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DOI 10.1007/s10820-006-9009-x

Journal of Computer-Aided Materials Design (2006) 13:161–183© Springer 2006

Quantum, classical, and multi-scale simulation of silica–water

interaction: molecules, clusters, and extended systems

HAI-PING CHENGa,∗, LIN-LIN WANGa,b, MAO-HUA DUa,c, CHAO CAOa,

YING-XIA WANa, YAO HEa, KRISHNA MURALIDHARANa, GRACE

GREENLEEaand ANDREW KOLCHINa

aDepartment of Physics and Quantum Theory Project, University of Florida, Gainesville, FL, 32611,

USA

bNational Renewable Energy Research Laboratory, 1617 Cole Blvd., Golden, CO, 80401-3393, USA

cDeptartment of Materials Science and Engineering, University of Illinois at Urbana-Champaign, 1304

W. Green St., Urbana, IL, 61801, USA

Received 26 September 2005; Accepted 5 January 2006; Published online 25 July 2006

Abstract. Over the past 6years, we have engaged in a multi-faceted computational investigation of

water–silica interactions at the fundamental physical and chemical level. This effort has necessitated

development and implementation of simulation methods including high-accuracy quantum mechani-

cal approaches, classical molecular dynamics, finite element techniques, and multi-scale modeling. We

have found that water and silica can interact via either hydration or hydroxylation. Depending on

physical conditions, the former process can be weak (<0.2eV) or strong (near 1.0eV). Compared to

hydration, the latter process yields much larger energy gains (2–3eV/water). Some hydroxylated silica

systems can accept more water molecules and undergo further hydroxylation. We have also studied

the role of external stress, effects of finite silica system size, different numbers of water molecules,

and temperature dependences.

Keywords: amorphous silica, multi-scale modeling, silica cluster, silica nano-wire, water

1. Introduction

The ubiquity of both water and silica was summarized in the general introduction to

this collection of articles [1] and has been the subject of many, more detailed treat-

ments [2–5]. The great abundance of water and silicon dioxide inevitably results in

a rich array of phenomena of great significance in both nature and in the technol-

ogy of materials. Many of the natural phenomena occur in the Earth’s crust, hence at

elevated pressures and therefore are examples of chemo-mechanical processes. Among

the best known is hydrolytic weakening of silica, a problem of concern in geo-science

for almost a half century [6]. SiO2 surface hydrolysis and water molecule intrusion

in bulk silica via dissociation and breaking Si–O–Si bonds to form Si–OH hydroxyl

groups are two closely-related problem categories.

There exist both experimental [7–20] and computational [21–33] studies of the

absorption, diffusion, and penetration of H2O with respect to SiO2surfaces. Although

these have led to significant insight into the water–silica interaction at both

∗To whom correspondence should be addressed, E-mail: cheng@qtp.ufl.edu

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Hai-Ping Cheng et al.

macroscopic and microscopic levels, there remain questions of basic mechanisms.

Moreover, modern nano-fabrication techniques allow preparation of silica nano-

particles [34–46], nano-wires [47, 48], nano-tubes [49], nano-belts [50], and thin films

of nano-meter thickness [51, 52], as well as composite materials that consist of nano-

silica objects [39, 40, 44, 53]. These clearly do not behave as simple miniatures of

macroscopic systems, despite the fact that the regions of actual chemo-mechanical

activity in the macroscopic systems often are of nano-meter scale. Clearly a consis-

tent understanding requires that these size effects be accounted for.

For both bulk surfaces (e.g. quartz and cristobalites) and SiO2clusters, first-princi-

ples quantum mechanical (QM) calculations (“electronic structure calculations”) have

provided detailed insight into interactions with water [26, 27, 31, 54]. There are

also classical molecular dynamics (MD) studies of amorphous silica [22, 23, 25, 55].

Although there are many clever schemes to mimic reactivity with classical poten-

tials for MD, chemical reactions are intrinsically QM. Eventually therefore, all such

schemes will fail to be predictive because they cannot handle the chemistry com-

ponent of a chemo-mechanical process. Nevertheless, as discussed in Section 1 to

this collection of articles, the intrinsic complexity of the amorphous SiO2 network

and the richness of crystalline phases, makes it impossibly costly computationally to

perform exhaustive first-principles electronic structure calculations on suitably sized

samples. The best that can be achieved by pure QM is selected computations on

presumably paradigmatic systems. Therefore, a hybrid quantum and classical model

is not just superior to conventional approaches for amorphous systems (and their

comparison with ordered ones). Rather, such methods are unavoidable. In this vein,

there are a few multi-scale simulation methods [30, 32, 56] that combine QM and

classical descriptions. As also mentioned in Section 1, however, we have found it

important to rethink the multi-scale scheme from the perspective of the essential

QMs.

What happens when a water molecule adsorbs on a silica surface or interaction

with a small cluster or a nano-particle (a cluster has an extremely large surface to

volume ratio)? Experiments [3, 4, 7] show that a wide range of phenomena can occur.

Water can dissociate and break Si–O bonds, or, sometimes the surface appears to be

hydrophobic. Surfaces undergo dehydroxylation under heat treatment [7]. The com-

plexity of amorphous silica and nano-size silica provides a rich array of sites that

respond differently to water attack. The dissociation process can involve single water

or multiple water molecules as well as configuration re-arrangement after dehydroxy-

lation. This richness of behaviors and features presents difficult challenges to model-

ers and simulationists. However, advances in modern computers and computational

science techniques have made it possible to provide an accurate, qualitative, and

quantitative description of the microscopic pictures underlying experimental observ-

ables. In this paper, we review our investigations of water–silica surface interactions

using various simulation methods and discuss pertinent aspects of method develop-

ment. We address issue of energetics, reaction path and barriers, and proton transfer

processes. Our studies include finite clusters and nano-wires, as well as surface sites

involving two-membered (2M) rings, which have been identified as an important and

unique feature [12, 13] of surfaces.

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2. Method and method development

2.1. Born–oppenheimer dft molecular dynamics method

What was originally called the “Born–Oppenheimer local spin density molecular

dynamics (BO–LSD–MD)” method is a first-principles method to drive molecular

dynamics with QM forces. It was developed in the framework of density functional

theory (DFT) [57] by Barnett and Landman [58]. Because the method does not

depend on the local spin density approximation (LSDA) in DFT, here we refer to

the scheme as BO–DFT–MD. Its purpose is to obtain the geometry and energy of

clusters in their DFT ground electronic state. In BO–DFT–MD, the Hamiltonian of

the dynamical system (nuclei or ions) is written as

H =

?

I

|PI|2

2mI

+

?

I?J

ZIZJ

|RI−RJ|+Eelec({RI;r}),

(2.1)

where upper case letters represent nuclear quantities, and {r} are the electron coor-

dinates. Eelec({RI;r}) is the total ground-state electronic energy, hence consists of the

electron kinetic energy, plus electron–electron and electron–nuclear interaction ener-

gies. In the classical limit, the nuclei follow Newtonian dynamics such that the tra-

jectories of the ions are on the ground state potential energy surface,?

equations of motion for the nuclei are

I?J

ZIZJ

|RI−RJ|+

Eelec({RI;r}), and each term in Eq. (2.1) is obtained by numerical integration. The

MIdR2

I

dt2=−∇RI

?

I?J

ZIZJ

|RI−RJ|−∇RIEelec.

(2.2)

In BO–DFT–MD, the electronic energy Eelec({RI;r}) consists of the Hartree energy,

the exchange-correlation energy evaluated via a suitable approximation (nowadays,

the generalized gradient approximation or GGA [59]), and the electron–nuclear

interaction energy. The Kohn–Sham (KS) equations for all the electron orbitals of

the system are solved self-consistently at a given time step tn for a given nuclear

configuration {RI.}. The energy and forces on each nucleus are evaluated once the

iterative KS equation solution has converged. The KS wave functions are expanded

in plane–wave basis sets in conjunction with pseudo-potentials [58] to remove highly

oscillatory core KS states. The advantage of the plane–wave basis is its independence

of the nuclear positions, as a result of which the only non-zero term in the gradient

of Eelec is from the derivative of the Hamiltonian operator of the electronic sys-

tem. Compared to atomic-site-specific basis sets commonly used in quantum chem-

istry, this is a considerable simplification of the inter-atomic force calculation. With

a sufficiently large cut-off energy being the only parameter, the plane–wave, pseudo-

potential approach also reduces substantially the complication in testing the quality

of the basis set.

In applications to water–silica systems, the Troullier–Martin [60, 61] pseudo-

potential for electron–nuclear interaction and the Perdew, Burke, and Ernzerhof

(PBE) GGA [59] exchange-correlation approximation were used, with a cut-off

energy of 62Ry for the plane–wave basis functions. With this combination of poten-

tials and energy cut-off, the precision in binding energy is about 1.0kcal/mol for H,

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Hai-Ping Cheng et al.

Si, and O atoms. The Newtonian dynamics is solved via the Verlet algorithm [62]

with a typical time step of 2.0fs.

It should be mentioned that, in addition to studies in which the system parti-

cles follow Newtonian dynamics, the BOMD–DFT–MD scheme is very useful for

performing structural optimization. Three algorithms for structural optimization are

often used in combination: a modified steepest-descent method, simulated annealing,

and a mixture of the first two approaches. According to the physical conditions, con-

straints can be applied to the systems in both optimization procedures and dynamical

processes. Typically, several initial configurations for each cluster are used to search

for the ground state geometry in conjunction with optimization algorithms.

2.2. Hybrid quantum and classical molecular dynamics

Pioneering work regarding combinations of quantum forces with classical dynamics

can be traced back to at least as early as the 1970s [63, 64]. In the last 15years,

researchers have modernized the idea and rejuvenated the subject [65–78], inspired

by modern computer technologies and advancements in computational methods and

capabilities and motivated by the need for insight into complicated material systems.

Compared to the effort on bio-molecules, [65–78], rather fewer groups have

reported applications of such methods in materials research [30, 56, 79, 80]. The

approach is described below in terms of SiO2systems but is general. As discussed in

other papers of this collection as well, we focus on obtaining an accurate description

of inter-atomic forces rather than the total energy in order to model the mechani-

cal properties of the system properly. Our work also emphasizes minimization of the

artificial effects in treating the interface between classical and quantum domains. The

illustration given here is in the framework of DFT, but can be extended to other

forms of QM approximation. This development is a crucial step towards multi-scale

simulation of nano-indentation and crack propagation [81] in silica under the influ-

ence of water molecules.

In general, the Hamiltonian of a system that is partitioned into QM and CM

regions can be written as follows:

?

+

IC

where Eelec= Te+ EeI+ Eee, is the sum of the electron kinetic energy, electron-ion

interaction, and electron–electron interaction. IQ and IC are indices for particles in

the QM and CM regions, respectively. R∗

the link-atoms or pseudo-atoms, respectively, and m is the index of the link/pseudo

atoms. In our first attempt, a link atom is placed on a straight line between a QM O

atom and a CM Si atom at the QM–CM interface. The distance between the O and

link atoms is held at a constant of 1.82a0 (Bohr), a value obtained from reproduc-

ing the equilibrium structure of a training molecule, H6Si2O7. The first three terms

of Eq. (2.3) form the energy of the QM region in the presence of a group of link-

atoms. The fourth and fifth terms are the kinetic and potential energies of the CM

H =

IQ

|PIQ|2

2mIQ

+

?

+U

IQ>JQ

ZIQZJQ

|RIQ−RJQ|+Eelec

?

??RIQ,R∗

??,

m

?;ρ [r,r∗]?

?

|PIC|2

2mIC

{RIC}

?

+U??RIQ,RIC

(2.3)

m, r∗are nuclear and electron coordinates for

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Multi-scale simulation of silica–water interaction

165

region particles. U ({RIC}) is calculated by summing up all pair interactions between

any two CM particles (and three-body terms if they exist in the CM potential). The

sixth term U??RIQ,RIC

tions as for the CM–CM ion interactions. In other words, a CM ion does not distin-

guish between ions in the QM and CM regions. In our work, a pair-wise potential

function of the van Beest, Kramer, and van Santen (BKS)/TTAM form [82] as dis-

cussed at length in other articles of this collection,

??is the sum of all interactions between QM and CM ions.

The CM–QM ion interactions are chosen to be the same classical potential func-

φI,J(|RI−RJ|)=qIqJ

RIJ

+aIJe−bIJRIJ−cIJ

R6

IJ

(2.4)

is chosen to describe the classical interactions. Clearly this is not a necessity.

The detailed specification of a QM–CM interface always involves a certain degree

of arbitrariness depending on the specific nature of the application. In our work, we

adopt the principle of optimizing the continuity of forces across the interface as the

way of addressing that arbitrariness. We also neglect the constraint forces (that come

from the constraints imposed on the link atoms) on QM atoms at the boundary. This

approximation causes the energy associated with Eq. (2.3) to be a non-conserving

quantity but provides correct forces and dynamics within the limit of a classical force

field.

The link atoms in the system do not carry kinetic energy; they are used only to

terminate the wave functions of a finite subsystem. Unlike some other methods, in

which link atoms play a role in the QM–CM interaction directly, our approach elim-

inates the direct influence they would assert on the dynamics of the system had they

been allowed to participate as real particles. In a way, one can view an O–H unit as

a special pseudo-atom whose mass is all on the O atom site.

In BO–MD–DFT, the gradients ∇RIQEelec

straightforwardly by taking the gradients of the Hamiltonian matrix elements with

respect to the ionic position without basis-set corrections. Since Teand Eeeare inde-

pendent of the ion coordinates, the forces are from ∇RIQEeI, which contains deriva-

tives of both local and non-local terms in the pseudo-potentials, i.e.,

?Elc

Elc

eI=

I

?

where ρ is the total electronic charge density Vlc

are the coefficients and the kernel of the Kleinman–Bylander expansion [83], fj,σ is

the Fermi distribution function, and ψj,σ is the KS orbital with spin index σ. The

parameters in the classical potential for SiO2 are the BKS values [82], discussed at

length in earlier papers of this collection.

We have tested this idea on H6Si2O7, a molecule often used to represent a proto-

type defect structure on silica surfaces. This system has two isomeric forms, D2d and

C2v, both having bridging oxygen atoms in the center. We performed both QM and

??RIQ,R∗

m

?;ρ [r,r∗]?

can be evaluated

∇RIQEeI=∇RIQ

?

eI+Enlc

?

?

eI

?,

(2.5a)

d3rρ (r)

Vlc

I(|r−RI|),

????

(2.5b)

Enlc

eI=

j,σ

fj,σ

I,l,m

FI

l

?

d3rKI

lm

?r−RI

?ψj,σ(r)

????

2

,

(2.5c)

I

is the local potential, FI

land KI

lm

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Hai-Ping Cheng et al.

Figure 1. Hydroxylated silica clusters with a water molecule: (a) and (b) and (d)–(f) are ring structures

(c) is a chain-structured cluster.

CM simulations to minimize the total energies. Consistent with the findings of our

colleagues, who used more sophisticated quantum chemical methods [84], the QM

calculations yield nearly the same energy for the two isomers while the classical force

field gives a slightly lower energy to D2d. The subsequent studies are based on the

D2disomer (Figure 1b).

The QM–CM model also was applied to a H6Si3O9 ring cluster (Figure 1(d)–

(f)). The optimized structure demonstrates again the transparency of the interface.

The structure parameters, generated by the QM–CM hybrid method, are in excellent

agreement with the results from QM calculations in the quantum region as well as

with the results from classical calculations in the classical region.

2.3. Interface between finite element and MD

The reason to do atomic-continuum modeling is twofold. First, there are multiple

phenomena that couple strongly on different physical scales. For example, in crack

propagation, the bond breaking at the crack tip depends on the deformation of the

surrounding material, which in turn depends on the long-range strain field. Con-

versely, the dissipation of strain energy is through dynamical processes at the crack

tip including bond breaking, plastic deformation, and emission of elastic waves. All

of these processes happen at the same time when the crack propagates. So a success-

ful description of crack propagation requires simultaneous resolution at both atom-

istic and continuum length scales. Second, it is not possible to compute all the rele-

vant dynamical processes in the most accurate and intensive model with a reasonable

computational cost. The essential perspective of multi-scale modeling is to find a suit-

able balance between accuracy and efficiency, i.e., let the most dramatically changing

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Multi-scale simulation of silica–water interaction

167

region be dealt with via the most accurate method and treat the broad, surrounding

regions by less accurate and less computationally costly methods.

The multi-scale combination of MD and the finite element (FE) method is much-

studied as a means of investigating fracture and crack propagation in macroscopic

materials. One group of studies focuses on deriving the FE method from the underly-

ing atomistic model rather than from the traditional continuum model. In the quasi-

continuum technique proposed by Tadmor and co-workers [85–88], the energy of

each element is computed from an underlying atomistic Hamiltonian, such that non-

linear elastic effects can be included. In the coarse-grained MD proposed by Rudd

and Broughton [89], a similar idea is pursued. The interpolation functions in the FE

mesh are assembled from the atomistic model. The other group of studies focuses on

combining MD and FE through an interface. Kohlhoff et al. [90] introduced an inter-

face plane between the MD and FE regions to pass the displacements as boundary

conditions for the two regions. Abraham et al. [91, 92] and Broughton et al. [93] used

a scheme based on coupling through forces. In it, the FE elements sitting at the inter-

face plane can have forces of MD nature. Smirnova et al. [94] extended the imaginary

interface plane to a finite size. In our work, we proposed an improved MD/FE inter-

face with gradual coupling of force and used it to study the mechanical behavior of

a SiO2nano-wire.

Amorphous silica is the major constituent in optical fiber. Quartz is the mate-

rial for timing in electronic circuits. Other crystalline silica, such as cristobalite, can

be found in Si and SiO2 interfaces in microchips [95, 96]. Since the discovery of

carbon nano-tubes, different types of nano-tubes and nano-wires have been stud-

ied both in theory and experiment. For example, a SiSe2 nano-wire has been

proposed and studied with MD [97–99]. In experiments, both SiO2sheathed Si nano-

wires [100] and pure SiO2 nano-wires [101–103] have been found. SEM and TEM

images show that the nano-wires are several µm long with diameters of 10–50nm.

Electron diffraction on SiO2sheathed Si nano-wires shows that the core is crystalline

Si and the cover is amorphous SiO2. In a pure SiO2nano-wire, however, the structure

is also amorphous. While the structures of crystalline silica are well understood as

different arrangements of corner-sharing SiO4tetrahedral, the structure of the amor-

phous silica surface is still an open problem [104, 105]. Since we do not have a

well-defined structure for the amorphous SiO2nano-wire to start with, a closed crys-

talline structure should be used. Besides amorphous structures, silica can have as

many as 40 crystalline structures in nature [106]. Among these different silica poly-

morphs, only quartz (α,β) and cristobalite (α,β) are stable at atmospheric pressure.

As remarked above and elsewhere in this collection, the density of β-cristobalite is

closest to that of amorphous silica. So β-cristobalite often is used as a preliminary

model for amorphous silica. As the first step to understand the mechanical proper-

ties of an amorphous SiO2nano-wire, we constructed an α-cristobalite nano-wire and

used the combined MD/FE method to study its amorphization and fracture under

tensile stretch. As we will demonstrate, during the initial part of tensile stretching,

a phase transition occurs from α-cristobalite to β-cristobalite. With further tensile

stretching, the nano-wire become amorphous before it starts to fracture.

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Hai-Ping Cheng et al.

2.3.1. Finite element method

The FE method is a general approach to solve a differential equation approxi-

mately [107]. A continuous system has infinite degree of freedom. The FE method

uses a finite number of degrees of freedom to approximate the continuous solution.

When a continuous system is divided into a FE mesh, the displacement field uc(r)

within an element can be interpolated by the local displacement on the nodes of that

element ue

ias

uc(r)=

Ns

?

i=1

Hi(r)ue

i,

(2.6)

where Hi(r) is the interpolation or shape function and 1≤i≤Ng, with Ng, the num-

ber of nodes in each element. The continuous strain within an element can be defined

symmetrically as [108]:

εc

µν(r)=

?

1−1

2δµν

? Ng

?

i=1

?∂Hi(r)

∂rν

ue

i,µ+∂Hi(r)

∂rµ

ue

i,ν

?

,

(2.7)

where all quantities are written in Cartesian components and, 1 ≤ µ,ν ≤ Nf, with

Nf the number of degrees of freedom of each node. These equations usually are

expressed in matrix form

uc(r)=H(r)ue

and

(2.8)

εc(r)=D(r)ue,

where the local displacement ueis written as a vector of Ng×Nf dimensions. The

matrix D(r) is the strain–displacement matrix as defined from Eq. (2.7).

In solid mechanics, the FE method is introduced as a minimization of the total

potential functional [107]

?

where σ is the stress, p the body force per unit volume, and q is the applied surface

force per unit area. The FE method often is used in the elastic regime, which means

small strain, harmonic response, and no plasticity. The corresponding stress–strain

relation is linear,

(2.9)

?=1

2

?

?εd?+

?

?

?uTpd?+

?

S

uTqdS,

(2.10)

σ =Cε=CDu,

where C is the elastic matrix. As the system is divided into a FE mesh, the elastic

potential energy functional can be written as

?

=

e

?e

?e

(2.11)

?=

e

?e

?

?1

2

?

ueTDTCDued?+

?

ueTHTpd?+

?

Se

ueTHTqdS

?

.

(2.12)

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Multi-scale simulation of silica–water interaction

169

Energy minimization to determine the equilibrium configuration of the system gives

??

from which

0=∂?

∂ue=

?

DTCD)d?

?

ue+

?

?

HTpd?+

?

∫

TTqdS

(2.13)

Keue+Fe=0,

where

?

is the local stiffness matrix and Feis the nodal force that results from the last two

terms in Eq. (2.13). There are as many as Ne equations like the one in Eq. (2.14),

since Neis the total number of elements in the system. These equations are coupled

through ue. They can be assembled as one global matrix equation,

(2.14)

Ke=

?

DTCDd?

(2.15)

Ku+F=0,

where K is the global stiffness matrix and u is the generalized displacement matrix.

To this point, we have only elastic statics, perhaps the most common application

of the conventional FE method. To consider elastic dynamics, we have to introduce

the kinetic energy for an element,

?

=1

2

?e

=1

where

?

is the local mass matrix and ρ (r) is the density of the material. If we construct the

Lagrangian and use the variational principle as before, a global dynamical equation

can be obtained

(2.16)

Te=1

2

?e

ρ (r) ˙ ue(r)2d?

?

?H˙ ue?Tρ (r)?H˙ ue?d?

2˙ ueTMe˙ ue,

(2.17)

Me=

?e

ρ (r)HTHd?

(2.18)

M ¨ u+Ku+F=0.

With the condition of no external force and free boundary condition, the dynamical

equation becomes that of free response,

(2.19)

M ¨ u+Ku=0.

The harmonic solution u=u(0)eiωtgives an eigenvalue equation

??−ω2M+K??=0,

(2.20)

(2.21)

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Hai-Ping Cheng et al.

which is the same as the dynamical matrix equation for phonons in a crystal lattice

[109]. With the condition of no external force and strain applied on the boundary of

the system, the dynamical equation is now

¨ u =−M−1Ku.

Generally M, the consistent-mass matrix, is non-diagonal. With a set of interpolation

functions satisfying

(2.22)

HTH=I,

where M is diagonal and is called the lumped-mass matrix.

The dynamical equation can be solved in the central difference method (Verlet)

or Newmark’s method [107]. The former method is simple and very computation-

ally efficient when combined with the lumped-matrix approximation. Each individ-

ual node vibrates linearly. The latter method is more stable numerically but involves

implicit solution of a matrix equation, hence is more computationally demanding.

(2.23)

2.3.2. Hybrid MD/FE: new gradual coupling

In our approach, the system is partitioned into three regions, to wit, the core MD

(CMD), dilute FE (DFE) and transition (TRN) regions. The mesh of FE nodes in

the TRN region matches the crystal lattice sites. The total Hamiltonian of the sys-

tem is

Htot=HCMD({r,˙ r} ∈CMD)+VCMD/TRN({r}∈CMD/TRN)

+HDFE({u(r), ˙ u(r)}∈DFE)+VDFE/TRN({u(r)}∈DFE/TRN)

+HTRN({u(r), ˙ u(r),r}∈TRN).

In HCMD, HDFE and HTRN, we include the kinetic energy from each region and the

contribution of the potential energy between any two particles or connecting nodes if

they both are in the same region. In VCMD/TRNand VDFE/TRN, we include the interac-

tion between two particles or connecting nodes which are in adjacent regions. Inside

the TRN region, we have

(2.24)

HTRN({u(r), ˙ u(r),r}∈TRN)=TTRN(˙ u(r))+w(|r|) VMD(r)

+[1−w(|r|)] VFE(u(r)),

(2.25)

where the weight function, w(r), is determined by the distance of the nodes from the

CMD and DFE regions. The forces between two FE nodes (or two particles) in the

TRN region are calculated by both FE and MD according to a chosen weight func-

tion. The relative weight is determined by the distance of the nodes/particles from the

CMD and DFE regions. So the hybrid force in the TRN region can change from the

long-ranged MD interaction (in the case that the nodes are very close to the CMD

region), to the nearest neighbor FE force (in the case that the nodes are close to

the DFE region). For the studies discussed here, we also used the BKS inter-atomic

potential; see above.

We used a two-dimensional FE grid formed by dividing the system into iso-para-

metric triangular elements with linear interpolation functions, Ng=3 and Nf=2. The

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171

third dimension is treated as uniform, continuum medium. The total numbers of ele-

ments and nodes are Ne and Nd, respectively. The potential energy is

VFE=1

2

Ne

?

m

6

?

p,q=1

um

pKm

pqum

q

(2.26)

and the stiffness matrix is

L

4Am[Dm]T[C][Dm],

where L is the thickness in the third dimension and Amis the area of the element.

For such a system the elastic constant [C], is a (3×3) matrix with the third dimen-

sion decoupled from the other two,

00

C14

[K]m=

(2.27)

[C]=

C11

C12

C12

C11

0

0

.

(2.28)

The strain–displacement matrix [D] consists of coordinate differences of the nodes on

each element. It is of dimension (3×6). The force on each element is

Fm

q,

p=Km

pqum

(2.29)

where the indices p and q run from one to six. For obtaining the dynamics of the

FE nodes, it is necessary to project the forces in Eq. (2.29) onto each FE node and

combine them with forces from atomistic region,

Fn

i=

Ne

?

m

3

?

l=1

δnmlFm

p,

(2.30)

where the index i runs from one to two. The kinetic energy is

TFE=1

2

Ne

?

m

6

?

p,q=1

m ˙ um

pMm

pq˙ um

q=

?

Nd

?

n=1??˙ un??2

(2.31)

and the mass matrix is

Mn=ρL

Ne

?

m

6

?

p,q=1

δnmAm

3,

(2.32)

where ρ is the bulk density of the material and we have used the lumped-mass

approximation. After the forces are calculated from different contributions, we use

the Verlet algorithm to integrate the dynamical equations for both MD and FE. The

FE equation is written as

Mn¨ un=−w(|rn|)

∂

∂rn

?

s?=n

s∈TRN

φ(rn−rs)+

?

1−w(|rn|)

?Ne

?

m

3

?

l=1

δnmlKm

pqum

q,

(2.33)

where φ is the inter-atomic potential defined in Eq. (2.4).

Page 12

172

Hai-Ping Cheng et al.

Table 1. Hydration energy of SiO and SiO2 with 1–4 water molecule (all values in eV)

ClustersSiO+H2O SiO+2(H2O)SiO+3(H2O)SiO+4(H2O)

SiO

SiO2

0.30

1.04

0.80

0.82

0.30

0.42

0.05

Hydroxylation 2.16

3. Results

3.1. SiO and SiO2molecule and their interactions with water

In gas phase, both SiO and SiO2 can be observed. The comparison between SiO

and SiO2molecules is very interesting. Both can interact with water either via hydra-

tion or hydroxylation. In former process, a water molecule does not dissociate, in

the latter, it does. From calculations, SiO is more stable than SiO2 when interacting

with water. Both hydration and hydroxylation energies for one SiO2are higher than

those for one SiO. Table 1 lists the calculated energies of these two molecules using

BO–MD–DFT with the PBE GGA; recall prior discussion.

One sees that the hydration energy of a SiO with one to four waters is always

higher than the corresponding energy for one SiO2. When hydrated by four water

molecules, a SiO2 will react spontaneously with water, behavior which was not

observed in the SiO system. The hydroxylation gain of SiO–H2O is 1.67eV, an energy

gained by dissociation of a water and formation of two O-H bonds. This energy is

consistently over 2eV in pure SiO2systems.

3.2. SiO2and SinOmHl clusters: size dependence and interaction with water

We have performed a systematic investigation on the size dependence of (SiO2)nOmHl

systems, which can take many isomeric forms [31]. These clusters can be classified

into two groups: hydrophobic ones and hydrophilic ones. We emphasize here that this

specific classification only applies to hydration and not hydroxylation, i.e., the process

that does not involve water dissociation. Figure 1 depicts a water interacting with a

special family of silica clusters that are already hydrated. For dry silica, i.e., clusters

that do not have an O-H group (one may construct such clusters by removing an

O–H group from a Si atom and an H atom from one oxygen atom), the hydration

energy is often close to 1eV (see Table 2), with some exceptions for “magic number”

clusters. For example, Figure 2 shows various positions at which a water can interact

with a (SiO2)36nano-rod (the 108 nano-rod); the highest hydration energy is less than

0.2eV (Table 3). Hydrophobic clusters often have hydration energies under 0.3eV, or

even lower. This is the case for all clusters in which oxygen bonds are saturated in

the Si–O network or by terminal O–H units. Tables 2 and 3 give detailed energetic

information on both clusters and the 108-atom nano-rod.

3.3. SiO2crystalline structures and surface studies

To obtain a comparison between clusters and bulk systems, we have performed

a thorough investigation of a few common crystal structures (Figure 3). The

Page 13

Multi-scale simulation of silica–water interaction

173

Table 2. Cohesive energy Eb and hydration energy Ehyd (4th and 4th columns), all in eV. The two

numbers in parenthesis in the last column are energies for dissociated water states. Strictly speaking,

they are not hydration energies but reaction energies. Note that, for a given n in the third column,

there can be more than one m

n

(SiO2)nEb/atom(SiO2)n(OH2)mEb/atom (SiO2)n+ H2O(SiO2)n(OH2)m + H2O

1

2

3

3

108 nano-rod

4.35

5.02

5.20

5.30

6.01

4.17 (m=2)

4.63 (m=2)

4.97 (m=2)

4.82 (m=3)

1.04

0.90

0.90

0.95

0.20 (1.20)

0.26

0.25 (−0.08)

0.18

Table 3. Hydration energy of rings and 108-atom nano-rod for various sites (as shown in Figure 2)

(SiO2)n(OH2)n

n=2

4.6

0.20

0.03

n=3

4.82

0.25

0.05

n=4

4.84

0.12

0.11

n=5

4.83

0.27

0.18

n=6

4.85

0.20

0.019

Eb/atom (eV)

Hydration E

H.E. for rod (a)–(e)

Figure 2. Interaction of a water molecule with a nano-rod at various sites (see Table 3 also).

configuration of atoms in a unit cell as well as lattice constants are fully optimized to

minimize the cohesive energy. Table 4 gives the calculated structure parameters and

cohesive energies for various plane–wave pseudo-potential codes in comparison with

experimental data and with the all-electron LDA calculation of our colleagues on

α-quartz [110]. The calculated cell parameters do not vary more among various the-

oretical treatments any more than is common for GGA versus LDA DFT exchange-

correlation (XC) models and compare reasonably well with experimental value. The

cohesive energies from PWSCF-GGA calculations are also in good agreement with

the experimental values, with a discrepancy of less than 2% compared with 3–4%

for VASP-GGA. With LDA, the errors are around 16%, consistent with the well-

known over binding tendency of LDA. Notice that both the VASP and PWSCF LDA

cohesive energies for α-quartz are in close agreement with the all-electron result from

Ref. [109]. As is often the case, LDA gives the right relative energies among different

Page 14

174

Hai-Ping Cheng et al.

Figure 3. (a)–(e) are α-quartz, α-cristobalite, β-quartz, and β–cristobalite

silica crystalline structures, a result that also has been found by other groups [114,

115]. It should be pointed out that Demuth et al. [114] found much higher silica

crystal binding energies using the VASP code, e.g., 23.83 eV/SiO2 with GGA for

α–quartz. We have obtained a similar value before including the correction for the

atomic spin state. As discussed in the cluster section, the true ground states for Si

and O atoms are triplets. The spin-state corrections to cohesive energies in PWSCF

GGA and LDA are 3.99 and 3.68eV/SiO2, respectively. In VASP, the corrections are

3.85 and 3.69eV/SiO2for GGA and LDA, respectively. After adding the corrections,

the binding energies are much closer to the experimental values. It should also be

pointed out that the cohesive energies from SIESTA are not as good those from

PWSCF or VASP. Thus, the correct energy ordering of crystalline phases given by

SIESTA (GGA) could be misleading. Among the SIESTA, VASP, and PWSCF cal-

culations, the PWSCF results should be most accurate, since the Troullier–Martin

pseudo-potential [60, 61] with a large cut-off energy was used. With the GGA XC

approximation, both VASP and PWSCF yield an incorrect sequence of structures but

by very small energy differences. The correct ordering given by SIESTA could very

well be accidental, and one should be very cautious in drawing any conclusion based

on these numbers. Furthermore, the differences in energy among all the crystalline

structures are so small that it is very possible that none of the DFT codes with pres-

ent XC approximations can actually make prediction with such precision even though

numerical convergence has been achieved in the calculations.

One of our major efforts in recent years is to study the water–SiO2surface using the

multi-scale model described in Section 2.2 [32]. Figure 4a depicts a physical model of an

amorphous surface. That surface is prepared by a well-documented method [55] using

classical molecular dynamics. We then divide the surface into a classical and a quan-

tum region. The quantum region is described, as before, by BO-MD-DFT [58] with the

GGA XC approximation. Three different sizes of quantum region are used to exam-

ine the energy convergence with respect to what is a methodological choice. For the

investigation of reaction barriers, we have used a 31-quantum-atom region (plus link

atom) embedded in a 104particle classical matrix (see Figure 4b). The results indi-

cate a zero-barrier reaction when we use more than one water molecule. For a single

water molecule, the reaction barrier is found to be 0.4eV. Compared to an isolated

cluster model (using the same 31-atom system but without the matrix), which shows

a very small hydration energy, the multi-scale model provides a more realistic descrip-

tion of the boundary condition of the quantum region and thus a better estimation

Page 15

Multi-scale simulation of silica–water interaction

175

Table 4. Calculated structure parameters and cohesive energies, with the PBE GGA, for four crys-

talline structures of silica. Values in parenthesis are from LDA calculations. Experimental values are

as found in previous publications [111–113]

α-quartz

α-cristobalite

β-quartz

β-cristobalite

a(˚A) EXPTL

a(˚A) SIESTA

a(˚A) PWSCF

a(˚A) all-electron LDAa

c/a EXPTL

c/a SIESTA

c/a all-electron LDAa

c/a PWSCF

Ec (eV/SiO2) EXPTL

SIESTA

VASP

PWSCF

all-electron LDAa

4.92

5.02

5.06 (4.88)

(4.931)

1.10

1.10

(1.087)

1.11 (1.10)

19.23

21.34

19.98 (22.27)

19.56 (22.37)

(21.74)

4.96

4.93

5.13 (5.00)

5.00

5.18

5.13 (5.02)

1.39

1.41

1.09

1.09

1.40 (1.39)

19.20

21.30

20.01 (22.25)

19.58 (22.35)

1.09 (1.10)

19.18

21.29

19.97 (22.25)

19.55 (22.34)

21.13

19.99 (22.21)

19.55 (22.31)

aRef. [110].

Figure 4. Model for multi-scale simulations: (a) is a cross-sectional view of the silica surface and (b)

is a close-up of part of the quantum region that reacts with two water molecules. A concurrent pro-

ton transfer is observed.

of hydration energies. Table 5 shows the results of water binding/dissociation energies

for various model surfaces. Note that two potential energy function BKS and Watan-

abe [116] have been used; the latter gives better energetics while the former focuses on

better forces.

3.4. Stress–strain relation for bulk silica and silica nano-wires

To understand size effects upon mechanical strength in silica, we have conducted

two groups of simulations comparing bulk silica and nano-wires. The first group was

done in conjunction with FE embedding, while the second used periodic conditions