Entropic modulation of divalent cation transport
Yechan Noh
Department of Physics, University of Colorado Boulder, Boulder, CO 80309, USA and
Applied Chemicals and Materials Division, National Institute of Standards and Technology, Boulder, CO 80305, USA
Demian Riccardi and Alex Smolyanitsky
∗
Applied Chemicals and Materials Division, National Institute of Standards and Technology, Boulder, CO 80305, USA
(Dated: September 23, 2025)
Aqueous cations permeate subnanoscale pores by crossing free energy barriers dominated by
competing enthalpic contributions from transiently decreased ion-solvent and increased ion-pore
electrostatic interactions. This commonly accepted view is rooted in the studies of monovalent
cation transport. Divalent cations, however, have significantly higher desolvation costs, requiring
considerably larger pores to enable retention of the first hydration shell and subsequently transport.
We show that this scenario gives rise to a strong enthalpy-entropy competition. Specifically, the
first hydration shell is shown to undergo rotational ordering inside the pore, resulting in a tight
transition state. Our results shed light on the basic mechanisms of transport barrier formation for
aqueous divalent cations permeating nanoporous 2D membranes.
Ion transport through solvated nanoporous membranes
is a fundamental process underlying complex natural and
engineered systems. The thermodynamics of ion trans-
port are broadly understood in terms of ion solvation and
energetic penalties incurred during ion transitions, but
the molecular-level decomposition of these phenomena
remain unclear. Furthermore, thermally accessible trans-
port depends on ion identity and valence. Here, we inves-
tigate the molecular thermodynamics of monovalent and
divalent cation transport across nanoporous 2D mem-
branes separating an aqueous ionic bath [1–4]. These
model systems comprise two identical aqueous regions,
minimally perturbed by nanoporous interfaces. Within
this framework, transport barriers are informed by the
electrostatic features of the pore and the membrane ma-
terial; describing ion transport through such pores then
reduces to a detailed understanding of all contributions
to the corresponding ion-specific barriers.
The importance of nano- and sub-nanoporous 2D solids
is far beyond their ability to serve as illustrative mod-
els of transport barrier formation. Recent advances in
fabrication make multivacancy pores, resulting from no
more than a dozen or so atomic sites ejected from the
host 2D lattice, a reality [5–10]. In aqueous environ-
ments, permeant-specific barriers underlie unique trans-
port properties, potentially promising to a wide range
of applied areas, including molecular and ionic separa-
tion [11, 12], sensing of biomolecules [13, 14] and me-
chanical strain [3, 4, 15–17], power generation [18], and
nanofluidics-based computing [19–21].
Because transport of alkali salt cations is most com-
monly studied, theoretically and experimentally, our un-
derstanding of transport barrier formation is broadly
based on the corresponding physics of monovalent
cations. For subnanoscale pores with locally dipolar
edges, permeation occurs one ion at a time, and the
underlying mechanisms are relatively straightforward.
FIG. 1: A simplified sketch of the ion-pore and ion-water
interactions as a function of the ion transport coordinate (de-
noted t.c. on the left) in the direction perpendicular to the
membrane plane.
Upon traversing the subnanoscale pore confinement,
cations transiently lose a significant portion of their first
hydration shell while gaining the energy of electrostatic
interactions with the pore region [1, 2, 22]. Depending
on the dehydration peak height in relation to the corre-
sponding ion-pore well depth, the overall barrier can then
be attractive or repulsive, as sketched in Fig. 1. Ubiq-
uitous in biological and artificial nanofluiodic/nanoionic
systems, this apparent competition between the enthalpic
ion-water and ion-pore contributions to the transport
barrier was pointed out as a potentially interesting bridge
with coordination chemistry [2, 5], which describes ion in-
teractions with entities such as crown ether molecules in
aqueous environment [23].
Divalent cations interact with water significantly more
strongly than their monovalent counterparts; for com-
parison, the standard enthalpies of hydration for K
+
and Mg
2+
are -322 kJ/mol and -1921 kJ/mol, respec-
tively [24]. This fact makes the permeation mechanism
outlined above far less probable for divalent cations under
a realistic electrostatic bias. For measurable transport of
divalent ions to occur, wider pores are therefore funda-
mentally required to allow retention of the entire first
