Melting and thermodynamic properties of rare gas nanocrystals

A self-consistent statistical method [Phys. Rev. B66, 054302 (2002)] is used to describe thermodynamic properties of free rare gas nanocrystals using thin plates as examples. It is shown that size influence on thermodynamic properties of nanocrystals is caused by size-dependent quantization of the v...

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Date:	2009
Main Authors:	Karasevskii, A.I., Lubashenko, V.V.
Format:	Article
Language:	English
Published:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2009
Series:	Физика низких температур
Subjects:	7th International Conference on Cryocrystals and Quantum Crystals
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Cite this:	Melting and thermodynamic properties of rare gas nanocrystals / A.I. Karasevskii, V.V. Lubashenko // Физика низких температур. — 2009. — Т. 35, № 4. — С. 362-370. — Бібліогр.: 50 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1171082025-06-03T16:26:43Z Melting and thermodynamic properties of rare gas nanocrystals Karasevskii, A.I. Lubashenko, V.V. 7th International Conference on Cryocrystals and Quantum Crystals A self-consistent statistical method [Phys. Rev. B66, 054302 (2002)] is used to describe thermodynamic properties of free rare gas nanocrystals using thin plates as examples. It is shown that size influence on thermodynamic properties of nanocrystals is caused by size-dependent quantization of the vibration spectrum affecting the parameters of a statistical distribution function of atomic displacements and, thus, governing size dependence of average values of energetic contributions to the Gibbs free energy of the system. For Xe nanocrystals, we present calculated size dependences of the Debye temperature, heat capacity, interatomic distance, melting temperature, etc. This work was supported in part by Award no. 28/08-H in the framework of the Complex Program of Fundamental Investigations «Nanosized systems, nanomaterials, nanotechnology» of National Academy of Sciences of Ukraine. 2009 Article Melting and thermodynamic properties of rare gas nanocrystals / A.I. Karasevskii, V.V. Lubashenko // Физика низких температур. — 2009. — Т. 35, № 4. — С. 362-370. — Бібліогр.: 50 назв. — англ. 0132-6414 PACS: 64.70.D–, 65.80.+n https://nasplib.isofts.kiev.ua/handle/123456789/117108 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	7th International Conference on Cryocrystals and Quantum Crystals 7th International Conference on Cryocrystals and Quantum Crystals
spellingShingle	7th International Conference on Cryocrystals and Quantum Crystals 7th International Conference on Cryocrystals and Quantum Crystals Karasevskii, A.I. Lubashenko, V.V. Melting and thermodynamic properties of rare gas nanocrystals Физика низких температур
description	A self-consistent statistical method [Phys. Rev. B66, 054302 (2002)] is used to describe thermodynamic properties of free rare gas nanocrystals using thin plates as examples. It is shown that size influence on thermodynamic properties of nanocrystals is caused by size-dependent quantization of the vibration spectrum affecting the parameters of a statistical distribution function of atomic displacements and, thus, governing size dependence of average values of energetic contributions to the Gibbs free energy of the system. For Xe nanocrystals, we present calculated size dependences of the Debye temperature, heat capacity, interatomic distance, melting temperature, etc.
format	Article
author	Karasevskii, A.I. Lubashenko, V.V.
author_facet	Karasevskii, A.I. Lubashenko, V.V.
author_sort	Karasevskii, A.I.
title	Melting and thermodynamic properties of rare gas nanocrystals
title_short	Melting and thermodynamic properties of rare gas nanocrystals
title_full	Melting and thermodynamic properties of rare gas nanocrystals
title_fullStr	Melting and thermodynamic properties of rare gas nanocrystals
title_full_unstemmed	Melting and thermodynamic properties of rare gas nanocrystals
title_sort	melting and thermodynamic properties of rare gas nanocrystals
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2009
topic_facet	7th International Conference on Cryocrystals and Quantum Crystals
url	https://nasplib.isofts.kiev.ua/handle/123456789/117108
citation_txt	Melting and thermodynamic properties of rare gas nanocrystals / A.I. Karasevskii, V.V. Lubashenko // Физика низких температур. — 2009. — Т. 35, № 4. — С. 362-370. — Бібліогр.: 50 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT karasevskiiai meltingandthermodynamicpropertiesofraregasnanocrystals AT lubashenkovv meltingandthermodynamicpropertiesofraregasnanocrystals
first_indexed	2025-11-24T04:19:06Z
last_indexed	2025-11-24T04:19:06Z
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fulltext	Fizika Nizkikh Temperatur, 2009, v. 35, No. 4, p. 362–370 Melting and thermodynamic properties of rare gas nanocrystals A.I. Karasevskii and V.V. Lubashenko G.V. Kurdyumov Institute for Metal Physics, 36 Vernadsky Str., Kiev 03142, Ukraine E-mail address: akaras@imp.kiev.ua vilu@imp.kiev.ua Received December 12, 2008 A self-consistent statistical method [Phys. Rev. B66, 054302 (2002)] is used to describe thermodynamic properties of free rare gas nanocrystals using thin plates as examples. It is shown that size influence on ther- modynamic properties of nanocrystals is caused by size-dependent quantization of the vibration spectrum affecting the parameters of a statistical distribution function of atomic displacements and, thus, governing size dependence of average values of energetic contributions to the Gibbs free energy of the system. For Xe nanocrystals, we present calculated size dependences of the Debye temperature, heat capacity, interatomic distance, melting temperature, etc. PACS: 64.70.D– Solid–liquid transitions; 65.80.+n Thermal properties of small particles, nanocrystals, and nanotubes. Keywords: rare gas nanocrystals, thermodynamic properties. 1. Introduction The effect of size dependence of the melting tempera- ture was discovered in the middle of the last century in electron microscopy studies of granular metallic films [1]. It was found that the melting temperature of free nanocrystals is substantially depressed as the crystal size decreases. The temperature can be changed by hundreds degrees, and this effect is pronounced even in the me- soscopic size range. Later, it was also established that this phenomenon has universal nature. It is inherent in any crystalline matter: metallic and semiconductor nanocrys- tals [2–5], rare gas [6,7] and ionic solids. It should be noted that depression of the melting point is observed only in free nanocrystals. If a nanoparticle is embedded into some medium, it may be considerably superheated [8,9]. Recently it was reported that thermodynamic proper- ties of nanocrystals, such as cohesion energy [10,11], Debye temperature [10,12,13], activation energy of diffu- sion [14,15], vacancy formation energy [16,17] etc. dis- play also size dependence. These experimental facts point to size influence on statistical characteristics of atoms in a nanoparticle. In early theoretical studies of thermodynamics of na- nocrystals, authors attempted to explain the strong size dependence of the melting temperature. Because of the lack of a common melting theory, these models took into account a variety of size-dependent factors affecting the transition between solid and liquid nanophases. In ther- modynamic consideration of an equilibrium between these phases, an important role was ascribed to the sur- face of the nanoparticle [3,5,18–20]. Size dependence of the chemical potentials of both phases was described by introducing the capillary pressure, caused by surface curvature, and excess surface energy. Some phenomeno- logical models of melting of free nanoparticles employed the effect of surface melting inherent in the most solids. For example, the liquid-layer model [2,21] postulates for- mation of a quasi-liquid layer at the surface of a nano- crystal at temperature below its melting point. Among recent theoretical approaches to description of size-dependent melting of nanocrystals, we would like to mention the liquid-drop model [22] taking into account reduction of the cohesion energy of atoms of a nanoparticle with reduction of its size and using a relation between the melting temperature and the cohesion energy [23]. A considerable advance in computational capacity made for the last decade has enabled first-principles simu- lation of dynamical behavior and energetic characteristics of systems consisting of more than 10 5 atoms [24–32], i.e. nanocrystals in the so-called mesoscale regime. � A.I. Karasevskii and V.V. Lubashenko, 2009 To our opinion, one of the most comprehensive molec- ular dynamics (MD) studies of thermodynamics of nano- sized systems was carried out in Ref. 29 for spherical Cu nanocrystals. The results obtained in Ref. 29 demonstrate that both melting temperature and the latent heat of fusion decrease smoothly as particle radius decreases. The anal- ysis of energetic and structural properties of nanocrystals revealed existence of a rather thin surface layer where av- erage potential energy and root-mean-square (rms) dis- placements of atoms are largely different from those of the core region. According to results obtained in Ref. 29, the behavior of the average potential energy of atoms of a nanocrystal is characterized by essentially nonlinear rise near Tm. As shown in Refs. 33, 34 for the bulk solids, such a behavior is attributed to evolution of anharmonic insta- bility of the phonon system of the crystal which is directly related to the melting transition. In this work we study influence of size-dependent modification of the phonon spectrum on thermodynamic properties of rare gas nanocrystals using a thin plate of nanosized thickness as an example. Similar consideration for metallic nanoplates was carried out in Ref. 35. We show that taking into account the finite size of the nano- crystal results in quantization of its vibrational spectrum which becomes essentially discrete. Formation of a dis- crete vibrational spectrum affects parameters of statistical distribution of atomic displacements in the nanocrystal and, therefore, the average values of its energetic cha- racteristics. As it will be shown, this is the chief mecha- nism responsible for depression of the melting point and change of thermodynamic properties of free nanocrystals with decreasing their size. 2. Free energy of a nanocrystal To examine size influence on thermodynamics of nanocrystals, we use a self-consistent statistical method [36] developed to compute thermodynamic properties of simple solids. The method consists, first, in derivation of a binary distribution function of atomic coordinates and, second, in a variational procedure of computation of in- teratomic distance and effective parameters of quasi-elas- tic bonds between atoms of the crystal. It was shown that the phonon spectrum of the crystal determines parameters of the distribution function and, therefore, the average values of energetic characteristics of the crystal. There- fore, this approach allows to clarify how size-dependent modification of the phonon spectrum of a small crystal af- fects its thermodynamic properties. The method has been applied to description of thermal characteristics of the rare gas crystals (RGC) [36–38] and some simple fcc met- als [39]. In the framework of this method we also com- puted the formation energy of a vacancy in the RGC as a function of temperature and demonstrated that approach- ing the melting point (which is assumed to be directly re- lated to the point of anharmonic instability) is accompa- nied by dramatic reduction of the energy required to cre- ate structure defects of the lattice [33,34,40]. As in our previous studies, we assume the interatomic interaction to be pairwise and approximated by the Morse potential u r A r R r R ( ) [ ] ( ) ( )� �� e e 2 0 02 � � . (1) The parameters A, R0, and � for RGC have been deter- mined in Ref. 33 so that calculated values of interatomic distance, bulk modulus, and sublimation energy at zero temperature fitted the corresponding experimental data. In this work we restrict ourselves to the high-tempera- ture limit when the distribution function of atomic dis- placements in a simple crystal may be presented as a pro- duct of one-particle Gaussian functions [33] given by f C c n qi i( ) exp ( ) q � � � � � � � � � 2 2 2 , (2) where q i is the displacement of an atom from the ith lat- tice site, � k T AB / is the reduced temperature, c is a dimensionless effective parameter of quasi-elastic bond of neighboring atoms, and the coefficient n( ) determines contribution of the phonon spectrum to the distribution width. At high temperature it is expanded into a power se- ries as [36] n n cl l l l ( ) ( ) � � � � � � � � � � 1 0 2� , (3) where � � �� MA (4) is the de Boer parameter for the Morse potential, M is atomic mass. The condition for the applicability of the high-temperature approximation is given by inequality � c�. Coefficients nl are determined by the phonon spectrum of the crystal [36]. For example, n dj j jx0 2 21 2 � �� ~ ( ) ( )� K K Ke , (5) n dj j jx1 4 21 24 � �� ~ ( )� K Ke , (6) where K is reduced wave vector [36], ~ ( ) ( ) / � � �j j M A c k k� � � � � � 2 2 1 2 (7) is the reduced frequency of a phonon with wave vector k and branch j, calculated from the dynamical matrix of the lattice, and e jx ( )k is the projection of a phonon polariza- Melting and thermodynamic properties of rare gas nanocrystals Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 363 tion vector on the x axis. The integration in (5) and (6) is carried out over the first Brillouin zone. For a bulk fcc crystal, n0 2� , n1 5 6� / . Equations (5) and (6) were de- rived in Ref. 36 for the bulk crystals, but, according to Ref. 41, such expressions are applicable to any finite sys- tem of bound harmonic oscillators. In the high-temperature approximation the Gibbs free energy� of a simple fcc crystal consisting of N atoms can be written as [33,36] g p c b AN c z b n c ( , , , ) ( ) � � �� 3 3 2 2 2 2 log e 2 1 4 2 3 2 6 2 2 2 2 e e e � � � � � � � � � � � � � � � b n c b n c b a c ( ) ( ) �� 2 2 6 3 2n c l A R p v ( ) . (8) Here R is the nearest neighbor distance, b R R� ��( )0 is a reduced lattice expansion, z �12 is the coordination number, p P A� / ( )� 3 is reduced external pressure, and v R� 3 2/ is volume per atom. The first term in (8) determines the entropy part of the free energy of atomic vibrations in the crystal, the second term represents the average potential energy of interac- tion of neighboring atoms. The third term determines a contribution to the free energy of the crystal due to the cubic anharmonicity of collective atomic vibrations, evaluated in the second order of the perturbation theory ( .a3 2 31� for the RGC [33,34]). The next term takes into account the long-range attraction between atoms of the crystal, with � and � being the parameters of the Len- nard–Jones potential and � l � 4 91. for the fcc lattice [33]. 3. Vibrational spectrum of a thin plate To determine size contribution to the free energy of a nanocrystal of size h (h R�� ), we proceed from the as- sumption that the size-dependent modification of the vi- brational spectrum is most pronounced in its long-wave part, with the wave vectors k h~ /� or kR � 1. Such a case corresponds to the elastic vibrations of the medium and is described by wave equations of theory of elasticity. The consideration of the size dependence of thermodynamic properties of nanocrystals will be carried out for the sim- plest type of nanocrystals, a thin plate. In this case, the dispersion relations have a rather simple appearance. A consistent consideration of elastic vibrations and propa- gation of elastic waves in plates was originally performed by Lamb [42] in the framework of general theory of elasticity. Let us consider a plate of thickness h, with free sur- faces parallel to the ( , )x y plane and the origin taken at the center of the plate. The z axis is normal to the free sur- faces z h� � / 2. The system is assumed to be of macro- scopic size in the ( , )x y plane, so that vibrations can prop- agate as plane waves in the x and y directions. Note that we do not consider vibrations of the plate as a whole. The components of the stress tensor vanish at the free sur- faces, � zx h� � /2 0 , � zz h� � /2 0 , (9) � zy h� � /2 0 . For such a system, the displacement vector u can be pre- sented as a superposition of waves of the horizontal polar- ization, with u ux z� � 0, u uy � , and waves of the verti- cal polarization with u x 0, u z 0, and u y � 0. It is important that these two types of waves do not mix at the boundaries, so that they can be considered separately (see, e.g., Refs. 42,43). Waves of the horizontal polarization are described by ordinary wave equations for the displacement u y , ! ! � ! ! � � 2 2 2 2 2 0 u x u z u y y y� , (10) where � �� / ct , � is an eigenfrequency, ct is transverse sound velocity. A solution of (10) in a particular case of a plane wave propagating along the x direction is given by u C zy i x� �cos ( ) ." # $e (11) Parameters " and # are found from both the boun- dary conditions (9) and the symmetry reasons allow- ing two types of the solutions, a symmetrical one, u z h u z hy y( / ) ( / )� � � �2 2 , and an antisymmetrical one, u z h u z hy y( / ) ( / )� � � � �2 2 . These solutions lead generally to a dispersion relation given by �t t nc k� , (12) with k n h nn � � � �$ % �2 2 2 1 2 3( / ) , , , ,� Here $ and % are, respectively, projections of the wave vector on the x and y axes for plane waves propagating in the xy plane. The projection of the wave vector on the z axis takes on discrete values, k n h z � � . In the case of waves of vertical polarization, the dis- placement is expressed in terms of scalar & and vector � potentials, 364 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 A.I. Karasevskii and V.V. Lubashenko u �' �& rot �. (13) The potential � can be chosen so that ( (y � and ( (x z� � 0. Then & and ( satisfy the wave equations, )& &� �k 2 0 , (14) )( (� ��2 0 , (15) where k cl�� / , cl is longitudinal sound velocity. The so- lutions of Eqs. (14), (15) are given by & � # $� �C z l i xcos ( ) e , (16) ( � �D z t i xcos ( )" # $e , (17) where � and" are real values given by � $� �k 2 2 , (18) " � $� �2 2 . (19) In terms of these notations, the boundary conditions (9) are rewritten as ! ! � * * * � �� ( z ip z h & /2 0 , (20) ! ! � * * * � �� & z ip z h ( /2 0 , (21) where p � �( ) / ( )$ " $2 2 2 . Hence we get the general dis- persion relation for the acoustic part of the vibrational spectrum of the nanocrystal, tan tan � # " # �" h h p l t 2 2 0 0 2 �� . (22) From the symmetry reasons, the projections of the dis- placement vectors u x and u z should be either symmetri- cal or antisymmetrical. At the symmetrical conditions # � # �l tm n m n0 0 2 1 2 0 1 2� � � � +, ( ) / , , , , , the dispersion relation (22) appears as cot tan � " �" h h p 2 2 2 � � . (23) The antisymmetrical condition is specified by # � # �l tn m m n0 02 1 2 0 1 2� � � � +( ) / , , , , , , leading to tan cot � " �" h h p 2 2 2 � � . (24) Note that satisfying the symmetrical condition for u x im- plies satisfying the antisymmetrical condition for u z , and viceversa. In Fig. 1 we plotted dispersion curves for the longitu- dinal characteristic vibrations of a thin plate obtained from (23). At a rather small thickness h of the plate, the long-wave part (kR � 1) of the vibrational spectrum of the nanocrystal differs substantially from that of the bulk crystal, described by the linear relation between the fre- quency and the wave vector, � $j jc� ( j l t� , ). The spec- trum of the thin plate splits into separate vibration bran- ches. For each branch, there is an integer number of wavelengths going across the plate thickness. The disper- sion curves are well approximated by parabolas and can be analytically represented as � $ % � � $ %n n b n lc2 2 2 2 2( , ) ( ).,� � � (25) Here�n b, is the minimal value of the frequency of the nth vibration branch, � n �1. For each of both symmetrical (23) and antisymmetrical (24) cases we have two types of solutions for �n b, corresponding to longitudinal and transverse characteristic vibrations that are generally written as � � n b j j n h c , � , where j l t� , . As in the case of waves of the horizontal po- larization (12), the wave vector for the both longitudinal and transverse standing waves of the vertical polariza- tions is given by Eq. (12). A similarity of the expression for ( ) , � n b j 2 to the for- mula for the energy of a quantum particle in a one-dimen- sional potential well [44] is explained by the wave nature of both quantum particles and vibrational modes, exhibit- Melting and thermodynamic properties of rare gas nanocrystals Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 365 20 15 10 5 0 5 10 15 20 $h � h /c 1 Fig. 1. Dispersion curves of longitudinal vibrations of a thin plate for n �1 5, ..., . The dotted line corresponds to the disper- sion relation � $� cl for the bulk crystal. ing discreteness of the eigenvalues in the case of a small size of the system. For the bulk fcc crystals, the first Brillouin zone has the form of a truncated octahedron which may be well ap- proximated by a sphere of radius k vs 0 2 1 36� ( / ) /� . The ra- dius k s 0 is determined by the requirement that the number of independent values of the wave vector falling within the sphere is equal to the number of atoms of the simple lattice. In the case of nanocrystals, evaluation of the radius k s of the spherical Brillouin zone requires taking into ac- count the discrete character of k z in replacing summation over k by integration. This can be done using the Euler–Maclaurin formula [45] f k f t dt f k f k k k k k ( ) ( ) [ ( ) ( )] min max min max min max� �� 1 2 + (27) Then we get k k k R R h s s s � � � � � � � � 0 0 1 4 � ( ) . (28) Presence of a size-dependent factor in (28) results in ap- pearance of similar corrections to the coefficients nl in (3), n h n R h l l l( ) ,� �� 0 1 , (29) where nl 0 is the bulk value of nl , the first two coefficients , l are , 0 0 68� . and ,1 0 74� . . As will be shown below, the size dependence of the coefficients n hl ( ) leads to a num- ber of physical effects observed in nanocrystals. It should be noted that the above procedure of calcula- tion of the coefficients , l seems to give overestimated values, because Eq. (28) for the quasi-discrete wave vec- tor is valid, strictly speaking, only in the range kR � 1, sen- sible to the particle size. Near the boundaries of the Brillouin zone, the values of k are determined by the structure of the crystal lattice and are slightly dependent of the crystal size. However, the structure of Eq. (28) should remain, on the whole, the same, so that , l may be considered as parameters of the theory. 4. Thermodynamic properties of nanocrystals The equilibrium values of the quasi-elastic bond pa- rameter c0 and reduced lattice expansion b0 are deter- mined from minimization of the Gibbs function (8) with respect to c and b, ! ! * * * � ! ! * * * � g c g bp b p c , , , , , .0 0 (30) The condition for the equilibrium values of the va- riational parameters to exist is that D g g g g cc cb bc bb � -- -- -- -- * * * * * *�det 0 . (31) The sufficient condition D � 0 defines the point of anhar- monic instability of the crystal where the minimum of g p c b( , , , ) with respect to c and b disappears. In the case of nanocrystals, the Gibbs free energy de- pends, besides temperature and pressure p, on the crys- tal size h representing an additional independent thermo- dynamic variable (while volume per atom is unchanged). As a result, a number of thermodynamic characteristics of the crystal display size dependence. For example, the Debye temperature is directly proportional to the equilib- rium value of the parameter c p h0( , , ) and is written as [37,38] . �D j BA c p h k� / 0( ) ( , , ) / ,/6 2 2 1 3 0� � k (32) where coefficients � �j j kRk k� / ( ) account for different polarizations of the acoustic waves, and / 0 �� jk 0 67. for the fcc crystals. The volume and temperature dependence of .D determines the Gr=�neisen parameter, ,G D T D V / V / T � � ! ! � ! ! ( ln ln ) ( ln ln ) . .1 . (33) The equilibrium value of b p h0( , , ) governs the inter- atomic distance, the coefficient of thermal expansion, and the bulk modulus of the nanocrystal. All these properties would depend on the crystal size. If the melting temperature Tm is assumed to be identi- cal to the temperature of anharmonic instability of the crystal [33,34], then it can be shown in the framework of the present statistical model that Tm is related to the factor n( ) as m B mk T A a n � ~ ( ) 1 3 3 . (34) Hence it follows that even a slight increasing of n( ) due to reduction of the crystal size results in appreciable de- pression of the melting temperature of the nanocrystal. Since R h/ 11 1, it follows from (34) that T T h m m b � �1 3 " , (35) where Tm b is the melting temperature of the bulk material, and " ,� 9 0R. This is the well-known relationship be- tween the relative melting temperature of a planar nanocrystal and its inverse size (see, e.g., Ref. 22). In Figs. 2 and 3 we plotted, respectively, the inter- atomic distance and the reduced coefficient of thermal ex- pansion � � db/d in Xe nanocrystals versus particle size 366 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 A.I. Karasevskii and V.V. Lubashenko at three values of temperature calculated from (30). Here- after all theoretical results are presented for zero external pressure. A nanocrystal of size h � 218. nm melts at T �125 K, which explains some nonlinear rise of �( )h in the top curve. While the interatomic distances in the considered nanocrystals show no marked dependence on the crystal size, the effective parameter c p h0( , , ) of the quasi-elas- tic bond varies substantially with h. Such dependence calculated for Xe nanocrystals is illustrated with Fig. 4 which demonstrates that c0 decreases nonlinearly as h de- creases even at temperatures much lower than the corre- sponding melting temperature of the nanocrystal. Size dependence of the Debye temperature .D of small crystals was observed in a number of experimental studies [10,12,13]. In the framework of the present ap- proach, .D is proportional to the quasi-elastic bond pa- rameter c0 (32). The .D h( ) dependence of Fig. 5 agrees qualitatively with that obtained experimentally, e.g., for gold nanoparticles [12]. In many bulk solids, the melting transition is preceded by anomalous behavior of temperature derivatives of thermodynamic functions (isobaric heat capacity, thermal expansion coefficient, Gr�neisen parameter) due to evo- lution of the anharmonic instability in the phonon subsys- tem of the crystal as T Tm2 [33,34]. As seen from Fig. 6, the isobaric heat capacity C P behaves itself in a similar way when the melting point is approached via reduction of the crystal size at constant temperature (the top curve). In other words, approaching the melting point by chang- ing a thermodynamic variable corresponding to the size h of the nanocrystal is in some way similar to variation of temperature or pressure of the crystal. In Fig. 7 we show the melting temperature of a free Xe thin plate as a function of its thickness calculated from Eqs. (30) and the condition D � 0, compared with that ob- tained from Eq. (34). Melting and thermodynamic properties of rare gas nanocrystals Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 367 h, nm T = 125 K T = 90 K T = 30 K 4.42 4.40 4.38 4.36 4.34 4.32 0 10 20 30 40 Fig. 2. Size dependence of the interatomic distance of Xe nanocrystals at different temperatures. T = 125 K T = 90 K T = 30 K 0 20 40 60 80 h, nm 0.30 0.25 0.20 0.15 � Fig. 3. Size dependence of the coefficient of thermal expan- sion of Xe nanocrystals at different temperatures. 1.8 1.7 1.6 1.5 T = 125 K T = 90 K T = 30 K 0 20 40 60 80 h, nm c 0 Fig. 4. Size dependence of the effective quasi-elastic bond pa- rameter c h0( , ) of Xe nanocrystals at different temperatures. 58 56 54 52 50 48 46 44 5 10 15 20 h, nm . D , K T = 125 K T = 90 K T = 30 K Fig. 5. Size dependence of the Debye temperature of Xe nanocrystals at different temperatures. At present, the only generally recognized melting cri- terion is the empirical Lindemann rule suggesting that a solid melts when the rms displacement of atoms reaches a characteristic fraction of the interatomic distance. In terms of notations accepted in this work, the Lindemann ratio # is given by # � 2 2 2 0 2 2 3 4 � / 0 � q R c R gt( )( ) . (36) Here gt � 0 77. is the correlation smearing of the distribu- tion width of an atom at a lattice site [36]. The results presented in Figs. 4 and 8 allow us to treat the physical meaning of the Lindemann criterion in terms of the solid state instability. The evolution of the anhar- monic instability is always accompanied by nonlinear re- duction of the quasi-elastic bond parameter c p h0( , , ) which reaches its minimal value at the instability point corresponding either a critical temperature c or a critical size of the particle (Fig. 4). According to (36), such a reduction of c0 in the critical range results in dramatic in- creasing of the rms displacement of atoms, so that its ratio to the average interatomic distance rises up to # � 01. . In addition to increasing the rms displacement, the critical range is characterized by a nonlinear rise in the isobaric heat capacity (Fig. 6), the coefficient of thermal expan- sion etc. as well as by a sharp drop in the formation energy of the structural lattice defects, as was shown for bulk solids [33,34,40], thus promoting a transition to a struc- turally disordered phase. Reduction of the crystal size involves, along with a shift of the temperature range of the anharmonic instabil- ity towards lower temperatures, a respective temperature shift of peculiarities of thermodynamic properties. This effect is illustrated in Fig. 9 presenting the temperature dependence of the Gr�neisen parameter computed for dif- ferent crystal sizes. 368 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 A.I. Karasevskii and V.V. Lubashenko T = 125 K T = 90 K T = 30 K 5 10 15 20 h, nm C /3 k N P B 1.5 1.4 1.3 1.2 1.1 1.0 0.9 Fig. 6. Size dependence of the isobaric heat capacity of Xe nanocrystals at different temperatures. Xe h, nm 0 10 20 30 40 170 160 150 140 130 120 110 T , K m Fig. 7. Melting temperature of a free Xe thin plate versus its thickness calculated by minimization of the Gibbs free energy (8) (solid curve) and from Eq. (34) (dotted curve). h, nm 0 10 20 30 40 0.12 0.10 0.08 0.06 0.04 # T = 125 K T = 90 K T = 30 K Fig. 8. The relative rms displacement of atoms (the Lindemann ratio) of Xe thin plate versus its thickness at different tempera- tures. 1 2 — h = 2.18 nm — h = 2.91 nm — h = 4.36 nm — h = 14.53 nm 3 4 1 2 3 4 40 60 80 100 120 140 160 T, K 3.1 3.0 2.9 2.8 2.7 2.6 , G Fig. 9. The temperature dependence of the Gr�neisen parame- ter of Xe nanocrystals of different values of thickness. In conclusion, one should note that influence of dis- creteness of the spectrum of eigenvalues on energetic characteristics of many-particle finite systems was con- sidered previously for atomic nuclei [49] and nano- particles of degenerated semiconductors [50]. In these studies it was demonstrated that corrections due to the discreteness of the spectrum are proportional to h�1. 5. Discussion So far there is a large amount of experimental data con- cerning size influence on the bulk properties of nanoparticles, such as cohesion energy [10,11], Debye temperature [10,12,13], and activation energy of diffu- sion [14,15]. One of the most pronounced manifestations of change of thermodynamic properties observed in nanosized crystalline systems is the effect of reduction of the melting temperature of free nanocrystals [1–9]. So the principal problem of statistical description of thermody- namics of nanocrystals is elucidation of the mechanism of size effect on statistical characteristics of atoms in such systems. Relying on the results of MD simulations of thermodynamic properties and melting of nanocrystals [29,30], it is natural to conclude that a thin (relative to the particle size) surface layer has only a minor influence upon the bulk properties of the particles; moreover, in the case of spherical particles an additional capillary pressure should contribute to increasing of both the melting temperature and the Debye temperature. It is shown in this work that the principal size-depen- dent mechanism governing thermodynamic behavior of a nanocrystal is quantization of its vibrational spectrum. It leads to increasing of the parameter n( ) of the statistical distribution function of atomic coordinates (2) and chang- ing, as a consequence, the average value of the interaction energy of atoms in the nanocrystal. One emphasizes that such an impact on a crystalline system mediated by direct change of the statistical distribution function is inherent only in nanosystems. In this connection, it is worth noting a peculiar character of thermodynamic response of a na- nocrystal to variation of its size. As the crystal's size de- creases, the parameter c p h0( , , ) of quasi-elastic bond of its atoms decreases (Fig. 4), as well as the Debye tempera- ture (32), while interatomic distance remains nearly con- stant (Figs. 2, 3). The melting temperature (34) of the nanocrystal is markedly decreased (Fig. 7). Independ- ently of the crystal size, the melting transition occurs when the Lindemann criterion is satisfied (Fig. 8). Along with a size-dependent shift of the melting temperature, there is also a corresponding shift of the range of the an- harmonic instability where thermodynamic properties (isobaric heat capacity, coefficient of thermal expansion etc.) display a peculiar behavior. Acknowledgement This work was supported in part by Award no. 28/08-H in the framework of the Complex Program of Fundamen- tal Investigations «Nanosized systems, nanomaterials, nanotechnology» of National Academy of Sciences of Ukraine. 1. M. Takagi, J. Phys. Soc. Jpn 9. 359 (1954). 2. C.R.M. Wronski, Br. J. Appl. Phys. 18, 1731 (1967). 3. C.J. Coombes, J. Phys. F: Met. Phys. 2, 441 (1972). 4. N.T. Gladkich, R. Neidermayer, and K. Spiegel, Phys. Sta- tus Solidi 15, 181 (1966). 5. Ph. Buffat and J.-P. Borel, Phys. Rev. A13, 2287 (1976). 6. F. Celestini, R.J.-M. Pellenq, P. Bordarier, and B. Rous- seau, Z. Phys. D37, 49 (1996). 7. K.K. Nanda, Chem. Phys. Lett. 419, 195 (2006). 8. H. Saka, Y. Nishikawa, and T. Imura, Philos. Mag. A57, 895 (1988). 9. L. Zhang, Z.H. Jin, L.H. Zhang, M.L. Sui, and K. Lu, Phys. Rev. Lett. 85, 1484 (2000). 10. C.C. Yang and S. Li, Phys. Rev. B75, 165413 (2007). 11. W.H. Qi and M.P. Wang, J. Mater. Sci. Lett. 21, 1743 (2002). 12. G. K�stle, H.-G. Boyen, A. Schr�der, A. Plettl, and P. Ziemann, Phys. Rev. B70, 165414 (2004). 13. C.C. Yang, M.X. Xiao, W. Li, and Q. Jiang, Solid State Commun. 139, 148 (2006). 14. T. Shibita, B.A. Bunker, Z. Zhang, D. Meisel, C.F. Vardeman II, and J.D. Gezelter, J. Am. Chem. Soc. 124, 11989 (2002). 15. Q. Jiang, S.H. Zhang, and J.C. Li, J. Phys. D37, 102 (2004). 16. T. Korhonen, M.J. Puska, and R.M. Nieminen, Phys. Rev. B51, 9526 (1995). 17. C.J. Zhang and A. Alavi, J. Am. Chem. Soc. 127, 9808 (2005). 18. P. Pawlow, Z. Phys. Chem. 65, 1 (1909); ibid. 65, 545 (1909). 19. K.-J. Hanszen, Z. Phys. 157, 523 (1960). 20. P.R. Couchman and W.A. Jesser, Nature (London) 269, 481 (1977). 21. K.F. Peters, J.B. Cohen, and Y.-W. Chung, Phys. Rev. B57, 13430 (1998). 22. K.K. Nanda, S.N. Sahu, and S.N. Behera, Phys. Rev. A66, 013208 (2002). 23. J. Tateno, Solid State Commun. 10, 61 (1972). 24. Y. Wang, S. Teitel, and Christoph Dellago, Chem. Phys. Lett. 394, 257 (2004). 25. Y. Qi, T. Cagin, W.L. Johnson, and W.A. Goddard III, J. Chem. Phys. 115, 385 (2001). 26. L. Hui, F. Pederiva, B.L. Wang, J.L. Wang, and G.H. Wang, Appl. Phys. Lett. 86, 011913 (2005). 27. U. Tartaglino, T. Zykova-Timan, F. Ercolessi, and E. Tosatti, arXiv:cond-mat / 0504680 v2: 3 Aug 2005. 28. F. Ercolessi, W. Andreoni, and E. Tosatti, Phys. Rev. Lett. 66, 911 (1991). 29. F. Delogu, Phys. Rev. B72, 205418 (2005). 30. F. Delogu, Phys. Rev. B75, 235421 (2007). Melting and thermodynamic properties of rare gas nanocrystals Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 369 31. S.J. Zhao, S.Q. Wang, D.Y. Cheng, and H.Q. Ye, J. Phys. Chem. B105, 12857 (2001). 32. F. Baletto and R. Ferrando, Rev. Mod. Phys. 77, 371 (2005). 33. A.I. Karasevskii and V.V. Lubashenko, Phys. Rev. B71, 012107 (2005). 34. A.I. Karasevskii, W.B. Holzapfel, and V.V. Lubashenko, J. Low Temp. Phys. 139, 609 (2005). 35. A.I. Karasevskii and V.V. Lubashenko, Eur. Phys. J. B66, 375 (2008). 36. A.I. Karasevskii and V.V. Lubashenko, Phys. Rev. B66, 054302 (2002). 37. A.I. Karasevskii and W.B. Holzapfel, Phys. Rev. B67, 224301 (2003). 38. A.I. Karasevskii and W.B. Holzapfel, Fiz. Nizk. Temp. 29, 951 (2003) [Low Temp. Phys. 29, 711 (2003)]. 39. A.I. Karasevskii and V.V. Lubashenko, Phys. Status Solidi B241, 1274 (2004). 40. A.I. Karasevskii and V.V. Lubashenko, Fiz. Nizk. Temp.33, 758 (2007) [Low Temp. Phys. 33, 578 (2007)]. 41. R.P. Feynmann, Statistical Mechanics, Benjamin, New York (1972). 42. H. Lamb, Proc. Math. Soc., London, 98, 205 (1920). 43. A.E.H. Love, Treatise on the Mathematical Theory of Elasticity, New York, Dover (1944). 44. L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Per- gamon Press, Oxford (1965). 45. E. Madelung, Mathematical Apparatus of Physics, Sprin- ger, Berlin (1957). 46. Physical acoustics. Principles and Methods, W.P. Mason (ed.), Academic Press, New York and London (1968), vol. 4, Applications to Quantum and Solid State Physics. Part B. 47. K.K. Nanda, S.N. Behera, and S.N. Sahu, J. Phys.: Con- dens. Matter 13, 2861 (2001). 48. W.H. Qi and M.P. Wang, J. Nanopart. Res. 7, 51 (2005). 49. S.A. Gurvits, A.A. Migdal, and A.M. Polyakov, Zh. Eksp. Teor. Fiz. 46, 213 (1964). 50. M.A. Krivoglaz and A.I. Karasevskii, Fiz. Tverd. Tela 17, 2565 (1975) [Sov. Phys. - Solid State 17 1709 (1976)]. 370 Fizika Nizkikh Temperatur, 2009, v. 35, No. 4 A.I. Karasevskii and V.V. Lubashenko

Melting and thermodynamic properties of rare gas nanocrystals

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