Chapter IX: Coupled and Decoupled Dynamics of Stern and Diffuse Layers
9.6 Appendix
Contributions of bulk oxide defects and surface roughness to SHG phase and angle measurment
Data and content in this section have been published as Ref. [110]. The author participated in conducting computations, discussing the computation results, and preparing a draft.
Recently, we proposed the existence of a nonlinear susceptibility term that is as- sociated with a 90o phase shift to recapitulate experimental data in non-resonant second harmonic generation phase and amplitude measurements [110]. To do so, we needed to show minor contributions of (i) bulk oxide defects and (ii) oxide surface roughness, both of which are supported by computations as is explained below.
Fused silica contains impurities, for which we considered using one-dimensional finite element calculations (0.01 nm steps, see computational methods below) of a solid having a relative permittivity of 3.8 [99], and 100 ppb charged impurities [2]
with vacuum on one side and a 0.1 M aqueous salt solution on the other side. The surface charge density is set to -0.015 C m−2, and the relative permittivity of the 0.1
Figure 9.7: Calculated electric field for model silica:water interfaces. (a, b) Electro- static field across the fused silica:water interface from finite element calculations.
The silica contains 100 ppm of charged impurities and the aqueous phase is at 0.1 M [NaCl]. The relative permittivity in the aqueous phase is 78. (c, d) Height vs.
position profiles from several atomic force microscopy lines scans across the flat side of our fused silica hemispheres, primitive cell used for finite element calcula- tions shown in black box nested in between its periodic images, and electrostatic field across an atomically smooth (black line) and rough (purple line for L=H=1 nm features, blue line for L=H=2 nm features) fused silica:water interface. Horizontal dashed line depicts the E-field value used for the 1-nm wide rough region. The rel- ative permittivity of the aqueous phase is 78. The surface charge density is -0.015 C m−2.
M salt solution is taken to be 78 [9]. Averaged over 200 nm of solid, and 106random impurity realizations, the model shows that the electric field is -100 kV m−1±2 MV m−1, i.e. statistically indistinguishable from zero (Fig. 9.7A-B). Nevertheless, using the point estimate of -100 kV m−1 between z=0 nm, the laser focus position, to z=- b=-0.0127 m, the hemisphere’s edge, we obtain 𝜒(2)
𝑑𝑐,𝑆𝑖𝑂2 =8×10−25×𝑒−145𝑖m2V−2,
which is again much smaller than the 𝜒(3)
𝑤 𝑎𝑡 𝑒𝑟 contribution. Note that we can also integrate 𝐸𝐷𝐶(𝑧) obtained from the finite element calculations over 200 nm, 2 x the coherence length in our experiments, which yields -0.013 V, three times larger than the result obtained using the field point estimate of -100 kV m−1, arriving at the same conclusion. Our two analyses indicate that the bulk silica contribution is not enough to explain the pH-dependent SHG phase observed at short Debye lengths (0.5 M [NaCl]), at least within the electric dipole approximation.
We also considered the impact on the third-order contribution due to the finite roughness of the surface of the fused silica hemispheres. Atomic force microscopy shows the surface to have a root-mean-square roughness of 0.4 nm, with +/- 1 nm variation in surface height (Fig. 9.7C). We therefore assigned the aqueous:solid interface a width, 𝜛, of 1 nm, as opposed to being atomically flat. We treat this 1 nm wide region with a third-order susceptibility taken to be the geometric mean of 𝜒(3)
𝑤 𝑎𝑡 𝑒𝑟 and 𝜒(3)
𝑆𝑖𝑂2, i.e. 𝜒𝜛(3) =6×10−22m2V−2. The second-order dc-contribution is then
𝜒(2)
𝑑𝑐,𝜛= 𝜒(3)
𝜛
∫ 0
𝜛=−1𝑛𝑚
𝐸𝐷𝐶(𝑧)𝑒𝑖Δ𝑘𝑧𝑧𝑑 𝑧, (9.9) where 𝐸𝐷𝐶(𝑧) can be approximated, again guided by finite element calculations (Fig. 9.7D), to be constant at𝐸𝐷𝐶 (𝑧) =− Φ(0)
1×10−9Vm−1over the 1-nm wide rough in- terfacial region. The second-order dc contribution to the second order nonlinear sus- ceptibility then becomes 𝜒(2)
𝑑𝑐,𝜛 =−Φ(0) × 6×10−22m2V−2
1×10−9𝑚
∫0 𝜛=−1𝑛𝑚
𝐸𝐷𝐶(𝑧)𝑒𝑖Δ𝑘𝑧𝑧𝑑 𝑧=
−Φ(0) ×6×10−22×𝑒−0.3𝑖m2V−1, having, again, an imaginary part that is insufficient to explain the observed pH-dependent SHG phase shift at high ionic strength.
Finite-element calculation methods
The one-dimensional Poisson equation was numerically solved for an oxide in contact with ionic water of 0.1 mol L−1NaCl using the Newton-Raphson iteration method [22, 60].
𝑑2Ψ(𝑥) 𝑑𝑥2
=
( −𝑒[𝛿(𝑥𝑝)−𝛿(𝑥𝑛)]
𝜖𝑜 𝑥𝜖0 ,if𝑥 ≤ 𝐿𝑜𝑥
− 𝑒 𝜌𝑏
𝜖𝑤𝜖0
𝑒−𝑒 𝛽Ψ+𝑒𝑒 𝛽Ψ
,if𝐿𝑜𝑥 < 𝑥 < 𝐿
(9.10) Here, Ψ(𝑥) is the one-dimensional electrostatic potential, and 𝐿 (= 220 nm) is the total length of the system, including a 200 nm thick oxide and a 20 nm thick water region. The water-oxide interface is located at x = 𝐿𝑜𝑥 = 200 nm. The vacuum permittivity is𝜖0=8.854 10−12F m−1, the relative dielectric permittivities are𝜖𝑜𝑥=4 for the oxide and𝜖𝑤=78 for the ionic water,𝑒is the elementary charge, and𝛽= 𝑘𝐵𝑇−1
Figure 9.8: Finite element calculation model for a corrugated oxide:water interface.
The rough oxide:water interface is modeled using a nanoscale dendrite of height (H) and length (L), represented by the yellow region. Bright green elements are the dendrite corners where the oxide:water boundary is along both the x- and y-axes.
with T being the temperature (300 K) and 𝑘𝐵 being the Boltzmann constant. The bulk ion density in the ionic water is 𝜌𝑏 = 0.1 mol L−1 with monovalent cations and anions whose thermal motion is incorporated using the Boltzmann factor. The oxide is modeled to include a pair of positively and negatively charged point defects that decrease the electric field inside the oxide. A positively charged defect (𝑥 =𝑥𝑝) is randomly placed inside the oxide region, followed by a random insertion of a negatively charged defect (𝑥 = 𝑥𝑛) according to a Poisson distribution having an average of 100 nm, so the distance between the defects is 100 nm on average to recapitulate the 100 ppm defect density in commercially available silica.
Three boundary conditions are applied in our model using the fact that the oxide sur- face in contact with water is lightly charged and both the oxide and water boundaries are electrically grounded. We employ (i) the Dirichlet condition withΨ(𝑥 =0)=0 at the oxide end, (ii) the Dirichlet condition withΨ(𝑥 = 𝐿) = 0 at the water end, and (iii) the Neumann condition at the aqueous oxide interface (𝑥 =𝐿𝑜𝑥) as follows:
𝜖𝑜𝑥
𝑑Ψ(𝑥) 𝑑𝑥
𝑥=𝐿𝑜 𝑥
−𝜖𝑤
𝑑Ψ(𝑥) 𝑑𝑥
𝑥=𝐿𝑜 𝑥
= 𝑄𝑠 𝜖0
, (9.11)
where 𝑄𝑠 (= -0.015 C m−2) is the oxide surface charge density. The differential equation is solved by discretizing the space with finite elements whose size is 0.01 nm along the x-axis. Note that 0.01 nm is short enough to properly recapitulate the Debye screening length (0.39 nm at 1 mol L−1 NaCl) in the water region. The
iteration is terminated once 𝐿2-norm of the solution is less than 10−6. The electric potential and the associated electric field is averaged over 106realizations with the random insertion of the defects inside the oxide.
The two-dimensional linearized Poisson-Boltzmann equation was numerically solved for rough oxide:water interfaces (Fig. 9.8), modeled by a small dendrite of height (𝐻) and length (𝐿):
𝑑2 𝑑𝑥2
+ 𝑑2 𝑑𝑦2
Ψ(𝑥 , 𝑦)=
( 0,in the oxide region
𝜅2Ψ(𝑥 , 𝑦),in the water region, (9.12) where𝜅(=
q2𝑒2𝛽 𝜌𝑏
𝜖𝑤𝜖0 )is the inverse of Debye screening length andΨ(𝑥 , 𝑦)is the two- dimensional electrostatic potential. Here, the length of each region is fixed with a 10 nm thick oxide and a 20 nm thick water region. The height of the oxide region varies according to the height of the dendrite: 𝐻𝑜𝑥𝑖 𝑑𝑒 =2𝐻. Other parameters are kept the same as in the one-dimensional model above. Boundary conditions are applied as in the one-dimensional model. We employ (i) the Dirichlet condition with Ψ(𝑥 , 𝑦) = 0 at both oxide and water ends (blue line in Fig. 9.8), and (ii) the Neumann condition at the corrugated aqueous oxide interface, 𝜕, (yellow line in Fig. 9.8) as follows:
𝜖𝑜𝑥 𝑑
𝑑𝑥 + 𝑑
𝑑𝑦
Ψ𝑜𝑥(𝑥 , 𝑦) 𝜕
·𝑛ˆ − 𝜖𝑤 𝑑
𝑑𝑥 + 𝑑
𝑑𝑦
Ψ𝑤(𝑥 , 𝑦) 𝜕
·𝑛ˆ = 𝑄𝑠 𝜖0
. (9.13) Here, ˆ𝑛 is the unit normal vector pointing the water region from the oxide region.
The differential equation is solved by discretizing the space with finite elements whose size is 0.1 nm along both x- and y-axes. Note that 0.1 nm is still short enough to properly recapitulate the Debye screening length (0.39 nm at 0.1 mol L−1NaCl) in the water region. The linear equation is solved using Jacobi and Gauss-Seidel methods with successive over-relaxation [60]. The relaxation method is terminated once𝐿2-norm of the solution is less than 10−12.
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