theoretical study of silver nanostructures using … · 2016. 6. 10. · nanostructures and through...

International Journal of Physics

and Research (IJPR)

ISSN 2250-0030

Vol. 3, Issue 2, Jun 2013, 33-44

© TJPRC Pvt. Ltd.

THEORETICAL STUDY OF SILVER NANOSTRUCTURES USING GUPTA POTENTIAL

DINESH KUMAR1, VEENA VERMA

2, KEYA DHARAMVIR

3 & H. S. BHATTI

1

1Department of Physics, Punjabi University, Patiala, Punjab, India

2Department of Physics, Government College Naya Nangal Distt, Ropar, Punjab, India

3Centre for Advance Study in Physics, Punjab University, Chandigarh, India

ABSTRACT

Using Gupta potential silver nanostructures have been studied. The Silver nanowires with icosa structure are

found to be most stable among the studied structures. Cohesive energy, Young’s Modulus and Shear Modulus values have

been computed and comparisons of these values (except Poisson ratio) exceed that of bulk silver. Another striking

observation about silver nanostructures is that Young’s modulus increases with tube radius whereas shear modulus

decreases. This study can give more fundamental understanding of nanoscale machines from atomistic motions and

contribute to the design, manufacture and manipulation of nano-devices. The knowledge of the structure and stability of

silver nanocrystals is of great importance and to model the silver clusters, we make use of Gupta Potential.

KEYWORDS: Ag structures, U, nearest neighbor distance (b), between two adjacent rings (b’), Young’s Modulus (Y),

Shear Modulus (G), Poisson’s ratio (ν)

INTRODUCTION

Metal nanowires find increasing application as structural components in molecular electronic devices [1-2] and

nanoelectromechanical systems [3-4], the mechanical properties of which have attracted tremendous attention [5-7] due to

the gaining relevance in controlling functionality of nanodevices. An efficient and simple analysis of mechanical properties

of metal NWs is critical because of the need for nanotechnology and the understanding of the basic physics underlying

mechanical phenomenon encountered in nanomaterials. General theoretical studies on mechanical response of metal

nanostructures and through corrections to continuum mechanics theory [7-9] or molecular dynamics simulations [5,6,10].

The dependence of the mechanical properties on the geometric size of 1D nanomaterials is very important factor in

affecting the performance of those 1D nanomaterials in the NEMS devices. So for several different fashions of size

dependence of the elastic properties of nanomaterials have been revealed (1) Young’s modulus increases with decreasing

size e.g. in Ag and Pd nanowires. [8, 11] (2) Young’s modulus decreases with the decreasing size e.g. in Cr and Si

nanocantilivers [12-13] and (3) Young’s modulus shows little dependence of the size of the nanomaterials such as Au

nanowires [14]. Moreover, for a same nanomaterial, different research groups could observe and report the opposite size

dependence of its elastic properties.

In 2005, Xiao Cong-wen et al. [15] studied large scale silver nanowires with controlled aspect ratio were

synthesized via reducing silver nitrate with 1, 2-propanediol in presence of PVP.SEM, TEM and XRD were employed to

characterize these silver nanowires. The diameter of the silver nanowire can be controlled in the range of 100-400 nm by

varying the experimental conditions.

Bulk silver ([Kr], 4d10

, 5s1) has nearest neighbor distance equal to 2.89Å [16] and cohesive energy of

2.95eV/A.Based on first principle calculations on extended AgSWNTs and NWs, Elizondo et al. [17] studied most

energetically favorable and least favorable structures and found (3,2) and (3,3) AgSWNTs to be highest in total energy and

therefore, is the least favorable structure and (7,5) most energetically favorable structure. They used triangular lattice for

34 Dinesh Kumar, Veena Verma, Keya Dharamvir & H. S. Bhatti

generation of coordinates of atoms in AgNTs.We also make use of triangular lattice for generation of coordinates of

AgNTs of different diameters and also generate AgNWs related to icosahedral cluster. We will make use of well known

Gupta Potential to compute cohesive energy of these nanostructures of Ag.

Figure 1: Atomic Structure of the (4, 4) AgSWNT, Obtained by a Cylindrical Folding of the 2D Triangular Lattice,

Whose Two Basis Vectors are a1 and a2. Here, the Chiral Vector Ch=4a1+4a2

THE N- BODY GUPTA POTENTIAL

The Gupta Potential [18] for the metal bond is based on the second moment approximation of electron density of

states in tight binding model [19-20]. It contains an attractive and effective band term (due to large d band density of the

states) and short-range repulsive pair potential. It is expressed as

embedpair EEE

ij i

i j i

(r ) F( )

(1)

where is a short range pair potential and F is a many body embedding or glue function, i being an atomic

density function.

ij

ij

j o

r(r ) Aexp[ p( 1)]

r (1a)

j

ijI r )(

ij

i

j o

r[exp{ 2q( 1)}]

r

F( ) (1b)

and the overall expression for the glue function becomes,

Theoretical Study of Silver Nanostructures Using Gupta Potential 35

)}]1(2[exp{*)( o

ij

j r

rqsqrtF (1c)

(2)

The first term in equation [2] represents the Born-Mayer repulsion and the second n- body cohesive energy of the

metallic bonding. The parameters p and q depend on the atom type and are obtained by fitting to the bulk equilibrium

distance and elastic constants [19]. The value is obtained by minimizing the bulk fcc cohesive energy. The potential was

originally derived for modeling transition metals where the d band is most important but it has also been shown to be valid

for noble metals, where s-d hybridization would require including higher moments by simply extending the range of

potential [21].The potential has been used extensively in modeling transition and noble metal clusters [22-24] as well as

bulk metals [25].As n body Gupta like potential has also been recently parameterized for alkali metal clusters [26]

The dependence of the Gupta potential on atomic separation for dimer molecule of silver metal is shown in figure

2.The range of attractive interaction is defined by the parameter q and repulsive core by the parameter p.

The parameters of this potential for Ag as given by [27] are

A= 0.10310eV p= 10.85 q= 3.180 = 1.1895eV ron=2.89Å

We make use of two bond lengths in figure 2 for study of silver nanostructures i.e. 2.53 Å and 2.89 Å.The smaller

bond length was chosen because we might expect the bond length to be smaller in these structures as compared to bulk as

evidenced in previous studies with finite silver nanostructures. [28-29]

Figure 2: Gupta Potential for Ag (Two Atom Cluster) the Dotted Line is Plotted with ro=2.53 Å

and Solid Line with ro=2.89 Å

Cohesive energies for the clusters calculated using the above methods also agree as far as hierarchy of structures

is concerned. Thus the Gupta Potential is expected to yield realistic results.

STRUCTURE OF ARMCHAIR (OR ZIGZAG) SILVER NANOWIRE AND ICOSAHEDRAL

NANOTUBE

We consider two different types of nanowire – armchair (zigzag) silver nanotubes (AgNT) and icosahedral

nanowires (NW). The former structures are hollow tubes, and only the ones with small diameters are expected to be stable

2

1

1)( 0

2

1)( 0

12exp[1expn

ji n

ijn

ji n

ij

jr

rq

r

rpAV


while the latter has pentagonal cross-section locally, and has a single-atom chain running through the axis of the wire.

Coordinates of armchair, i.e., (n, n) or zigzag, i.e., (n,0) nanotubes are generated starting from the two dimensional

triangular lattice of silver atoms shown in figure 1 and figure 3. The primitive vectors are shown as a1 and a2 and nearest

neighbor distance is taken as the bond length, b. When this two dimensional lattice of silver is rolled to form a cylinder, it

forms an AgNT of a particular diameter depending on the number of atoms on the circumference of its cross section.

Figure 3: Triangular Network of Silver Atoms. Basis Vectors are Designated as a1 and a2. Each Tube is Labeled By

Two Integers, (n1,n2), Defined By The Rollup Vector R

sin

*5.0 bR , (3)

where

mSin

, (3a)

and b is the bond length (nearest neighbor distance). Although this distance is called ‘radius’, it is actually the center-to-

atom distance in the m-sided polygon, which is the shape of a cross-section of the NT. Because of this, the area of cross-

section of the wire is not equal to πR2. Smaller the diameter of the AgNT, larger is the departure of the cross-sectional area

from this value (πR2). This fact will be relevant when Young’s modulus is calculated. For the triangular lattice used here,

unlike the hexagonal one which is used to generate the carbon nanotubes [30], there is no difference between zigzag and

armchair directions. Similar observations have been made by Verma et.al. [31]. For example, to generate a (4, 4) AgNT,

the silver sheet of figure 1 is rolled into a cylinder in such a way that the solid line shown in the figure forms the

circumference of a normal cross-section of the wire. The radius R of such a tube, described as an (m, m) AgNT is given

by equation 3.

Silver Nanowire Icosa Structure

Figure 4: Silver Nanowire Icosa Structure


A unit cell of this structure consists of two regular pentagons one above the other, with one of the pentagons

rotated through 180º and with one silver atom at the center of these ten atoms; in addition there is one atom just below (or

above) it at the same distance as interplanar separation between the two pentagons. So there are total 13 atoms and 20

triangles , which is the same as one surface atom taken off the well-known icosahedral cluster Ag13 , which according to

Michaelian et.al.[32] others is the global minimum for a 13-cluster of silver. This wire can also be described as a (5,5) and

(3,3) armchair (or zigzag) tube with an extra line running through the centre. This structure (AgNW with pentagonal cross

section) has also been studied by J. Reyes-Gasga et.al. [33]

MINIMISATION OF ENERGY

After generating the coordinates of silver atoms on an AgNT or NW, the neighbors of each atom are found within

a range of 15.0 Å, which we find to be the effective cutoff radius for the potential being used. Energy of interaction of the

atom with each of these neighbors is calculated using the potential described in equation 2. Next, the coordinates of each

atom are modified in very small steps and the calculated energy is compared with the previous energy. The modified

coordinate is accepted if this energy is smaller than the previous one, otherwise we continue with the previous value of the

coordinate. This is done successively for all the coordinates (= total number of Ag atoms in the tube or wire); and

such a cycle is repeated several times till the energy of the tube is minimized. This is the process of relaxation under the

given potential.

RESULTS

Cohesive Energy and Structural Parameters

The value of cohesive energy obtained by us is –3.3504 eV/A for both short (25 atoms. length =12.16 Å (10.0 Å

minimized length) and long (49 atoms length = 24.32 (20.1Å minimized length) icosa NW. Each atom of this NW has the

same binding energy, except for those at the ends which have slightly less energy (-2.96 eV). The bond lengths were also

measured and it was found that the bond length between atoms on circumference varies from 2.96 Å to 3.21 Å and the

atoms on the wire axis were 2.49Å distance apart. The structure was found to have enhanced stability when the length was

increased. Similar observations have been made for armchair (or zigzag) tubes as well. For (4, 4) AgNT, the end ring

atoms have four neighbors each having bond length equal to 2.96 Å, 2.99 Å, 3.213 Å, 3.213 Å. respectively. All other

atoms have six nearest neighbors at 2.96 Å., 3.00 Å., 2.69 Å., 2.60 Å., 2.87 Å. and 2.92 Å. Energy of all the atoms (expect

end atoms) is almost constant (-3.29 eV/A) whereas end atoms have slightly less energy (-2.35 and -3.0 eV/A).For all the

(n,n) structures under study, the nearest neighbor distances were found to be between 2.73 Å to 2.705 Å.

Table 1: Cohesive Energy (U), b is the Nearest Neighbor Distance within the Ring, b’ between Two Adjacent Rings

in Case of AgNT and for AgNW it Represents Distance between Chain and Ring Atoms, Y, ν and G for Various

AgNTs

AgNT R (Å) No of

Atoms U (eV/A) b(Å) b'(Å) Y (GPa) ν

G

(GPa)

Bulk silver - - -2.95 [34] 82.7[35] 0.37[35] 30[35]

79[36] 0.38[36]

Icosa NW 2.453 49 -3.3504 2.9 2.96 230.9 0.19 234

(3,3) AgNT 1.665 30 -3.254 2.81 2.81 114.6 0.34 139.8

(4,4) AgNT 2.039 42 -3.2874 2.96 2.84 128 0.35 179.3

(5,5) AgNT 2.453 51 -3.2817 2.79 2.8 134.6 0.37 67.1

(6,6) AgNT 2.884 60 -3.2905 2.76 2.72 179.1 0.38 66.4

(7,7) AgNT 3.323 72 -3.2794 2.73 2.71 188 0.39 59

(8,8) AgNT 3.768 81 -3.272 190 0.41


Figure 5: Energy versus Length of Icosahedron NW (Ag 49)

Figure 6: U verses Length of AgNT (Triangular Lattice)

Figure 7: U versus Twist in Radian for AgNT (3, 3) (Triangular Lattice)



Young’s Modulus

After obtaining the relaxed position coordinates of all the atoms of an AgNT/ AgNW, we consider it under

longitudinal stress. This stress is simulated by keeping the mutual distance between the end rings fixed at slightly larger

(elongation) or smaller (compression) than the normal length. The coordinates of the rest of the atoms of the tube are

varied till minimum energy, or the relaxed value of energy at the given strain, is obtained. The force on an extended/

compressed wire is given by

l

UF

The small force F required to produce a certain amount of extension l, is given by

2

2

UF l

l

The standard expression for Young’s modulus is,

0

F aY

l l

(4a)

where l0 is the length of the tube and a is the cross sectional area. We take this area, to be the product of

circumference of the tube of radius R and the wall thickness t, i.e. 2πRt. We are assuming the AgNT to be cylindrical in

shape. Rolling of a triangular lattice into a (4,4) AgNT results in a tube with local cross section in the shape of a square of

thickness ‘t’. As we go along the tube, orientation of successive squares change, alternating, such that if one looks down

the wire, one perceives an octagonal cross section. As n increases (in an (n,n) tube) the tube attains a shape that is closer to

a cylinder and the area of cross section considered here is justified. For icosa NW the cross section considered is that of a

solid tube, not hollow.

As pointed out by Kumar et.al. [30] and Verma et.al. [31], for tubes with cross-sections of atomic dimensions, the

definition of cross-sectional area can be controversial and is non-trivial. We assume t to be equal to interatomic distance,

taken to be uniformly equal to 2.89 Ǻ (the bulk value) in this work. However, this also introduces uncertainty, since the


electron cloud of atoms is not distributed uniformly over the circumference, nor is the inter-atomic distance same for all

NWs or NTs.

Around the minimum of the U-l curve,

2)( lkU (4b)

where k is a constant

0

2

2

2

1

l

Uk . Using this, we obtain, from eq. 4a,

0

2

2

l

U

a

lY o

(4c)

The cohesive energy plotted against length for the AgNW and the AgNTs as shown in figures 5 and 6. The

calculated values of Young’s moduli are listed in table I for both varieties of nanotubes.For hollow AgNTs the thickness

of cylinder is taken as 2.58 Å as size of (Ag ion is 1.29 Å) for calculating Young’s modulus. The computed values of

Young’s modulii are listed in table 1 for both AgNW and other AgNTs.

Poisson Ratio

Another mechanical property of interest is the Poisson ratio, , which is given by the variation of the radius of

SWNT resulting from applying the axial strain on the tube.

o

os

o

o rrr

ll

rr

strainallongitudin

strainlateral

(5)

where

strainaxiall

l

0

,

rs is the radius of the strained tube and ro is the radius of the unstrained tube. . The larger diameter tubes have

larger Poisson ratio showing they are laterally more strainable/expandable as expected. We have calculated for the tubes

under study values are tabulated in Table I.

Shear Modulus

A lateral force, in the form of a twist, is applied to the nano tube and nanowire , keeping one of its ends fixed.

The total torsion that has to be given to the AgNW has been distributed equally along its whole length and again the

coordinates of the tube in minimum energy configuration are obtained. Based on the theory of elasticity, shear modulus at

the macroscopic scale is given by

)(tJ

TlG o

, (6)

where T, lo, and J(t) stand for the torque acting at the end of the AgNT / AgNW, the length , the total twist that

is applied and the cross sectional polar moment of inertia of the AgNT or AgNW respectively. The polar inertia J(t) is a

function of wall thicknes and for a AgNT with radius ro and wall thickness t is given by


44

222)(

tr

trtJ oo

(6a)

and for AgNW, a solid wire, 'thickness' = r0 + t/2 and 'radius' = thickness/2 and the polar inertia turns out to be

4

22)(

trtJ o

(6b)

Using these equations, the shear modulii of Ag nanowire and AgNTs have been obtained and the values are given

in table I.

Figure 12: Variation of Y and G with Radius of AgNT

CONCLUSIONS

In conclusion, using Gupta potential isolated silver nanoclusters have been studied and we have found essentially

equal structural stability for AgNTs and AgNW icosa structure. Various modulii of silver nanostructers have been

computed and results are encouraging. Therefore, we expect further experimental and theoretical efforts in the near future,

to confirm the above predictions and provide a complete characterization of silver nanoclusters. The possible existence of

novel physical and chemical properties of small silver and other metal nanoparticles provide motivation for further

theoretical and experimental studies on these systems, since they would be useful in the fabrication of new materials based

on these nanostructures.

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