Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects
Energy-related problems of solid nanoparticles and nanoscale materials concerning their stability and structure are investigated with a specifically targeting on nanocrystalline metallic systems. The new concept based on the atomic hypothesis about the size dependence of nearest atom—atom interactio...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2015
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irk-123456789-1118842017-01-16T03:03:10Z Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects Shirinyan, Aram S. Строение и свойства наноразмерных и мезоскопических материалов Energy-related problems of solid nanoparticles and nanoscale materials concerning their stability and structure are investigated with a specifically targeting on nanocrystalline metallic systems. The new concept based on the atomic hypothesis about the size dependence of nearest atom—atom interaction energy and co-ordinated actions of atoms is offered. The verification is done for metallic thin films and nanoparticles on the basis of experimental results, theoretical approach, and molecular static simulations. As shown, for nanomaterials, the concepts of size-dependent interatomic interaction energies can be used for description of thermodynamic and kinetic properties. Досліджено питання, пов’язані з енергією твердих наночастинок і нанорозмірних матеріялів, щодо їх стійкости та структури з акцентом на нанокристалічних металевих системах. У роботі пропонується новий підхід, який базується на атомній гіпотезі про розмірну залежність енергії міжатомової взаємодії найближчих атомів та узгоджені колективні дії атомів. Перевірку гіпотези було здійснено для тонких металевих плівок і наночастинок на основі експериментальних результатів, теоретичного наближення і молекулярно-статичного моделювання. Показано, що для наноматеріялів можна використовувати концепцію розмірної залежности енергії міжатомової взаємодії для опису термодинамічних і кінетичних властивостей. Исследованы вопросы, связанные с энергией твёрдых наночастиц и наноразмерных материалов, относительно их устойчивости и структуры с акцентом на нанокристаллических металлических системах. В работе предлагается новый подход, основанный на атомной гипотезе о размерной зависимости энергии межатомного взаимодействия ближайших атомов и согласованных коллективных действиях атомов. Гипотеза была проверена для тонких металлических плёнок и наночастиц на основе экспериментальных результатов, теоретического приближения и молекулярно-статического моделирования. Показано, что для наноматериалов можно использовать концепцию размерной зависимости энергии межатомного взаимодействия для описания термодинамических и кинетических свойств. 2015 Article Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects / Aram S. Shirinyan // Металлофизика и новейшие технологии. — 2015. — Т. 37, № 4. — С. 475-486. — Бібліогр.: 48 назв. — рос. 1024-1809 PACS: 05.70.Ce, 34.20.Cf, 61.46.Hk, 64.75.Jk, 66.30.Pa, 81.07.Bc, 82.60.Qr http://dspace.nbuv.gov.ua/handle/123456789/111884 en Металлофизика и новейшие технологии Інститут металофізики ім. Г.В. Курдюмова НАН України |
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Строение и свойства наноразмерных и мезоскопических материалов Строение и свойства наноразмерных и мезоскопических материалов |
| spellingShingle |
Строение и свойства наноразмерных и мезоскопических материалов Строение и свойства наноразмерных и мезоскопических материалов Shirinyan, Aram S. Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects Металлофизика и новейшие технологии |
| description |
Energy-related problems of solid nanoparticles and nanoscale materials concerning their stability and structure are investigated with a specifically targeting on nanocrystalline metallic systems. The new concept based on the atomic hypothesis about the size dependence of nearest atom—atom interaction energy and co-ordinated actions of atoms is offered. The verification is done for metallic thin films and nanoparticles on the basis of experimental results, theoretical approach, and molecular static simulations. As shown, for nanomaterials, the concepts of size-dependent interatomic interaction energies can be used for description of thermodynamic and kinetic properties. |
| format |
Article |
| author |
Shirinyan, Aram S. |
| author_facet |
Shirinyan, Aram S. |
| author_sort |
Shirinyan, Aram S. |
| title |
Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects |
| title_short |
Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects |
| title_full |
Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects |
| title_fullStr |
Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects |
| title_full_unstemmed |
Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects |
| title_sort |
concept of size-dependent atomic interaction energies for solid nanomaterials: thermodynamic and diffusion aspects |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| publishDate |
2015 |
| topic_facet |
Строение и свойства наноразмерных и мезоскопических материалов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/111884 |
| citation_txt |
Concept of Size-Dependent Atomic Interaction Energies for Solid Nanomaterials: Thermodynamic and Diffusion Aspects / Aram S. Shirinyan // Металлофизика и новейшие технологии. — 2015. — Т. 37, № 4. — С. 475-486. — Бібліогр.: 48 назв. — рос. |
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Металлофизика и новейшие технологии |
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2025-07-08T02:50:57Z |
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2025-07-08T02:50:57Z |
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| fulltext |
475
STRUCTURE AND PROPERTIES OF NANOSCALE
AND MESOSCOPIC MATERIALS
PACS numbers:05.70.Ce, 34.20.Cf,61.46.Hk,64.75.Jk,66.30.Pa,81.07.Bc, 82.60.Qr
Concept of Size-Dependent Atomic Interaction Energies
for Solid Nanomaterials: Thermodynamic and Diffusion Aspects
Aram S. Shirinyan
Physicochemical Materials Science Centre of
the Taras Shevchenko Kyiv National University and
the National Academy of Science of Ukraine,
64 Volodymyrska Str.,
01601 Kyiv, Ukraine
Energy-related problems of solid nanoparticles and nanoscale materials con-
cerning their stability and structure are investigated with a specifically target-
ing on nanocrystalline metallic systems. The new concept based on the atomic
hypothesis about the size dependence of nearest atom—atom interaction energy
and co-ordinated actions of atoms is offered. The verification is done for metal-
lic thin films and nanoparticles on the basis of experimental results, theoreti-
cal approach, and molecular static simulations. As shown, for nanomaterials,
the concepts of size-dependent interatomic interaction energies can be used for
description of thermodynamic and kinetic properties.
Досліджено питання, пов’язані з енергією твердих наночастинок і наноро-
змірних матеріялів, щодо їх стійкости та структури з акцентом на нанок-
ристалічних металевих системах. У роботі пропонується новий підхід,
який базується на атомній гіпотезі про розмірну залежність енергії міжа-
томової взаємодії найближчих атомів та узгоджені колективні дії атомів.
Перевірку гіпотези було здійснено для тонких металевих плівок і наночас-
тинок на основі експериментальних результатів, теоретичного наближення
і молекулярно-статичного моделювання. Показано, що для наноматеріялів
можна використовувати концепцію розмірної залежности енергії міжато-
мової взаємодії для опису термодинамічних і кінетичних властивостей.
Исследованы вопросы, связанные с энергией твёрдых наночастиц и нано-
размерных материалов, относительно их устойчивости и структуры с ак-
центом на нанокристаллических металлических системах. В работе пред-
лагается новый подход, основанный на атомной гипотезе о размерной зави-
симости энергии межатомного взаимодействия ближайших атомов и согла-
сованных коллективных действиях атомов. Гипотеза была проверена для
тонких металлических плёнок и наночастиц на основе экспериментальных
Металлофиз. новейшие технол. / Metallofiz. Noveishie Tekhnol.
2015, т. 37, № 4, сс. 475—486
Оттиски доступны непосредственно от издателя
Фотокопирование разрешено только
в соответствии с лицензией
2015 ИМФ (Институт металлофизики
им. Г. В. Курдюмова НАН Украины)
Напечатано в Украине.
476 Aram S. SHIRINYAN
результатов, теоретического приближения и молекулярно-статического
моделирования. Показано, что для наноматериалов можно использовать
концепцию размерной зависимости энергии межатомного взаимодействия
для описания термодинамических и кинетических свойств.
Key words: atom—atom interaction, size effect, thermodynamic properties,
diffusion in nanomaterials, nanocrystalline structures.
(Received January 27, 2015)
1. INTRODUCTION
Many quantities and physicochemical properties of nanosize metals
and nanosolids, such as their melting, solubility, lattice parameter,
and Young’s modulus, remain no longer constant when their size be-
comes small [1—3]. One of the most commonly known size effect is re-
lated to the melting (transition) temperature Tm shift [4—6]. In a gen-
eral form, the condition for the melting temperature can be written as:
Tm T{1 /L}, (1)
where T is the bulk melting temperature of the material, L is a charac-
teristic size of a nanosystem (the radius R or the diameter d of a nano-
particle or the thickness h of the film, etc.). The coefficient depends
on the material data [6].
Nanomaterials’ research related to the size effect is usually based on
the increase of the excess surface energy of small systems (large part of
grain boundaries and surface atoms). The decrease of the size leads to
the situation when the phase with smaller surface free energy becomes
more probable and stable. Such behaviour is found for many solid
chemical elements and metallic systems (Мо, W, Y, Gd, Tb, Ho, Tm,
etc. [7, 8]).
Due to the large part of grain boundary and surface atoms in solid
nanomaterials, one should pay attention to the existence of defects
(such as dislocations, dirt) that also leads to the difference in interac-
tion between such defects and the bulk and surface atoms and results in
size effects. Hereby, the main interest is related to the surface as the
source of sink and barrier for moving defects [9—11] and self-
organization of the matter like the self-cleaning, etc.
There is one more fundamental reason for size effects in multicom-
ponent nanomaterials, where the first-order phase transformation
starts from a nucleation and includes a change of composition or densi-
ty. This is the less known so-called ‘depletion effect’ related to the con-
servation of matter. In the usual treatment of phase transition, it is
assumed that there is no problem of matter supply during new phase
formation. In a nanosystem, the amount of one of the components in
SIZE-DEPENDENT ATOMIC INTERACTION ENERGIES FOR NANOMATERIALS 477
the entire nanosystem may be not sufficient for the construction of a
critical nucleus of the different composition [12—14]. Moreover, in
nanosystems, the well-known concept of equilibrium phase diagram
has to be revised, due to the fact that the amount of matter is limited
[15, 16]. Similar arguments may be applied for the case of grain
boundary segregation problem as a successful approach to stabilize
nanocrystalline materials against grain growth [17—19].
Another reason of the size effect in nanosolids appears because of
quantum mechanics. The surface state of electrons may give the ener-
gy local levels yielding the size quantization, surface waves and the
difference of the interaction with the light, dispersion and reflection
changes, etc. The change of the dimension of a nanomaterial is also the
reason for size effects explained mainly due to the quantum mechani-
cal behaviour and dimension quantization. Basically, the quantum-
mechanical explanation relies on the quantization of the phonon ener-
gy and anharmonicity of vibrations and atomic interaction around the
nanomaterial crystal-lattice sites, which increases with the increase of
the temperature and vice versa [20, 21]. This approach gives size ef-
fects within 0.1% for interatomic distances in intervals from about 1
nm to 10 nm at high temperatures (above the Debye temperature) and
disappearing as the temperature becomes lower. Thus, the traditional
quantum-mechanical scheme does not work at least for low tempera-
tures.
Most of the physical quantities, such as the Debye temperature, the
heat capacity, etc., are related to the cohesive energy of atoms. That is
why it is topical to check the atomic hypothesis related to the crystal
energy optimization, collective effect of a group of atoms and space
redistribution of atoms in a solid nanosystem even at the zero tempera-
tures when the vibrations are absent. It is worth noting that such type
of properties and behaviour can exist in many disciplines, say: in eco-
nomics, there exist the so called ‘scale economy’ and ‘regional econom-
ics’ where one considers the effect of firm size on financial reliability
and performance [22, 23]; in biology and descriptive statistics, one
considers the statistical significance [24]; in mathematics education,
relationship between class size and student achievements, etc. [25, 26].
In present communication, we show that there exist in physics and
chemistry the similar fundamental reasons for size effects and one can
offer the idea of the size-dependence of the cohesive energy on atomic
level: the idea of the dependence of the energy of atom—atom interac-
tions on the size of a nanosystem.
2. THEORY AND CALCULATIONS
In present work, we pay attention to one more fundamental reason for
size effects in nanocrystalline materials–the size-dependent atomic
478 Aram S. SHIRINYAN
interaction energy, which means the dependence energy interaction of
atoms with the co-ordination environment and the lattice on the size
(for example, on the thickness of a solid nanofilm/nanolayer or the ra-
dius of a nanocrystalline particle). In other words, physicochemical
properties become size-dependent and one can resume the atomic hy-
pothesis: ‘the nearest atom—atom interactions and co-ordinated ac-
tions of atoms in a nanomaterial will be different depending on the size
L: the change of a grain or crystal in size can lead to a redistribution of
atoms in the lattice both on the surface and in the bulk and to the cor-
responding change in the interaction energy E of each atom on the sur-
face and in the bulk with neighbouring atoms’.
The atomic hypothesis for nanomaterials can be described in the
formula:
E f(L). (2)
Physical mechanism of proposed atomic hypothesis that leads to the
change of the interatomic distance and dimensional change of the po-
tential energy of interatomic interaction is based on the collective be-
haviour of atoms in nanosolids and existence of the non-symmetrical
tails in the potential profile. The long-range part of the tail in nanosol-
ids is cut off and leads to the changes of the number of atoms in inter-
action spheres and the distances between the atoms.
Recently, Liang [27] deduced that the cohesive energy of nanoparti-
cles decreases with a reduction in size d. Therefore, nearly all thermo-
dynamic quantities can be approximately expressed as a linear func-
tion of 1/d by general thermodynamics [28]. Noteworthy is the fact
that the linear function of 1/d cannot be fully substantiated by the hy-
pothesis of size dependence of atom—atom interaction energy. It may
be the result of increasing the percentage of surface atoms in nano-
materials and averaging procedure. That is why it is important to
proof that the proposed hypothesis is true not only for the surface at-
oms of a nanomaterial, but also for the internal ones.
Additionally, there are other factors such as the atomic radius and
the shear modulus, which can modify the cohesive energy [29]. Howev-
er, the nanoscale size seems to be the main factor influencing the in-
teraction energy of atoms.
Let us verify the atomic hypothesis for nanomaterials on the basis of
experimental results and our theoretical approach and simulations.
2.1. Theoretical Approach Based on Experimental Data
and Microscopic Model
Let us first look at the experimental results on lattice parameter in
nanocrystalline solids and give simple theoretical estimation. In most
SIZE-DEPENDENT ATOMIC INTERACTION ENERGIES FOR NANOMATERIALS 479
experiments, an f.c.c. structure is observed, the densities of nano-
materials and the lattice parameters have been found changed in the
range of up to 1—2% [30—33]. With the sufficient degree of accuracy,
one can write the approximation function of hyperbolical type for the
crystal lattice parameter, observed in many experiments:
a(L) a(1 b/L). (3)
Here, a is an asymptotic value of the lattice parameter for macroscop-
ic sizes, b is a fitting parameter.
Let us now make the theoretical estimation for the potential energy
of atomic interactions. To describe the behaviour in metals, the pair
Morse potential can be used [34—36]:
U(r) U0[exp{2(r r0)} 2exp{(r r0)}]; (4)
here, U(r) is the potential of atom—atom interaction, U0 U(r0) is the
parameter of the dissociation energy, and is the degree of anharmon-
icity; r0–equilibrium distance between two atoms, r–the distance be-
tween the atoms and can be ranged from 0 to an unlimited distance.
The Morse potential function (4) correctly describes vibrational lev-
els of b.c.c., f.c.c., and h.c.p. metals, covalent molecules, and defects
in cubic crystals [37, 38]. Using the Morse pair potential (4) and the
experimental result (3), we can estimate the dependence of the poten-
tial energy of the interaction between nearest atoms within the f.c.c.
nanosystem on the size L. It is well known that the distance between
two nearest atoms ropt in the f.c.c. metals is equal to the lattice parame-
ter divided by square root of two [39]. It means that we can use the re-
lation ropt a(L)/ 2 with the condition a(L) according to formula (3).
The simple algebra leads to the energy function of atom—atom interac-
tion E(L) in a first co-ordination sphere:
E(L) U0[exp{2( b/L)} 2exp{( b/L)}] (5)
with parameters a/ 2 , 1 r0 2 /a. The energy of the inter-
action of a single atom with all atoms in a first co-ordination sphere
can be found due to the summation and is equal to the quantity of
W ZE(L), where Z is the co-ordination number and W0 ZU0.
For instance, for the f.c.c. Cu nanoparticles, U0 0.3249 eV,
1.3588·1010
m
1, r0 2.86610
10
m, a 3.61510
10
m, Z 12 [31—33]
yield the estimation values: b 2.110
11
m, L R, W0 6.246510
19
J,
3.474, 0.1212, and the energy E(R):
E(R) 5.20510
20{exp(6.947[0.1212 0.021/R])
2exp(3.474[0.1212 0.021/R])}.
480 Aram S. SHIRINYAN
Here, the energy E(R) is calculated in Joules, the radius R of Cu parti-
cle is in nm. Graphical visualization is presented in Fig. 1. Thus, the
result (5) explains the proposed atomic hypothesis on size-dependent
physical properties of nanomaterials and presents new and important
concept for nanoscience.
2.2. Verification by Molecular Static Simulations
The pair Morse potential function does not take into accounts the ion—
electron—ion component of interaction. That is why we verified the
proposed hypothesis by performing molecular static study based on the
Sutton—Chen potential of atomic interactions for the solid monatomic
thin films and nanoparticles of different metals in order to find out the
stable structures and average atomic pair interaction energies inside
the nanosystems and at the surfaces [40—42]. The Sutton—Chen poten-
tial is the many-body potential and, due to the reasonable agreement
with experiments, is often employed for the description of the interac-
tion between metallic atoms inside a metallic cluster or a nanoparticle
[40, 43]. In Figure 2, we present the simulation results for Cu nanofilm
of the thickness h, obtained due to molecular static simulations of en-
ergy optimization procedure for the whole nanofilm.
Let us look at the simulation results on thin Cu film (Fig. 2). One can
Fig. 1. Results of estimated calculations using Morse potential (5) and exper-
imental data for lattice parameter (3) of isolated Cu nanoparticles: the de-
pendence of the energy E(R) on the radius of the nanoparticle.
SIZE-DEPENDENT ATOMIC INTERACTION ENERGIES FOR NANOMATERIALS 481
see that the potential energy of atomic interaction demonstrates the
size effect up to 20 nm. This is true for nanofilms and it is stronger for
the case of nanoparticles, where we obtained the size effect for atoms
up to 30 nm. Hence, simulations confirm the proposed concept of size-
dependent energies of atomic interaction for nanomaterials.
3. RESULTS AND DISCUSSION
3.1. Application to the Thermodynamics of Nanosystems
As the example of the application of atomic hypothesis, we show the
quantitative treatment emphasizing new physics: the methodology of
phase diagram construction of binary nanosystems that may be fur-
bished. Nanophase diagrams in this work mean the size-dependent iso-
baric temperature—composition phase diagrams of coexistence of dif-
ferent new nanosize phases.
For the first example, for a two-component nanosystem consisting A
and B atoms, let us use well-known regular solution model and find the
phase separation temperature Тtr (temperature, at which the binary
Fig. 2. Results of computer simulations using Sutton—Chen potential [40, 41]
for Cu thin f.c.c. film in vacuum: the dependence of the energy of the interac-
tion W of the internal atom within the nanofilm with neighbouring atoms on
the different thicknesses.
482 Aram S. SHIRINYAN
solid solution separates on two different phases) [12, 39].
Due to the abovementioned size effect for interatomic interaction
energy within the scope of the thermodynamic approach for nanosol-
ids, one can use the potential energy of the atom—atom interaction (5)
in the form of a Taylor series, and, in the simplest case, it may be as-
sumed by the following formula:
E(L) E{1 /L}. (6)
Here, E–potential energy of atomic interaction in bulk phase and is
the energy fitting parameter. Then, for the two-component nanosys-
tem, we can use the corresponding formula of the potential energies, E:
EAA(L) EAA(1 AA/L), EBB(L) EBB(1 BB/L),
EAB(L) EAB(1 AB/L). (7)
Hereby, the values EAA(L), EBB(L) and EAB(L) are the corresponding in-
teraction energies of each sort of atoms (in a first co-ordination sphere)
in a nanomaterial indicated by the corresponding subindexes A and B.
The mixing energy
Emix(L) {0.5(EBB(L) EAA(L)} EAB(L) (8)
becomes size-dependent so that the phase separation temperature
Тtr 2ZEmix(L)/kB (9)
depends on the size L of a nanosystem and may vary from a few to hun-
dreds Kelvins similar to equation (1). In equation (9), kB is the Boltz-
mann constant. Simple algebra gives the condition for the phase tran-
sition temperature:
Тtr Т{1 /L}, (10)
where Т 2ZEmix/kB is the phase separation temperature, and Emix is
mixing energy in a bulk material, coefficient {0.5[EAAAA EBBBB]
EABAB}/Emix.
For the second example, we apply the relations (6) and (7) to recon-
struct the Gibbs free energy dependence for each nanophase in the na-
nomaterial. Using the Gibbs method of geometric thermodynamics
(when the equilibrium concentrations in phases are determined by the
rule of the common tangent drawn from curves of concentration de-
pendences of the Gibbs free energy) and commonly used CALPHAD data
for bulk phases and quantities EAA, EBB, EAB, one can draw nanophase di-
agrams in a wide range of sizes and compositions, calculate the size de-
SIZE-DEPENDENT ATOMIC INTERACTION ENERGIES FOR NANOMATERIALS 483
pendences of phase transition temperatures, solubilities of components,
etc. Due to such technique, we calculated the nanophase diagram for
thin Bi—Sn solid film giving adequate and coinciding with experiments
results [44]. The size dependence of phase transition temperatures are
found as following: Te 161.5/L 412 K–the eutectic temperature of
Bi—Sn thin film, Tm
Bi
279.5/L 543 K–the melting temperature of
thin Bi film, Tm
Sn
187.3/L 504.8 K–the melting temperature of
thin Sn film (the thickness L of the Bi—Sn film is taken in nm, and tem-
peratures are in Kelvins). The approximation formulae for limit solubil-
ities (solvus line) in Bi—Sn nanofilms have been found in the following
form:
0.96
Bi
0.1 0.131x L –for Bi in Sn and
3.4 3
Sn
61.8 2 10x L
–for Sn in Bi. It is worth noting that the limit solubility of Sn in Bi is
equal to 0.024 for the 9 nm film at 395 K and 0.002 in a bulk material
[45].
3.2. Application to the Diffusion in Nanomaterials
The aforementioned reasons allow us to make the conclusion that a size
dependence of the kinetic coefficients in nanomaterials exists. It is ev-
ident from the consideration of the diffusion in terms of the energy-
activated process, which follows the Arrhenius dependence:
D D0exp{Q/(kBT)}, (11)
where D0 is the frequency factor and Q is the activation energy for the
diffusion process.
If we return to the idea that the size is important for bonding ener-
gy, then it becomes understandable that, first, depending on the diffu-
sion mechanism (interstitial diffusion, or vacancy diffusion, etc.)
bonding to the surrounding atoms may be different and, second, due to
the relations (5)—(7) bonding energy and activation energy Q depend on
the size of the nanomaterial.
Moreover, it can be understood from other viewpoint. On the one
hand, it is well established that the correlation between the activation
energy Q and the melting temperature Tm has the form:
Q 18kBTm. (12)
The diffusion parameters D0 and Q are empirically correlated with the
melting temperature and the proportionality factor is nearly a con-
stant for a specific class of materials [46]. On the other hand, there ex-
ists a size dependence of the melting temperature in the form of equa-
tion (1). Taking into account the conditions (11), (12) and equation (1),
one can find out the size dependence of the diffusion coefficient in the
nanomaterial, which may be written as
484 Aram S. SHIRINYAN
D D0exp{18T(1 /L)/T}. (13)
One can see from equation (12) that the diffusion coefficient be-
comes a decreasing function of the size: it increases when the size L de-
creases and vice versa. The expression (13) explains the fast diffusion
in nanoscale materials as compared with the bulk cases.
In addition, in polycrystalline materials compared to single crystals,
diffusion is faster because of the grain boundaries [47]. In many cases,
effective transport properties may be written due to diffusion coeffi-
cients in the form:
D Dbulk Dgb/L, (14)
where Dgb is the grain boundary diffusion coefficient, Dbulk is the bulk
diffusion coefficient, and is the width of the grain boundary diffu-
sion channel. It is worth noting that due to the proposed concept of
size-dependent energies of atomic interaction for nanomaterials both
quantities Dgb and Dbulk become size-dependent similar to equation (13).
Furthermore, there are other physical reasons for the existence of the
size dependence of the kinetic coefficients related to the solubility limits
in nanomaterials, etc. In this respect, the critical revision of many reac-
tive diffusion problems (where a new phase appears due to interdiffu-
sion processes) with regard for the size dependent diffusion at the initial
stages of evolution (when the new phases are in a nm range) leads to new
nonmonotonic solutions for rates of reactions and a time exponent and
gives the new understanding of the old problems [48].
4. CONCLUSIONS
The paper presents the atomic concept in the following simple form:
‘the nearest bonding and co-ordinated actions of atoms in a nano-
material are different depending on the size of it’. This is caused by the
fact that in the nanosystem one has to take into account the change of a
density and the limited number of atoms, while in a bulk material the
last one is assumed to be infinite. We find it useful to compare size ef-
fect magnitudes using the pair Morse potential, experimental data on
lattice parameter, and Sutton—Chen many-body potential. For the case
of the solid thin film and nanoparticles, the effect of size on the energy
of atomic interaction is found up to a few tens nm. The outlined in the
paper nanophase diagram construction technique provides an accepta-
ble explanation of the first order phase transition phenomenon. Our
findings suggest that the size of the nanomaterial is important for dif-
fusion of atoms and proposed atomic hypothesis may be the reason of
the size-dependent diffusion in nanosize structures. It might be fruit-
ful to further investigations of the atomic mechanisms in different
SIZE-DEPENDENT ATOMIC INTERACTION ENERGIES FOR NANOMATERIALS 485
nanomaterials, through which the size translates into macroscopic be-
haviour and properties.
We believe that one can use the proposed concept to solve the inverse
problem: the modification of the well-known pair potentials for real
metals and chemical elements in nanosize states due to simultaneous
taking into account the theoretical approach (4)—(7), experimental da-
ta, and computer simulations.
It is worth noting that new questions appear, say, when accepting
system-size-dependent energy between nearest-neighbouring atoms,
one should assume also position-dependence of the energy in the sam-
ple and include directional terms. This would be necessary to obtain
end effects, say, in nanowires, whiskers, and transition metals.
ACKNOWLEDGEMENTS
Author thanks Yuriy Bilogorodskyy (Cherkasy, Ukraine) for simula-
tions. The part of the work has been done in the framework of German—
Ukraine DAAD collaboration Project (reference code A/14/02389).
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
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(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
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/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
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/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
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/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
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/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|