The process of coalescence of various metal nanoparticles has been studied by the Monte Carlo method.
The interaction of nanoparticles is described by a multiparticle Gupta type potential. An algorithm of recognizing
and estimating a dihedral angle at the neck is developed. The dihedral angle between metal nanoparticles
during their sintering is estimated

Bogatov, A.A.

Bogdanov, S.S.

Kolosov, A.Yu.

Komarov, P.V.

Myasnichenko, V.S.

Sdobnyakov, N.Yu.

Sokolov, D.N.

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 9 No 5, 05042(4pp) (2017) Том 9 № 5, 05042(4cc) (2017)   2077-6772/2017/9(5)05042(4) 05042-1  2017 Sumy State University Estimation of the Dihedral Angle Between Metal Nanoparticles During Their Coalescence  A.Yu. Kolosov, D.N. Sokolov, N.Yu. Sdobnyakov*, P.V. Komarov, S.S. Bogdanov, A.A. Bogatov, V.S. Myasnichenko  Tver State University, 35, Sadovyi Per., 170002 Tver, Russia  (Received 19 June 2017; revised manuscript received 18 August 2017; published online 16 October 2017)  The process of coalescence of various metal nanoparticles has been studied by the Monte Carlo method. The interaction of nanoparticles is described by a multiparticle Gupta type potential. An algorithm of rec-ognizing and estimating a dihedral angle at the neck is developed. The dihedral angle between metal na-noparticles during their sintering is estimated.  Keywords: Nanoparticles of gold, silver, and copper; Coalescence; Neck, dihedral angle; Monte Carlo method; Gupta potential.  DOI: 10.21272/jnep.9(5).05042 PACS numbers: 61.46. + w, 66.30. – h, 02.70.Uu  1. INTRODUCTION  The properties of nanostructured materials are de-termined predominantly by the specific features of in-teratomic interaction in their surface layers and the quantum confinement effects. Nanosized structures are of considerable interest in both fundamental and ap-plied science. The interest in this field of science, which is among the most extensively developing fields, is re-lated to both the fundamental physical issues and phe-nomena and the prospects of creating new quantum devices and systems with broad possibilities for nanoe-lectronics, metrology, and measurement techniques. The synthesis of nanoparticles and nanostructures, as well as experimental research in this field, is usually difficult and expensive. Theoretical approaches to the investigation of nanosystems are yet not sufficiently elaborated. This circumstance makes the development of analytical and numerical methods for predicting the properties of nanoparticles and nanomaterials as well as for monitoring processes in these systems an espe-cially topical task. Nanoclusters in a condensed state possess physical characteristics different from those of the correspond-ing macroscopic crystals. In addition, nanoclusters ex-hibit new optical and magnetic properties dependent not only on their size, but also on the mode of organiza-tion or self-organization. In this context, there appear tasks related to reduction in sizes of systems and con-trol over their structures. In particular, a decrease in dimensions of elements for microelectronics leads to problems related to the creation of nanocontacts and nanowires [1, 2], which must possess both high struc-tural stability and sufficient electric conductivity so as to meet requirements of the new stage of technology development. Creation of new magnetic memory devic-es, nanodiodes, nanowires, etc. poses the problem of controlled synthesis and organization of nanoclusters. In this respect, the ability of predicting and controlling the size and structure of nanoparticles becomes very important, since the arrangement of atoms is a key factor that determined the properties of these particles and influences the strength and elasticity of related nanostructures. Coalescence is among the processes that can be used for controlling the structure and di-mensions of particles.  There are three mechanisms involved in the process of nanoparticle formation in an inert gas medium: co-agulation, coalescence, and aggregation on a small clus-ter-nucleus. The knowledge of a dominating mechanism and peculiarities of each one under given external con-ditions provides a basis that allows us to control the structure and size of particles.  According to Arcidiacono et al. [3], the coalescence of two nanoparticles can be characterized by the two main stages: (i) The first stage is very fast and involves the for-mation of a neck region between particles and its growth by the exponential law:       ,nmp px Btr r (1)  where pr  is the initial particle radius, x  is the neck radius, t  is the time, B  is the temperature-dependent constant coefficient, and n, m are power exponents. This process depends neither on the radii of particles nor on the initial temperature. The first stage duration amounts up to 0.3 ns.  (ii) The second stage involves disappearance of the neck with the formation of a coalesced particle.  In order to explain the process of neck formation and growth during coalescence, Asoro et al. [4] suggest-ed that the neck growth at early stage proceeds due to decreasing chemical potential at the point of contact between particles. By measuring a dihedral angle at the neck, it is possible to calculate the driving force of coalescence and define the onset of particle growth via coalescence. In the equilibrium state, the dihedral angle can be ex-pressed as follows      2 cos( 2),gb s   (2)  where gb is the energy of the grain boundary and s is the surface energy.  In the case under consideration, the neck growth proceeds as long as   eq, while the growth by grain boundary motion proceeds until the dihedral angle will reach its equilibrium value that is attained at the final stage of coalescence (Fig. 1).  A. YU. KOLOSOV, D. N. SOKOLOV ET AL J. NANO- ELECTRON. PHYS. 9, 05042 (2017)   05042-2   Fig. 1 – Scheme of the coalescence of two spherical particles: r  is the particle radius; L  is the distance between centers of particles; x  is the neck radius;   is the dihedral angle  2. METHOD OF INVESTIGATION   The present study was aimed at modeling the coales-cence of two spherical particles of various metals and determining the dihedral angle during their sintering. For this purpose, we have developed a special algorithm of recognizing a dihedral angle at the neck and estimat-ing its value. This algorithm is briefly as follows:  1. The system is centered so that one particle would occur in the negative part, and the other in the positive part of axis X ; 2. Plane z  0 is considered, and all atoms occurring in its certain vicinity are recognized and projected onto this plane; 3. Highest atoms in the left and right cluster are de-termined, and only atoms occurring within the interval of axis X  between these highest atoms are considered; 4. Attempt at recognizing the dihedral angle is made by selecting pairs of atoms in each cluster and drawing lines via them, so that all atoms would be be-neath these lines. Then, the angle between lines pass-ing over the left and right clusters is calculated. To eliminate ambiguity, the angles obtained for various pairs of lines are averaged.  5. Clusters are rotated about axis X  by certain an-gle x , steps 1-4 are repeated, and statistics for final estimation of the dihedral angle is thus accumulated.  In the present work, the process of coalescence (in-cluding the parameters of growing neck region) of nanoclusters having various sizes was studied for na-noparticles of gold, silver, and copper by Monte Carlo method with the interatomic interaction described by a multiparticle Gupta potential [5]. The evolution of na-noparticles with varying temperature and diameter was numerically simulated using a computational scheme developed previously [6, 7] based on the Me-tropolis algorithm [8]. 3. ANALYSIS OF RESULTS  The numerical simulations of coalescence were per-formed for three systems of nanoparticles: 381 381Au Au , 381 381Cu Cu , and 381 381Au Ag . The dihedral angles were determined at temperatures close to the melting points, which were set to be 894 K for distances within 0.0-0.3 nm and 1026 K for distances within 0.6-0.9 nm. This differ-ence is related to the fact that the coalescence at greater distances takes place at higher temperatures because only nanoparticles whose spacing does not exceed the satura-tion radius of a metallic potential ( 6 Å) can fall into the contact zone. The dihedral radius was analyzed using the algorithm described above and the accumulated statistics was averaged for each system. Fig. 2 shows an example of the construction and analysis of the dihedral angle between nanoparticles in the 381 381Cu Cu  system at a temperature of 894 K.  Fig. 2 – Analysis of the dihedral angle between 381Cu  nano-particles at a temperature of 894 K (dihedral angle,   109.9  ; initial distance between nanoparticles, 0.0 nm)  In all the three cases studied, the process of neck formation proceeded in a similar manner despite the difference of metals in these systems. The average square deviation of the dihedral angle in all systems amounted to   12   An increase in the initial spacing of nanoparticles led to growth in the dihedral size and in the relative size of the neck region. Tables 1-3 pre-sent the parameters of all systems of two nanoparticles during coalescence. An analysis of these data shows that increase in the relative size of the neck is accom-panied by decreasing ratio of the grain boundary   Table 1 – Calculated values of the relative size of the neck and dihedral angle at the neck for the 381 381Cu Cu  system in the vi-cinity of the melting temperature  Parameter  0.0d  nm  0.3d  nm  0.6d  nm  0.9d  nm x r  0.52 0.54 0.43 0.67  ,  113 115 117 135 gb/s 1.1 1.07 1.05 0.77  Table 2 – Calculated values of the relative size of the neck and dihedral angle at the neck for the Au381 – Au381 system in the vi-cinity of the melting temperature  Parameter  0.0d  nm  0.3d  nm  0.6d  nm  0.9d  nm x r  0.6 0.61 0.64 0.68  ,  118 120 142 144 gb/s 1.03 0.99 0.65 0.62  ESTIMATION OF THE DIHEDRAL ANGLE BETWEEN METAL J. NANO- ELECTRON. PHYS. 9, 05042 (2017)   05042-3 Table 3 – Calculated values of the relative size of the neck and dihedral angle at the neck for the 381 381Au Au  system in the vicin-ity of the melting temperature  Parameter  0.0d  nm  0.3d  nm  0.6d  nm  0.9d  nm x r  0.55 0.56 0.61 0.64  ,  113 114 117 122 gb/s 1.1 1.08 1.05 0.97    Fig. 3 – Instantaneous configurations of the 381 381Cu Cu  system for various initial distances between nanoparticles, with indicated results of structure analysis. Green, blue, and red atoms correspond to fcc, bcc, and hcp lattices, respectively  energy to surface energy. Fig. 3 shows instantaneous configurations of the 381 381Cu Cu  system for various initial distances between nanoparticles, with indicated dihedral angles. In all the three systems studied, nanoparticles had the crystalline lattice orientation (001). However, it should be noted that variations in the lattice orienta-tion play significant role in the process of neck for-mation. однако. In the sphere-sphere system with (101) lattice orientation, the melting temperature of the system decreases on the average by 30 K, but the system energy upon coalescence increases. In this case, the neck is formed slower and at higher temperatures, with decreasing contact area and dihedral angle (see Fig. 4).  Fig. 4 – Comparison of dihedral angles in the 381 381Cu Cu  system for different lattice orientations in one of the nanopar-ticles. Initial distance between nanoparticles,0.3 nm. Green, blue, and red atoms correspond to fcc, bcc, and hcp lattices, respectively   For some other systems, in particular, those with lattice orientations (001), (101) and (011) in one parti-cle, the energy decreases at distances where the coales-cence proceeds prior to melting. This is due to the ap-pearance of surface defects and the resulting fracture of the crystalline lattice. The process is usually accom-panied by the rotation of either one particle or the whole system. In these cases, the neck is formed faster, with significant increase in the area of contact and di-hedral angle between nanoparticles. Fig. 5 shows an example of this behavior for the 627 627Cu Cu  system. Here, the lattice orientation does not influence the fact that the ratio of the grain boundary energy to the sur-face energy decreases with increasing relative size of the neck.    Fig. 5 – Instantaneous configurations of the 627 627Cu Cu  system during coalescence at temperatures (a) 293 K and (b) 909 K. Initial distance between nanoparticles, 0.3 nm  4. CONCLUSION   Analysis of the obtained results showed that the orientation of crystalline lattices and the initial dis-tance between nanoparticles influence the subsequent coalescence process and the strength of a neck formed in the system, which is an important factor for nano-contacts. Geometric parameters of the neck (dihedral angle, effective length) depend on the distance between nanoparticles in the initial configuration and on the orientation of crystalline lattices. In addition, these factors also influence the temperature at which the neck is formed and the temperature at which complete sintering of nanoparticles takes place.  In the framework of this research, we have created and described an algorithm for estimation of the dihe-dral angle and used this algorithm to calculate the rel-ative neck size and dihedral angles for systems of spherical gold, silver, and copper nanoparticles. It is established that an increase in the relative neck size is accompanied by decrease in the ratio of the grain boundary energy to the surface energy and that this dependence is not significantly influenced by orienta-tion of the crystalline lattices of nanoparticles.  It should be noted that the rate of the coalescence process depends on the interfacial tension and the cor-responding dimensional effects [9]. In the framework of our approach described previously [10, 11], it was shown that the presence of interfaces is additional fac-tor favoring the stability of such nanosystems.  In a similar example [12], dominating mechanisms of sintering have been investigated using molecular  A. YU. KOLOSOV, D. N. SOKOLOV ET AL J. NANO- ELECTRON. PHYS. 9, 05042 (2017)   05042-4 dynamics simulations based on the embedded-atom method for copper nanoparticles. One of the main re-sults in [12] is that the particle rotation during sinter-ing is mainly induced by grain boundary torque with grain boundary diffusion and rarely by dislocation. This coincides with the result obtained in our simula-tions by Monte-Carlo method. The problem of descrip-tion of the shape of a neck formed in the coalescence process has been discussed in our work [13] and in a later work [14]. The models used in [13, 14] provide, in our opinion, a rather simple analytical description of coalescence between nanoparticles, which is independ-ent of their constituent elements or crystal orienta-tions. Thus, the proposed methodology of recognizing and estimating a dihedral angle at the neck, together with results presented in [13, 14], allow us to more ac-curately describe the dynamics of changes in shape of the neck during coalescence.  ACKNOWLEDGMENTS  This research was supported by grant of the Rus-sian Foundation for Basic Research, project no. 16-33-00742-mol-a.  REFERENCES 1. A.L. Klavsyuk, S.V. Kolesnikov, E.M. Smelova, A.M. Saletsky, Phys. Sol. State 53, 2356 (2011). 2. A.L. Klavsyuk, S.V. Kolesnikov, E.M. Smelova, A.M. Saletsky, J. Exp. Theor. Phys. Let. 91, 158 (2010). 3. N.R. Arcidiacono, S. Bieri, D. Poulikakos, C.P. Grigoropoulos, Int. J. Multiph. Flow 30, 979 (2004). 4. M.A. Asoro, D. Kovar, Y. Shao-Horn, L.F. Allard, P.J. Ferreira, Nanotechnology 21, 025701 (2010). 5. R.P. Gupta, Phys. Rev. B 23, 6265 (1981). 6. A.Yu. Kolosov, N.Yu. Sdobnyakov, V.S. Myasnichenko, D.N. Sokolov, J. Surf. Invest.: X-ray, Synchrotron Neutron Tech. 10, 1292 (2016). 7. A.Yu. Kolosov, N.Yu. Sdobnyakov, P.V. Komarov, N.V. Novozhilov, A.N. Bazulev, D.N. Sokolov, Phys. Chem. Asp. Res. Clust., Nanostr. and Nanomat.: Interuniv. Coll. Sci. Papers 5, 134 (2013). 8. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.N. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953). 9. A.N. Bazulev, V.M. Samsonov, N.Yu. Sdobnyakov, Russ. J. Phys. Chem. A. 76, 1872 (2002). 10. V.M. Samsonov, N.Yu. Sdobnyakov, Centr. Eur. J. Phys. 1, 344 (2003). 11. V.M. Samsonov, N.Yu. Sdobnyakov, A.N. Bazulev, Surf. Sci. 532-535, 526 (2003). 12. Y. Seong, Y. Kim, R. German, S. Kim, S.-G. Kim, S.J. Kim, H.J. Kim, S.J. Park, Comp. Mat. Sci. 123, 164 (2016). 13. N.Yu. Sdobnyakov, D.N. Sokolov, E.M. Kozlova, T.M. Neverova, A.Yu. Kolosov, Herald of Tver St. Univer. Series: App. Math. 1, 5 (2014). 14. P. Grammatikopoulos, E. Toulkeridou, K. Nordlund, M. Sowwan, Model. Simul. Mat. Sci. Engin. 23, 015008 (2015). 

Estimation of the Dihedral Angle Between Metal Nanoparticles During Their Coalescence

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 9 No 5, 05042(4pp) (2017) Том 9 № 5, 05042(4cc) (2017) 
 
 
2077-6772/2017/9(5)05042(4) 05042-1  2017 Sumy State University 
Estimation of the Dihedral Angle Between Metal Nanoparticles During Their Coalescence 
 
A.Yu. Kolosov, D.N. Sokolov, N.Yu. Sdobnyakov*, P.V. Komarov, S.S. Bogdanov, A.A. Bogatov, 
V.S. Myasnichenko 
 
Tver State University, 35, Sadovyi Per., 170002 Tver, Russia 
 
(Received 19 June 2017; revised manuscript received 18 August 2017; published online 16 October 2017) 
 
The process of coalescence of various metal nanoparticles has been studied by the Monte Carlo method. 
The interaction of nanoparticles is described by a multiparticle Gupta type potential. An algorithm of rec-
ognizing and estimating a dihedral angle at the neck is developed. The dihedral angle between metal na-
noparticles during their sintering is estimated. 
 
Keywords: Nanoparticles of gold, silver, and copper; Coalescence; Neck, dihedral angle; Monte Carlo 
method; Gupta potential. 
 
DOI: 10.21272/jnep.9(5).05042 PACS numbers: 61.46. + w, 66.30. – h, 02.70.Uu 
 
1. INTRODUCTION 
 
The properties of nanostructured materials are de-
termined predominantly by the specific features of in-
teratomic interaction in their surface layers and the 
quantum confinement effects. Nanosized structures are 
of considerable interest in both fundamental and ap-
plied science. The interest in this field of science, which 
is among the most extensively developing fields, is re-
lated to both the fundamental physical issues and phe-
nomena and the prospects of creating new quantum 
devices and systems with broad possibilities for nanoe-
lectronics, metrology, and measurement techniques. 
The synthesis of nanoparticles and nanostructures, as 
well as experimental research in this field, is usually 
difficult and expensive. Theoretical approaches to the 
investigation of nanosystems are yet not sufficiently 
elaborated. This circumstance makes the development 
of analytical and numerical methods for predicting the 
properties of nanoparticles and nanomaterials as well 
as for monitoring processes in these systems an espe-
cially topical task. 
Nanoclusters in a condensed state possess physical 
characteristics different from those of the correspond-
ing macroscopic crystals. In addition, nanoclusters ex-
hibit new optical and magnetic properties dependent 
not only on their size, but also on the mode of organiza-
tion or self-organization. In this context, there appear 
tasks related to reduction in sizes of systems and con-
trol over their structures. In particular, a decrease in 
dimensions of elements for microelectronics leads to 
problems related to the creation of nanocontacts and 
nanowires [1, 2], which must possess both high struc-
tural stability and sufficient electric conductivity so as 
to meet requirements of the new stage of technology 
development. Creation of new magnetic memory devic-
es, nanodiodes, nanowires, etc. poses the problem of 
controlled synthesis and organization of nanoclusters. 
In this respect, the ability of predicting and controlling 
the size and structure of nanoparticles becomes very 
important, since the arrangement of atoms is a key 
factor that determined the properties of these particles 
and influences the strength and elasticity of related 
nanostructures. Coalescence is among the processes 
that can be used for controlling the structure and di-
mensions of particles.  
There are three mechanisms involved in the process 
of nanoparticle formation in an inert gas medium: co-
agulation, coalescence, and aggregation on a small clus-
ter-nucleus. The knowledge of a dominating mechanism 
and peculiarities of each one under given external con-
ditions provides a basis that allows us to control the 
structure and size of particles.  
According to Arcidiacono et al. [3], the coalescence 
of two nanoparticles can be characterized by the two 
main stages: 
(i) The first stage is very fast and involves the for-
mation of a neck region between particles and its 
growth by the exponential law: 
 
 
 
 
 
 
,
n
m
p p
x Bt
r r
 (1) 
 
where 
pr  is the initial particle radius, x  is the neck 
radius, t  is the time, B  is the temperature-dependent 
constant coefficient, and n, m are power exponents. 
This process depends neither on the radii of particles 
nor on the initial temperature. The first stage duration 
amounts up to 0.3 ns.  
(ii) The second stage involves disappearance of the 
neck with the formation of a coalesced particle.  
In order to explain the process of neck formation 
and growth during coalescence, Asoro et al. [4] suggest-
ed that the neck growth at early stage proceeds due to 
decreasing chemical potential at the point of contact 
between particles. 
By measuring a dihedral angle at the neck, it is 
possible to calculate the driving force of coalescence 
and define the onset of particle growth via coalescence. 
In the equilibrium state, the dihedral angle can be ex-
pressed as follows 
 
    2 cos( 2),gb s   (2) 
 
where gb is the energy of the grain boundary and s is 
the surface energy.  
In the case under consideration, the neck growth 
proceeds as long as   eq, while the growth by grain 
boundary motion proceeds until the dihedral angle will 
reach its equilibrium value that is attained at the final 
stage of coalescence (Fig. 1). 
 A. YU. KOLOSOV, D. N. SOKOLOV ET AL J. NANO- ELECTRON. PHYS. 9, 05042 (2017) 
 
 
05042-2 
 
 
Fig. 1 – Scheme of the coalescence of two spherical particles: 
r  is the particle radius; L  is the distance between centers of 
particles; x  is the neck radius;   is the dihedral angle 
 
2. METHOD OF INVESTIGATION  
 
The present study was aimed at modeling the coales-
cence of two spherical particles of various metals and 
determining the dihedral angle during their sintering. 
For this purpose, we have developed a special algorithm 
of recognizing a dihedral angle at the neck and estimat-
ing its value. This algorithm is briefly as follows:  
1. The system is centered so that one particle would 
occur in the negative part, and the other in the positive 
part of axis X ; 
2. Plane z  0 is considered, and all atoms occurring 
in its certain vicinity are recognized and projected onto 
this plane; 
3. Highest atoms in the left and right cluster are de-
termined, and only atoms occurring within the interval 
of axis X  between these highest atoms are considered; 
4. Attempt at recognizing the dihedral angle is 
made by selecting pairs of atoms in each cluster and 
drawing lines via them, so that all atoms would be be-
neath these lines. Then, the angle between lines pass-
ing over the left and right clusters is calculated. To 
eliminate ambiguity, the angles obtained for various 
pairs of lines are averaged.  
5. Clusters are rotated about axis X  by certain an-
gle x , steps 1-4 are repeated, and statistics for final 
estimation of the dihedral angle is thus accumulated.  
In the present work, the process of coalescence (in-
cluding the parameters of growing neck region) of 
nanoclusters having various sizes was studied for na-
noparticles of gold, silver, and copper by Monte Carlo 
method with the interatomic interaction described by a 
multiparticle Gupta potential [5]. The evolution of na-
noparticles with varying temperature and diameter 
was numerically simulated using a computational 
scheme developed previously [6, 7] based on the Me-
tropolis algorithm [8]. 
3. ANALYSIS OF RESULTS 
 
The numerical simulations of coalescence were per-
formed for three systems of nanoparticles: 381 381Au Au , 
381 381Cu Cu , and 381 381Au Ag . The dihedral angles were 
determined at temperatures close to the melting points, 
which were set to be 894 K for distances within 0.0-0.3 nm 
and 1026 K for distances within 0.6-0.9 nm. This differ-
ence is related to the fact that the coalescence at greater 
distances takes place at higher temperatures because only 
nanoparticles whose spacing does not exceed the satura-
tion radius of a metallic potential ( 6 Å) can fall into the 
contact zone. The dihedral radius was analyzed using the 
algorithm described above and the accumulated statistics 
was averaged for each system. 
Fig. 2 shows an example of the construction and 
analysis of the dihedral angle between nanoparticles in 
the 381 381Cu Cu  system at a temperature of 894 K. 
 
Fig. 2 – Analysis of the dihedral angle between 
381Cu  nano-
particles at a temperature of 894 K (dihedral angle,   109.9  ; 
initial distance between nanoparticles, 0.0 nm) 
 
In all the three cases studied, the process of neck 
formation proceeded in a similar manner despite the 
difference of metals in these systems. The average 
square deviation of the dihedral angle in all systems 
amounted to   12   An increase in the initial spacing 
of nanoparticles led to growth in the dihedral size and 
in the relative size of the neck region. Tables 1-3 pre-
sent the parameters of all systems of two nanoparticles 
during coalescence. An analysis of these data shows 
that increase in the relative size of the neck is accom-
panied by decreasing ratio of the grain boundary  
 
Table 1 – Calculated values of the relative size of the neck and dihedral angle at the neck for the 381 381Cu Cu  system in the vi-
cinity of the melting temperature 
 
Parameter  0.0d  nm  0.3d  nm  0.6d  nm  0.9d  nm 
x r  0.52 0.54 0.43 0.67 
 ,  113 115 117 135 
gb/s
 
1.1 1.07 1.05 0.77 
 
Table 2 – Calculated values of the relative size of the neck and dihedral angle at the neck for the Au381 – Au381 system in the vi-
cinity of the melting temperature 
 
Parameter  0.0d  nm  0.3d  nm  0.6d  nm  0.9d  nm 
x r  0.6 0.61 0.64 0.68 
 ,  118 120 142 144 
gb/s 1.03 0.99 0.65 0.62 
 ESTIMATION OF THE DIHEDRAL ANGLE BETWEEN METAL J. NANO- ELECTRON. PHYS. 9, 05042 (2017) 
 
 
05042-3 
Table 3 – Calculated values of the relative size of the neck and dihedral angle at the neck for the 
381 381Au Au  system in the vicin-
ity of the melting temperature 
 
Parameter  0.0d  nm  0.3d  nm  0.6d  nm  0.9d  nm 
x r  0.55 0.56 0.61 0.64 
 ,  113 114 117 122 
gb/s 1.1 1.08 1.05 0.97 
 
 
 
Fig. 3 – Instantaneous configurations of the 381 381Cu Cu  
system for various initial distances between nanoparticles, 
with indicated results of structure analysis. Green, blue, and 
red atoms correspond to fcc, bcc, and hcp lattices, respectively 
 
energy to surface energy. Fig. 3 shows instantaneous 
configurations of the 381 381Cu Cu  system for various 
initial distances between nanoparticles, with indicated 
dihedral angles. 
In all the three systems studied, nanoparticles had 
the crystalline lattice orientation (001). However, it 
should be noted that variations in the lattice orienta-
tion play significant role in the process of neck for-
mation. однако. In the sphere-sphere system with 
(101) lattice orientation, the melting temperature of 
the system decreases on the average by 30 K, but the 
system energy upon coalescence increases. In this case, 
the neck is formed slower and at higher temperatures, 
with decreasing contact area and dihedral angle (see 
Fig. 4).
 
 
Fig. 4 – Comparison of dihedral angles in the 381 381Cu Cu  
system for different lattice orientations in one of the nanopar-
ticles. Initial distance between nanoparticles,0.3 nm. Green, 
blue, and red atoms correspond to fcc, bcc, and hcp lattices, 
respectively  
 
For some other systems, in particular, those with 
lattice orientations (001), (101) and (011) in one parti-
cle, the energy decreases at distances where the coales-
cence proceeds prior to melting. This is due to the ap-
pearance of surface defects and the resulting fracture 
of the crystalline lattice. The process is usually accom-
panied by the rotation of either one particle or the 
whole system. In these cases, the neck is formed faster, 
with significant increase in the area of contact and di-
hedral angle between nanoparticles. Fig. 5 shows an 
example of this behavior for the 627 627Cu Cu  system. 
Here, the lattice orientation does not influence the fact 
that the ratio of the grain boundary energy to the sur-
face energy decreases with increasing relative size of 
the neck. 
 
 
 
Fig. 5 – Instantaneous configurations of the 627 627Cu Cu  
system during coalescence at temperatures (a) 293 K and (b) 
909 K. Initial distance between nanoparticles, 0.3 nm 
 
4. CONCLUSION  
 
Analysis of the obtained results showed that the 
orientation of crystalline lattices and the initial dis-
tance between nanoparticles influence the subsequent 
coalescence process and the strength of a neck formed 
in the system, which is an important factor for nano-
contacts. Geometric parameters of the neck (dihedral 
angle, effective length) depend on the distance between 
nanoparticles in the initial configuration and on the 
orientation of crystalline lattices. In addition, these 
factors also influence the temperature at which the 
neck is formed and the temperature at which complete 
sintering of nanoparticles takes place.  
In the framework of this research, we have created 
and described an algorithm for estimation of the dihe-
dral angle and used this algorithm to calculate the rel-
ative neck size and dihedral angles for systems of 
spherical gold, silver, and copper nanoparticles. It is 
established that an increase in the relative neck size is 
accompanied by decrease in the ratio of the grain 
boundary energy to the surface energy and that this 
dependence is not significantly influenced by orienta-
tion of the crystalline lattices of nanoparticles.  
It should be noted that the rate of the coalescence 
process depends on the interfacial tension and the cor-
responding dimensional effects [9]. In the framework of 
our approach described previously [10, 11], it was 
shown that the presence of interfaces is additional fac-
tor favoring the stability of such nanosystems.  
In a similar example [12], dominating mechanisms 
of sintering have been investigated using molecular 
 A. YU. KOLOSOV, D. N. SOKOLOV ET AL J. NANO- ELECTRON. PHYS. 9, 05042 (2017) 
 
 
05042-4 
dynamics simulations based on the embedded-atom 
method for copper nanoparticles. One of the main re-
sults in [12] is that the particle rotation during sinter-
ing is mainly induced by grain boundary torque with 
grain boundary diffusion and rarely by dislocation. 
This coincides with the result obtained in our simula-
tions by Monte-Carlo method. The problem of descrip-
tion of the shape of a neck formed in the coalescence 
process has been discussed in our work [13] and in a 
later work [14]. The models used in [13, 14] provide, in 
our opinion, a rather simple analytical description of 
coalescence between nanoparticles, which is independ-
ent of their constituent elements or crystal orienta-
tions. Thus, the proposed methodology of recognizing 
and estimating a dihedral angle at the neck, together 
with results presented in [13, 14], allow us to more ac-
curately describe the dynamics of changes in shape of 
the neck during coalescence. 
 
ACKNOWLEDGMENTS 
 
This research was supported by grant of the Rus-
sian Foundation for Basic Research, project no. 16-33-
00742-mol-a.
 
 
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Estimation of the Dihedral Angle Between Metal Nanoparticles During Their Coalescence

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