀorts have been made for several decades todevelop proton conductive polymer electrolytes beyond Naon;4
however, Naon still reigns as the most commonly used polymer electrolyte membrane for fuel cells and some other applications.3 Proton conduction in Naon can be understood using
a simple 2D nanochannel model composed of a water nanochannel sandwiched between hydrophobic walls (Fig. 1a).5,6
There are three proton conduction mechanisms in the nanochannel. One is the Grotthuss mechanism,7 which is a mechanism to transport protons via the formation/cleavage ofhydrogen-bonding networks of water molecules in the bulk
water region, namely a water pool, in the center of the nanochannel. The second is the en masse diﬀusion mechanism,
which is the mechanism whereby hydrated oxonium ionsmigrate in the water pool. The third one is the surface proton
hopping conduction (SPHC) mechanism where there are proton
hops (PHs) from an SO3− site on the surface of the hydrophobic
wall to the neighboring SO3− site via bridging water moleculesaDepartment of Biotechnology and Life Science, Tokyo University of Agriculture andTechnology, Naka-cho, Koganei, Tokyo, 184-8588, Japan. E-mail: t-ichi@cc.tuat.ac.jpb

Neutron Science and Technology Center, Comprehensive Research Organization for
Science and Society, Tokai, Ibaraki, 319-1106, Japan† Electronic supplementary information
https://doi.org/10.1039/d4sc01211a

7034 | Chem. Sci., 2024, 15, 7034–7040(ESI)

available.

See

DOI:

between them. The proton conductivities
in each mechanismP
are represented by sG, sE, and s , respectively. The total proton
conductivityof the polymer electrolyte membranes is sG + sE +P
s . Because of the signicantly high proton conductivity in the
Grotthuss mechanism, it has been common sense that theconstruction of macroscopically-continuous water pools, toFig. 1 (a) Schematic representation of a proton conductive waternanochannel with a thickness of several nanometers. Grotthuss
conduction (sG) is the dominant proton conduction mechanism. It isa useful model for conventional proton conductive polymer electrolytes, such as Naﬁon. (b) Schematic representation of a proton
conductive water nanosheetwith a thickness of <1 nm. Surface protonPhopping conduction (s ) is the dominant conduction mechanism.
Electrostatic potentials in the plane parallel to the hydrophobic wall
surface are described.

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enhance the contribution of sG, is the most important designprinciple for creating advanced proton conductive polymer
electrolytes.
Here, we propose a totally new design principle departingfrom the common sense idea for the creation of proton
conductive polymer electrolytes showing high proton conductivity. We focused on the theoretical foresight on the SPHC
mechanism. In 2001, Eikerling et al. revealed that distances
between the neighboring SO3− groups (lSO3−–SO3−) on a hydrophobic wall surface strongly inuence the electrostatic potential
barrier in a direction parallel to the hydrophobic wall surface, j(Fig. 1a).5 According to their estimates, when lSO3−–SO3− = 15 Å, j
is about 30 kJ mol−1 while it reduces exponentially to 2–
3 kJ mol−1 as lSO3−–SO3− decreases to 5 Å. Since lSO3−–SO3− values inNaon are relatively large at approximately 12 to 20 Å,3,5 the P
j inNaon is considered to be very large and consequently the s ofNaon appears as a very small value. These insights have led
many researchers to consider the SPHC mechanism to be trivial;
however, we envision that the dense and highly periodic alignment of SO3− sites on a continuous surface should enable anultrafast SPHC mechanism with an extremely small electrostatic
potential barrier, resulting in the exhibition of a high proton
conductivity on the order of 10−2 to 10−1 S cm−1, that is,comparable to sG in Naon and/or bulk water (Fig. 1b).
Recently, the number of studies on the design of advanced
proton conductive materials focusing on the alignment of theSO3− groups has been increasing.8–15 Moreover, there are somestudies that focus on the importance of the SPHC mechanism.16,17 There are also some studies that have employed thepotential utility of zwitterions as ion conductive materials.18–20
Even bearing these previous studies in mind, the present workis the rst study in which high proton conductivity is induced
solely by the SPHC mechanism with an extremely-small activation energy.

Chemical Science(a) Molecular structure of amphiphilic zwitterion monomer GZ.
(b) Preparation scheme for a gyroid nanostructured polymer ﬁlm(Film-G). (c) A photograph of Film-G and 3D image of the gyroid
nanostructure in Film-G. (d) 3D image of a cubic cell in Film-G. A waternanosheet is constructed along a gyroid minimal surface (G-surface)shown in blue. The positions of the SO3− groups of GZ are displayed asorange spheres.
Fig. 2

We expect that SPHC mechanisms in Film-G with lSO3−–SO3− =5.9 Å can be roughly classied into four types depending on the
degree of hydration of the SO3− groups (Fig. 3a–d). In the driest
state, SPHC is composed only of very slow PHs due to the strong

Results and discussion
Aiming for the construction of polymer electrolytes with a denseand highly periodic alignment of SO3− sites for the SPHC
mechanism, we focused on our previously-designed liquidcrystalline (LC) amphiphile GZ (Fig. 2a).21 GZ has an ability toco-organize into bicontinuous cubic (Cubbi) phases with someacids, such as bis(triuoromethane)sulfonimide (HTf2N)(Fig. 2b) and a small amount of water. In the Cubbi phase, all of
the SO3− groups of GZ are aligned on a 3D gyroid minimal
surface (G-surface) in a cubic cell with a lattice parameter(LCubic) of approximately 9–10 nm. A signicant point is that it ispossible to preserve its unique molecularly assembled nanostructure by in situ polymerization, which yields a self-standing
polymer lm, Film-G, with gyroid nanostructures (Fig. 2c). We
roughly calculated lSO3−–SO3− in Film-G containing X wt% ofwater (Film-G(X)) based on structural analysis and its physicochemical properties (Fig. S2†). lSO3−–SO3− in Film-G(16.4) was
estimated to be roughly 5.9 Å (Fig. 2d and S3†), which is
predominantly shorter than that in Naon. This is a value forwhich one can expect the exhibition of an ultrafast SPHC
mechanism with a very small electrostatic potential barrier, j.

© 2024 The Author(s). Published by the Royal Society of Chemistry

Schematic illustration of the SPHC mechanism on the Gsurface in Film-G with various water contents: (a) in the driest state; (b)in low water content; (c) in moderate water content; and (d) in highwater content.

Fig. 3Chem. Sci., 2024, 15, 7034–7040 | 7035

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electrostatic interaction between the SO3− anion and a proton(Fig. 3a). In the low-water-content state, these PHs are slightly
accelerated due to the hydration of the SO3− groups that largely
weakens SO3−–proton interactions, while the PH speed is stillslow because lSO3−–SO3− = 5.9 Å is a large value for a proton to hop
directly (Fig. 3b). A further increase in water content leads to thesituation where two hydrated SO3− groups are connected via
bound water molecules. PHs via these hydrated SO3− groups are
greatly accelerated, while PHs via those with fewer bound watermolecules are still slow (Fig. 3c). When the water content reaches a situation where all the SO3− groups are connected bybound water molecules (Fig. 3d), all PHs are highly activatedand an eﬀective SPHC mechanism is enabled at a macroscopic
scale. This expectation is consistent with the dependence of
proton conductivity (sH+) of Film-G on water content that wasreported in our previous study.21
When the SPHC is dominant in proton conductivity, its sH+
can be described by
sHþ ¼

3i F 2  S S 
D þC þ
q RT H H(1)

where F is Faraday's constant, R is the universal gas constant,
DH+S (cm2 s−1) and CH+S (mol cm−3) are the diﬀusion coeﬃcientand the concentration of H+ on the surface, T (K) is temperature,3i (dimensionless) is the volume fraction occupied by water, andq (dimensionless) is a tortuosity factor. Detailed discussion
about these parameters is provided in the ESI.† We will discuss
below whether the high sH+ of Film-G can be explained solely by
the SPHC mechanism.
According to the view of random walk diﬀusion, DH+S is givenby the Einstein–Smoluchowski equation22,23DSHþ ¼

lS2
ksSD

(2)

where k is a constant value that depends on the dimensionalityof random proton hops, lS is the hopping length, and sSD is themean jump time for a PH. sSD can be expressed as e;0 DGS
(3)
sSD ¼ v0 1 exp
kB T
where v0 is the thermal frequency, kB is the Boltzmann constant,and DGe,0
S is the eﬀective Gibbs free energy of activation forkB T
surface hops. v0 is expressed as, where h is Planck's
he,0
constant. The main component of DGS is the potential barrierin the SPHC mechanism.5 Assuming that the proton conductionin Film-G(X) is solely based on the SPHC mechanism, DGe,0
S wasestimated to be 13.3 kJ mol−1 for Film-G(15.2) using theexperimental value of sH+ = 8.5 × 10−3 S cm−1 (Tables S4 and−1S5†). To conrm the validity of DGe,0
S = 13.3 kJ mol , weexamined the activation energy (Ea) of Film-G(X). The temperature dependences of sH+ of Film-G(15.2), Film-G(9.9), and FilmG(6.0) are shown in Fig. 4 using an Arrhenius plot. The
measurements were performed in a narrow temperature rangeof 30 to 40 °C under a constant relative humidity of 90% withthe aim of reducing the change in water content during themeasurements. Ea was found to be 17.2 kJ mol−1 for Film-

7036 | Chem. Sci., 2024, 15, 7034–7040

Fig. 4 Experimentally-obtained sH+ values of Film-G(15.2), FilmG(9.9), and Film-G(6.0) plotted against the reciprocal of absolutetemperature.

−1
G(15.2), which was close enough to DGe,0
S = 13.3 kJ mol . Since
there is a small increase in the water content in Film-G(15.2)
upon heating from 30 to 40 °C when the sample is kept under
RH = 90% due to the increase in the absolute humidity, Ea =17.2 kJ mol−1 is expected to be an overestimate. The exact valueof Ea should be slightly smaller than 17.2 kJ mol−1, meaning−1that our estimate of DGe,0is very reasonable.
S = 13.3 kJ mol
These results strongly support the validity of our assumptionthat the SPHC mechanism is solely-dominant for the proton
conduction mechanism in Film-G(15.2). Conversely, larger
values of Ea = 26.8 and 35.3 kJ mol−1 were found for Film-G(9.9)
and Film-G(6.0), respectively, with lower water contents. Thiscan be explained by the situation, where the SO3− groups are
partially bridged with bound water molecules, as shown in
Fig. 3c, and some PHs with slower speed exist along the Gsurface. This indicates that a suitable amount of water molecules is required to exclude slow PHs and yield only fast PHs,
leading to the exhibition of high proton conductivity at
a macroscopic scale.
Here, we compare the estimated physical property values ofFilm-G with those of Naon. Using the value of DGe,0=S
13.3 kJ mol−1, the DH+S of Film-G(15.2) is calculated to be 6.99 ×
10−6 cm2 s−1, which is 70 times larger than that in Naon.6 The
ultrafast SPHC in Film-G(15.2) is ascribed to the formation ofa considerably reduced electrostatic potential barrier on the Gsurface by the correlated barrier hopping model24 that was
realized for the rst time by creating a unique situation, where
the SO3− groups are placed on a continuous surface in a denseand highly-periodic manner.
The experimental results on the water composition dependence of the sH+ of Film-G indicated that it is necessary to
introduce a suitable amount of water molecules on the SO3−aligned surface to enable the SPHC mechanism with a reduced
j. We expected that the elucidation of the water state, dynamics
and their composition dependence should be a signicant way
to understand the proton conduction mechanism in Film-G(X).
The state of water in Film-G(X) was examined by DSC
measurements (Fig. 5 and S11†). The DSC thermograms oncooling are shown in Fig. 5. There is no exothermic peak corresponding to the crystallization of water when the watercontent X is 17.3 wt% or less, while a small exothermic peak is© 2024 The Author(s). Published by the Royal Society of ChemistryView Article Online

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DSC thermograms of Film-G(X) on cooling. X is the watercontent in the ﬁlm (wt%).

Fig. 5observed when X = 22.2 wt%. These results clearly suggest thatthe water molecules in Film-G(X) exist as non-freezing boundwater at least when X # 17.3, suggesting that the ultrafast SPHC
mechanism in Film-G(15.2) is assisted only by non-freezingbound water.
With the aim of further investigating the dynamics of waterin Film-G(X), we focused on quasi-elastic neutron scattering(QENS) measurements. QENS is a powerful technique forinvestigating the dynamics of water molecules in the picosecond to nanosecond timeframe and Ångstrom length-scaledue to the large incoherent scattering cross-section ofhydrogen. It has been employed for various materials includingNaon to reveal the dynamics of water molecules in it.25,26
Unlike Naon that is mainly composed of C, O, S, and F
elements, Film-G contains many hydrogen atoms. Thus, toobtain information on the dynamics of the water molecules in
Film-G, we planned to examine the diﬀerence between the
QENS spectra of Film-G containing normal water (H2O) andthose of Film-G containing heavy water (D2O). Two types ofFilm-G samples with moderate water content (9.0 wt% (FilmG(9.0))) and high water content (16.4 wt% (Film-G(16.4))) wereprepared by putting them under controlled relative humidity
conditions. Resultant polymer lms contain 3.0 and 6.0 watermolecules per SO3− group, respectively. Below they aredescribed as Film-G/mH2O (m = 3 and 6). Using D2O, Film-G/
mD2O (m = 3 and 6) were also prepared.
QENS experiments were carried out for these four samples at
300, 280, 260, and 240 K. The same experiment was performed
at 10 K for resolution. Contour maps of QENS data of Film-G/
3H2O and Film-G/3D2O at 300 K are shown in Fig. 6a and b. Xand Y-axes are momentum transfer (Q) and energy transfer (DE),respectively. DE reects the quantity of energy exchange
between an incident neutron and the sample. If a sample showsa relaxation phenomenon, a broad peak is observed as QENS
around the elastic position (DE = 0). The width of the QENS
broadening is inversely proportional to the relaxation time.27
There is a strong sharp scattering intensity at DE = 0, anda moderate broad scattering intensity centered at DE = 0,
indicating that elastic and quasi-elastic components exist in thelms. To extract the information on water molecules in Film-G,
a diﬀerential contour map between the two was created usinga scaling factor derived from the weight ratio of the two samples
in the dry state (Fig. 6c). In the diﬀerential contour map,

© 2024 The Author(s). Published by the Royal Society of ChemistryFig. 6 Contour maps of QENS data of (a) Film-G/3H2O and (b) FilmG/3D2O at 300 K. (c) A diﬀerential contour map between Film-G/3H2O
and Film-G/3D2O. Contour maps of QENS data of (d) Film-G/6H2O
and (e) Film-G/6D2O at 300 K. (f) A diﬀerential contour map between
Film-G/6H2O and Film-G/6D2O.

a signicant diﬀerence is found mostly only in the DE =0 region, indicating that the dynamics of the water molecules in
Film-G/3H2O have almost stopped or are slower than thetimescale represented by the instrumental energy resolution.
Conversely, completely diﬀerent results were obtained when m= 6. The contour maps of the QENS data of Film-G/6H2O andFilm-G/6D2O and their diﬀerential data are shown in Fig. 6d–f.
It is noteworthy that a broad scattering intensity was found in
Fig. 6f, indicating that there is a certain amount of mobile watermolecules in Film-G/6H2O.
To evaluate the dynamics of the water molecules in Film-G
more quantitatively, we analyzed the QENS proles of Film-G/
mD2O and Film-G/mH2O (m = 3 and 6). The QENS proles of m
= 3 were integrated in the Q direction due to the small contribution from hydrated water. The obtained results were tted by
the following phenomenological equation()
nXSðQ; EÞ ¼ A0 dðEÞ þ
Ai Li ðGi ; EÞ 5RðQ; EÞ(4)i¼1

where A0d(E) denotes the elastic component with coeﬃcient A0
and delta-function d(E), Ai is the coeﬃcient of the i-th
Lorentzian-function, Li(Gi,E), where Gi is the half width at half
maximum (HWHM) of the Lorentzian-function, and R(Q,E)
refers to the instrumental resolution function, which was obtained at 10 K. Fig. 7a and b show tting results for the QENS
prole of Film-G/3D2O and Film-G/6D2O at T = 300 K. It is
successfully tted with the sum of a delta-function and two
Lorentzian-functions (n = 2 in eqn (4)), which are drawn withlines in red, green, and blue, respectively. Since D2O hardly
contributes to the QENS prole, the observed QENS broadeningreects the dynamics of Film-G, namely the alkyl chain motionof GZ (Fig. S16–S18†). In contrast, the QENS prole of Film-G/
mH2O (m = 3 and 6) includes dynamics of not only Film-G but
also mH2O. This diﬀerence between Film-G/mH2O and Film-G/
mD2O (Fig. 6c and f) leads us to consider that an additional
Lorentzian-function, representing the mobile water molecules,
is required for tting of the QENS prole of Film-G/mH2O.

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Fig. 7 (a) Fitting of the observed QENS proﬁle of Film-G/3D2O withthe sum of a delta function and two Lorentzian-functions convoluted
with the resolution function. (b) Fitting of the observed QENS proﬁle ofFilm-G/6D2O at Q = 0.78 Å−1 and 300 K with the sum of a delta
function and two Lorentzian-functions convoluted with the resolutionfunction. (c) Fitting of the QENS proﬁle of Film-G/3H2O when theparameters of two Lorentzian-functions were ﬁxed by the results ofFilm-G/3D2O. (d) Fitting of the QENS proﬁle of Film-G/6H2O at Q =0.78 Å−1 and 300 K when the parameters of two Lorentzian-functions
were ﬁxed by the results of Film-G/6D2O.

Therefore, it was tted by eqn (4) with three Lorentzianfunctions (n = 3), where parameters L1 and L2 corresponding
to Film-G were xed using the results of Film-G/mD2O, and thethree parameters A0, A3, and G3 were adjusted. This approach
substantially reproduced observed data (Fig. 7c and d).
In the case of m = 3, the number of immobile and mobilehydrogen atoms (NH(immob) and NH(mob)) in Film-G/3H2O wasestimated as follows,NH ðimmobÞ ¼

A0 ðFilm-G=3H2 OÞ  A0 ðFilm-G=3D2 OÞ
ðA0 ðFilm-G=3H2 OÞ  A0 ðFilm-G=3D2 OÞ þ A3 Þ 12(a) Number of immobile and mobile hydrogen atoms (NH(immob)
and NH(mob)) in 3H2O. (b) Q dependence of EISF of 6H2O in Film-G/
6H2O. The solid lines are results ﬁtted with eqn (8). (c) Temperaturedependence of the radius of the conﬁned sphere obtained from theﬁtted results of the EISF (b). (d) Number of immobile and mobilehydrogen atoms (NH(immob) and NH(mob)) in 6H2O.
Fig. 8the Q dependence of the parameter L3 was examined to extract
spatial information about the observed dynamics. The Q
dependence of G3 at 300 K did not approach zero at Q = 0 Å−1(Fig. S20†). Although obtained data was scattered, G3 was foundto be almost constant with respect to Q. These results suggestthat the observed dynamics corresponding to L3 is a local mode,such as rotation and/or local jumps of the water molecules. A
similar trend was also observed at other temperatures. Thismeans the scale of the localized mode is within a scale of severalÅngstroms. For further consideration, the Q dependence of theelastic incoherent scattering factor (EISF) was calculated asfollows,EISFðQÞ ¼

A0 ðFilm-G=6H2 OÞ  A0 ðFilm-G=6D2 OÞ(7)
ðA0 ðFilm-G=6H2 OÞ  A0 ðFilm-G=6D2 OÞ þ A3 Þ

(5)NH ðmobÞ ¼

A3
ðA0 ðFilm-G=3H2 OÞ  A0 ðFilm-G=3D2 OÞ þ A3 Þ
 12

(6)

where 12 indicates the number of hydrogen atoms in the watermolecules per GZ molecule. The obtained NH(immob) and NH(mob)are plotted in Fig. 8a. It is notable that immobile water
predominates in Film-G/3H2O. In the case of m = 3, thehydration degree on the G-surface is expected to be the situation
shown in Fig. 3b or c. Therefore, many of the water moleculesare xed on the SO3− groups, while some of them near H+ areuctuating.
In contrast, in the case of m = 6, since it was revealed thata certain amount of the water molecules are in the mobile state,7038 | Chem. Sci., 2024, 15, 7034–7040

The Q dependence of EISF, as plotted in Fig. 8b, was well
reproduced by a diﬀusion model in the conned sphere,28 asfollows,
2
3j1 ðQrÞ
(8)EISFðQÞ ¼ ð1  fmob Þ þ fmob
Qr
where fmob, r, and j1 are the mobile fraction of hydrogen corresponding to water, the radius of the conned sphere, anda rst-order spherical Bessel function, respectively. The value ofr was 4.2 Å at 300 K (Fig. 8c), which covers enough neighboringsites of SO3− (lSO3−–SO3−). The space decreased with decreasing
temperature. NH(immob) and NH(mob) were estimated from 24 ×
(1 − fmob) and 24 × fmob, respectively (Fig. 8d). In contrast toFilm-G/3H2O, the mobile water predominates in Film-G/6H2O.
In the case of m = 6, the degree of hydration reaches the© 2024 The Author(s). Published by the Royal Society of ChemistryView Article Online

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manuscript. All authors have given approval to the nal versionof the manuscript.

Conﬂicts of interest
There are no conicts to declare.

Acknowledgements
Fig. 9 Relaxation time of mobile water molecules (sQENS) estimatedfrom the HWHM of L3 in 6H2O. sSD values of Film-G(15.2) at various
temperatures are also shown in the yellow bar graph.

situation shown in Fig. 3d, where the presence of H+ causes not
only the near water molecules to wobble but also distant onesconnected through hydrogen-bonding networks, which consequently puts a large part of the water molecules in a mobile
state.
The relaxation time of these mobile water molecules (sQENS)
was evaluated using the following formula,sQENS ¼ ħ

1
G3

(9)

where ħ is h/2p. The value sQENS was estimated to be 37 ± 4 ps at
300 K, which is of the same order as sSD calculated when−1
assuming DGe,0(Table S5†). Consistency of
S = 13.3 kJ mol
S
theoretical sD and experimental sQENS is also found for results atother temperatures (Fig. 9). These results suggest that the
dynamics of the mobile water molecules are deeply involved in
the SPHC mechanism.

Conclusions
We have succeeded in demonstrating the potential utility of thesurface proton hopping conduction (SPHC) mechanism for thedevelopment of advanced proton conductive polymer electrolytes. These ndings will contribute to the development ofadvanced fuel cells, such as anhydrous-drive-type and/or lowtemperature-drive-type ones. From a comprehensive viewpoint, present ndings suggest that control over selforganization should be a key strategy for creating new functional materials that support next-generation technologies.

Data availability
The datasets supporting this article have been uploaded as partof the ESI.†

Author contributions
T. I. supervised this project. T. I. and T. K. designed materials. T.
I., N. A., Y. M., K. S., T. K., and T. Y. performed the experiments
and analyzed the data. T. I. and T. Y. wrote the manuscript. All
authors discussed the results and commented on the© 2024 The Author(s). Published by the Royal Society of ChemistryThis work was supported by the Japan Science and Technology
Agency (JST) FOREST (JPMJFR223C). This work was also supported by JSPS KAKENHI numbers JP21H02010, JP22H04526,
and JP23K17937 from the Japan Society for the Promotion of
Science. This work was partially supported by Nippon Sheet
Glass Foundation for Materials Science and Engineering. The
QENS experiments at the Materials and Life Science Experimental Facility of the J-PARC was performed under a user
program (Proposal No. 2022A0005 and 2023A0072).

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