Criterion of Crystal–Liquid Phase Transition for NanoSystem

M.N. Magomedov

Institute for Geothermal Research of Daghestan Scientific Center RAS,
Chamil str., 39a, Makhachkala, 367030, RUSSIA, E-mail: mahmag@dinet.ru

At this work a localization criterion for the solid–liquid phase transition (S-LPT) is suggested. According to this criterion the PT starts when the fraction of delocalization atoms reaches a boundary value: xd(S-L) = 10 – 2, above which a liquid phase, and below which a solid phase is located in the phase diagram. It is shown that this localization criterion is equitable both for macro-, and for nano- systems too. As was shown in [1], the fraction of atoms that pass from the localized state (when an atom vibrates in the cell formed by the nearest environment) into the delocalized state (i.e., acquiring a kinetic energy that is higher than a certain threshold value Ed) is described as:
xd(Yd)  =  Nd / N  =  ( 2 / 1/2 )t1/2 exp( – t) d t  ,
where Yd = Ed / kb T ;  Ed is the energy of the delocalization of an atom:
Ed(T, c)    =   ( 27 / 128 2 kp2/3 ) m fy ( c kb  /  ) 2  .                           (1)
Here, T is the temperature; c is the distance between centers of nearest atoms; kp is the packing fraction of structure; kb and  are the Boltzmann and Planck constants; m is the atomic mass;  is Debye temperature, fy = (2/y)[1–exp(– y)]/[1+exp(– y)], y = 3/4T.
As was shown in [1], at the definite value of Yd starts both melting of macro-crystal at Tm , and crystallization macro-liquid at TN: 
Yd(S-L)  =  Ed(S) / kb Tm  =  Ed(L) / kb TN    5.67.                                 (2)
We use for description of nanocrystal with free surface the model of rectangular parallelepiped with a square base, which consists of N atom, and whose shape can be varying by the shape parameter [2]. Proceeding from Eq. (1), it is shown that functions (2) and xd(Tm) have not of a size dependencies, whatever herewith the “surface pressure”. Therefore, the localization criterion for S-LPT (2) has it the same value, independently neither from size nor from shape of nanocrystal.
This work was supported by RFBR (Project No. 09 – 08 – 96508), and by Programs of the Presidium of RAS (Project No. 12.1.19).

References

[1] M.N. Magomedov. The Physics of Metals and Metallography. 105 (2008) 116.
[2] M.N. Magomedov. Technical Physics Letters. 33 (2007) 212.