SURFACE TENSION PROBLEM FOR MICRO-AND NANOWIRES

An analytical solution for the Gibbs–Tolman–Koenig–Buff equation for microwire and nanowire surfaces has been obtained. Analysis has been performed for a cylindrical surface in terms of the linear and nonlinear Van der Waals theory.


Introduction
In this paper, the surface tension in microwires prepared by the Taylor-Ulitovsky method is studied. Surface tension is a fundamental thermodynamic parameter that significantly affects the formation of micro-and nanowires.
The study aims at derivation and detailed analysis of expressions for the surface tension of microwires in thermodynamic equilibrium in terms of the Gibbs-Tolman-Koenig-Buff equation method and the Van der Waals theory.
The given theory can find application in microwire production technology.  It is evident from the figures that cylindrical and conical surfaces should be studied.

Modeling of Surface Energy for Microwires in Terms of the Gibbs-Tolman-Konig-Buff Theory
The Gibbs-Tolman-Koenig-Buff differential equation (for a cylinder) [2][3][4][5][6] will be used to describe the surface tensions (σ i ) of micro-and nanowires [1]: , (1) where R i are the radii of micro-and nanowires (metallic kernel radius R m or the total radius of glass R g ). Non-negative parameters (δ i ) characterize the thickness of the interfacial layer (for example, between the glass and the metal).
In surface thermodynamics, Tolman length is used as a parameter that is equal to the distance between the surface of tension and the equimolar surface. The numerical values of the parameter-"Tolman length" analog-for micro and nanowires lie in a range of 0.1-1 µm.
The integral in (1) (if δ i = const) can be taken exactly. The final result is as follows [11]: . (2) The well-known Tolman`s formula (for a cylinder) is in a special case of   R for formula (2): In the case of : . (3b) We get the Rusanov`s linear formula (3b) for the cylindrical surface (see [5,11]).  (3) and (bold line) solutions (10)- (14). Experimental data for the surface tension of metal-glass as a function of metallic kernel radius R m .

Modeling of Surface Energy for Micro-and Nanowires
The basic equation of the linear Van der Waals theory of an inhomogeneous medium (see [1-3, 11, 27] for details) can be written as follows: , where n (x) is the function proportional to the volume density N(x) (x = r/δ, n o = const), r is the radial variable measured from the center of a nanoparticle, and  is the Tolman length [1-3, 11, 27].
The general solution to Eq. (4) has the form where are modified Bessel and Hankel functions. We will accept for the volume density function N(r/δ): , .
Substituting solution (5) into expression (7) and integrating, we obtain [11,27]: Solution (8) can be used for calculating adsorption, which is defined as the excess number of atoms or molecules in the surface layer of the nanoparticle per unit area: , Taking into account adsorption (9), we obtain the differential equation [11,27] . (10) we obtain 1 1 ln (see Eqs. (2) and (3a)); where  = 1.781 is the Euler constant, we obtain . (14) This equation is integrated numerically.

Modeling of Surface Energy for Micro-and Nanowires in Terms of the Nonlinear Theory
The nonlinear equation can be written as follows [27]: , Simple volume density function N can be determined as follows [27]: , (16) . The results obtained have a physical meaning only as long as function N 1 is positive. The resulting density profile (see Fig. 5 and (16), (17)) significantly differs from the results of the linear theory (see Fig. 4 and (8)) and, therefore, the Gibbs-Tolman-Koenig-Buff theory (see (2), (3a), (3b)).
Micro-and nanowires will be produced with a limited metallic kernel R m [27].

Modeling of Surface Energy of Micro-and Nanowires for a Particular Case of the Theory
The equation can be written (for a particular case of the theory) as follows .
A particular solution to Eq. (18) can have the form .

Conclusions
A feature of micro-and nanowires is that these objects consist of an amorphous alloy core (metal conductor) with a diameter of 0.1-50 µm, which is covered with a Pyrex-like coating with a thickness of 0.5-20 µm. Therefore, the main technological parameter for the preparation of glass-isolated micro-and nanowires is the surface tension of micro-and nanowires.
According to the previous analysis [1], the geometry of this microwire is most significantly affected by the glass properties. Microwire radius R g (outer radius of the glass shell) is estimated as follows [1]: (21) where k is a parameter dependent on a casting rate (0 < k < 1), V d is the casting rate, and σ s is the surface tension.
The metallic radius R m can be estimated approximately: (22) σ sm is the surface tension of metal-glass(0 < k m < 1). Thus, it has been confirmed that surface tension, which is defined as excess free energy per unit surface area, determines the radius of micro-and nanowires.
Acknowledgments. This work was supported by a Moldavian national project.