Surface energy for nanowire
ISSN: 2689-7636
Annals of Mathematics and Physics
Review Article       Open Access      Peer-Reviewed

Surface energy for nanowire

Serghei A Baranov1,2*

1Institute of Applied Physics, str. Academiei 5, Chisinau, MD-2028, Republic of Moldova
2Shevchenko Pridnestrov’e State University, str. 25 Oktyabrya 128, Tiraspol, Republic of Moldova
*Corresponding author: Serghei A Baranov, Laboratory of Electrical and Electrochemical Treatment of Materials, Institute of Applied Physics, str. Academiei 5, Chisinau, MD-2028, Republic of Moldova, Tel: ++ 373-22-731725; Fax ++373-22-738149; E-mail: baranov@phys.asm.md
Received: 11 March, 2022 | Accepted: 07 July, 2022 | Published: 08 July, 2022
Keywords: Gibbs–Tolman–Koenig–Buff theory; Tolman length; Van der Waals theory

Cite this as

Baranov SA (2022) Surface energy for nanowire. Ann Math Phys 5(2): 081-085. DOI: 10.17352/amp.000043

Copyright Licence

© 2022 Baranov SA. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

The theory of surface phenomena in the production of micro-and nanocylinder for important cases is considered. Analytical solution to Gibbs–Tolman–Koenig–Buff equation for nanowire surface is given. Analytical solutions to equations for case the cylindrical surface for the linear and nonlinear Van der Waals theory are analyzed. But for a nonlinear theory, this correspondence is absent.

Introduction

In this article, the surface tension in nanowires production by the Taylor–Ulitovsky method is studied. Surface tension is a fundamental thermodynamic parameter that significantly influences the creation of nanowires.

The chemical and physical properties of interphase boundaries in nanowires, as well as for nanoparticles, have been studied in a huge number of publications (see [1], fundamental monographs [2-5], and literature [6-21], and also my researches [22-28]. We can single out the following theoretical approaches: Gibbs–Tolman–Koenig–Buff equation method ends the linear and nonlinear Van der Waals theory.

The study aims at derivation and detailed analysis of expressions for the surface tension for the microwire in thermodynamic equilibrium on the Gibbs–Tolman–Koenig–Buff equation method and on the Van der Waals theory.

The given theory can find application in microwire production technology Figures 1,2.

As you can see from the figures, we must study cylindrical and conical surfaces.

Modeling of surface energy for microwires in Gibbs–Tolman– Konig–Buff`s theory

We will use the Gibbs – Tolman – Koenig – Buff differential equation [2-6] (for a cylinder) to describe the surface tensions, σi, of nanowires [1]:

dln σ i dln R i = 2 δ i R i + ( δ i R i ) 2 2+ 2 δ i R i + ( δ i R i ) 2        (1) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8339@

Where Ri is the radii of micro-and nanowires (the radius of its metallic kernel, Rm, or the total radius of glass, Rg).

Non-negative parameters (Tolman length), δi, characterize the thickness of the interfacial layer (for example, between glass and glass-metal).

In surface thermodynamics, the 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 parameter the analog


"Tolman length" for micro and nanowire is in the range from 0.1 to 1 µm.

The integral in (1) (if δi =const.) can be exactly taken. The final result has the form [7,10]:

σ/ σ () = R δ 2 2 (R/δ) 2 +2R/δ+1 exp( arctg( 1 1+2R/δ ) )      (2) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacqaHdpWCcaGGVaGaeq4Wdmxcfa4aaWbaaSqabeaajugqbiaacIcacqGHEisPcaGGPaaaaiaaykW7cqGH9aqpjuaGdaWcaaGcbaqcLbuacaWGsbaakeaajugqbiabes7aKbaajuaGdaGcaaGcbaqcfa4aaSaaaOqaaKqzafGaaGOmaaGcbaqcLbuacaaIYaGaaGPaVlaacIcacaWGsbGaai4laiabes7aKjaacMcajuaGdaahaaWcbeqaaKqzafGaaGOmaaaacqGHRaWkcaaIYaGaamOuaiaac+cacqaH0oazcqGHRaWkcaaIXaaaaaWcbeaajugqbiGacwgacaGG4bGaaiiCaKqbaoaabmaakeaajugqbiabgkHiTiaabggacaqGYbGaae4yaiaabshacaqGNbqcfa4aaeWaaOqaaKqbaoaalaaakeaajugqbiaaigdaaOqaaKqzafGaaGymaiabgUcaRiaaikdacaaMi8UaamOuaiaac+cacqaH0oazaaaakiaawIcacaGLPaaaaiaawIcacaGLPaaajuaGcaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabMcaaaa@776C@

The well-known Tolman formula (for cylinder) is a special case R>>δ MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGsbGaeyOpa4JaeyOpa4JaeqiTdqgaaa@3D40@ for this formula (2)

σ/ σ () ~ 1 1+ δ R     (3a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacqaHdpWCcaGGVaGaeq4Wdmxcfa4aaWbaaSqabeaajugqbiaacIcacqGHEisPcaGGPaaaaiaaykW7caGG+bqcfa4aaSaaaOqaaKqzafGaaGymaaGcbaqcLbuacaaIXaGaey4kaSscfa4aaSaaaOqaaKqzafGaeqiTdqgakeaajugqbiaadkfaaaaaaKqbakaabccacaqGGaGaaeiiaiaabccacaqGOaGaae4maiaabggacaqGPaaaaa@509B@

In case R<<δ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGsbGaeyipaWJaeyipaWJaeqiTdqgaaa@3B26@ :

σ/ σ () ~0.645(R/δ)     (3b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacqaHdpWCcaGGVaGaeq4Wdmxcfa4aaWbaaSqabeaajugqbiaacIcacqGHEisPcaGGPaaaaiaac6haqaaaaaaaaaWdbiaaicdacaGGUaGaaGOnaiaaisdacaaI1aGaaiika8aacaWGsbGaai4laiabes7aKjaacMcacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGZaGaaeOyaiaabMcaaaa@4E92@

We represent the Rusanov linear formula [5,11] for the cylindrical surface Figure 3.

Modeling of surface energy for Micro- and Nanowires in linear Van der Waals theory

The basic equation of the linear Van der Waals theory of an inhomogeneous medium (see [1-3] for details) can be written in the form:

n + n r 1 δ 2 (n n 1,2 )=0      (4) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaaMc8UabmOBayaagaGaey4kaSscfa4aaSaaaOqaaKqzafGabmOBayaafaaakeaajugqbiaadkhaaaGaeyOeI0scfa4aaSaaaOqaaKqzafGaaGymaaGcbaqcLbuacqaH0oazjuaGdaahaaWcbeqaaKqzafGaaGOmaaaaaaGaaiikaiaad6gacqGHsislcaWGUbqcfa4aaSbaaSqaaKqzafGaaGymaiaacYcacaaIYaaaleqaaKqzafGaaiykaiabg2da9iaaicdacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeinaiaabMcaaaa@55AE@

Where n (x) is the function when proportional to the volume density N(x) (x = r/δ, no=const.), δ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacqaH0oazaaa@3847@ is the radial variable measured from the center of a nanoparticle, is the Tolman length [1-3].

The general solution to Eq. (4) has the form

n(r)= n 1,2 +A I 0 (r/δ)+B K 0 (r/δ)      (5) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGUbGaaiikaiaadkhacaGGPaGaeyypa0JaamOBaOWaaSbaaSqaaKqzafGaaGymaiaacYcacaaIYaaaleqaaKqzafGaey4kaSIaamyqaiaaykW7caWGjbGcdaWgaaWcbaqcLbuacaaIWaaaleqaaKqzafGaaiikaiaadkhacaGGVaGaeqiTdqMaaiykaiabgUcaRiaadkeacaaMc8Uaam4saOWaaSbaaSqaaKqzafGaaGimaaWcbeaajugqbiaacIcacaWGYbGaai4laiabes7aKjaacMcacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeynaiaabMcaaaa@5D9E@

Where I 0 (r/δ), K 0 (r/δ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaaMc8UaamysaKqbaoaaBaaaleaajugqbiaaicdaaSqabaqcLbuacaGGOaGaamOCaiaac+cacqaH0oazcaGGPaGaaiilaiaaykW7caWGlbqcfa4aaSbaaSqaaKqzafGaaGimaaWcbeaajugqbiaacIcacaWGYbGaai4laiabes7aKjaacMcaaaa@4B10@ is a modification to Bessel and Hankel functions?

n(0)=n(R)= n 1 ,n(+)= n 2     (6) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGUbGaaiikaiaaicdacaGGPaGaeyypa0JaamOBaiaacIcacaWGsbGaaiykaiabg2da9iaad6gakmaaBaaaleaajugqbiaaigdaaSqabaGccaGGSaqcLbuacaWGUbGaaiikaiabgUcaRiabg6HiLkaacMcacqGH9aqpcaWGUbGcdaWgaaWcbaqcLbuacaaIYaaaleqaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG2aGaaeykaaaa@51DA@

We will accept the volume density function, N(r/δ). We get:

N(r/δ)n(0)n(R)=1,N( + )n(+)=0      (7) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuaqaaaaaaaaaWdbiaad6eapaGaaiika8qacaWGYbGaai4laiabes7aK9aacaGGPaaccaGae8NKH4QaamOBaiaacIcacaaIWaGaaiykaiab=fMi6kaad6gacaGGOaGaamOuaiaacMcacqGH9aqpcaaIXaGaaiila8qacaWGobqcfa4damaabmaakeaajugqbiabgUcaRiabg6HiLcGccaGLOaGaayzkaaqcLbuacqGHsgIRcaWGUbGaaiikaiabgUcaRiabg6HiLkaacMcacqGH9aqpcaaIWaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabEdacaqGPaaaaa@5DB4@

Substituting solution (5) into expression (7) and integrating, we obtain:

N(r/δ)={ 1,rR, K 0 (r/δ) K 0 (R/δ) ,r>R,       (8) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@720E@

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:

Γδ K 1 (x) K 0 ( x 0 )      (9) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacqqHtoWrcqGHsgIRcqaH0oazcaaMc8UaaGzaVlaaykW7juaGdaWcaaGcbaqcLbuacaWGlbqcfa4aaSbaaSqaaKqzafGaaGymaaWcbeaajugqbiaacIcacaWG4bGaaiykaaGcbaqcLbuacaWGlbqcfa4aaSbaaSqaaKqzafGaaGimaaWcbeaajugqbiaacIcacaWG4bqcfa4aaSbaaSqaaKqzafGaaGimaaWcbeaajugqbiaacMcaaaqcfaOaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeyoaiaabMcaaaa@5700@

(x = r/δ, x0 = R/δ).

Taking into account adsorption (9), we obtain the differential equation

dlnσ dlnx = 1 x{ K 0 (x)/ K 1 ( x 0 )}+1       (10) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaKqzafGaamizaiaaykW7caaMi8UaciiBaiaac6gacqaHdpWCaOqaaKqzafGaamizaiaayIW7caaMc8UaciiBaiaac6gacaWG4baaaiabg2da9KqbaoaalaaakeaajugqbiaaigdaaOqaaKqzafGaamiEaiaaykW7caGG7bGaam4saKqbaoaaBaaaleaajugqbiaaicdaaSqabaqcLbuacaGGOaGaamiEaiaacMcacaGGVaGaam4saKqbaoaaBaaaleaajugqbiaaigdaaSqabaqcLbuacaGGOaGaamiEaKqbaoaaBaaaleaajugqbiaaicdaaSqabaqcLbuacaGGPaGaaiyFaiabgUcaRiaaigdaaaqcfaOaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqGWaGaaeykaaaa@6740@

if х >> 1

K 0 ( x ) K 1 ( x 0 ) 1       (11) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaKqzafGaam4saKqbaoaaBaaaleaajugqbiaaicdaaSqabaqcfa4aaeWaaOqaaKqzafGaamiEaaGccaGLOaGaayzkaaaabaqcLbuacaWGlbqcfa4aaSbaaSqaaKqzafGaaGymaaWcbeaajuaGdaqadaGcbaqcLbuacaWG4bqcfa4aa0baaSqaaKqzafGaaGimaaWcbaaaaaGccaGLOaGaayzkaaaaaKqzafGaeyOKH4QaaGymaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabgdacaqGPaaaaa@525E@

We obtain

dlnσ dlnx = 1 x+1      (12) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaajugqbiaadsgacaaMc8UaaGjcVlGacYgacaGGUbGaeq4WdmhakeaajugqbiaadsgacaaMi8UaaGPaVlGacYgacaGGUbGaamiEaaaacqGH9aqpkmaalaaabaqcLbuacaaIXaaakeaajugqbiaadIhacqGHRaWkcaaIXaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabkdacaqGPaaaaa@53E5@

(see formula (2) and (3a));

and if x << 1

K 0 ( x ) K 1 ( x ) xln 2 γx      (13) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaajugqbiaadUeakmaaBaaaleaajugqbiaaicdaaSqabaGcdaqadaqaaKqzafGaamiEaaGccaGLOaGaayzkaaaabaqcLbuacaWGlbGcdaWgaaWcbaqcLbuacaaIXaaaleqaaOWaaeWaaeaajugqbiaadIhaaOGaayjkaiaawMcaaaaajugqbiabgIKi7kaadIhaciGGSbGaaiOBaOWaaSaaaeaajugqbiaaikdaaOqaaKqzafGaeq4SdCMaaGPaVlaadIhaaaGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaae4maiaabMcaaaa@56A2@

Where γ = 1,781 is Euler constant, we obtain

dlnσ dlnx = 1 xln 2 γx +1      (14) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaKqzafGaamizaiaaykW7caaMi8UaciiBaiaac6gacqaHdpWCaOqaaKqzafGaamizaiaayIW7caaMc8UaciiBaiaac6gacaWG4baaaiabg2da9KqbaoaalaaakeaajugqbiaaigdaaOqaaKqzafGaamiEaiGacYgacaGGUbqcfa4aaSaaaOqaaKqzafGaaGOmaaGcbaqcLbuacqaHZoWzcaaMc8UaamiEaaaacqGHRaWkcaaIXaaaaKqbakaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG0aGaaeykaaaa@5C5E@

This equation is integrated numerically.

Modeling of surface energy for Micro-and Nanowires in nonlinear theory

The nonlinear equation can be written in the form

n 1 + 1 r n 1 + 1 δ 1 2 exp{ n 1 }=0     (15) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaaMc8UaaOOBaKqbaoaaBaaaleaajugqbiaaigdaaSqabaqcfa4aaWbaaSqabeaaiiaajugqbiab=ndiYcaacqGHRaWkjuaGdaWcaaGcbaqcLbuacaaIXaaakeaajugqbiaadkhaaaGaaOOBaKqbaoaaBaaaleaajugqbiaaigdaaSqabaqcfa4aaWbaaSqabeaajugqbiab=jdiIcaacqGHRaWkjuaGdaWcaaGcbaqcLbuacaaIXaaakeaajugqbiabes7aKLqbaoaaBaaaleaajugqbiaaigdaaSqabaqcfa4aaWbaaSqabeaajugqbiaaikdaaaaaaiGacwgacaGG4bGaaiiCaiaacUhacqGHsislcaGIUbqcfa4aaSbaaSqaaKqzafGaaGymaaWcbeaajugqbiaac2hacqGH9aqpcaaIWaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabwdacaqGPaaaaa@63FD@

The simple volume density function, N, may be determined:

N 1 =1+2ln[1 X 1 2 ]     (16) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuaqaaaaaaaaaWdbiaad6eakmaaBaaaleaajugqbiaaigdaaSqabaqcLbuapaGaeyypa0JaaGymaiabgUcaRiaaikdaciGGSbGaaiOBaiaacUfacaaIXaGaeyOeI0IaamiwaOWaaSbaaSqaaKqzafGaaGymaaWcbeaakmaaCaaaleqabaqcLbuacaaIYaaaaiaac2facaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeOnaiaabMcaaaa@4CA5@

X 1 =r/ (2 2 δ 1 )     (17) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGybqcfa4aaSbaaSqaaKqzafGaaGymaaWcbeaajugqbiabg2da9KqbaoaalyaakeaajugqbiaakkhaaOqaaKqzafGaaOikaiaakkdajuaGdaGcaaGcbaqcLbuacaaIYaaaleqaaKqzafGaeqiTdqwcfa4aaSbaaSqaaKqzafGaaGymaaWcbeaajugqbiaacMcaaaqcfaOaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaae4naiaabMcaaaa@4DC5@

The results obtained have a physical meaning only as long as the function N1 is positive.

The resulting density profile (see Figure 4 and (16), (17)) is very different from the results of the linear theory (see Figure 5 and (8)), and therefore the GTKB theory (see (2), (3a), (3b)).

Micro and nanowire will only be produced for a limited metallic kernel, Rm.

The density profiles in [13] (see Figure 6) are very different too from the results in Figure 4.

Modeling of surface energy for Micro-and Nanowires in the pure case theory

The equation can be written (in the pure case theory) in the form:

n 2 + 1 r n 2 =0     (18) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaaMc8UaaOOBaKqbaoaaBaaaleaajugqbiaaikdaaSqabaqcfa4aaWbaaSqabeaaiiaajugqbiab=ndiYcaacqGHRaWkjuaGdaWcaaGcbaqcLbuacaaIXaaakeaajugqbiaadkhaaaGaaOOBaKqbaoaaBaaaleaajugqbiaaikdaaSqabaqcfa4aaWbaaSqabeaajugqbiab=jdiIcaacqGH9aqpcaaIWaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabIdacaqGPaaaaa@50BB@

A particular solution for equation (18) can have the form:

n 2 =cln (R / δ)       (19) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGUbGcdaWgaaWcbaqcLbuacaaIYaaaleqaaKqzafGaeyypa0Jaam4yaiGacYgacaGGUbGcdaWcgaqaaKqzafGaaiikaiaakkfaaOqaaGGaaKqzafGae8hTdqMaaiykaaaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeyoaiaabMcaaaa@49AD@

We will accept the initial values

N 2, 0 =1     (20) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaGIobqcfa4aaSbaaSqaaKqzafGaaGOmaiaacYcaaSqabaqcfa4aaSbaaSqaaKqzafGaaGimaaWcbeaajugqbiabg2da9iaaigdacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGYaGaaeimaiaabMcaaaa@44E8@

and get

N 2 =1+ln (R / δ)     (20a) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGobGcdaWgaaWcbaqcLbuacaaIYaaaleqaaKqzafGaeyypa0JaaGymaiabgUcaRiGacYgacaGGUbGcdaWcgaqaaKqzafGaaiikaiaakkfaaOqaaGGaaKqzafGae8hTdqMaaiykaaaakiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabcdacaqGHbGaaeykaaaa@49D8@

Function graph N2 is shown in Figure 7.

Conclusion

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, covered with a Pyrex-like coating with a thickness of (0.5...20) µm. Therefore, the main technological parameter for the production of glass micro-and nanowires is the surface tension of the surfaces of micro-and nanowires.

According to the previous analysis [1], the most significant effect on the geometry of such microwires comes from the glass properties. The microwire radius Rg (the outer radius of the glass shell) is estimated as follows [1]:

R g ~ η 2k V d k σ s 1k      (21) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGsbqcfa4aaSbaaSqaaKqzafGaam4zaaWcbeaajugqbiaac6hajuaGdaWcaaGcbaqcLbuacaqG3oqcfa4aaWbaaSqabeaajugqbiaaikdacqGHsislcaWGRbaaaaGcbaqcLbuacaWGwbqcfa4aa0baaSqaaKqzafGaamizaaWcbaqcLbuacaWGRbaaaiaabo8ajuaGdaqhaaWcbaqcLbuacaWGZbaaleaajugqbiaaigdacqGHsislcaWGRbaaaaaacaaMc8UaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabgdacaqGPaaaaa@5628@

Where k is the parameter, which is dependent on a casting rate (0 < k < 1); Vd is the casting rate; σs is the surface tension.

The metallic radius, Rm, is possible to estimate approximately:

R m ~ σ sm V d 2 k m       (22) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGsbqcfa4aaSbaaSqaaKqzafGaamyBaaWcbeaajugqbiaac6hajuaGdaWcaaGcbaqcLbuacqaHdpWCjuaGdaqhaaWcbaqcLbuacaWGZbGaamyBaaWcbaaaaaGcbaqcLbuacaWGwbqcfa4aa0baaSqaaKqzafGaamizaaWcbaqcLbuacaaIYaGaeyOeI0Iaam4AaKqbaoaaBaaameaajugqbiaad2gaaWqabaaaaaaajugqbiaaykW7caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabkdacaqGPaaaaa@5478@

σsm is the surface tension of metal – glass (0 < km< 1).

We thus confirm that surface tension, defined as excess free energy per unit surface area, determines the radius of micro-and nanowires.

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