Supplemental material for “Size-dependent elasticity of amorphous silica nanowire:
a molecular dynamics study”
Fenglin Yuan and Liping Huanga)
Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy,
New York 12180, USA
a) Electronic mail: huangL5@rpi.edu
1. Simulation Details
All simulations were conducted in the Large-scale Atomic/Molecular Massively
Parallel Simulator (LAMMPS) package (http://lammps.sandia.gov/) using a modified
version of van Beest, Kramer, and van Santen (BKS) potential1 with a short range cutoff
of 0.55 nm and a long-range Columbic cutoff of 1.0 nm. The Velocity-Verlet algorithm
was used for integrating the equations of motion and the Nose-Hoover thermostat2,3 and
barostat4 were used to control the system temperature and pressure when necessary. The
initial bulk silica liquid was obtained by heating cristobalite silica to 7000 K and
equilibrating it for at least 1 ns. This well-equilibrated silica liquid was then used as the
common starting point for preparing all three types of nanowires (NWs). The asquenched bulk silica glass sample is 5 nm by 5 nm by 8 nm in size and duplicated three
times along the x and y axis for preparing large NWs. Four parallel samples were
generated from the high-temperature liquid equilibrated for different amount of time.
a) Preparation of Cast NWs
1
For cast NWs, a cylindrical repulsive wall was applied after the bulk liquid was cut
into the desired geometry at 7000 K5. The repulsive wall applies a central repulsive force
on atoms to prevent the collapse of the system at high temperature. Four parallel cast
NWs were prepared for each radius.
b) Preparation of Cut NWs
The bulk liquid was quenched at a cooling rate of 10 K/ps from 7000 K to 300 K to
generate the bulk silica glass. The bulk silica glass was then cut into cylindrical shapes
with different radii and followed by 1 ns relaxation to allow for the surface relaxation and
reconstruction. Four parallel samples for the bulk silica glass were prepared and
correspondingly four parallel cut NWs were obtained for each radius.
c) Preparation of Cut-Strained NWs
After relaxation to reduce the surface energy and the total energy, cut NWs in
equilibrium shrank with respect to the stress-free bulk counterpart. They were then
stretched slowly along the axial direction to recover the original length cut from the bulk
silica glass, followed by 1 ns relaxation to obtain the cut-strained NWs. The same number
of cut-strained NWs was obtained from the cut NWs.
2. Computation of Nanowire Radius
An accumulative number density distribution (ANDD) method was developed to
estimate the cross-section area of a nanowire in order to compute the uniaxial stress and
subsequently the Young’s modulus accurately. The ANDD method is based on the idea
2
that the effective cross section area of a nanowire is equal to a circular cross section with
certain radius. The specific procedures are listed below:
1) For each nanowire, we first divide the nanowire along the axial length into multiple
sections, given that each section is thick enough (e.g., at least 1 nm in thickness).
2) For each section, we find the central axis by averaging over all atoms’ x and y
coordinates in this section (z-axis is the axial direction).
3) For each section, we calculate each atom’s distance from the central axis and bin
atoms according to the distance by a well-chosen bin-size dr to obtain the number
density distribution function f(r).
4) For each section, we then plot the accumulative number density distribution function
H(r) vs. r2 curve by integrating f(r) for all bins that are within certain distance r.
5) For each section, we fit the H(r) vs. r2 curve to a linear function and make sure the
non-linear part is excluded from the fitting process.
6) From the linear fitting, we can calculate the slope to be 1/R2, where R is the estimated
radius for this section.
7) Repeat steps 2)6) for each section and average over all sections to get a final
estimated radius for the nanowire.
8) One example for a cut nanowire with a radius of 2 nm is given here to illustrate how
the method works. Before relaxation, as shown by red circles in Fig. S1(a), H(r) vs. r2
is a perfect linear line and the slope indicates a radius of 2 nm. However, after
relaxation, due to the surface relaxation, the slope gradually changes to zero, as
indicated by blue squares. A linear region for the relaxed nanowire is shown by the
black line in Fig. S1(a). Depending on the range of data (the cutoff fraction in H(r))
3
used for the linear fitting, the computed radius changes slightly as seen in Fig. S1(b).
We set the cutoff fraction to be 0.90, the nanowire radius can be estimated to be 1.96
nm in this case.
4
FIG. S1. (a) H(r) versus r2 for a cut nanowire before and after relaxation (notes: only the
linear region is used for fitting shown as the black line), (b) the computed nanowire
radius as a function of the cutoff fraction.
3. Computation of Zero-strain Young’s Modulus of Nanowire
In order to obtain a good estimate of the zero-strain Young’s modulus, the strain
range from which it is extracted should be small enough considering the strong
nonlinearity effect in silica glass6. We conducted a tension test within 00.5% uniaxial
strain and a compression test within -0.50% uniaxial strain. Then the uniaxial stress vs.
strain curve was fitted linearly within the -0.5%0.5% strain range to calculate the zerostrain Young’s modulus. Error bars were obtained from four parallel samples.
4. Eigenstress Elasticity Model in Nanowire
FIG. S2. Two-step relaxation for a nanowire cut from a stress-free bulk sample.
The derivation of the eigenstress model in nanowire similar to Zhang et al.’s work on thin
film7 can be divided into two parts: a) description of the after-cutting relaxation b)
description of linear response to external loading. For part a), following Zhang et al.’s
5
approach7, we separate the relaxation process into radial and parallel relaxations as seen
in Fig. S2. In the radial relaxation, the nanowire is only allowed to relax along the radial
direction under a fixed axial length. After the radial relaxation, the radial dimension
changes from 2  R0 to 2  R=2  (R0+R), meaning the surface has a radial displacement
of R (could be positive or negative depending on the nanonwire is expanded or
contracted after the radial relaxation). After the radial and parallel relaxation, the
nanowire with original dimensions of 2  R0 by L0 cut from a stress-free bulk sample
changes into 2  Rini by Lini, with Lini=L0+L and 2  Rini=2  (R0+R+R), where R is
due to the Poisson’s ratio effect. The contracting strain after relaxation can be defined by
L0 and Lini:
 C  ln(Lini / L0 ),
(S.1)
The total energy of the nanowire can be expressed as:
UL  UR  WS  WC ,
(S.2)
where UR is the total energy after the radial relaxation, WS and WC are the work done by
the surface and the core during the parallel relaxation, respectively. Since
WS 
Lini
 (
S
 2    R)dL,
(S.3)
   R 2 )dL,
(S.4)
L0
and
WC 
Lini
 (
C
L0
where R is the radius of the nanowire, S is the surface stress (N/m) and C is the core
stress (N/m2). At equilibrium, the force must be self-balanced, such that:
6
F
U L
|L  Lini  0,
L
(S.5)
so we can obtain
( C  R  2   S ) |L  Lini  0,
(S.6)
 Cini  R ini  2   Sini  0,
(S.7)
i.e.,
The above analysis shows that after relaxation to minimize the surface energy and the
total energy, a free-standing nanowire subjected to no external loads is at equilibrium and
deformed with respect to its stress-free bulk counterpart.
For part b), if we apply a small external load to the equilibrated nanowire, the
nominal Young’s modulus Ya can be defined by
Ya 
 a
,
 a
(S.8)
where a is the applied stress and a is the strain. The force at any applied strain can be
derived by following the same procedure as for the parallel relaxation:
F   C    R 2  2   S    R,
(S.9)
then the stress is:
  F / (  R 2 )   C  2   S / R,
(S.10)
For small deformation, the core stress  C and the surface stress  S under an applied strain
 a can be written as:
 C   Cini  YCa*   a ,
(S.11)
 S   Sini  YSa*   a ,
(S.12)
and
7
where YCa* and YSa* are the Young’s modulus of the core and surface, respectively, at the
initial state (after the parallel relaxation). Then the nominal Young’s modulus can be
derived from Eqns. S.812 as:
Ya 
 a  ( C  2   S / R)

 YCa*  2  YSa* / R,
 a
 a
(S.13)
Furthermore, by taking into account of the nonlinear elasticity in both the core and the
surface, we can express YCa* and YSa* with respect to the state after the radial relaxation:
YCa*  YC*  2  YC1*   C  3  YC2*   C 2 ,
(S.14)
YSa*  YS*  2  YS1*   C  3  YS2*   C 2 ,
(S.15)
and
where  C is the contract strain along the axial direction after the parallel relaxation. YC2*
and YS2* are needed in S.14 and S.15, because measurements using high strength silica
fibers at high strains show that Young’s modulus of silica glass initially increases with
tensile strain, reaches a maximum then decreases at higher strains8. By combining Eqns.
S.13S.15, we finally can obtain:
Y a  YC*  2  YC1*   C  3  YC2*   C 2  2  (YS*  2  YS1*   C  3  YS2*   C 2 ) / R.
(S.16)
8
FIG. S3. Young’s modulus versus the inverse of the radius in (a) cut-strained and (b) cast
amorphous silica NWs.
9
FIG. S4. Fraction of 4-fold coordinated Si in cast, cut, cut-strained amorphous silica NWs
(the bulk limit is shown as dashed line for comparison).
10
FIG. S5. Ring size distribution in (a) cut and (b) cast amorphous silica NWs in
comparison with that in bulk silica glass.
11
FIG. S6. Two (blue) and three-membered ring center (red) distribution in cut (left) and
cast (right) nanowire of 5 nm in radius.
12
FIG. S7. Angle between the plane normal of two membered ring and the radial direction
of (a) cut and (b) cast nanowire of 5 nm in radius.
13
FIG. S8. Density profile in cut and cast nanowire of 5 nm in radius (the bulk limit is
shown as dashed line for comparison).
14
FIG. S9. Dependence of density on the radius of nanowire (the bulk limit is shown as
dashed line for comparison).
15
FIG. S10. Young’s modulus versus density for amorphous silica NWs and bulk silica
sample.
References:
1
K. Vollmayr, W. Kob, and K. Binder, Phys. Rev. B 54, 15808 (1996).
S. Nose, Mol. Phys. 52, 255 (1984).
3
S. Nose, J. Chem. Phys. 81, 511 (1984).
4
W. Shinoda, M. Shiga, and M. Mikami, Phys. Rev. B 69, 134103 (2004).
5
F. Yuan and L. Huang, J. Non-Cryst. Solids 358, 3481 (2012).
6
P.K. Gupta and C.R. Kurkjian, J. Non-Cryst. Solids 351, 2324 (2005).
7
T.-Y. Zhang, Z.-J. Wang, and W.-K. Chan, Phys. Rev. B 81, 195427 (2010).
8
J. Krause, L. Testardi, and R. Thurston, Phys. Chem. Glasses 20, 135 (1979).
2
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