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Refining Glass Structure in Two Dimensions
Mahdi Sadjadi∗
Department of Physics, Arizona State University, Tempe, AZ 85287-1604
Bishal Bhattarai†and D.A. Drabold‡
Department of Physics and Astronomy, Ohio University, Athens, OH 45701, United States
M.F. Thorpe§
Department of Physics, Arizona State University, Tempe, AZ 85287-1604 and
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, U.K
Mark Wilson¶
Department of Chemistry, Physical and Theoretical Chemistry Laboratory,
University Of Oxford, South Parks Road,Oxford OX1 3QZ, U.K
Recently determined atomistic scale structures of near-two dimensional bilayers of vitreous silica (using scan-
ning probe and electron microscopy) allow us to refine the experimentally determined coordinates to incorporate
the known local chemistry more precisely. Further refinement is achieved by using classical potentials of varying
complexity; one using harmonic potentials and the second employing an electrostatic description incorporating
polarization effects. These are benchmarked against density functional calculations. Our main findings are that
(a) there is a symmetry plane between the two disordered layers; a nice example of an emergent phenomenon,
(b) the layers are slightly tilted so that the Si-O-Si angle between the two layers is not 180◦as originally thought
but rather 175 ±2◦and (c) while interior areas that are not completely imagined can be reliably reconstructed,
surface areas are more problematical. It is shown that small crystallites that appear are just as expected sta-
tistically in a continuous random network. This provides a good example of the value that can be added to
disordered structures imaged at the atomic level by implementing computer refinement.
The atomic structure of covalent network glasses has been
a subject of both experimental and theoretical interest since
the introduction of the Continuous Random Network (CRN)
model by Zachariasen [1]. Almost all of these studies have
focused on the Pair Distribution Function (PDF) which is the
Fourier transform of a diffraction pattern [2]. Experimental
diffraction studies offer useful information, in particular re-
garding pair-wise ordering [3]. However, simulation models
can greatly aid the interpretation of these data as the atom po-
sitions are known unequivocally. As a result, information such
as the ring statistics, which is in many ways a natural language
for discussing network structure [4–6], is directly accessible.
While this work has been very informative and clearly estab-
lished the correctness of the CRN model for materials like vit-
reous silica, it is not accurate enough to distinguish between
different models with varying ring statistics etc. This situa-
tion has changed recently with the direct imaging of bilayers
of silica [7, 8] that has provided detailed information regard-
ing atomic positions.
Silica, SiO2, represents an archetypal network-forming ma-
terial. At ambient pressure the crystalline and amorphous
structures can be considered as constructed from corner-
sharing SiO4tetrahedral coordination polyhedra (CP) which
link to form a network. The complex linking of the CP may re-
sult in significant ordering on length-scales beyond the short-
range ordering imposed by the system electrostatics (effec-
tively controlled by the relative atom electronegativities) [9–
14].
Recently developed synthetic pathways have allowed thin
films of SiO2to be deposited on either metallic [7, 15, 16] or
graphitic [8] substrates whilst advances in imaging techniques
allow for true atomic resolution of the surface structure. Al-
beit, because the bilayer is a glassy material, it is not com-
mensurate with any substrate, and so we do not include the
substrate here.
Some of the thinnest films deposited are bilayers of corner-
(a)
(b)
(c)
FIG. 1. (a) A small piece of silica bilayer in which oxygen atoms
(red) form a tetrahedral network while silicons (blue) are located at
the center of tetrahedra. (b) The top view of the silica bilayer where
O and Si atoms are projected into the plane, with O forming a net-
work of corner-sharing triangles. (c) An alternative view where Si
atoms form a network of edge-sharing polygons (rings), while oxy-
gens are removed for clarity. This view is stressed in Fig. 2.
arXiv:1711.00596v1 [cond-mat.dis-nn] 2 Nov 2017
2
FIG. 2. The Cornell hnetwork viewed perpendicular to the plane
containing the bilayer with the O atoms removed for clarity, and only
the top layer of Si atoms shown as vertices. The atoms associated
with the blue and magenta bonds were not directly imaged but have
been added in the computer refinement. The blue and magenta bonds
highlight bonds reconstructed within the main body of the sample
and at the surface, respectively. Dashed lines highlight small sec-
tions in which an under-coordinated Si atom was required for filling.
The intensity of the red highlights the difference between the config-
uration relaxed with the spring and PIM potentials. The green circles
show small crystallites.
sharing SiO4CP in which all of the Si and O atoms obtain
their full (four- and two- respectively) coordination numbers.
Amorphous and crystalline films have been grown with both
states characterized by the presence of a mirror plane (which
houses a layer of O atoms which act as bridges between the
two monolayers [17]). Critically, the pseudo-two-dimensional
nature of these systems allows the ring structures to be directly
observed and hence offers a potentially unique insight into the
origin of any ordering on long length-scales. Silica can be
considered as a network of silicon atoms in which the nearest-
neighbor Si-Si pairs are dressed with O atoms. As a result,
the crystalline system can be considered as constructed exclu-
sively from a net of six-membered (Si-Si-Si...) rings, whilst
the amorphous systems are constructed from a distribution of
4- to 10-membered rings (Fig. 1). However, this new ex-
perimental information, whilst ground-breaking, is naturally
imperfect as the location of each atom has associated with it
a natural uncertainty which translates into an uncertainty in
atom-atom separations.
In this Letter, we show how value can be added by com-
bining the experimental image with computer refinement that
builds in the known local chemistry. Whilst no refinement of
the experimental data is required in order to obtain, for ex-
ample, accurate ring statistics, refinement is required in order
to address the geometrical issues associated with the network.
For example, value can be added on the effect of the pres-
ence of significant unimaged regions as well as on the subtle
variations in the structure perpendicular to the resolved plane
containing the bilayer.
In this Letter we focus on a single large sample of a bilayer
of vitreous silica imaged by the Cornell group [8] which we
will refer to as sample h, shown in Fig. 2, to distinguish it
from previous smaller experimental and computer-generated
samples [18]. The sample is ∼270 ×270 ˚
A2in area con-
taining 19,330 O and 9,492 Si atoms, and is the largest such
sample imaged at the atomic level of which we are aware.
Importantly, we are using the whole experimental sam-
ple, including voids, rather than selecting a more rectangu-
lar shaped section without voids, which would have thrown
out most of the experimental data. This is also significant as
the full configuration shows a number of interesting features.
For example, there are several regions which may be con-
sidered nanocrystalline showing relatively large numbers of
neighbouring six-membered rings (highlighted by green cir-
cles with a diameter of 9 ˚
A). Such regions are to be expected
staistically in a CRN and from previous studies [6] we find
that about 50% of all rings are sixfold and of these about 2%
are surrounded by 6 sixfold rings leading to a little microcrys-
tallite of 7 sixfold rings. The total number of rings in the
Cornell hsample is 1811, where we exclude surface rings that
do not have their full compliment of neigboring rings. Thus
we expect 1811 ×0.5 ×0.02 ≈18 of such regions which
is fortuitously exactly the number of regions shown by green
circles. So this certainly cannot be taken as any evidence for
microcrystallites as has been postulated at various times since
the original ideas of Lebedev and coworkers [19].
More obviously the configuration shows three relatively
large regions which were unable to be imaged (of approxi-
mate dimensions 160×40 ˚
A2, 50×20 ˚
A2and 10×10 ˚
A2respec-
tively) which resist reasonable attempts at computational fill-
ing (see below). A potential implication is that the underlying
surface (on which the bilayer has been grown) in some way
distorts the bilayer thus preventing effective imaging or per-
haps the network was never formed in these regions because
of surface roughness.
To construct the bilayer from the experimental image, O
atoms (which are not imaged) are placed midway between Si
atoms (which are imaged) thus forming a network of corner-
sharing O3triangles (each of which has an Si atom at the cen-
tre). The Si and O atoms planes are then separated, forming
trigonal pyramids with Si atoms at the apices. A mirror image
of these pyramids is joined to the original via O-atom bridges
to form the completed bilayer, resulting in an initial set of
180oSi-O-Si bond angles centered around the O atoms in the
mirror plane (Fig. 1). An important question involves the ex-
perimental length metric to ensure the correct calibration of
the image. We calculated the mean average length of the im-
aged nearest neighbour Si-Si distances as 3.097 ˚
A, which is
3
close to the expected value of 3.100 ˚
A for glassy silica struc-
tures [20], confirming the overall accuracy of the experiment,
and alleviating the need for any length rescaling. To recon-
struct the unimaged regions, we use mean bond length and
internal angles of rings to find the correct local topology. The
subsequent relaxation of the bilayer will fix the geometry en-
suring the proper bond length and angles.
This relaxation is carried out using model potentials of
increasing complexity. In the simplest case, the nearest-
neighbour O-O bonds are mimicked by harmonic springs with
lengths set as the mean average (2.645 ˚
A). This ensures that
the system does not have any internal degrees of freedom and
is minimally rigid or isostatic [21, 22]. A hardcore poten-
tial is added to prevent overlap of O atoms from different
tetrahedra as well as an RMSD (Root-Mean-Square Devia-
tion) term which penalizes deviation from the experimental
coordinates. This RMSD term involves the sum of squares
of the refined minus the experimental atomic positions and
is important as this maintains the overall area and alleviates
the need for additional boundary conditions to maintain the
sample area. Although proper boundary conditions for finite
pieces of amorphous systems can be designed [23], this sim-
ple potential can account for structural information extracted
in this Letter. Maintaining the configurational area is criti-
cal in avoiding, for example, unphysical overlaps in nearest-
neighbour tetrahedra in the absence of formal (electrostatic)
repulsions. The balance of the surface extension and the inter-
tetrahedral repulsions define an effective flexibility window of
acceptable structural solutions, of the type commonly associ-
ated with zeolites [24]. As a result, samples with irregular
boundary conditions are not a problem.
A second classical model used is a polarizable-ion model
(PIM) [25], specifically the TS potential [26] which utilises
pair potentials to model the Coulomb, short-range (overlap)
and dispersive interactions. The potential employs a combina-
tion of reduced ion charges and anion dipole polarisation (as
described in reference [25]). The results from the harmonic
potential model are used as the input with the PIM further re-
fining the results.
The most sophisticated method applied uses Density
Functional Theory (DFT). However, the method is too
computationally-demanding to apply to the experimental Cor-
nell hconfiguration. Therefore, a relatively small 1200 atom
periodic computer-generated model (with 200 Si atoms in
each monolayer) of a vitreous silica bilayer [27] was used.
Density functional calculations were undertaken with the code
SIESTA [28], with single-zeta basis and the local density ap-
proximation. Relaxation with a variable cell area resulted in
very little change. Stability of the relaxed model was also ver-
ified [29].
The result of the PIM refinement of Cornell his shown in
Fig. 2. The blue (bulk) and magenta (surface) bonds have been
computer-reconstructed, as described earlier. The interior re-
construction was deemed to be successful, as the differences
between the spring and PIM models were minor. These dif-
ferences are shown by the red shading where the darkest red
100 120 140 160 180
θSiOSi [o]
0
0.05
0.1
0.15
0.2
n(θ)
Expt.
Harmonic
PIM
DFT
Bulk
FIG. 3. The Si-O-Si bond angle distributions determined from the
original experimental configuration and from the bilayers obtained
using models of increasing complexity as well as for the bulk glass.
The peak at θSiOSi ∼145◦arises from the “in-plane” tetrahedral links
whilst the peak at ∼180◦arises from the central bridging oxygen
atoms between the two planes. The unrefined experimental result for
the Cornell hsample is shown in black where it was assumed that
the central bridging angle was exactly 180◦. The DFT calculation is
on a computer-generated periodic sample and acts as the best guide
for what to expect. The other two results are for the refined Cor-
nell hsample using both the harmonic model and the polarizable-ion
model as described in the text. Both show significant tilting as ex-
pected from the results of DFT, while maintaining the central sym-
metry plane.
corresponds to an atomic displacement of ∼0.5˚
A from the
original (unrefined) coordinates. This strongly suggests that
the network existed in these interior areas but was not imaged
reliably, rather than the networks growing around a pillar or
avoiding surface roughness on the substrate and never exist-
ing. At the surface, the difference between the spring and PIM
models was much greater as the reconstruction was not con-
tained within a small closed exterior perimeter.
In addition to in-plane information, refinement can provide
valuable information in perpendicular direction. As a bench-
mark of our model potentials, we have studied the Si-O-Si
angle, θSiOSi, as this contains important information on how
the tetrahedra are linked. Figure 3 shows the distributions of
θSiOSi for three models. The experimental structure (in which
linear Si-O-Si bridges between the two monolayers are im-
posed) shows a bond angle of θSiOSi ∼140.3o(with a FWHM
of ∆θ∼5.2◦) in the bilayer plane. All of the models generate
bimodal distributions in which the peak at θSiOSi ∼145omay
be assigned to the Si-O-Si triplets in the bilayer plane whilst
the peaks at θSiOSi >175ocorrespond to the triplets centred
around the bridging O atoms in the mirror plane (i.e. perpen-
dicular to the bilayer plane). There is not much latitude in the
in-plane values of this angle as they must be consistent with
the measured area and the known Si-O bond lengths, which
leads to a single peak in the θSiOSi ∼145◦. Fig. 3 shows
that the harmonic model reproduces the important high-angle
peak at θ∼178.5◦. The lower-angle peak is at θ∼140.9◦
4
FIG. 4. A section of the Cornell hnetwork shown along the plane
containing the bilayer with O atoms shown in red, and with Si atoms
at the center of the yellow tetrahedra. Note the symmetry plane of
the central O atoms and also the tilting of the tetrahedra away from
the vertical about the central plane.
(∆θ∼3.8◦) and some way below the DFT result.
The figure also shows the analogous distribution obtained
from the bulk glass at ambient pressure using PIM, which is
similar to distributions observed in bulk silicates [30, 31]. The
bulk distribution is significantly broader than those generated
for the bilayer with θ∼145◦and ∆θ∼36◦. The require-
ment for the relatively obtuse bond angles which characterise
the links between the two layers constrains the in-plane bond
angles to a relatively narrow range. For the intra-layer angles
all of the models show peaks at θ∼140 −142◦with the
harmonic potential showing a far sharper peak retaining the
symmetry plane.
However the bridging O angle is tilted and reduced to about
175.1◦. A Si-O-Si angle of 180osits on a local energy max-
imum [32] and, as a result, tilting is inevitable. A tilt in the
inter-layer bond angle is observed in all the models. At the
simplest level (harmonic potential) a relatively small deviation
from linear (θ∼178.5◦) is shown. As greater detail is added
to the models these angles become more acute with both the
PIM and DFT results showing peaks at θ∼175o. Figure 4
shows the configuration perpendicular to the plane contain-
ing the bilayer relaxed using the PIM and clearly showing the
tilted corner-sharing tetrahedra, with a peak at θ∼174.9◦.
At first sight this suggests an incompatibility with the ex-
perimental results where only a single layer is seen, with the
second layer of Si tetrahedra being exactly behind and under-
neath the first. However this can be maintained if there is a
symmetry plane involving the central O atoms, such that the
upper and lower tetrahedra tilt and pucker in the same way and
there is not a second image when the bilayer is imaged from
above, as shown in Figure 4. This conclusion is supported by
an entropy argument in which the bilayer with a mirror plane
is able to explore configurational space more effectively than
one without [17]. There are more degrees of freedom with
the symmetry plane present, thus increasing the entropy and
lowering the free energy, and hence leading to this unexpected
emergent phenomena. Thus symmetry is induced in a system
which at first sight seems a canonical example of a system
without symmetry. This argument is confirmed both by de-
tailed atomic computer modeling and by experiment, where
no shadow is seen beside each atom imaged, so that the sec-
ond layer must be exactly behind the first layer.
A feature to notice from Fig. 2 is that the polygons with
silicon atoms at the corners appear regular, having areas close
to that of regular polygons as has been previously noted [18].
This feature has been absent in computer generated models of
vitreous silica bilayer as the Si-O-Si angle of around 145◦in
the plane is hard to achieve in models while maintaining the
maximal convexity of Si polygons. Nature has found a way
and we need to understand better how this is achieved. Note
there is no difficulty in achieving regular polygons in samples
of amorphous As [27] where there are no bridging atoms to
contend with.
In this Letter we have described how computer-refinement
can add value to experimental images of disordered structures
at the atomic level. Although this is the first time this has
been attempted with an amorphous structure, with advances
in imaging, many more such systems are expected to be im-
aged in the near future. This somewhat parallels the proce-
dures employed to rationalise protein structure where the lo-
cal chemistry, via bond lengths etc is included to produce the
best possible structure [33]. We have shown that simple po-
tentials are adequate here, and as well as producing refined
coordinates for the bilayer (available upon request), we have
shown that the two layers are tilted while maintaining a flat
central symmetry plane of O atoms between the upper and
lower parts of the bilayer. It is remarkable that such symme-
try can exist in disordered system and this can be viewed as a
nice clean example of an emergent phenomena.
Future work will help determine how ubiquitous bilayer
structures of this type may be. It is possible, for example,
that forming such structures for systems such as GeO2may
be more problematic as a significantly larger tilt (θ180◦)
would have to be accommodated [32].
We should like to thank Berlin and Cornell groups for the
coordinates of their networks and for useful discussions. This
work used the Extreme Science and Engineering Discovery
Environment (XSEDE), which is supported by National Sci-
ence Foundation grant number ACI-1548562 [34]. Support
through NSF grants # DMS 1564468 (MFT) and # DMR
1506836 (DAD) is gratefully acknowledged.
∗mahdisadjadi@asu.edu
†bb248213@ohio.edu
‡drabold@ohio.edu
§mft@asu.edu
¶mark.wilson@chem.ox.ac.uk
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