Quantum, classical, and multiscale simulation of silica–water interaction: molecules, clusters, and extended systems
ABSTRACT Over the past 6years, we have engaged in a multifaceted computational investigation of water–silica interactions at the fundamental physical and chemical level. This effort has necessitated development and implementation of simulation methods including highaccuracy quantum mechanical approaches, classical molecular dynamics, finite element techniques, and multiscale 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.

Article: Working at the interface
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ABSTRACT: In this paper, a few problems based on my work at QTP are selected and organized with a focus on physical systems and processes that involve interaction between extended states in solids and localized states in molecules. Such interactions are ubiquitous in interfacial processes that stir an intense interest in the science community.Molecular Physics 01/2010; 108:32353248. · 1.67 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The hydration of fullerenelike silica molecules was studied by the density functional method (exchangecorrelation functional B3LYP, basis set 631G**). It was demonstrated that completely coordinated structures transform to more stable hydroxylated ones during hydrolysis. These in turn react with H2O molecules with the formation of hydrogen bonds. Keywordshydration–fullerenelike silica molecules–density functional methodRussian Journal of Physical Chemistry 01/2011; 85(5):841844. · 0.39 Impact Factor  SourceAvailable from: Vladyslav V Lisnyak[Show abstract] [Hide abstract]
ABSTRACT: This article describes the genesis of amorphous silica under highheat conditions from SiO2 molecules through protoparticles, primary particles, and aggregates to agglomerates using vibrational spectra and quantum chemical simulations data. The impact of small molecules (water, HCl, CO2) is also discussed. The article also explains the nature of the pyrogenic silica amorphism.Critical Reviews in Solid State and Material Sciences 04/2011; 36:4765. · 5.95 Impact Factor
Page 1
DOI 10.1007/s108200069009x
Journal of ComputerAided Materials Design (2006) 13:161–183© Springer 2006
Quantum, classical, and multiscale simulation of silica–water
interaction: molecules, clusters, and extended systems
HAIPING CHENGa,∗, LINLIN WANGa,b, MAOHUA DUa,c, CHAO CAOa,
YINGXIA 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, 804013393, USA
cDeptartment of Materials Science and Engineering, University of Illinois at UrbanaChampaign, 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 multifaceted computational investigation of
water–silica interactions at the fundamental physical and chemical level. This effort has necessitated
development and implementation of simulation methods including highaccuracy quantum mechani
cal approaches, classical molecular dynamics, finite element techniques, and multiscale 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, multiscale modeling, silica cluster, silica nanowire, 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 chemomechanical processes. Among
the best known is hydrolytic weakening of silica, a problem of concern in geoscience
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 closelyrelated 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, Email: cheng@qtp.ufl.edu
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HaiPing Cheng et al.
macroscopic and microscopic levels, there remain questions of basic mechanisms.
Moreover, modern nanofabrication techniques allow preparation of silica nano
particles [34–46], nanowires [47, 48], nanotubes [49], nanobelts [50], and thin films
of nanometer 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 chemomechanical
activity in the macroscopic systems often are of nanometer 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, firstprinci
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 chemomechanical 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 firstprinciples 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 multiscale 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 multiscale 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 nanoparticle (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 nanosize 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 rearrangement 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 nanowires, as well as surface sites
involving twomembered (2M) rings, which have been identified as an important and
unique feature [12, 13] of surfaces.
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Multiscale simulation of silica–water interaction
163
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 firstprinciples 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
PI2
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 groundstate 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 exchangecorrelation 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 selfconsistently 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 pseudopotentials [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 nonzero term in the gradient
of Eelec is from the derivative of the Hamiltonian operator of the electronic sys
tem. Compared to atomicsitespecific basis sets commonly used in quantum chem
istry, this is a considerable simplification of the interatomic force calculation. With
a sufficiently large cutoff 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] exchangecorrelation approximation were used, with a cutoff
energy of 62Ry for the plane–wave basis functions. With this combination of poten
tials and energy cutoff, the precision in binding energy is about 1.0kcal/mol for H,
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HaiPing 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 steepestdescent 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 biomolecules, [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 interatomic 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 multiscale
simulation of nanoindentation 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, electronion
interaction, and electron–electron interaction. IQ and IC are indices for particles in
the QM and CM regions, respectively. R∗
the linkatoms or pseudoatoms, 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
PIQ2
2mIQ
+
?
+U
IQ>JQ
ZIQZJQ
RIQ−RJQ+Eelec
?
??RIQ,R∗
??,
m
?;ρ [r,r∗]?
?
PIC2
2mIC
{RIC}
?
+U??RIQ,RIC
(2.3)
m, r∗are nuclear and electron coordinates for
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Multiscale simulation of silica–water interaction
165
region particles. U ({RIC}) is calculated by summing up all pair interactions between
any two CM particles (and threebody 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 pairwise 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 nonconserving
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 pseudoatom 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 basisset corrections. Since Teand Eeeare inde
pendent of the ion coordinates, the forces are from ∇RIQEeI, which contains deriva
tives of both local and nonlocal terms in the pseudopotentials, 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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HaiPing Cheng et al.
Figure 1. Hydroxylated silica clusters with a water molecule: (a) and (b) and (d)–(f) are ring structures
(c) is a chainstructured 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 atomiccontinuum 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 longrange 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 multiscale modeling is to find a suit
able balance between accuracy and efficiency, i.e., let the most dramatically changing
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Multiscale 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 multiscale 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 coworkers [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 coarsegrained 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 SiO2nanowire.
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 nanotubes, different types of nanotubes and nanowires have been stud
ied both in theory and experiment. For example, a SiSe2 nanowire has been
proposed and studied with MD [97–99]. In experiments, both SiO2sheathed Si nano
wires [100] and pure SiO2 nanowires [101–103] have been found. SEM and TEM
images show that the nanowires are several µm long with diameters of 10–50nm.
Electron diffraction on SiO2sheathed Si nanowires shows that the core is crystalline
Si and the cover is amorphous SiO2. In a pure SiO2nanowire, however, the structure
is also amorphous. While the structures of crystalline silica are well understood as
different arrangements of cornersharing SiO4tetrahedral, the structure of the amor
phous silica surface is still an open problem [104, 105]. Since we do not have a
welldefined structure for the amorphous SiO2nanowire 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 βcristobalite is
closest to that of amorphous silica. So βcristobalite often is used as a preliminary
model for amorphous silica. As the first step to understand the mechanical proper
ties of an amorphous SiO2nanowire, we constructed an αcristobalite nanowire 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 αcristobalite to βcristobalite. With further tensile
stretching, the nanowire become amorphous before it starts to fracture.
Page 8
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HaiPing 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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Multiscale 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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HaiPing 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 consistentmass matrix, is nondiagonal. With a set of interpolation
functions satisfying
(2.22)
HTH=I,
where M is diagonal and is called the lumpedmass 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 lumpedmatrix 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
longranged 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 interatomic
potential; see above.
We used a twodimensional FE grid formed by dividing the system into isopara
metric triangular elements with linear interpolation functions, Ng=3 and Nf=2. The
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Multiscale simulation of silica–water interaction
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 lumpedmass
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 interatomic potential defined in Eq. (2.4).
Page 12
172
HaiPing 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 OH 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 OH 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)36nanorod (the 108 nanorod); 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 108atom nanorod.
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
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Multiscale 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 nanorod
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 108atom nanorod 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 nanorod 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 pseudopotential codes in comparison with
experimental data and with the allelectron LDA calculation of our colleagues on
αquartz [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 PWSCFGGA calculations are also in good agreement with
the experimental values, with a discrepancy of less than 2% compared with 3–4%
for VASPGGA. 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 αquartz are in close agreement with the allelectron result from
Ref. [109]. As is often the case, LDA gives the right relative energies among different
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174
HaiPing Cheng et al.
Figure 3. (a)–(e) are αquartz, αcristobalite, βquartz, 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 spinstate 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
pseudopotential [60, 61] with a large cutoff 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
multiscale model described in Section 2.2 [32]. Figure 4a depicts a physical model of an
amorphous surface. That surface is prepared by a welldocumented 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 BOMDDFT [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 31quantumatom region (plus link
atom) embedded in a 104particle classical matrix (see Figure 4b). The results indi
cate a zerobarrier 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 31atom system but without the matrix), which shows
a very small hydration energy, the multiscale model provides a more realistic descrip
tion of the boundary condition of the quantum region and thus a better estimation
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Multiscale 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]
αquartz
αcristobalite
βquartz
βcristobalite
a(˚A) EXPTL
a(˚A) SIESTA
a(˚A) PWSCF
a(˚A) allelectron LDAa
c/a EXPTL
c/a SIESTA
c/a allelectron LDAa
c/a PWSCF
Ec (eV/SiO2) EXPTL
SIESTA
VASP
PWSCF
allelectron 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 multiscale simulations: (a) is a crosssectional view of the silica surface and (b)
is a closeup 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 nanowires
To understand size effects upon mechanical strength in silica, we have conducted
two groups of simulations comparing bulk silica and nanowires. The first group was
done in conjunction with FE embedding, while the second used periodic conditions