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Page 6 of 28 Choi et al. Energy Mater. 2025, 5, 500106 https://dx.doi.org/10.20517/energymater.2025.50

where D is the diffusion coefficient, D is the thermodiffusion coefficient, n is the ion concentration, and T
T
i
is the absolute temperature. The Eq. 1 describes ion transport driven by both a concentration gradient
(kinetic diffusion; first term) and a temperature gradient (thermodiffusion; second term). At equilibrium
state (J = 0) with no external force, the thermodiffusion-induced potential difference follows :
[56]
i

The Soret effect can be denoted as:

where D /D = S is the Soret coefficient. The Soret coefficient describes the formation of an ion
T
t
concentration gradient in response to temperature gradient. If cations and anions exhibit different Soret
coefficients, a local net charge distribution will form under a temperature gradient, thereby generating an
electric potential (∇V).

∇V = -S ∇T (5)
T

where S represents the Seebeck coefficient of the hydrogel-based i-TE system. In the hydrogel-based i-TE
T
system, a higher S indicates greater thermodiffusion-induced charge separation, leading to increased
T
potential difference under a given temperature gradient. In addition, the ion total flux (J) under steady state
condition ( = 0, D ≠ 0) with no external force can also be expressed as follows [58-60] :

where J is the diffusive flux, J is the electromigrative flux, J is the thermodiffusive flux, q is the charge of
E
i
D
T
the ion, is the Eastman transfer entropy, and k is the Boltzmann constant. The concentration profile of
B
each ion species reaches a quasi-equilibrium driven by the balance between thermodiffusive and
electrostatic forces. As a result, ∇n will be several orders of magnitude lower than other terms. Then, the S
T
i
of hydrogel-based i-TE materials can be expressed as, based on Onsager transport theory [59,60] :

According to Onsager transport theory and assuming symmetric electrolytes (q = q and n = n ), S can be
-
0+
+
T
0-
simplified as [21,61] :

where the n and n represent the concentrations of cations and anions, respectively, and e is the electron
+
-
charge. Therefore, the Seebeck coefficient of i-TE materials is higher than that of e-TE materials, which can