Chemically Selective Alternatives to Photoferroelectrics for Polarization‐Enhanced Photocatalysis: The Untapped Potential of Hybrid Inorganic Nanotubes

Linear‐scaling density functional theory simulation of methylated imogolite nanotubes (NTs) elucidates the interplay between wall‐polarization, bands separation, charge‐transfer excitation, and tunable electrostatics inside and outside the NT‐cavity. The results suggest that integration of polarization‐enhanced selective photocatalysis and chemical separation into one overall dipole‐free material should be possible. Strategies are proposed to increase the NT polarization for maximally enhanced electron–hole separation.


Potential step in a co-axial cylindrical capacitor
As shown in Figure 3a, in spite of the NT-wall polarization, and owing to the NT cylindrical symmetry and overall charge-neutrality, the NTs present a flat electrostatic potential [ ( ) inside and outside the NT-cavity. Since the electrostatic field ( E ) is given by the negative gradient of the electrostatic potential [ ], no electrostatic field is present inside and outside the NT. As a result, it is possible to model the NT electrostatics on the basis of an overall neutral co-axial (hollow) cylindrical capacitor (Figure S1). on the inner and outer hollow cylinder, respectively. L is the length of the inner and outer hollow cylinders. The three cylindrical Gaussian surfaces of radius ' r , ' ' r , and ' ' ' r are also indicated.
Gauss' theorem relates the flux of the electrostatic field ( E ) across a closed surface (S) to the charge Q contained inside the closed surface: where 0 ε is the electric permittivity of vacuum, and s dˆa vector of unitary module locally normal to the infinitesimal surface element. Eq. S1 allows definition of three electrostatic regions with different in the co-axial capacitor in Figure S1: Region 1. For in R r < , the electric field is zero ( 0 = E ) since the Gaussian surface of radius ' r does not contain any net charge ( 0 = Q ).

Region 2. For
, the electric field is not zero ( 0 ≠ E ) since the Gaussian surface of radius ' ' r does contain a net charge ( 0 < − Q , see Figure S1). As the hollow cylinders are taken to be in electrostatic equilibrium, with no net transfer of charge, r EÊ = i.e. the electrostatic field must lie parallel to the tube radius, with zero components along the tube (lateral) surface.
Region 3. For out R r > , the electric field is zero ( 0 = E ) since the Gaussian surface of radius ' ' ' r does not contain any net charge ( 0 = Q ), being the cylindrical capacitor overall neutral.
We thus focus in Region 2 to calculate the electric field and potential difference between in R and out R by Gauss' flux theorem. We start by expanding both sides of Eq. S1 as: Where in the right-side term we have taken advantage of E being locally parallel to s dˆ and that the E between the cylinders has to be directed parallel to the surface normal with zero components along the cylinder axis.
Eq. S2 can be rearranged to read: which allows the integration of the electrostatic potential between in R and out R as: It is worth noting that, in Eq. S4, the overall negative sign of ] is consistent with E being directed from the outer (positively charged) to the inner (negatively charged) cylinder ( Figure S1).
In analogy with the treatment for the surface dipole density ( σ µ ) due to two charged surfaces (of surface charge-density σ ) locally parallel and separated by a distance d: For infinitesimally small separation R ∆ , leading to out in R R = and , the separation in surface charge density (σ ) between the inner and outer cylinders (Figure S2) can be described via a surface dipole-density as: which in turn can be used to write: Figure S2. Separation of charge density (σ ) between the co-axial cylinders, leading to a surface dipole-density σ µ .
Although Eq. S5 is strictly verified for infinitesimally small separations between the chargelayers (d), it is routinely used in the modelling of potential steps across atomically heterogeneous bi-dimensional junctions (of finite thickness) between different materials (see, for instance, Ref. [S2] and references therein). Therefore, by using Eq. S6, we resort to the same approximation in computing the potential step across the interface dipole at the NT-wall of atomically finite thickness 0 ≠ ∆R . Accordingly, and based on Eq. S7, Eq. S4 can be rearranged into: where we have used the fact that in atomic units π ε 4 1 0 = .
For consistency with the convention in some DFT-codes of calculating the electrostatic potential (Figure 3a) using the (negatively charged) electron as test charge, leading to lower (higher) electrostatic potential for electron-rich (poor) regions, the sign of Eq. S8 needs to be changed leading to: This correctly describes regions of high (low) electrostatic potential for the electron-rich (poor) side of the NT-cavity (Figure 3a). Eq. S8 allows computation of dipole-density from the step in the electrostatic potential across the NT-wall. Given the solution to the DFT problem via discretized grids, [S3] the non-homogeneous electrostatic potential inside (and immediately outside) any material, and in analogy with standard procedure for planar dipole densities, [S2] it is convenient to angularly and longitudinally average the electrostatic potential (expressed in cylindrical coordinates): This last equation is used to compute σ µ on the basis of the potential step ( V ∆ ) between the electrostatically derived in R and out R ( Figure 3).

It is worth noting that, for increasingly large
), the cylindrical capacitor asymptotically tends to a planar one, and Eq. S11 asymptotically recovers the established π µ σ 4 = ∆V relationship ( Figure S3) for the potential step due to a planar dipole density. [S1] Eq. S10 allows exploring the role of the geometric factors and the interplay between in R and R ∆ in damping the relationship between surface dipole-density σ µ and potential step across the NT-wall V ∆ . As shown in Figure S4, Large in R and small R ∆ values allow maximization of the potential difference ( V ∆ ) for a given surface dipole-density ( σ µ ).
Conversely, smaller V ∆ values can be obtained for the same σ µ provided in

Band structure calculations
Band structure calculations were performed via the Projected Augmented Wave (PAW) method as implemented in the VASP program [S4], with the PBE XC-functional [S5], a 400 eV plane wave energy cutoff, 0.1 eV Gaussian smearing, and 10 k-points along the reciprocal periodic direction of the NTs.
As common practice [S6], effective electron (hole) mass were computed via parabolic fitting at the bottom (top) of the computed conduction (valence) band, with wavevector (k) fitting ranges small enough to ensure fitting errors of less than 0.5%. Table S1. Average atom-resolved diameters and standard deviation (Å) for the optimized NTmodels and considered XC-functionals. The adopted labeling corresponds to the atom element and the subscript-suffix numbers the radial layer (see also Figure 1a). N is the number of radially non-equivalent Al-atoms contained within the NT circumference.  Table S2. Average Layer-resolved bond lengths and their standard deviations (Å) for optimized NT-models and considered XC-functionals. The adopted labeling corresponds to the atom element and the subscript-suffix numbers the radial layer (see also Figure 1a). N is the number of radially non-equivalent Al-atoms in the NT circumference. The PBE results for the pristine AlSi 24 NT are reported for comparison.      [14f] for hydroxylated (not methylated) NTs with different XC-functionals (PW91, BLYP, B3LYP), high-spin (magnetic moment per Fe-atom: 3.7 µ B ) ferromagnetic ordering is computed to be favored by more than 1.3 eV and 1.6 eV eV over ferromagnetic low-spin (magnetic moment per Fe-atom: 1 µ B ) and anti-ferromagnetic (magnetic moment per Fe-atom: ±1 µ B ) solutions, respectively. The absence of details on whether different magnetic solutions were explored and converged in Ref.

Supplementary results
[14f] prevents further elaboration on these deviations.    Figure 1a for the adopted layerlabeling. Figure S10. Vacuum-aligned PBE-E total DOS plot (filled grey) and layer resolved LDOS plots for the AlSi 28 -Me (left) and AlSi 34 -Me (right) NTs. See Figure 1a for the adopted layerlabeling. Figure S11. Vacuum-aligned PBE-D2 total DOS plot (filled grey) and layer resolved LDOS plots for the AlSi 28 -Me (left) and AlSi 34 -Me (right) NTs. See Figure 1a for the adopted layerlabeling. Figure S12. Vacuum-aligned VDWDF total DOS plot (filled grey) and layer resolved LDOS plots for the AlSi 28 -Me (left) and AlSi 34 -Me (right) NTs. See Figure 1a for the adopted layerlabeling. Figure S13. Vacuum-aligned OPTPBE total DOS plot (filled grey) and layer resolved LDOS plots for the AlSi 28 -Me (left) and AlSi 34 -Me (right) NTs. See Figure 1a for the adopted layerlabeling. Figure S14. Vacuum-aligned OPTB88 total DOS plot (filled grey) and layer resolved LDOS plots for the AlSi 28 -Me (left) and AlSi 34 -Me (right) NTs. See Figure 1a for the adopted layerlabeling. Figure S15. The calculated difference in the radially averaged electrostatic plateau inside and outside the NT cavity ( V ∆ ) as a function of NGWF radius (bohr, a 0 ) for NTs containing 24, 28 and 36 Al-atoms within their circumference. Figure S16. The calculated difference in the radially averaged electrostatic plateau inside and outside the NT cavity ( V ∆ ) as a function of the number of Al-atoms within the tube circumference for 8 a 0 (purple) and 12 a 0 (green) NGWFs.  Figure S17. The PBE optimized geometry (left) and real-space separation (right) between the VBE (green) and CBE (red) of the (a) N=24 and (b) N=36 AlSi N -CF 3 NTs. Same coloring scheme as in Figure 1, with the additional F-atoms being colored purple.  Figure S18. Vacuum-aligned PBE total DOS plot (filled grey) and layer resolved LDOS plots for the AlSi 24 -CF 3 (left) and AlSi 36 -CF 3 (right). See Figure 1a and S17 for the adopted layerlabeling. The energy scale has been referenced to the VB-maximum (0 eV). VB-maxima not at the centre of the Brilluoin zone (Γ-point) have previously been reported for other inorganic (ionic) nanotubes [S6]. The energy scale has been referenced to the VB-maximum (0 eV). The computed results are in accordance with previously published PBE results for pristine (aluminosilicate) imogolite NTs [S7].