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The dependence of surface tension of solid nanoscale films on their thickness
Physica B 406 (2011) 4124–4128
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Physica B journal homepage: www.elsevier.com/locate/physb
The dependence of surface tension of solid nanoscale films on their thickness Alfred P. Chernyshev n Department of General Physics, Novosibirsk State Technical University, pr. Karl Marks 20, Novosibirsk 630092, Russia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 January 2011 Received in revised form 28 July 2011 Accepted 2 August 2011 Available online 7 August 2011
This paper presents a theoretical method to calculate the surface energy dependence on thickness of nanofilms based on the Landau theory. For the first time, it is shown that the surface energy of thin films having free surfaces is greater than the surface energy of macroscopic objects. For nano-objects having free surfaces, it is stated that their interior order parameter is always less than that of macroscopic solids of the same composition. It is obtained that the surface energy of thin films increases with decrease in their thickness passing its maximum meaning. A further decrease in the solid film thickness leads to a monotonic decrease in the surface energy. & 2011 Elsevier B.V. All rights reserved.
Keywords: Thin films Surface energy Second-order phase transition First-order phase transition Order parameter
1. Introduction
2. Model
Elements of microelectronics are becoming smaller and smaller. It is now widely discussed on properties of nano-objects with a typical size below 10 nm [1,2]. The nanoscale range has a number of features. In particular, the dependence of the thermodynamic properties on the characteristic size of solid nanoobjects becomes significant, e.g. the melting temperature, melting entropy and melting enthalpy of nanoparticles having free surfaces are less than those of macroscopic bodies. When nanoparticles are in a solid matrix, the melting temperature may be larger or smaller than those of massive bodies. It depends on the nature of the interaction between surface atoms of nanoparticles and matrix atoms [3,4]. If solid nanoparticles are of small sizes, their surface tension significantly differs from the surface tension of macroscopic solids [5–8]. To account for this dependence, several theoretical models were proposed; molecular dynamics simulations and experimental studies were carried out. The theoretical models predict a decrease in the surface tension with decrease in the characteristic size of nanoparticles [5,6]. The results of the molecular dynamics simulations predict that the surface tension is either size-independent or this dependence is weaker than theoretical [7,8]. The experimental studies show, on the contrary, that the surface tension of macroscopic solids is less than that of nanoparticles [9,10]. The aim of this paper is a theoretical investigation of this dependency for crystalline thin films.
Consider a thin crystal film having thickness L in the nanoscale dimensions, i.e. Lo100 nm. We assume that the film has macroscopic dimensions in the other directions, and in the film the second-order phase transition occurs at a critical temperature Tc. As a model of this film, we consider a two-dimensional object. Its order parameter has a dependence, which is qualitatively depicted in Fig. 1. For the sake of simplicity, the dependence is assumed to be symmetrical about the point x¼ 0. In the center of the film, the order parameter Z takes a value ZV; on the film surfaces, its value is ZS. In different matrixes, the thin film may have different types of matrix-film boundaries. Thus, the surface order parameter may also take different values in dependence on the type of a matrix. Moreover, the dependence Z(x) may be asymmetric if the film separates two different matrixes.
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2.1. Thin films with the second-order phase transition In the framework of the Landau model [11], the density of the Gibbs energy for solids with the second-order phase transition can be defined as gðZÞ ¼ g0 þ
A 2 C 4 Z þ Z 2 4
Let us write the Landau functional # Z þ L=2 " 2 J dZ þgðxÞ dx GðLÞ ¼ 2 dx L=2
ð1Þ
ð2Þ
Here J is assumed to be a constant. Using the analogy with mechanics, one can immediately write the analog of the law of
A.P. Chernyshev / Physica B 406 (2011) 4124–4128
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It is evident that if Z-Z0, then 9x9-N. Therefore, the relation Z ¼ Z0 can be fulfilled in the only case where the object is macroscopic, i.e. L-N. Let us consider a one-dimensional nano-object with the firstorder phase transition in the same way. According to the Landau model for such a system, the density of the Gibbs energy has the following form [12]:
(x) V
gðZÞ ¼ g0 þ
s
A 2 B 3 C 4 Z Z þ Z 2 3 4
ð11Þ
According to Eq. (6), one can obtain that
ZðABZ þ C Z2 Þ ¼ 0
x -L/2
0
L/2
Fig. 1. Schematic dependence of the order parameter Z on coordinate x (red curve). L is the thickness of the solid film; ZV ¼ Z(0); ZS ¼ Z(7L/2). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
energy conservation as 2 J dZ gðZÞ ¼ gðZV Þ 2 dx
ð3Þ
For deriving this relation, it has been taken into account that the symmetry of the problem (see Fig. 1) shows that the first derivative of Z with respect to x equals zero at x¼0. Now Eq. (3) can be transformed to ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðgðZÞgðZV ÞÞ=J, 0 o x o L=2 dZ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ dx L=2 o x o0 2ðgðZÞgðZV ÞÞ=J, In the upper part of the formula, the minus sign is due to the fact that the order parameter decreases when x runs from 0 to L/2. Let us select a small region in the center of the nano-object such that the order parameter can be considered constant within this region. From a physical point of view, the constancy of Z can be stated as follows: dZðxÞ dZðxÞ b max ð5Þ dx x-7 L=2; dx xAD x A ½L=2, þ l=2
Firstly, we assume that the order parameter in the center of the nano-object is size-independent. The equilibrium value of the order parameter in the bulk of macroscopic sample is determined by Eq. (1) and by the condition dg ¼0 dZ
ð6Þ
which gives the minimum of the Gibbs energy. One can obtain from Eqs. (1) and (6) that AZ þ C Z3 ¼ 0
ð7Þ
Solving it, we get the solutions
Z1 ¼ 0;
Z20 ¼
A C
ð8Þ
From Eq. (8) it follows that A ¼ C Z20 . Using the last relation and Eq. (1), one can transform Eq. (4) to the form ( pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 C=ð2JÞðZ0 Z Þ, 0 o x oL=2 dZ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð9Þ dx L=2 ox o 0 C=ð2JÞðZ0 Z Þ, Integrating Eq. (9) then gives rffiffiffiffiffiffi 1 J Z0 Z þconst, x A 0, L=2 ln x¼ Z 2C Z þ Z 0
0
ð10Þ
ð12Þ
The obvious solution is Z1 ¼0 that corresponds to the symmetric phase. From the expression in the parentheses we obtain the equation for A: A ¼ BZ0 C Z20
ð13Þ
Substituting Eqs. (11) and (13) into Eq. (4) gives after obvious transformations a differential equation of the form 1=2 1=2 dZ 2 C B ¼7 ðZ0 ZÞ ðZ0 þ ZÞ2 ðZ0 þ2ZÞ ð14Þ J 4 6 dx The sign in front of the right-hand side of Eq. (14) is determined by the same argument as in Eq. (9). It is easy to see that the integral of Eq. (14), as that of Eq. (7), diverges if Z ¼ Z0. This shows that the value of the order parameter Z0 is achieved only in macroscopic objects. Thus, we have proved that the order parameter inside nano-objects is always less than the order parameter of macroscopic objects at the same thermodynamic conditions. Let us assume that Z(0) ¼ ZV and Z(L/2)¼ ZS. Correspondingly, these are the bulk and surface order parameters. In this case Eq. (9) takes the form 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > C=ð2JÞðZ2 Z2 Þð2Z2 Z2 Z2 Þ, 0 o x oL=2 V V 0 dZ < ð15Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx > : C=ð2JÞðZ2V Z2 Þð2Z20 Z2V Z2 Þ, L=2 ox o 0 Now calculate the surface energy gS. To do this, one can subtract g(ZV) from the total Gibbs energy (2). Here g(ZV) is a homogeneous density of the Gibbs energy # Z þ L=2 " 2 Z þ L=2 J dZ gS ¼ þ gðZÞgðZV Þ dx ¼ 2 ðgðZÞgðZV ÞÞdx 2 dx L=2 L=2 ð16Þ where we used Eq. (3). Substituting Eq. (15) into Eq. (16) gives pffiffiffiffiffiffiffiffi Z ZV qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðZ2V Z2 Þð2Z20 Z2V Z2 ÞdZ ð17Þ gS ¼ 2JC ZS
The integral in Eq. (17) was doubled because g(x) is the even function. After performing the integration [13], this equation becomes 3=2 2 gS ¼ gk2 1 þ k2 (
1 1 1 zð1k2 z2 Þ1=2 ð1z2 Þ1=2 þ 2 ðk2 1ÞFðz,kÞ þ 2 ð1 þ k2 ÞEðz,kÞ k k zS ð18Þ To derive it, we introduced the following notation: z ¼ Z=ZV ; z0 ¼ Z0 =ZV ; k2 ¼ 1=2z20 1; zS ¼ ZS =ZV qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi g ¼ Z30 2JC =3 ¼ 2J9A93 =ð3CÞ E(z,k) is the incomplete elliptic integral of the second kind. Since L-N as ZV-Z0, it follows from Eq. (18) that the surface energy
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2.2. Thin films with the first-order phase transition
of a macroscopic solid is gsm ¼ gf2ð1Eðz1S ,1Þz1S ð1z21S Þg ¼ gð23z1S þz31S Þ
ð19Þ
where z1S ¼ ZS/Z0. Now Eq. (18) can be represented as 3=2 gS 2 1 ¼ k2 f2ð1Eðz1S ,1ÞÞz1S ð1z21S Þg 2 gsm 1þ k 1 1 2 ðk2 1ÞðKðkÞFðzS ,kÞÞ þ 2 ð1 þ k2 Þ k k ðEðkÞEðzS ,kÞÞzS ð1k2 z2S Þ1=2 ð1z2S Þ1=2
ZV ZS
A 2 2 B 3 3 C ðZ ZV Þ ðZ ZV Þ þ ðZ4 Z4V Þ 2 3 4 Z2 þ Z2V C 2 ¼ ðZV Z2 Þ 2Z20 Z2V Z2 Zm 3Z0 ZV Z 4 Z þ ZV
gðZÞgðZV Þ ¼ o
ð20Þ
Here K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively. It should be noted that if both macroscopic and thin solid films have the same temperature and the same chemical composition, then the inequality Z0 4 ZV is held. Therefore, the second-order phase transition in nanosolids occurs at lower temperature than in relative macroscopic solids. The ratio z0 ¼ Z0/ZV is a monotonic function of the nano-object characteristic size: smaller the size of the nano-object, larger the value of z0. Eq. (20) is represented graphically in Fig. 2. This figure shows that a decrease of the thin film thickness gives a surface tension increase if z0 o1.15. For performing the calculations, it was assumed that the ratio
a¼
Let us consider the surface energy of a thin film having the first-order phase transition at any temperature. From the minimum conditions for the Gibbs energy density and the equilibrium condition g(Tm, Z1)¼g(Tm, Zm) at melting temperature Tm it follows that Zm ¼2B/(3C). Here Zm is the equilibrium order parameter of solids at Tm. First, using Eq. (12), we obtain
ð22Þ This relation can be reduced to the form k4 ð1z2 Þ gðZÞgðZV Þ ¼ C Z40 2 ðk þ 1Þ2 " !# rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 k2 þ 1 2z2 2 k2 z2 zm k 2 zþ1
ð23Þ
Here zm ¼ Zm/ZV. Substituting Eq. (23) into Eq. (4) then gives dz k ¼ dðx=L0 Þ ðk2 þ 1Þ1=2 ( " ð1z2 Þ k2 z2 zm
3 k
!#)0:5 rffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ1 2z2 2 2 zþ1 ð24Þ
ð21Þ
is independent of the characteristic size of nano-objects. This ratio depends on the chemical composition and on the structure of the crystal lattice. Hence, zS ¼z1S. The validity of this assumption is discussed below.
where we introduced the parameter L0 defined by relation pffiffiffiffiffiffiffi L0 ¼ 1=Z0 J=C . To select the macroscopic part of the surface energy (ZV ¼ Z0), it is necessary to rewrite Eq. (16) in the following form [13]: pffiffiffiffiffi Z Z0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gsm ¼ 2 2J gðZÞgðZ0 ÞdZ ZS
pffiffiffiffiffiffiffiffi Z ¼ 2JC Z30
1
ð1z1 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þ z1 Þ2 z1m ð1 þ 2z1 Þdz1
Z1S
pffiffiffiffiffiffiffiffi Z 1z1S 1=2 pffiffiffiffiffiffiffiffi ¼ 2JC Z30 xX dx ¼ 2JC Z30 0 ðz1m 2Þðz1m z1 1Þ 1=2 z1m ðz1m 2Þð1z1m Þ 3=2 X X Arsh ¼ 3 2 2 #1z1S pffiffiffiffiffiffiffiffi z1m z1 1 ¼ 2JC Z30 gðz1S Þ ð25Þ ðz1m ð1z1m ÞÞ1=2 0 Here x¼1 z1; X¼x2 2(2 z1m)xþ 4 3z1m, z1 ¼ Z/Z0, z1m ¼ Zm/Z0; g(z1S) is the relation in the braces. The surface energy can be represented by the reduced form 3=2 Z 1 gS 1 2k2 ¼ ð1z2 Þ1=2 2 gsm g k þ1 zS " !#1=2 rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 k2 þ 1 2z2 2 2 2 k z zm dz k 2 z þ1
Fig. 2. Dependence of the reduced surface energy gS/gsm on the thicknessdependent parameter z0. gS is the surface energy of the thin solid film; gsm is that of the macroscopic solid film (L b100 nm); z0 ¼ Z0/ZV; Z0 ¼ Z(0) and ZV ¼ Z(0) are the bulk order parameters of the macroscopic and thin solid films. This figure relates to thin films with the second order phase transition. Red, green and blue curves correspond to a ¼ 1.5, 1.8 and N, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ð26Þ
The results of the definite integration according to relation (26) are shown in Fig. 3. In Figs. 2 and 3 we used the same values of a. It is accepted that the parameter of order is identical at the melting point for all characteristic sizes of nanoparticles. Thus, zm/z1m ¼ z0. The maximum value of z1m and zm is equal to unity. The minimum value of z0 is achieved for macroscopic solids. The calculations were performed at z1m ¼0.4 and zm ¼z1mz0. The order parameter of macroscopic body Z0 decreases with an increase in the temperature. The parameter z0 of the macroscopic solids is of the minimum value, equal to unity. Both parameters z1m and zm are monotonic functions of temperature.
A.P. Chernyshev / Physica B 406 (2011) 4124–4128
Fig. 3. Dependence of the reduced surface energy gS/gsm on the thicknessdependent parameter z0. gS is the surface energy of the thin solid film; gsm is that of the macroscopic solid film (L b 100 nm); z0 ¼ Z0/ZV; Z0 ¼ Z(0) and ZV ¼ Z(0) are the bulk order parameters of the macroscopic and thin solid films. This figure relates to thin films with the first order phase transition. Red, green and blue curves correspond to a ¼ 1.5, 1.8 and N, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3. Discussion In this work, the theoretical model is developed to describe the experimentally observed fact [9,10] that the surface tension of nanoparticles is greater than that of macroscopic particles of the same substance. Qualitatively, this increase in the surface tension is explained as follows. As the characteristic size of a nano-object increases, the ratio
w¼
NS NV
ð27Þ
decreases. NS and NV are the number of surface and bulk atoms, respectively. The surface energy part of the nano-object energy increases proportionally to w. The lowest curve (see Figs. 2 and 3) demonstrates the case where the surface parameter of order is equal to zero. This is possible in practice if the surface layer has already passed into the symmetric phase while the inner layers are still in the asymmetric phase. In this case, the surface energy of the thin films, in which the second-order phase transition occurs at Tc, is always less than that of the analogous macroscopic objects (Fig. 2). In order to obtain the dependence of z0 on w, one must determine what physical quantity can serve as an order parameter for solving this problem. Corresponding models can involve a single scalar order parameter or multiple order parameters. In general, it might be difficult to associate the order parameter with a measurable quantity. For many solid–solid transformations, a set of order parameters naturally arises from the symmetry and orientation relationships between the phases [12,14]. A single scalar order parameter can be used to model solidification of a single-phase material. However, to employ such a simple description of the liquid–solid transition, one has to adopt a number of approximations [15,16]. A rigorous description of the liquid-tosolid transformation requires an infinite number of order parameters. One can imagine the lattice of the crystal extending into the interfacial region [16,17]. Then a Fourier series can be used to represent the local atomic density at each location. The Fourier coefficients in the series are functions of distance. However, a detailed description of the density on this scale would require a full Fourier series, which contains an infinite set of order
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parameters [15]. There exists a representation of the atomic density variation across an interface in reciprocal space [15,18]. This representation describes the approximations necessary to reduce the infinite number of order parameters to just one scalar. The first simplification is that a description of the liquid-to-crystal transition can be reduced to a small number of amplitudes that simply describe the probability of the occupancy at the lattice positions in the three-dimensional unit cell. The second simplification is that the amplitudes of this limited set of Fourier components are proportional to each other. Then a single scalar can describe the amplitudes as they vary across the interface. For single order parameter models, anisotropy must be introduced ad hoc through an orientation dependence of the gradient energy coefficients. Alternatively, one can keep the multiple-order parameter picture and naturally derive anisotropy [19,20]. We shall introduce a new alternative model in which we use a single-order parameter to describe the thermodynamic behavior of metal nanoparticles. Our physical interpretation of this scalar order parameter is close to [17]. Mikheev and Chernov [17] have showed that one possible physical interpretation of a scalar order parameter is the probability of finding an atom at a particular location. The atoms tend to be located at discrete atomic planes corresponding to the crystal. Obviously, the probability has maxima at these locations. In close proximity to the solid–liquid interface, the probability has the same average value but becomes less localized. Finally the probability achieves a constant value in the liquid indicating the absence of localization of atoms to specific sites. Our approach is based on a consideration of thermal vibrations of atoms of nano-objects. This kind of motion disturbs the regular localization of atoms in a crystal lattice. To describe this disturbance, one must have all information about the phonon spectrum of the lattice. This method requiring a full knowledge of the normal modes of nano-objects gives a finite set of order parameters. However, we can obtain a reduced description using a scalar order parameter by the following way. We can introduce the density of the probability of finding an atom at the point with radius vector r:r(r). To simplify the problem, we assume that each atom performs harmonic vibrations around the equilibrium position, i.e. we use the harmonic approximation. In this approximation, the vibrational motion of every atom of the crystal lattice can be represented as the superposition of normal oscillations. Thus, the total combinations of the normal modes contain all information about the position of the atom relative to the crystal lattice. In a crystal consisting of N primitive cells, each containing p atoms in each cell, there are 3p vibrational modes with frequencies oS(kt) (here, s ranging from 1 to p). A set of kt denotes the wave vectors (the index ranges from 1 to N). After averaging over time, we have the probability distribution r(r)dx dy dz of atom coordinates. Let us consider the coordinate distribution of the harmonic oscillator in the space. This distribution is given by the Bloch function [14] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mos ðkt Þtanhðyst =2Þ mos ðkt Þ 2 qst tanhðyst =2Þ exp rðqst Þ ¼ _ p_ _os ðkt Þ ð28Þ yst ¼ kB T Here m is the atomic mass, and _, kB are Planck’s and Boltzmann’s constants, respectively and qst is the coordinate of the atom. It directly follows from relation (28) that the mean square displacement is y _ st coth s2st ¼ ð29Þ 2mos ðkt Þ 2 Obviously, the width of distribution (28) is also characterized by (29), since it is the second moment of this distribution. Therefore, the reciprocal value of s2st can serve as a measure of ordering of the crystal lattice, i.e. it acts as order parameter. Other physical arguments for this choice are discussed below. Since the normal modes are statistically independent, the total second moment can be
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A.P. Chernyshev / Physica B 406 (2011) 4124–4128
obtained simply by summing 3p N 1 XX s2 ¼ s2 pN s ¼ 1 t ¼ 1 st
ð30Þ
After introduction of the phonon density of states g(o), Eq. (30) can be rewritten as Z 1 _2 s2 ¼ y1 gðyÞcothðy=2Þ dy ð31Þ 2mkB T 0 It follows from Eqs. (28) to (31) that the second moment of the distribution r(r) determines the mean square displacement of atoms from their equilibrium position. This allows us to determine the order parameter as a function of s2 [21,22]. An important property of the function r(r) is its symmetry, i.e. its group of coordinate transformations with respect to which r(r) is invariant. It is evident that this group determines the crystal symmetry too. We shall consider the isotropic liquids for which r(r) is constant. To determine the melting point of solids, we will also use the Lindemann criterion. Therefore it is convenient to use s2 instead of r(r). Certainly, the carried out averaging hides the majority of the special features of crystal structure, e.g. anisotropy of the thermal vibrations of atoms. Nevertheless, it has been shown that the Lindemann criterion is suitable to describe melting properties of nanoparticles [1,4,6]. Introduce the order parameter Z as any function of s2. Let it satisfy the following conditions [21]:
Z ¼ 0 if T ZTm , liquid phase Z ¼ 1 if T ¼ Tm , solid phase
ð32Þ
According to the Lindemann criterion, at the melting point the rootpffiffiffiffiffiffiffi mean-square displacement of atoms in solid substance, s2m , equals 2 the definite part x of the crystal lattice parameter a, i.e. s2m ¼ x a2 2 2 Therefore the melting occurs if s Z sm One of the functions, which meet both conditions (32) and the Lindemann criterion simultaneously, is
Z ¼ s2m =s2
Z-0 as s2 -1 ðliquid phase at Tm Þ ð34Þ
It is worth mentioning for a further justification of definition (33) that this definition gives the order parameter, which correlates with physical properties of materials [2,21,22]. Really, s2 is proportional to T in the high-temperature Debye approximation [14]. Thus, it directly follows from definition (33) that Z ¼Tm/T in this approximation. It is well known that similar materials (e.g. metals) having the same relation T/Tm, but different temperatures, demonstrate similar physical properties [23]. Of course, when the second-order phase transition ferromagnetic–paramagnetic is considered, it is relevant to use the magnetization as an order parameter. However, the structural transformations of nano-objects are adequately described using definition (32) [4,21,22]. Based on both definitions (21) and (33), the ratio a can be determined from the following relation:
a¼
ZV s2S ¼ ZS s2V
s2 ðw þ dwÞs2 ðwÞ ¼ ða1Þs2 ðwÞ dw
ð36Þ
After simple transformations of this equation, we find the exponential dependence
ZðwÞ ¼ Z0 exp½ða1Þw
ð37Þ
Here Z0 is the order parameter of a macroscopic body. Let us adopt that ZV E Z(w). From Eq. (37) it then follows that z0 Z0 =ZðwÞ ¼ exp½ða1Þw
ð38Þ
One can see from Figs. 2 and 3 that the surface tension approaches its maximal value at z0 E1.13 (see Fig. 2) and z0 E1.4 (Fig. 3). It corresponds to w ¼0.19 and w ¼0.52 at a ¼ 1.65. 4. Conclusions A large value of the surface tension of nanoparticles in comparison with the surface tension of macroscopic solids was discovered experimentally in Refs. [9,10], but still had no theoretical explanation. In the present paper, the new model, which describes such a dependence of the surface tension, has been developed for the first time. The model allowed us to obtain the adequate dependence of the surface energy of thin films on their thickness in the nanoscopic scale. For nano-objects having free surfaces, it has been shown that their interior order parameter ZV is always less than that of macroscopic solids of the same composition. It was obtained that the surface energy of thin films increases with decrease in their thickness approaching its maximum meaning. Then the surface energy decreases monotonically.
ð33Þ
where s2m is the mean square displacement of atoms from their equilibrium positions at Tm. In the case of the first-order transition, where the order parameter has a jump at Tm, the order parameter has to satisfy conditions (32). From relationships (32) and (33) it follows that
Z-1 as s2 -ðxaÞ2 ðsolid phase at Tm Þ
almost independent of w, i.e. a ¼constant for a given material. For metals, typical values of a are in the range from 1.5 to 1.7 [22]. To determine the form of the dependence of the order parameter on the degree of dispersion of nano-objects, one can use Shi’s equation [4]
ð35Þ
For the derivation of this formula, we have employed the Lindemann criterion. It implies that s2m ¼ const As shown in Refs. [4,24], a is
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Q.S. Mei, K. Lu, Prog. Mater. Sci. 52 (2007) 1175. A.P. Chernyshev, Phys. Lett. A 374 (2010) 4622. K. Lu, Z.H. Jin, Curr. Opin. Solid State Mater. Sci. 5 (2001) 39. Q. Jiang, C.C. Yang, Curr. Nanosci. 4 (2008) 179. M.N. Magomedov, High Temp 42 (2004) 221. H.M. Lu, Q. Jiang, J. Phys. Chem. B 108 (2004) 5617. T. Hawa, M.R. Zachariah, J. Chem. Phys. 121 (2004) 9043. V.V. Hoang, B.T. Khanh, J. Phys.: Condens. Matter. 21 (2009) 075103 9 pp. K.K. Nanda, A. Maisels, F.E. Kruis, H. Fissan, S. Stappert, Phys. Rev. Lett. 91 (2003) 106102 4 pp. K.K. Nanda, A. Maisels, F.E. Kruis, J. Phys. Chem. C 112 (2008) 13488. A.Z. Patashinskii, V.L. Pokrovskii, Fluctuation Theory of Phase Transitions, Elsevier, 1979. Y.A. Izyumov, V.N. Syromyatnikov, Phase Transitions and Crystal Symmetry, Springer-Verlag, Berlin, Heidelberg, New York, 2009. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, sixth ed., Academic Press, San Diego, 2000. L.D. Landau, E.M. Lifshitz, Statistical Physics, third ed.,Course of Theoretical Physics, vol. 5, Butterworth-Heinemann, Oxford, 1980. P. Harrowell, D.W. Oxtoby, J. Chem. Phys. 80 (1984) 1639. W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma, Annu. Rev. Mater. Res. 32 (2002) 163. L.V. Mikheev, A.A. Chernov, J. Cryst. Growth 112 (1991) 591. A.G. Khachaturyan, Philos. Mag. A 74 (1996) 3. L.-Q. Chen, Annu. Rev. Mater. Res. 32 (2002) 113. V.G. Vaks, K.Yu. Khromov, J. Exp. Theor. Phys. 109 (2009) 619. A.P. Chernyshev, Mater. Chem. Phys. 112 (2008) 226. A.P. Chernyshev, Mater. Lett. 63 (2009) 1525. A.W. Cahn, P. Haasen, Physical Metallurgy, fourth ed., North-Holland, Amsterdam, 1996. C.Q. Sun, Prog. Solid State Chem. 35 (2007) 1.
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The dependence of surface tension of solid nanoscale films on their thickness Alfred P. Chernyshev n Department of General Physics, Novosibirsk State Technical University, pr. Karl Marks 20, Novosibirsk 630092, Russia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 January 2011 Received in revised form 28 July 2011 Accepted 2 August 2011 Available online 7 August 2011
This paper presents a theoretical method to calculate the surface energy dependence on thickness of nanofilms based on the Landau theory. For the first time, it is shown that the surface energy of thin films having free surfaces is greater than the surface energy of macroscopic objects. For nano-objects having free surfaces, it is stated that their interior order parameter is always less than that of macroscopic solids of the same composition. It is obtained that the surface energy of thin films increases with decrease in their thickness passing its maximum meaning. A further decrease in the solid film thickness leads to a monotonic decrease in the surface energy. & 2011 Elsevier B.V. All rights reserved.
Keywords: Thin films Surface energy Second-order phase transition First-order phase transition Order parameter
1. Introduction
2. Model
Elements of microelectronics are becoming smaller and smaller. It is now widely discussed on properties of nano-objects with a typical size below 10 nm [1,2]. The nanoscale range has a number of features. In particular, the dependence of the thermodynamic properties on the characteristic size of solid nanoobjects becomes significant, e.g. the melting temperature, melting entropy and melting enthalpy of nanoparticles having free surfaces are less than those of macroscopic bodies. When nanoparticles are in a solid matrix, the melting temperature may be larger or smaller than those of massive bodies. It depends on the nature of the interaction between surface atoms of nanoparticles and matrix atoms [3,4]. If solid nanoparticles are of small sizes, their surface tension significantly differs from the surface tension of macroscopic solids [5–8]. To account for this dependence, several theoretical models were proposed; molecular dynamics simulations and experimental studies were carried out. The theoretical models predict a decrease in the surface tension with decrease in the characteristic size of nanoparticles [5,6]. The results of the molecular dynamics simulations predict that the surface tension is either size-independent or this dependence is weaker than theoretical [7,8]. The experimental studies show, on the contrary, that the surface tension of macroscopic solids is less than that of nanoparticles [9,10]. The aim of this paper is a theoretical investigation of this dependency for crystalline thin films.
Consider a thin crystal film having thickness L in the nanoscale dimensions, i.e. Lo100 nm. We assume that the film has macroscopic dimensions in the other directions, and in the film the second-order phase transition occurs at a critical temperature Tc. As a model of this film, we consider a two-dimensional object. Its order parameter has a dependence, which is qualitatively depicted in Fig. 1. For the sake of simplicity, the dependence is assumed to be symmetrical about the point x¼ 0. In the center of the film, the order parameter Z takes a value ZV; on the film surfaces, its value is ZS. In different matrixes, the thin film may have different types of matrix-film boundaries. Thus, the surface order parameter may also take different values in dependence on the type of a matrix. Moreover, the dependence Z(x) may be asymmetric if the film separates two different matrixes.
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2.1. Thin films with the second-order phase transition In the framework of the Landau model [11], the density of the Gibbs energy for solids with the second-order phase transition can be defined as gðZÞ ¼ g0 þ
A 2 C 4 Z þ Z 2 4
Let us write the Landau functional # Z þ L=2 " 2 J dZ þgðxÞ dx GðLÞ ¼ 2 dx L=2
ð1Þ
ð2Þ
Here J is assumed to be a constant. Using the analogy with mechanics, one can immediately write the analog of the law of
A.P. Chernyshev / Physica B 406 (2011) 4124–4128
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It is evident that if Z-Z0, then 9x9-N. Therefore, the relation Z ¼ Z0 can be fulfilled in the only case where the object is macroscopic, i.e. L-N. Let us consider a one-dimensional nano-object with the firstorder phase transition in the same way. According to the Landau model for such a system, the density of the Gibbs energy has the following form [12]:
(x) V
gðZÞ ¼ g0 þ
s
A 2 B 3 C 4 Z Z þ Z 2 3 4
ð11Þ
According to Eq. (6), one can obtain that
ZðABZ þ C Z2 Þ ¼ 0
x -L/2
0
L/2
Fig. 1. Schematic dependence of the order parameter Z on coordinate x (red curve). L is the thickness of the solid film; ZV ¼ Z(0); ZS ¼ Z(7L/2). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
energy conservation as 2 J dZ gðZÞ ¼ gðZV Þ 2 dx
ð3Þ
For deriving this relation, it has been taken into account that the symmetry of the problem (see Fig. 1) shows that the first derivative of Z with respect to x equals zero at x¼0. Now Eq. (3) can be transformed to ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðgðZÞgðZV ÞÞ=J, 0 o x o L=2 dZ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ dx L=2 o x o0 2ðgðZÞgðZV ÞÞ=J, In the upper part of the formula, the minus sign is due to the fact that the order parameter decreases when x runs from 0 to L/2. Let us select a small region in the center of the nano-object such that the order parameter can be considered constant within this region. From a physical point of view, the constancy of Z can be stated as follows: dZðxÞ dZðxÞ b max ð5Þ dx x-7 L=2; dx xAD x A ½L=2, þ l=2
Firstly, we assume that the order parameter in the center of the nano-object is size-independent. The equilibrium value of the order parameter in the bulk of macroscopic sample is determined by Eq. (1) and by the condition dg ¼0 dZ
ð6Þ
which gives the minimum of the Gibbs energy. One can obtain from Eqs. (1) and (6) that AZ þ C Z3 ¼ 0
ð7Þ
Solving it, we get the solutions
Z1 ¼ 0;
Z20 ¼
A C
ð8Þ
From Eq. (8) it follows that A ¼ C Z20 . Using the last relation and Eq. (1), one can transform Eq. (4) to the form ( pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 C=ð2JÞðZ0 Z Þ, 0 o x oL=2 dZ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð9Þ dx L=2 ox o 0 C=ð2JÞðZ0 Z Þ, Integrating Eq. (9) then gives rffiffiffiffiffiffi 1 J Z0 Z þconst, x A 0, L=2 ln x¼ Z 2C Z þ Z 0
0
ð10Þ
ð12Þ
The obvious solution is Z1 ¼0 that corresponds to the symmetric phase. From the expression in the parentheses we obtain the equation for A: A ¼ BZ0 C Z20
ð13Þ
Substituting Eqs. (11) and (13) into Eq. (4) gives after obvious transformations a differential equation of the form 1=2 1=2 dZ 2 C B ¼7 ðZ0 ZÞ ðZ0 þ ZÞ2 ðZ0 þ2ZÞ ð14Þ J 4 6 dx The sign in front of the right-hand side of Eq. (14) is determined by the same argument as in Eq. (9). It is easy to see that the integral of Eq. (14), as that of Eq. (7), diverges if Z ¼ Z0. This shows that the value of the order parameter Z0 is achieved only in macroscopic objects. Thus, we have proved that the order parameter inside nano-objects is always less than the order parameter of macroscopic objects at the same thermodynamic conditions. Let us assume that Z(0) ¼ ZV and Z(L/2)¼ ZS. Correspondingly, these are the bulk and surface order parameters. In this case Eq. (9) takes the form 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > C=ð2JÞðZ2 Z2 Þð2Z2 Z2 Z2 Þ, 0 o x oL=2 V V 0 dZ < ð15Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx > : C=ð2JÞðZ2V Z2 Þð2Z20 Z2V Z2 Þ, L=2 ox o 0 Now calculate the surface energy gS. To do this, one can subtract g(ZV) from the total Gibbs energy (2). Here g(ZV) is a homogeneous density of the Gibbs energy # Z þ L=2 " 2 Z þ L=2 J dZ gS ¼ þ gðZÞgðZV Þ dx ¼ 2 ðgðZÞgðZV ÞÞdx 2 dx L=2 L=2 ð16Þ where we used Eq. (3). Substituting Eq. (15) into Eq. (16) gives pffiffiffiffiffiffiffiffi Z ZV qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðZ2V Z2 Þð2Z20 Z2V Z2 ÞdZ ð17Þ gS ¼ 2JC ZS
The integral in Eq. (17) was doubled because g(x) is the even function. After performing the integration [13], this equation becomes 3=2 2 gS ¼ gk2 1 þ k2 (
1 1 1 zð1k2 z2 Þ1=2 ð1z2 Þ1=2 þ 2 ðk2 1ÞFðz,kÞ þ 2 ð1 þ k2 ÞEðz,kÞ k k zS ð18Þ To derive it, we introduced the following notation: z ¼ Z=ZV ; z0 ¼ Z0 =ZV ; k2 ¼ 1=2z20 1; zS ¼ ZS =ZV qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi g ¼ Z30 2JC =3 ¼ 2J9A93 =ð3CÞ E(z,k) is the incomplete elliptic integral of the second kind. Since L-N as ZV-Z0, it follows from Eq. (18) that the surface energy
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2.2. Thin films with the first-order phase transition
of a macroscopic solid is gsm ¼ gf2ð1Eðz1S ,1Þz1S ð1z21S Þg ¼ gð23z1S þz31S Þ
ð19Þ
where z1S ¼ ZS/Z0. Now Eq. (18) can be represented as 3=2 gS 2 1 ¼ k2 f2ð1Eðz1S ,1ÞÞz1S ð1z21S Þg 2 gsm 1þ k 1 1 2 ðk2 1ÞðKðkÞFðzS ,kÞÞ þ 2 ð1 þ k2 Þ k k ðEðkÞEðzS ,kÞÞzS ð1k2 z2S Þ1=2 ð1z2S Þ1=2
ZV ZS
A 2 2 B 3 3 C ðZ ZV Þ ðZ ZV Þ þ ðZ4 Z4V Þ 2 3 4 Z2 þ Z2V C 2 ¼ ðZV Z2 Þ 2Z20 Z2V Z2 Zm 3Z0 ZV Z 4 Z þ ZV
gðZÞgðZV Þ ¼ o
ð20Þ
Here K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively. It should be noted that if both macroscopic and thin solid films have the same temperature and the same chemical composition, then the inequality Z0 4 ZV is held. Therefore, the second-order phase transition in nanosolids occurs at lower temperature than in relative macroscopic solids. The ratio z0 ¼ Z0/ZV is a monotonic function of the nano-object characteristic size: smaller the size of the nano-object, larger the value of z0. Eq. (20) is represented graphically in Fig. 2. This figure shows that a decrease of the thin film thickness gives a surface tension increase if z0 o1.15. For performing the calculations, it was assumed that the ratio
a¼
Let us consider the surface energy of a thin film having the first-order phase transition at any temperature. From the minimum conditions for the Gibbs energy density and the equilibrium condition g(Tm, Z1)¼g(Tm, Zm) at melting temperature Tm it follows that Zm ¼2B/(3C). Here Zm is the equilibrium order parameter of solids at Tm. First, using Eq. (12), we obtain
ð22Þ This relation can be reduced to the form k4 ð1z2 Þ gðZÞgðZV Þ ¼ C Z40 2 ðk þ 1Þ2 " !# rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 k2 þ 1 2z2 2 k2 z2 zm k 2 zþ1
ð23Þ
Here zm ¼ Zm/ZV. Substituting Eq. (23) into Eq. (4) then gives dz k ¼ dðx=L0 Þ ðk2 þ 1Þ1=2 ( " ð1z2 Þ k2 z2 zm
3 k
!#)0:5 rffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ1 2z2 2 2 zþ1 ð24Þ
ð21Þ
is independent of the characteristic size of nano-objects. This ratio depends on the chemical composition and on the structure of the crystal lattice. Hence, zS ¼z1S. The validity of this assumption is discussed below.
where we introduced the parameter L0 defined by relation pffiffiffiffiffiffiffi L0 ¼ 1=Z0 J=C . To select the macroscopic part of the surface energy (ZV ¼ Z0), it is necessary to rewrite Eq. (16) in the following form [13]: pffiffiffiffiffi Z Z0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gsm ¼ 2 2J gðZÞgðZ0 ÞdZ ZS
pffiffiffiffiffiffiffiffi Z ¼ 2JC Z30
1
ð1z1 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þ z1 Þ2 z1m ð1 þ 2z1 Þdz1
Z1S
pffiffiffiffiffiffiffiffi Z 1z1S 1=2 pffiffiffiffiffiffiffiffi ¼ 2JC Z30 xX dx ¼ 2JC Z30 0 ðz1m 2Þðz1m z1 1Þ 1=2 z1m ðz1m 2Þð1z1m Þ 3=2 X X Arsh ¼ 3 2 2 #1z1S pffiffiffiffiffiffiffiffi z1m z1 1 ¼ 2JC Z30 gðz1S Þ ð25Þ ðz1m ð1z1m ÞÞ1=2 0 Here x¼1 z1; X¼x2 2(2 z1m)xþ 4 3z1m, z1 ¼ Z/Z0, z1m ¼ Zm/Z0; g(z1S) is the relation in the braces. The surface energy can be represented by the reduced form 3=2 Z 1 gS 1 2k2 ¼ ð1z2 Þ1=2 2 gsm g k þ1 zS " !#1=2 rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 k2 þ 1 2z2 2 2 2 k z zm dz k 2 z þ1
Fig. 2. Dependence of the reduced surface energy gS/gsm on the thicknessdependent parameter z0. gS is the surface energy of the thin solid film; gsm is that of the macroscopic solid film (L b100 nm); z0 ¼ Z0/ZV; Z0 ¼ Z(0) and ZV ¼ Z(0) are the bulk order parameters of the macroscopic and thin solid films. This figure relates to thin films with the second order phase transition. Red, green and blue curves correspond to a ¼ 1.5, 1.8 and N, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ð26Þ
The results of the definite integration according to relation (26) are shown in Fig. 3. In Figs. 2 and 3 we used the same values of a. It is accepted that the parameter of order is identical at the melting point for all characteristic sizes of nanoparticles. Thus, zm/z1m ¼ z0. The maximum value of z1m and zm is equal to unity. The minimum value of z0 is achieved for macroscopic solids. The calculations were performed at z1m ¼0.4 and zm ¼z1mz0. The order parameter of macroscopic body Z0 decreases with an increase in the temperature. The parameter z0 of the macroscopic solids is of the minimum value, equal to unity. Both parameters z1m and zm are monotonic functions of temperature.
A.P. Chernyshev / Physica B 406 (2011) 4124–4128
Fig. 3. Dependence of the reduced surface energy gS/gsm on the thicknessdependent parameter z0. gS is the surface energy of the thin solid film; gsm is that of the macroscopic solid film (L b 100 nm); z0 ¼ Z0/ZV; Z0 ¼ Z(0) and ZV ¼ Z(0) are the bulk order parameters of the macroscopic and thin solid films. This figure relates to thin films with the first order phase transition. Red, green and blue curves correspond to a ¼ 1.5, 1.8 and N, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3. Discussion In this work, the theoretical model is developed to describe the experimentally observed fact [9,10] that the surface tension of nanoparticles is greater than that of macroscopic particles of the same substance. Qualitatively, this increase in the surface tension is explained as follows. As the characteristic size of a nano-object increases, the ratio
w¼
NS NV
ð27Þ
decreases. NS and NV are the number of surface and bulk atoms, respectively. The surface energy part of the nano-object energy increases proportionally to w. The lowest curve (see Figs. 2 and 3) demonstrates the case where the surface parameter of order is equal to zero. This is possible in practice if the surface layer has already passed into the symmetric phase while the inner layers are still in the asymmetric phase. In this case, the surface energy of the thin films, in which the second-order phase transition occurs at Tc, is always less than that of the analogous macroscopic objects (Fig. 2). In order to obtain the dependence of z0 on w, one must determine what physical quantity can serve as an order parameter for solving this problem. Corresponding models can involve a single scalar order parameter or multiple order parameters. In general, it might be difficult to associate the order parameter with a measurable quantity. For many solid–solid transformations, a set of order parameters naturally arises from the symmetry and orientation relationships between the phases [12,14]. A single scalar order parameter can be used to model solidification of a single-phase material. However, to employ such a simple description of the liquid–solid transition, one has to adopt a number of approximations [15,16]. A rigorous description of the liquid-tosolid transformation requires an infinite number of order parameters. One can imagine the lattice of the crystal extending into the interfacial region [16,17]. Then a Fourier series can be used to represent the local atomic density at each location. The Fourier coefficients in the series are functions of distance. However, a detailed description of the density on this scale would require a full Fourier series, which contains an infinite set of order
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parameters [15]. There exists a representation of the atomic density variation across an interface in reciprocal space [15,18]. This representation describes the approximations necessary to reduce the infinite number of order parameters to just one scalar. The first simplification is that a description of the liquid-to-crystal transition can be reduced to a small number of amplitudes that simply describe the probability of the occupancy at the lattice positions in the three-dimensional unit cell. The second simplification is that the amplitudes of this limited set of Fourier components are proportional to each other. Then a single scalar can describe the amplitudes as they vary across the interface. For single order parameter models, anisotropy must be introduced ad hoc through an orientation dependence of the gradient energy coefficients. Alternatively, one can keep the multiple-order parameter picture and naturally derive anisotropy [19,20]. We shall introduce a new alternative model in which we use a single-order parameter to describe the thermodynamic behavior of metal nanoparticles. Our physical interpretation of this scalar order parameter is close to [17]. Mikheev and Chernov [17] have showed that one possible physical interpretation of a scalar order parameter is the probability of finding an atom at a particular location. The atoms tend to be located at discrete atomic planes corresponding to the crystal. Obviously, the probability has maxima at these locations. In close proximity to the solid–liquid interface, the probability has the same average value but becomes less localized. Finally the probability achieves a constant value in the liquid indicating the absence of localization of atoms to specific sites. Our approach is based on a consideration of thermal vibrations of atoms of nano-objects. This kind of motion disturbs the regular localization of atoms in a crystal lattice. To describe this disturbance, one must have all information about the phonon spectrum of the lattice. This method requiring a full knowledge of the normal modes of nano-objects gives a finite set of order parameters. However, we can obtain a reduced description using a scalar order parameter by the following way. We can introduce the density of the probability of finding an atom at the point with radius vector r:r(r). To simplify the problem, we assume that each atom performs harmonic vibrations around the equilibrium position, i.e. we use the harmonic approximation. In this approximation, the vibrational motion of every atom of the crystal lattice can be represented as the superposition of normal oscillations. Thus, the total combinations of the normal modes contain all information about the position of the atom relative to the crystal lattice. In a crystal consisting of N primitive cells, each containing p atoms in each cell, there are 3p vibrational modes with frequencies oS(kt) (here, s ranging from 1 to p). A set of kt denotes the wave vectors (the index ranges from 1 to N). After averaging over time, we have the probability distribution r(r)dx dy dz of atom coordinates. Let us consider the coordinate distribution of the harmonic oscillator in the space. This distribution is given by the Bloch function [14] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mos ðkt Þtanhðyst =2Þ mos ðkt Þ 2 qst tanhðyst =2Þ exp rðqst Þ ¼ _ p_ _os ðkt Þ ð28Þ yst ¼ kB T Here m is the atomic mass, and _, kB are Planck’s and Boltzmann’s constants, respectively and qst is the coordinate of the atom. It directly follows from relation (28) that the mean square displacement is y _ st coth s2st ¼ ð29Þ 2mos ðkt Þ 2 Obviously, the width of distribution (28) is also characterized by (29), since it is the second moment of this distribution. Therefore, the reciprocal value of s2st can serve as a measure of ordering of the crystal lattice, i.e. it acts as order parameter. Other physical arguments for this choice are discussed below. Since the normal modes are statistically independent, the total second moment can be
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A.P. Chernyshev / Physica B 406 (2011) 4124–4128
obtained simply by summing 3p N 1 XX s2 ¼ s2 pN s ¼ 1 t ¼ 1 st
ð30Þ
After introduction of the phonon density of states g(o), Eq. (30) can be rewritten as Z 1 _2 s2 ¼ y1 gðyÞcothðy=2Þ dy ð31Þ 2mkB T 0 It follows from Eqs. (28) to (31) that the second moment of the distribution r(r) determines the mean square displacement of atoms from their equilibrium position. This allows us to determine the order parameter as a function of s2 [21,22]. An important property of the function r(r) is its symmetry, i.e. its group of coordinate transformations with respect to which r(r) is invariant. It is evident that this group determines the crystal symmetry too. We shall consider the isotropic liquids for which r(r) is constant. To determine the melting point of solids, we will also use the Lindemann criterion. Therefore it is convenient to use s2 instead of r(r). Certainly, the carried out averaging hides the majority of the special features of crystal structure, e.g. anisotropy of the thermal vibrations of atoms. Nevertheless, it has been shown that the Lindemann criterion is suitable to describe melting properties of nanoparticles [1,4,6]. Introduce the order parameter Z as any function of s2. Let it satisfy the following conditions [21]:
Z ¼ 0 if T ZTm , liquid phase Z ¼ 1 if T ¼ Tm , solid phase
ð32Þ
According to the Lindemann criterion, at the melting point the rootpffiffiffiffiffiffiffi mean-square displacement of atoms in solid substance, s2m , equals 2 the definite part x of the crystal lattice parameter a, i.e. s2m ¼ x a2 2 2 Therefore the melting occurs if s Z sm One of the functions, which meet both conditions (32) and the Lindemann criterion simultaneously, is
Z ¼ s2m =s2
Z-0 as s2 -1 ðliquid phase at Tm Þ ð34Þ
It is worth mentioning for a further justification of definition (33) that this definition gives the order parameter, which correlates with physical properties of materials [2,21,22]. Really, s2 is proportional to T in the high-temperature Debye approximation [14]. Thus, it directly follows from definition (33) that Z ¼Tm/T in this approximation. It is well known that similar materials (e.g. metals) having the same relation T/Tm, but different temperatures, demonstrate similar physical properties [23]. Of course, when the second-order phase transition ferromagnetic–paramagnetic is considered, it is relevant to use the magnetization as an order parameter. However, the structural transformations of nano-objects are adequately described using definition (32) [4,21,22]. Based on both definitions (21) and (33), the ratio a can be determined from the following relation:
a¼
ZV s2S ¼ ZS s2V
s2 ðw þ dwÞs2 ðwÞ ¼ ða1Þs2 ðwÞ dw
ð36Þ
After simple transformations of this equation, we find the exponential dependence
ZðwÞ ¼ Z0 exp½ða1Þw
ð37Þ
Here Z0 is the order parameter of a macroscopic body. Let us adopt that ZV E Z(w). From Eq. (37) it then follows that z0 Z0 =ZðwÞ ¼ exp½ða1Þw
ð38Þ
One can see from Figs. 2 and 3 that the surface tension approaches its maximal value at z0 E1.13 (see Fig. 2) and z0 E1.4 (Fig. 3). It corresponds to w ¼0.19 and w ¼0.52 at a ¼ 1.65. 4. Conclusions A large value of the surface tension of nanoparticles in comparison with the surface tension of macroscopic solids was discovered experimentally in Refs. [9,10], but still had no theoretical explanation. In the present paper, the new model, which describes such a dependence of the surface tension, has been developed for the first time. The model allowed us to obtain the adequate dependence of the surface energy of thin films on their thickness in the nanoscopic scale. For nano-objects having free surfaces, it has been shown that their interior order parameter ZV is always less than that of macroscopic solids of the same composition. It was obtained that the surface energy of thin films increases with decrease in their thickness approaching its maximum meaning. Then the surface energy decreases monotonically.
ð33Þ
where s2m is the mean square displacement of atoms from their equilibrium positions at Tm. In the case of the first-order transition, where the order parameter has a jump at Tm, the order parameter has to satisfy conditions (32). From relationships (32) and (33) it follows that
Z-1 as s2 -ðxaÞ2 ðsolid phase at Tm Þ
almost independent of w, i.e. a ¼constant for a given material. For metals, typical values of a are in the range from 1.5 to 1.7 [22]. To determine the form of the dependence of the order parameter on the degree of dispersion of nano-objects, one can use Shi’s equation [4]
ð35Þ
For the derivation of this formula, we have employed the Lindemann criterion. It implies that s2m ¼ const As shown in Refs. [4,24], a is
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Q.S. Mei, K. Lu, Prog. Mater. Sci. 52 (2007) 1175. A.P. Chernyshev, Phys. Lett. A 374 (2010) 4622. K. Lu, Z.H. Jin, Curr. Opin. Solid State Mater. Sci. 5 (2001) 39. Q. Jiang, C.C. Yang, Curr. Nanosci. 4 (2008) 179. M.N. Magomedov, High Temp 42 (2004) 221. H.M. Lu, Q. Jiang, J. Phys. Chem. B 108 (2004) 5617. T. Hawa, M.R. Zachariah, J. Chem. Phys. 121 (2004) 9043. V.V. Hoang, B.T. Khanh, J. Phys.: Condens. Matter. 21 (2009) 075103 9 pp. K.K. Nanda, A. Maisels, F.E. Kruis, H. Fissan, S. Stappert, Phys. Rev. Lett. 91 (2003) 106102 4 pp. K.K. Nanda, A. Maisels, F.E. Kruis, J. Phys. Chem. C 112 (2008) 13488. A.Z. Patashinskii, V.L. Pokrovskii, Fluctuation Theory of Phase Transitions, Elsevier, 1979. Y.A. Izyumov, V.N. Syromyatnikov, Phase Transitions and Crystal Symmetry, Springer-Verlag, Berlin, Heidelberg, New York, 2009. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, sixth ed., Academic Press, San Diego, 2000. L.D. Landau, E.M. Lifshitz, Statistical Physics, third ed.,Course of Theoretical Physics, vol. 5, Butterworth-Heinemann, Oxford, 1980. P. Harrowell, D.W. Oxtoby, J. Chem. Phys. 80 (1984) 1639. W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma, Annu. Rev. Mater. Res. 32 (2002) 163. L.V. Mikheev, A.A. Chernov, J. Cryst. Growth 112 (1991) 591. A.G. Khachaturyan, Philos. Mag. A 74 (1996) 3. L.-Q. Chen, Annu. Rev. Mater. Res. 32 (2002) 113. V.G. Vaks, K.Yu. Khromov, J. Exp. Theor. Phys. 109 (2009) 619. A.P. Chernyshev, Mater. Chem. Phys. 112 (2008) 226. A.P. Chernyshev, Mater. Lett. 63 (2009) 1525. A.W. Cahn, P. Haasen, Physical Metallurgy, fourth ed., North-Holland, Amsterdam, 1996. C.Q. Sun, Prog. Solid State Chem. 35 (2007) 1.