Joule Heating Effects on Transport-Induced-Charge Phenomena in an Ultrathin Nanopore

Transport-induced-charge (TIC) phenomena, in which the concentration imbalance between cations and anions occurs when more than two chemical potential gradients coexist within an ultrathin dimension, entail numerous nanofluidic systems. Evidence has indicated that the presence of TIC produces a nonlinear response of electroosmotic flow to the applied voltage, resulting in complex fluid behavior. In this study, we theoretically investigate thermal effects due to Joule heating on TIC phenomena in an ultrathin nanopore by computational fluid dynamics simulation. Our modeling results show that the rise of local temperature inside the nanopore significantly enhances TIC effects and thus has a significant influence on electroosmotic behavior. A local maximum of the solution conductivity occurs near the entrance of the nanopore at the high salt concentration end, resulting in a reversal of TIC across the nanopore. The Joule heating effects increase the reversal of TIC with the synergy of the negatively charged nanopore, and they also enhance the electroosmotic flow regardless of whether the nanopore is charged. These theoretical observations will improve our knowledge of nonclassical electrokinetic phenomena for flow control in nanopore systems.


Computational domain sensitivity and mesh analysis
The size of the computational domain is determined by comparing the simulation results of different computational domain sizes. Because the left and right end boundaries are satisfied with the Dirichlet boundary conditions (fixed boundary condition) for voltage, solution concentration, and temperature (also the up silicon nitride thin layer boundary and up reservoir boundaries for temperature), the size of the computational domain directly affects the simulation results in the nanopore. In addition, different reservoir sizes affect the resistance ratio between the inner and outer parts of the nanopore, thus affecting the electric potential distribution. Therefore, a sufficiently large computational domain is required to reduce the influence of the reservoir resistance on the electric potential distribution of the system.
Considering Case (iv) in Section 3.1 for comparison (a negatively charged nanopore with Joule heating), we simulate nine cases with different reservoir sizes, where the length of the reservoir varies from 200 to 1000 nm. The ion concentration ± , electric potential , and temperature | axis on the centerline ( -axis: = 0, −50 nm < < 50 nm), and temperature | middle on the middle line ( -axis: 0 < < 50 nm, = 0) of the nanopore are shown in Figure S1a-e. We observe that the changes in the simulation results become smaller as increases, as seen by the results of 800 nm and 1000 nm , which are almost completely overlapped in the figure, such that the simulation result of 1000 nm can be regarded as an accurate convergence result. We define the relative difference in quantity between the cases of and 1000 nm as max �� − 1000 � × 100%�, where and 1000 are the corresponding values of on the same coordinate point in the two cases, and is the reference variation range of . We consider the ion concentration as 0.4 M (external solution concentration bias), the electric potential as 2 V (external electric potential difference), and the temperature as 7.15 K (typical temperature change between the midpoint of the nanopore and the system boundary).
Under this definition, the relative differences among the simulation results of different and exceeds 600 nm (not more than 1000 nm), the differences between the simulation results under different conditions are less than 2.5%. Such small differences are the boundary effects that can be considered marginal when the size of the reservoir exceeds 600 nm.
Here, we determine the ion concentration difference + − − (proportional to induced charge concentration) and axial velocity on the centerline, and temperature | axis and | middle in the system for evaluating the quality of different meshes. The results of three meshes with different densities for set as 600 nm are shown in Figure S2, in which mesh 2 doubles the structured cells of mesh 1 (which means mesh 2 covers a larger area surrounding the nanopore using structured cells), and mesh 3 has a higher number of total cells, approximately 1.3 times that of meshes 1 and 2 (detailed information regarding meshes is listed in Table S1). The simulation results indicate that these three meshes are almost identical. In the design of meshes, the influence of the thickness of EDL on the simulation results has been considered already. Usually the characteristic thickness of EDL is measured by Debye length Considering all the above factors and for practical concerns, we adopted = 600 nm in our simulations, providing accurate results and comparing to the required simulation time when is 800 and 1000 nm; in addition, mesh 2 was adopted, considering that the velocity direction in the nanopore and on the -axis is almost always parallel to the -axis.    Total CPU time / s 3.5 × 10 3 3.8 × 10 3 5.9 × 10 3 4.8 × 10 3 6.2 × 10 3

Comparison of the magnitudes of viscous dissipation and Joule heating
In this section, we discuss the heat generated from viscous dissipation − (∇ + (∇ ) T ) ∶ ∇ and that from Joule heating in the nanopore. It was found that the effect of viscous dissipation is far smaller than the Joule heating effects. For instance, for Case (iv) in Section 3.1, when Δ = 1, 2, and 3 V, the largest ratio of the heat from viscous dissipation and Joule heating inside the nanopore is only 2.0%, 1.2%, and 0.84% near the nanopore surface, respectively. Therefore, the effect of viscous dissipation is negligible in our model.

Information on correlation equations for pure water at .
The relationships of the water density , viscosity , static dielectric constant , specific isobaric heat capacity , and thermal conductivity to temperature are given by regression   Table S3; the parameters , , , ℎ, , and are in Table S4; other parameters are in Table S5 [15].