Supporting Information for

Origin of Facet Selectivity and Alignment in Anatase TiO2 Nanoparticles in

Electrolyte Solutions: Implications for Oriented Attachment in Metal Oxides

Maria L. Sushko* and Kevin M. Rosso

Computational Methods

Classical density functional theory (cDFT) was combined with Lifshitz theory of van der

Waals forces and used to evaluate forces acting between two nanoparticles immersed in

electrolyte solution. The total free energy functional has the form:

𝐹(𝑑) = 𝐹!!"!(𝑑)+ 𝐹!!!"#(𝑑)+ 𝐹!!"# (𝑑)

where the first two terms denote zero frequency (static) and high frequency (dynamic)

van der Waals interactions between two anatase nanoparticles immersed in electrolyte

solution and the last term describes all ion-mediated mesoscopic and microscopic

interactions described within classical density functional theory (see below) and d. is the

distance between nanoparticle surfaces.

To avoid double counting microscopic interactions, mean-field non-retarded Lifshitz

theory is employed for particle/particle van der Waals terms.1 In particular, the high

frequency van der Waals term is evaluated as

𝐹!!!"#(𝑑) =𝑘𝑇8𝜋𝑑! 𝑥𝑙𝑜𝑔 1− Δ!𝑒!! 1− Δ!𝑒!! 𝑑𝑥

!

!!

!

!!!

where 𝑟! = 2𝑑𝜀!!/!𝜉!/𝑐 is a dimensionless factor quantifying the retardation screening

and 𝜉! = 2𝜋𝑛𝑘𝑇/ℏ defines the eigenfrequencies at which dielectric function is

evaluated, c is the velocity of light in vacuum and functions Δ and Δ are defined as

Electronic Supplementary Material (ESI) for Nanoscale.This journal is © The Royal Society of Chemistry 2016

Δ =𝑠!𝜀!(𝑖𝜉!)− 𝑠!𝜀!(𝑖𝜉!)𝑠!𝜀!(𝑖𝜉!)+ 𝑠!𝜀!(𝑖𝜉!)

Δ =𝑠! − 𝑠!𝑠! + 𝑠!

𝑠! =𝑥𝑟!

!− 1−

𝜀!(𝑖𝜉!)𝜀!(𝑖𝜉!)

,      𝑠! =𝑥𝑟!

Note that since relative magnetic permeability of water and anatase TiO2 is close to 1,2

we ignore the magnetic contribution in the above equation.

In the non-retarded limit Δ = 0 and

Δ =𝜀!(𝑖𝜉!)− 𝜀!(𝑖𝜉!)𝜀!(𝑖𝜉!)+ 𝜀!(𝑖𝜉!)

and the high frequency van der Waals free energy is given by

𝐹!!!"# 𝑑 = −𝑘𝑇8𝜋𝑑!

Δ!!(𝑖𝜉!)𝑚!

!

!!!

!

!!!

The zero frequency contribution includes double-layer screening of zero frequency

fluctuations and is calculated as

𝐹!!"#(𝑑) =𝑘𝑇4𝜋 𝛽  𝑙𝑜𝑔 1− Δ!𝑒!!!" 𝑑𝛽

!

!

Δ ≡

1− 𝜅𝛽

!𝜀! − 𝜀!

1− 𝜅𝛽

!𝜀! + 𝜀!

where 𝜅 is the Debye screening length in electrolyte solution and 𝜅 ≤ 𝛽 < ∞.

Classical Density Functional Theory

Aqueous salt solutions are modeled as a dielectric medium with εw = 78.5 with certain

densities of positively and negatively charged, spherical particles, representing the ions

and neutral spherical particles representing water molecules. The density of spherical

“water molecules” was 55.5 M, which give an experimental water density. We used

experimental crystalline ionic diameters for mobile ions: σH+/H3O+ = 0.282 nm, σCl = 0.362

nm, and σwater = 0.275 nm.3 The ion charges were qH3O+ = +1, qCl = -1, and qwater = 0. All

simulations were performed at 298 K temperature.

To determine the equilibrium water and ion distributions via cDFT and ion-mediated

forces between nanoparticles, the total Helmholtz free energy functional is minimized

with respect to the densities of all the species in the presence of rigid nanopraticles. For

this optimization, it is convenient to partition the total free energy of the system into so-

called ideal (Fid) and excess components (Fex).4 The ideal free energy corresponds to the

non-interacting system and is determined by the configurational entropy contributions

from water and small ions,

𝐹!" = 𝑘𝑇 𝜌! 𝒓 𝑙𝑜𝑔𝜌! 𝒓 − 𝜌! 𝒓 𝑑𝒓!

!

!

where k is Boltzmann’s constant, T is the temperature, ρi is the density profile of ion and

water species i, N is the number of species, r ∈ Ω is the ion coordinate, and Ω is the

calculation domain. The excess free energy is generally not known exactly but can be

approximated by

𝐹!" = 𝐹!"#!" + 𝐹!!!" + 𝐹!"!" + 𝐹!!"#!" + 𝐹!"#_!"#!"  

where 𝐹!"#!" = 𝐹!!" + 𝐹!"!" is describes first-order electrostatics and includes direct

Coulomb term and image terms, 𝐹!!!" is the hard sphere repulsion term, 𝐹!"!" is the

electrostatic ion correlation term, 𝐹!!"#!" is the ion hydration term, and 𝐹!"#_!"#!" describes

ion-surface van der Waals interactions.

1. Poisson equation for first-order electrostatics

The electric double layer contribution to the free energy (𝐹!"#!" ) includes direct Coulomb

and image interactions and is evaluated through the solution of Poisson’s equation

−∇ ⋅ 𝜀 𝒓 ∇φ 𝒓 = 𝜌! 𝒓 + 𝑞!!

𝜌! 𝒓  

for the electrostatic potential, φ(r), where ρj(r) is the fixed charge density on nanoparticle

faces, ε(r) is the dielectric coefficient equal to 78.5 in solution and 48 in the

nanoparticles. The discrete distribution of charges on anatase faces was constructed on a

2D grid using trilinear interpolation.

2. Fundamental Measure Theory of excluded volume effects

Hard sphere repulsive interactions describes ion and water many-body interactions in

condensed phase due to density fluctuations. These interactions were described using a

Fundamental Measure theory (FMT).5 The approach is based on the solution of the

Ornstein-Zernike equation for direct correlation function using the Percus–Yevick

approximation and yields the following form of the corresponding component of the free

energy6:

𝐹!!!" = 𝑘𝑇 Φ!! 𝑛! 𝒓 𝑑𝒓

where the hard-sphere free energy density Φ!! is a functional of four scalar and two

vector weighted densities (𝑛! 𝒓 ) and has the form

Φ!! 𝑟 = −𝑛! ln 1− 𝑛! +𝑛!𝑛!1− 𝑛!

+1

36𝜋𝑛!!𝑙𝑛 1− 𝑛! +

136𝜋𝑛! 1− 𝑛! ! 𝑛!!

−𝒏! ∙ 𝒏𝟐1− 𝑛!

−1

12𝜋𝑛!!𝑙𝑛 1− 𝑛! +

112𝜋𝑛! 1− 𝑛! ! 𝑛! 𝒏! ∙ 𝒏𝟐

where scalar (α = 0, 1, 2, 3) and vector (β = 1, 2) weighted densities are defined as

𝑛! 𝒓 = 𝜌!(𝒓!)𝜔!!

!!

𝒓! − 𝒓 𝑑𝒓′

𝒏! 𝒓 = 𝜌!(𝒓!)𝝎!(!)

!!

𝒓! − 𝒓 𝑑𝒓′

The “weight functions” ω(α) and ω(β) characterizing the geometry of particles (hard sphere

with radius Ri for ion species i) are given by:

𝜔!! 𝒓 = 𝜃 𝒓 − 𝑅!

𝜔!! 𝒓 = ∇𝜃 𝒓 − 𝑅! = 𝛿 𝒓 − 𝑅!

𝝎!! 𝒓 = ∇𝜃 𝒓 − 𝑅! =

𝒓𝑟 𝛿 𝒓 − 𝑅!

𝜔!! 𝒓 = 𝜔!

! 𝒓 4𝜋𝑅!!

𝜔!! 𝒓 = 𝜔!

! 𝒓 4𝜋𝑅!

𝝎!! 𝒓 = 𝝎!

! 𝒓 4𝜋𝑅!

In the preceding formulae (70), θ is the Heaviside step function with θ(x) = 0 for x > 0

and θ(x) = 1 for x ≤ 0, and δ denotes the Dirac delta function.

3. Mean Spherical Approximation of ion-ion electrostatic correlations

To treat correlations resulting from electrostatic interactions between charged species on

the same footing as those resulting from hard sphere excluded volume interactions Mean

Spherical Approximation7,8 was employed to solve the Ornstein-Zernike equation with

respect to electrostatic direct correlation function. Taylor expansion of electrostatic free

energy was cut after the second order. Then the electrostatic correlation component of the

free energy (𝐹!"!") is

𝐹!"!" = 𝐹�!!" 𝜌!!"#$ − 𝑘𝑇 𝑑𝒓 ∆𝐶!! !" 𝜌! 𝒓 − 𝜌!!"#$ −

!!!,!

𝑘𝑇2 𝑑𝒓𝑑𝒓′ ∆𝐶!"

! !" 𝒓− 𝒓′ 𝜌! 𝒓 − 𝜌!!"#$

!,!!!,!

𝜌! 𝒓′ − 𝜌!!"#$

where 𝜌!!"#$ are the bulk densities of charged species and the first and second-order direct

correlation functions are defined as

∆𝐶!! !" = − 𝜇!!" 𝑘𝑇

∆𝐶!"! !" 𝒓− 𝒓′ =

− !!!!!!

!"#!!!!"− !!

!!"

!𝒓− 𝒓! − !

𝒓!𝒓!, 𝒓− 𝒓′ ≤ 𝜎!"

0,                                                                                                                                                       𝒓− 𝒓′ > 𝜎!"

with

𝐵 = 𝜉 + 1− 1+ 2𝜉 !/! /𝜉

𝜉! = 𝜅!𝜊!"! =𝑒!

𝜀𝑘𝑇 𝑞!!𝜌!!"#$

!

𝜊!"!

In the above equations, 𝜇!!" is the chemical potential of the mobile ions, 𝜅 is the inverse

Debye length and the contact distance 𝜎!" = (𝜎! + 𝜎!)/2.

In many cases these interactions lead to the overall attractive interactions between

like-charged surfaces 9-20 or polyelectrolytes 21-24 even in low salt conditions 25-28.

3. Ion hydration interactions

The short-range attractive hydration interactions between ions (denoted as “ion”) and

water “molecules” (denoted as “w”) in electrolyte solution are given by

𝐹!!!" =12 𝑑𝒓𝑑𝒓!

!𝜌!

!,!!!"#,!

𝒓!

𝜌! 𝒓! Φ!" 𝒓− 𝒓!  

where Φ!" 𝒓− 𝒓! is the square-well potential

Φ!" 𝒓− 𝒓! =∞, 𝒓− 𝒓′ < 𝜎!"                                    −𝜏, 𝜎!" ≤ 𝒓− 𝒓′ < 1.2𝜎!"    0, 𝒓− 𝒓′ ≥ 1.2𝜎!"                      

with 𝜎!" equal to the contact distance between species α and β and depth 𝜏 equal to the

scaled hydration enthalpy of the ions: 𝜏!" -= 0.0033166 eV, 𝜏!!!!= 0.01036 eV.

4. Ion-surface van der Waals interactions

Lifshitz theory was used to calculate ion-surface van der Waals interactions29,30. It links

ion dynamic polarizability and dielectric functions of the surface and the solvent giving

the following potential for these interactions

𝑉!!"#_!"# 𝑑 = −

ℏ(4𝜋)!𝑑! 𝑑𝜉

𝛼!∗ 𝑖𝜉𝜀! 𝑖𝜉

!

!𝜀! 𝑖𝜉

where 𝛼!∗ 𝑖𝜉 is the excess polarizability of ion in aqueous solution determined using

time dependent DFT.

5. Density profiles

Density profiles are calculated within cDFT via the minimization of the excess free

energy functional Fex with respect to the densities of all the species. The densities satisfy

the following equation

𝜌!(𝒓) = 𝜌!!"#$ 𝒓 𝑒𝑥𝑝 −𝑞!𝜑 𝒓𝑘𝑇 −

1𝑘𝑇

𝛿 𝐹!"!" + 𝐹!!!" + 𝐹!!"#!" + 𝐹!"#_!"#!"

𝛿𝜌!(𝒓)

where 𝜌!!"#$ is the bulk value for the atomic density of matrix and minor elements in

alloy and oxygen atomic density in oxide, respectively. We solve Poisson’s equation for

the electrostatic potential (φ(𝒓) ). The resulting system of equations was solved

iteratively to self-consistency using the numerical procedure described by Meng 31. In

particular, equilibrium ion density distributions were obtained using a relaxed Gummel

iterative procedure. Convergence was considered to be achieved when the maximum

difference between the input and the output density profiles between iterations was

smaller than 10-6.

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