Surface energy and surface tension of liquid metal nanodrops

A unitary approach has been proposed for the calculation of surface energy and surface tension of nanoparticle being in equilibrium with its saturated vapor on both flat and curved surfaces at given temperature. The final equations involve parameters dependent on the type of premelting structure: bcc, fcc or hcp. The surface energy u and the surface tension  are the most important thermodynamic characteristics of the layer between the coexistent phases. These quantities, in turn, allow one to obtain other characteristics of the interface boundaries such as work of adhesion, edge angle of wetting, coefficient of spreading etc. The values of u and  are of special interest when studying the properties of nanoobjects (particles, bubbles, thin films) because the increase of surface contribution to the properties of the whole system becomes intrinsic. This communication deals with the calculation of surface energy and surface tension of nanoparticle held in equilibrium with its saturated vapor at constant temperature on the basis of proposed unitary approach. Let the nanoparticle be sphere of radius r corresponding to the equimolar dividing surface at which the autoadsorption, as known, is nil ( 0 e Ã N    , 0 N  number of excess particles, e  molar area). The surface energy and surface tension shall be the excess energy e u and excess free energy e F per unit area of equimolar dividing surface respectively: ( ) ( ) ( ) ( ) 0 0 , e N N e e u u V u V u u               (1) ( ) ( ) ( ) ( ) 0 0 , V V e N N e e F f V f V F              (2) where u and F – internal and free energies of the system, ( ) u   and ( ) u   densities of energy in bulk phases  and  of volumes ( )0 N V   and ( ) 0 N V   , located each side from the equimolar dividing surface, ( )

The surface energy u and the surface tension  are the most important thermodynamic characteristics of the layer between the coexistent phases.These quantities, in turn, allow one to obtain other characteristics of the interface boundaries such as work of adhesion, edge angle of wetting, coefficient of spreading etc.The values of u and  are of special interest when studying the properties of nanoobjects (particles, bubbles, thin films) because the increase of surface contribution to the properties of the whole system becomes intrinsic.
This communication deals with the calculation of surface energy and surface tension of nanoparticle held in equilibrium with its saturated vapor at constant temperature on the basis of proposed unitary approach.
Let the nanoparticle be sphere of radius r corresponding to the equimolar dividing surface at which the autoadsorption, as known, is nil ( 0 e Ã N    , 0 N  -number of excess particles, e  -molar area).
The surface energy and surface tension shall be the excess energy e u and excess free energy e F per unit area of equimolar dividing surface respectively: where u and F -internal and free energies of the system, Let us introduce two more dividing surfaces conditioned by where ( ) appearing after the division of all the system by the surfaces ( ).
Simple transformations of ( 1) and (2) with account of (3) and (4) bring us to where eu e u r r r    and eF e F r r r    -distances from the equimolar dividing surface to the separating surfaces corresponding to the conditions of 0 u  and 0 F  .Switching to the planar case ( e r   ) and after some transformations we obtain   where lim The equations ( 9) and (10) are taken from [1], while (10) and (12) -from [2].If eu r  and eF r  are supposed to be independent from the radius r e (as Tolman accepted in his formula for (r) [3]), then it comes from (9)-( 12) that ) were performed for 20 metals in [1].Here we present calculation for 50 liquid metals, results for ( ) for the same 50 liquid metals obtained on the basis of experimental data of  are reported in [2] by us.Now we are going to define the relations for the u z  and z   .Following [2] we have where  -molar volume of the liquid and the constant B  is The value of u z  will be found self-consistently availing isotropic model for the liquid metal [4] and using Gibbs-Helmholtz formula The term in round brackets of the latter can be approximately estimated, following [3], as where L 0 -heat of evaporation of the overcooled liquid at absolute zero temperature (per mole), D -density (of mass) of liquid metal, dD/dT -temperature coefficient of the density, R -universal gas constant.Now, using ( 11) and ( 12) we have where The values of H, D, dD/dT, L 0 and B  entering (18) are known, the chemical potential can be found as ln ln[ / ( )] , where  -activity,   So, equations obtained here allow one to calculate the surface energy u and the surface tension  of liquid metals on flat as well as curved surfaces within the unitary approach.The final equations for u and  involve parameters u B and B  dependent on type of premelting structure being either bcc, fcc or hcp.i Author to whom all the correspondence should be sent, e-mail: sh-madina@mail.ru01027-p.4 free energy in mentioned phases.
and (12) one can, if knows the values of u  and   , calculate u z  and z   and then find dependences for ( ) u r and (r) according equations (13) and (14).Similar calculations of u z  on the basis of experimental data on u  (obtained from Gibbs-Helmholtz formula and on measured values of  and / d dT  10   for bcc, fcc and hcp -structures of premelting respectively, 8 0, 284 10 B     for liquid Hg possessing rombohedron structure of premelting.

Table 1 .
[4] values of L 0 were calculated on the known H according[4].The full data for the calculation of values related to the surface tension ( z  e u r and ( ) u z confirm the relevant proximity of the profiles of density and free energy as well as of density and potential energy in surface layer.Besides, the dividing surface corresponding to zero cohesion energy is closer to the condensed phase in comparison to the dividing surface determined by zero excess free energy in the transition layer, the physical boundary liquid-vapor being the same at melting point.01027-p.2LAM14

Table 1 .
The values of , z   , surface energy u  and surface tension   for the flat boundary liquid metal -vapor at melting point (the given are also input data on H, , dD/dT, d/dT) u z  [1] Web of ConferencesThe latter can be associated, as mentioned in[1], to the increasing diffusion and thickness of the transition layer when heating liquid metal.It comes from Table1that calculated values of surface energy( )The same comes from the calculated data of Table2for some liquid metals.The similar data for the surface tension of some other liquid metals were published in [2].

Table 2 .
The size dependence of the surface energy ( ) e u r and of surface tension ( )