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Molecular Dynamics Analysis of Hydroxide Ion Transport Mechanism in Anion Exchange Membrane

Author: Ono, Ryusei; Suzuki, Hiroto; Miyatake, Kenji; Tokumasu, Takashi
Publisher: Zenodo
DOI: 10.5281/zenodo.17244122
Source: https://zenodo.org/records/17244122/files/EFCF-2025_Paper_A0804_10860_IonTransport-AEM-MD_Ono_Ryusei_01.pdf
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A0804
Molecula Dynamics Analysis o Hyd oxide Ion
T anspo Mechanism in Anion Exchange Memb ane
Ryusei Ono* (1,2), Hi o o Suzuki (1,2), Kenji Miya ake (3), Takashi Tokumasu (2)
(1) G adua e School o Enginee ing, Tohoku Uni e si y, Sendai/Miyagi/Japan;
(2) Ins i u e o Fluid Science, Tohoku Uni e si y, Sendai/Miyagi/Japan;
(3) Clean Ene gy Resea ch Cen e , Uni e si y o Yamanashi, Ko u/Yamanashi/Japan;
*Con ac co esponding au ho s: www.EFCF.com/Con ac Reques
Abs ac
In o de o add ess global clima e change, clean ene gy sou ces ha educe CO2 emissions
mus be u gen ly de eloped. Hyd ogen has eme ged as a p omising ene gy ca ie due o
i s high ene gy densi y and en i onmen ally iendly combus ion, p oducing only wa e .
Among hyd ogen p oduc ion echnologies, g een hyd ogen, which is p oduced ia wa e
elec olysis powe ed by enewable elec ici y, is a key solu ion o deca boniza ion. In
pa icula , anion exchange memb ane wa e elec olysis is conside ed a cos -e ec i e
al e na i e because i allows he use o non-p ecious me al ca alys s and o e s he po en ial
o ope a ion wi h pu e wa e . Howe e , he ela i ely low conduc i i y o hyd oxide ions in
anion exchange memb ane (AEM) is a majo challenge in imp o ing sys em pe o mance.
To in es iga e hyd oxide ion anspo mechanisms a he molecula scale, his s udy
employs classical molecula dynamics (MD) simula ions using QPAF-4, an AEM con aining
pendan ime hylammonium g oups. Focusing on he ehicle mechanism, we explo e how
changes in wa e con en s (λ) a ec hyd oxide ion di usion. Ou MD simula ion e eals ha
he di usion coe icien o hyd oxide ion inc eases apidly wi h wa e con en a low λ,
pa icula ly be ween λ = 6 and 9, bu shows a mo e g adual ise a highe λ alues. This
beha io is conside ed o be in luenced by s uc u al changes in he sol a ion s uc u e
a ound he ime hylammonium g oups.
We de ined a “sol a ion numbe ” o quan i y he numbe o i s sol a ion shells su ounding
each hyd oxide ion and ca ego ized he local s uc u es acco dingly. A low λ, hyd oxide ions
we e p edominan ly apped in o e lapped a eas wi h s ong elec os a ic in e ac ions,
whe eas a high λ, hey mainly esided in isola ed a eas o he second sol a ion shell,
exhibi ing highe mobili y.
These esul s sugges ha inc eased wa e con en inc eases he dis ance be ween
ime hylammonium g oups and dec eases he o e lap o sol a ion shells, hus weakening
ion apping and p omo ing di usion. Ou esul s p o ide basic insigh s in o he ela ionship
be ween sol a ion s uc u e and ion anspo . These insigh s o e guidelines o he a ional
design o high-pe o mance AEM ma e ials o nex -gene a ion wa e elec olysis sys ems.
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In oduc ion
In esponse o he Pa is Ag eemen , many coun ies ha e accele a ed deca boniza ion
e o s, se ing goals such as ca bon neu ali y by 2050 and emission educ ions by 2030.
Majo economies, including he EU and he U.S., ega d enewable ene gy and g een
hyd ogen as essen ial o sus ainable g ow h. Hyd ogen, wi h i s high ene gy densi y and
ze o CO₂ emissions upon combus ion, has a ac ed a en ion as a nex -gene a ion ene gy
ca ie [1]. Howe e , mos cu en hyd ogen p oduc ion me hods ely on ossil uels o
biomass, esul ing in CO₂ emissions. In con as , g een hyd ogen p oduced by wa e
elec olysis powe ed by enewable elec ici y is a key echnology o ealizing a
deca bonized socie y [2]. Elec olyze s no only u ilize su plus enewable elec ici y bu also
unc ion as e ec i e ene gy s o age de ices [3], [4], [5].
Elec olyze s a e classi ied by memb ane ype in o p o on exchange memb ane wa e
elec olysis (PEMWE) and anion exchange memb ane wa e elec olysis (AEMWE).
PEMWE enables high-pu i y hyd ogen p oduc ion and apid dynamic esponse. Howe e , i
ope a es unde acidic condi ions, which equi e he use o expensi e p ecious me al
ca alys s. This subs an ially inc eases he o e all sys em cos . AEMWE, in con as ,
ope a es unde alkaline condi ions, enabling he use o non-p ecious me als and signi ican
cos educ ion [6]. Addi ionally, AEMWE can be ope a ed wi h low-concen a ion alkaline
solu ions and is expec ed o enable pu e-wa e ope a ion, imp o ing bo h sa e y and cos -
e ec i eness. Consequen ly, AEMWE is conside ed o be a p omising echnology o nex -
gene a ion wa e elec olysis [7], [8].
A ypical wa e elec olyze consis s o a s ack o uni cells, each con aining a memb ane
elec ode assembly (MEA) comp ising an anion exchange memb ane (AEM), ca alys laye s,
and gas di usion laye s [3]. In AEMWE, wa e is educed a he ca hode o p oduce
hyd ogen and hyd oxide ions (Eq. 1). The gene a ed hyd oxide ions mig a e h ough he
AEM o he anode, whe e hey a e oxidized o o m oxygen, wa e , and elec ons (Eq. 2).
The o e all wa e -spli ing eac ion is summa ized as ollows (Eq. 3):
ca hode:4H2O+4e−→2H2+4OH−(1)
anode:4OH−→O2+2H2O+4e−(2)
o al:2H2O→2H2+O2(3)
The e iciency o hyd ogen p oduc ion in AEMWE is di ec ly go e ned by hyd oxide ion
anspo h ough he AEM. The e o e, high hyd oxide ion conduc i i y is a c i ical
equi emen o memb ane de elopmen .
The ionic conduc i i y o AEMs is s ongly dependen on in e nal wa e con en . Al hough
hyd a ion enhances conduc i i y, elec o-osmo ic d ag o en leads o dehyd a ion nea he
ca hode. O e -hyd a ion can comp omise mechanical s eng h, while insu icien hyd a ion
lowe s ion anspo pe o mance. Thus, op imal wa e managemen is essen ial. Ope a ing
condi ions such as empe a u e and p essu e also signi ican ly in luence wa e dis ibu ion
in he memb ane. Molecula dynamics (MD) simula ions ha e been widely employed o
in es iga e such in e nal wa e dynamics.
Despi e ecen ad ances, hyd oxide ion conduc i i y in AEM emains lowe han p o on
conduc i i y in p o on exchange memb ane (PEM). Fo example, Na ion® PEM exhibi s
abou 100 mS/cm a 25 °C when ully hyd a ed in wa e , whe eas Tokuyama A201 AEM
shows abou 46 mS/cm a 22 °C unde simila ly ully hyd a ed condi ions in wa e [9], [10].
All-a om MD simula ions ha e been conduc ed o compa e he di usion coe icien s o ions
in polyme sys ems sha ing he same backbone, bu wi h di e en ion-exchange g oups—
sul onic acid (PEM) and qua e na y ammonium (AEM). These simula ions showed ha
hyd oxide ion di usion in AEMs co esponds o only 6–11% o he hyd onium ion di usion
in PEMs [11], [12]. These indings highligh ha enhancing ion anspo e iciency wi hin
AEMs is a key challenge o imp o ing he o e all pe o mance o AEMWE.
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Al hough AEM has been ac i ely s udied, he ela ionship be ween hyd oxide ion anspo
mechanisms and memb ane s uc u e emains insu icien ly unde s ood [13]. Two main
anspo mechanisms ha e been p oposed: he ehicle mechanism, in ol ing he physical
mig a ion o hyd oxide ions, and he G o huss mechanism, in ol ing p o on hopping h ough
hyd ogen-bonded ne wo ks [14], [15].
P e ious s udies ha e demons a ed ha wa e con en c i ically a ec s he dominan
mechanism. Ab ini io simula ions by Tama e al. showed ha he ehicle mechanism
p e ails unde low hyd a ion (λ = 2–4), while insu icien hyd a ion supp esses ion mobili y
[16]. In con as , mul iscale eac i e molecula dynamics simula ions by Chen e al. indica ed
ha a high hyd a ion (λ = 14), o e lapping sol a ion shells acili a e ehicle- ype anspo
ia con inuous wa e channels [17]. Foglia e al., using neu on sca e ing, ound a ansi ion
om G o huss o ehicle mechanism wi h inc easing hyd a ion [18]. These indings sugges
ha hyd oxide ion anspo is s ongly go e ned by bo h wa e con en and memb ane
mo phology.
In his s udy, we ocus p ima ily on he ehicle mechanism o elucida e he unde lying ion
anspo beha io wi hin AEMs. Di ec obse a ion o such nanoscale anspo phenomena
emains challenging wi h con en ional expe imen al echniques. While quan um chemical
me hods o e high accu acy, hey a e compu a ionally cos ly and imp ac ical o la ge
polyme sys ems. Classical MD simula ions, in con as , p o ide an e icien means o
analyzing s uc u al and dynamical p ope ies o memb anes ac oss a ious condi ions.
By applying MD simula ions o a model AEM sys em, his s udy aims o e eal he s uc u al
ac o s go e ning hyd oxide ion mobili y. The ob ained insigh s will suppo he a ional
design o AEM ma e ials wi h imp o ed pe o mance o sus ainable hyd ogen p oduc ion.
1. Simula ion me hods
1.1 Simula ion De ails
In his s udy, we conduc ed classical MD simula ions using he La ge-scale
A omic/Molecula Massi ely Pa allel Simula o (LAMMPS) o in es iga e he anspo
beha iou s o hyd oxide ions in AEM, wi h a pa icula ocus on he ehicle mechanism. As
he memb ane model, we adop ed he QPAF-4 polyme , which con ains ime hylammonium
unc ional g oups and exhibi s high hyd oxide conduc i i y, excellen mechanical s eng h,
and s abili y unde alkaline condi ions (Figu e 1) [19], [20], [21]. The polyme s uc u e and
cha ge assignmen we e based on quan um chemical calcula ions. We se he deg ee o
polyme iza ion o m = 25 and n = 15, co esponding o an ion exchange capaci y o 1.47
meq/g. This composi ion showed op imal hyd oxide conduc i i y (47.8 mS/cm) and
mode a e wa e up ake (105%) in expe imen s [19].
The simula ion sys em comp ised se en QPAF-4 chains andomly placed in a pe iodic cubic
box (600 × 600 × 600 ų). Each chain con ained 30 ime hylammonium g oups, esul ing in
Figu e 1: Molecula s uc u e o he QPAF-4 polyme [20]
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a o al o 210 g oups, and he same numbe o hyd oxide ions was added o cha ge
neu ali y. Wa e molecules we e in oduced o achie e wa e con en s o λ = 3, 6, 9, 12, 15,
and 18, whe e λ is de ined as he a io o he o al numbe o wa e molecules and hyd oxide
ions o he numbe o ime hylammonium g oups.
The sys em pe o med a se ies o he mal annealing ea men s o emo e a i icial
me as able s a es esul ing om he complex polyme s uc u e and andom ini ial
con igu a ions. Fi s , we pe o med ene gy minimiza ion, ollowed by high- empe a u e NPT
comp ession a 800 K and 200 ba , which we g adually educed o 300 K and 1 ba . Then,
we conduc ed 10 cycles o he ollowing wo s eps o u he equilib a e he sys em: (1) NVT
simula ion a 800 K o 100 ps, and (2) NPT simula ion wi h cooling om 800 K o 300 K
unde 1 ba o 100 ps. This p ocess enabled he sys em o o e come local ene gy minima
and s abilize. We hen pe o med a 5 ns equilib ium simula ion unde he NPT ensemble a
300 K and 1 ba . We used he esul ing s uc u e o pe o m p oduc ion uns unde he NVT
ensemble a 300 K wi h a 1 s imes ep, sa ing snapsho s e e y 1 ps o e a o al o 10 ns.
1.2 Analysis De ails
To e alua e he local s uc u e, we used he adial dis ibu ion unc ion (RDF), which gi es
he p obabili y o inding a ype B pa icle a a dis ance om a e e ence ype A pa icle:
𝑔𝐴−𝐵(𝑟)=(𝑛𝐵
4𝜋𝑟2𝛥𝑟)
(𝑁𝐵
𝑉)(4)
He e, nB is he numbe o B- ype pa icles ound wi hin a sphe ical shell o hickness Δ a
adius , NB is he o al numbe o B- ype pa icles in he sys em, and V is he sys em olume.
The coo dina ion numbe (CN), ep esen ing he numbe o su ounding pa icles wi hin a
speci ic in e ac ion ange, was compu ed as:
𝐶𝑁=∫ 𝑔
A-B
(𝑟)𝑑𝑟
𝑟𝑣
0(5)
In his equa ion, ep esen s he adial dis ance co esponding o he i s minimum
ollowing he RDF peak.
Fu he mo e, he di usion coe icien D o hyd oxide ions was calcula ed based on he slope
o he mean squa ed displacemen (MSD) as a unc ion o ime, using he ollowing Eins ein
ela ion: 𝐷=1
6𝑑
𝑑𝑡⟨|𝑟(𝑡)−𝑟(0)|2⟩ (6)
He e, he MSD was a e aged o e all hyd oxide ions and mul iple ime o igins, and he slope
was ex ac ed om he linea egion o he MSD cu e o de e mine he di usion coe icien .
To e alua e he dynamic e en ion o hyd oxide ions in speci ic a eas, he esidence ime
dis ibu ion was analyzed using he app oach p oposed by B uun e al [22]. The dis ibu ion
unc ion d(τk) is de ined by he ollowing ime co ela ion unc ion:
𝑟𝑡𝑑(𝜏𝑘)= ∑∑ 1
𝑀−𝑘𝜈𝑖(𝑡𝑗)
𝑁
𝑘=1
𝑀−𝑘
𝑗=1 𝜈𝑖(𝑡𝑗+𝜏𝑘),𝜏𝑘=𝑘∆𝑡,𝑘=0,1,2,…,(𝑀−1)(7)
He e, N is he numbe o hyd oxide ions, νi( j) is an indica o unc ion ha equals 1 when ion
i esides wi hin he de ined a ea a ime j , and 0 o he wise. Δ is he sampling in e al (1
ps), and M is he o al numbe o sampled ime ames. The a e age esidence ime τ was
hen ex ac ed by i ing he decay o d(τ) wi h an exponen ial unc ion using he ollowing
ela ion:
𝑙𝑛(𝑟𝑡𝑑(𝜏)
𝑟𝑡𝑑(𝜏𝑚𝑖𝑛))=−𝜏𝑟𝑡
−1𝜏(8)
This me hod was applied o de e mine he mean esidence imes o hyd oxide ions in a ious
a eas wi hin he memb ane.
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2. Resul s
2.1 Analysis o Hyd oxide Ion Di usion Coe icien s
Figu e 2 p esen s he di usion coe icien s o hyd oxide ions a di e en wa e con en
condi ions. As λ inc eases, he di usion coe icien shows a ise, a ibu ed o he g owing
numbe o wa e molecules su ounding he ime hylammonium g oups. This hyd a ion
educes elec os a ic in e ac ions, allowing ee hyd oxide ion mo emen wi hin he
memb ane. No ably, he inc ease is nonlinea o λ ≤ 9 and becomes linea a highe λ,
sugges ing a ansi ion in anspo beha io . This shi likely e lec s s uc u al changes such
as he eo ganiza ion o hyd a ion shells and he o ma ion o con inuous wa e channels,
which in luence he hyd oxide ion anspo mechanism depending on he wa e con en .
Al hough simila ends ha e been obse ed in o he AEMs, he di ec co ela ion be ween
polyme s uc u e and di usion enhancemen emains insu icien ly unde s ood.
To cla i y he molecula basis o his wa e con en dependence, he sol a ion s uc u es
speci ic o QPAF-4 is analyzed in he nex sec ion. By examining how hese con igu a ions
a ec hyd oxide mobili y, we aim o iden i y key anspo mechanisms and po en ial limi ing
ac o s unde lying he obse ed end.
2.2 Molecula S uc u e A ound T ime hylammonium G oups
This s udy examined he molecula en i onmen su ounding he ime hylammonium
g oups in QPAF-4. A each wa e con en , RDFs we e calcula ed be ween ni ogen a oms
o he ime hylammonium g oups and he oxygen a oms o wa e and hyd oxide ions. F om
he RDFs, we de ined he spa ial anges o he i s and second sol a ion shells: he i s
sol a ion shells we e de ined om he s a o he end o he i s RDF peak, and he second
om he end o he i s RDF peak o he end o he second RDF peak. These de ini ions
enabled a quan i a i e assessmen o how he elec os a ic ield o he ime hylammonium
g oups in luences local wa e and hyd oxide ion dis ibu ions.
We also analyzed RDFs be ween ni ogen a oms (N–N) o di e en ime hylammonium
g oups unde a ying wa e con en . Based on hese, we de ined he o e lapped a ea as
Figu e 2: Wa e con en dependence o hyd oxide ion di usion coe icien s in
QPAF-4 memb anes

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whe e i s sol a ion shells o mul iple ime hylammonium g oups in e sec , and he isola ed
a ea as whe e shells exis independen ly wi hou o e lap (Figu e 3). As wa e con en
inc eased, mo e wa e molecules en e ed be ween he ime hylammonium g oups,
expanding hei sepa a ion and dec easing he o e lapped olume.
Hyd oxide ions in o e lapped and isola ed a eas o he i s sol a ion shell, and in he second
sol a ion shell, expe ience dis inc elec os a ic en i onmen s. Consequen ly, hei di usion
beha io is expec ed o di e . The anspo cha ac e is ics associa ed wi h each a ea will
be discussed in he ollowing sec ion.
2.3 Residence Time Analysis o Hyd oxide Ions in Each Sol a ion A ea
We analyzed he p e e en ial localiza ion o hyd oxide ions in di e en sol a ion a eas unde
a ying wa e con en . To quan i y his, we de ined he sol a ion numbe (SN) as he numbe
o i s sol a ion shells om ime hylammonium g oups ha su ound each hyd oxide ion. A
hyd oxide ion wi h SN = 0 is loca ed ou side all i s sol a ion shells, SN = 1 co esponds o
an isola ed a ea in luenced by a single shell, and SN = 2 o 3 indica es o e lapped a eas
whe e he hyd oxide ion is simul aneously a ec ed by wo o h ee o e lapping shells,
espec i ely. These classi ica ions a e illus a ed in Figu e 4.
Fo simplici y, we ocused on wa e con en a λ = 3, 9, and 15 and calcula ed he pe cen age
o hyd oxide ions p esen a each SN (Figu e 5). SN ≥ 5 occu ed a ely (e.g., 0.009% a λ
= 3) and we e excluded. A λ = 3, 52.0% o hyd oxide ions we e in SN = 3 and 16.4% in SN
= 4. A λ = 15, SN = 0 and 1 we e dominan , accoun ing o 22.9% and 48.6%, espec i ely,
Figu e 3: Schema ic o i s and second sol a ion shells a ound ime hylammonium
g oups (λ = 15), illus a ing o e lapped and isola ed a eas wi hin he i s shell
Figu e 4: Schema ic illus a ion o sol a ion en i onmen s ca ego ized by sol a ion
numbe wi hin he i s sol a ion shell
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while SN = 4 d opped o 0.08%. We also e alua ed esidence imes by sol a ion a ea (Figu e
6). Due o he ex emely low p esence o ions wi h SN = 0 and 1 a λ = 3, and SN = 0 a λ =
6 (each below 1%), hese cases we e excluded om he analysis in Figu e 6 due o
insu icien s a is ical eliabili y. A low wa e con en (λ = 3, 6), hyd oxide ions esided mos ly
in highly coo dina ed s uc u es (SN = 3, 4), whe e s ong elec os a ic apping in
o e lapped a eas esul ed in long esidence imes. In con as , a high wa e con en (λ = 15,
18), educed SN in o e lapped a eas led o sho e esidence imes due o weakened
apping. Residence imes in isola ed a eas (SN = 1) we e ela i ely s able ac oss wa e
con en , likely due o consis en one- o-one elec os a ic in e ac ions. A sligh inc ease in
Figu e 5: P esence a io o hyd oxide ions as a unc ion o sol a ion numbe wi hin
he i s sol a ion shell unde di e en wa e con en s
Figu e 6: Mean esidence ime o hyd oxide ions in each a ea o he i s and
second sol a ion shells as a unc ion o wa e con en
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esidence ime om λ = 6 o 12 may e lec expanded isola ed a ea olume, as sugges ed
by inc eased N–N dis ance. Beyond λ = 12, he N–N dis ance s abilized (abou 8.90 Å), and
esidence imes emained unchanged.
2.4 T anspo Cha ac e is ics in Each Sol a ion A ea
We e alua ed he apping s eng h o hyd oxide ions a each SN using p esence a io
(Figu e 5) and he olume ac ion (Figu e 7). A low wa e con en (λ = 3), SN = 3 and 4
a eas accoun ed o only 5.70% and 1.09% o he o al olume, espec i ely, ye con ained
52.0% and 16.4% o he hyd oxide ions. This imbalance indica es s ong apping wi hin
o e lapped a eas. A λ = 15, he olume o SN = 2 dec eased o 9.91%, while SN = 3 and 4
d opped below 1%, e lec ing educed o e lap due o inc eased N–N dis ance. This shi
inc eased he olume and occupancy o SN = 1, as well as he p esence o ions in SN = 0.
These changes sugges weake apping a high wa e con en , allowing mo e mobile
hyd oxide ion beha io and suppo ing he linea inc ease in di usion coe icien obse ed
in Sec ion 2.1.
Fu he mo e, he ends obse ed in he p esence a io (Figu e 5) and olume ac ion
(Figu e 7) indica e ha highe SN s uc u es co espond o s onge ion con inemen . O e all,
a low wa e con en , hyd oxide ions a e s ongly apped in o e lapped a eas, esul ing in
longe esidence imes and es ic ed mobili y. In con as , a high wa e con en , educ ions
in bo h SN and o e lapped olume weaken apping s eng h, enhance di usi i y, and
di ec ly con ibu e o imp o ed hyd oxide ion conduc i i y.
3. Conclusion
In his s udy, classical molecula dynamics simula ions we e pe o med o elucida e he
wa e con en -dependen anspo beha io o hyd oxide ions in AEM, ocusing on he
ehicle mechanism. By employing QPAF-4, an AEM ma e ial wi h ime hylammonium
g oups, we sys ema ically analyzed how wa e con en a ec s sol a ion s uc u e, esidence
ime, and hyd oxide ion di usi i y.
The di usion coe icien inc eased nonlinea ly a low λ and linea ly a high λ, indica ing a
s uc u al ansi ion. A low λ, hyd oxide ions we e mainly con ined in o e lapped a eas
Figu e 7: Volume ac ion o each sol a ion numbe egion λ = 3, 9, and 15
EFCF 2025: Low-Temp. Fuel Cells, Elec olyse s & H2 P ocessing 1 – 4 July 2025, Luce ne Swi ze land
h ps://doi.o g/10.5281/zenodo.17244121 A0804 Page 9/10
o med by mul iple i s sol a ion shells, esul ing in s ong elec os a ic apping and limi ed
mobili y. As λ inc eased, expanded spacing be ween ime hylammonium g oups educed
o e lap, shi ing ion localiza ion owa d isola ed a eas and second sol a ion shells, whe e
mobili y was highe . We in oduced he SN o quan i y he numbe o o e lapping shells pe
hyd oxide ion and con i med i s co ela ion wi h con inemen s eng h. Highe SNs indica ed
s onge elec os a ic apping and limi ed mobili y.
These esul s o e molecula -le el insigh in o he ole o sol a ion s uc u e in wa e
con en -dependen hyd oxide ion anspo , p o iding a ounda ion o designing AEMs wi h
enhanced hyd oxide ion conduc i i y o e icien and du able AEMWE applica ions.
Acknowledgemen s
This wo k was suppo ed by he G eX P og am Japan (G an Numbe JPMJGX23H2) and
he JSPS Co e- o-Co e P og am (G an Numbe JPJSCCA20210005). The compu a ions
we e pe o med using he nex -gene a ion in eg a ed esea ch sys em a he Ins i u e o
Fluid Science, Tohoku Uni e si y. We would like o exp ess ou deepes g a i ude o all
indi iduals and suppo e s in ol ed in his esea ch.
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