Thermal stability of polycrystalline nanowires

Leonid Klinger; Eugen Rabkin

doi:10.1515/ijmr-2005-0193

Article

Thermal stability of polycrystalline nanowires

Leonid Klinger and Eugen Rabkin

Published/Copyright: February 3, 2022

Published by

Become an author with De Gruyter Brill

Author Information

From the journal International Journal of Materials Research Volume 96 Issue 10

Abstract

The equilibrium shape of a polycrystalline cylindrical nanowire with bamboo microstructure is determined. It is shown that for a certain ratio of grain boundary and surface energies a critical grain length exists above which the integrity of a nanowire cannot be preserved. Numerical calculations of the kinetics of nanowire shape evolution controlled by surface diffusion are performed. It is shown that the rate of thinning of unstable nanowires in the region of grain boundaries diverges as a break-up event is approached.

Keywords: Nanowires; Bamboo microstructure; Surface diffusion; Grain boundaries

Prof. Eugen Rabkin Department of Materials Engineering Technion-Israel Institute of Technology, 32000 Haifa, Israel Tel.: +972 4 829 4579 Fax: +972 4 829 5677

Dedicated to Professor Dr. Lasar Shvindlerman on the occasion of his 70th birthday

The authors would like to express their deep gratitude to Professor L. S. Shvindlerman for many years of inspiration, encouragement, and support.

References

[1] T.I. Kamins, in: H.-J. Fecht, M. Werner (Eds.), The Nano-Micro Interface: Bridging the Micro and Nano Worlds, Wiley-VCH, Weinheim, Germany (2004) 195.10.1002/3527604111.ch15Search in Google Scholar

[2] D. Raabe, J. Ge: Scripta mater. 51 (2004) 915.10.1016/j.scriptamat.2004.06.016Search in Google Scholar

[3] Lord Rayleigh: Proc. London Math. Soc. 10 (1878) 4.10.1112/plms/s1-10.1.4Search in Google Scholar

[4] P. Yang: MRS Bulletin 30 (2005) 85.10.1557/mrs2005.26Search in Google Scholar

[5] M.E. Toimil Molares, A.G. Balogh, T.W. Cornelius, R. Neumann,C. Trautmann: Appl. Phys. Lett. 85 (2004) 5337.10.1063/1.1826237Search in Google Scholar

[6] W.C. Carter: Acta metal. 36 (1988) 2283.10.1016/0001-6160(88)90328-8Search in Google Scholar

[7] W.W. Mullins: J. Appl. Phys. 28 (1957) 333.10.1063/1.1722742Search in Google Scholar

[8] L. Klinger, E. Rabkin: Scripta mater. 53 (2005) 229.10.1016/j.scriptamat.2005.03.041Search in Google Scholar

Appendix 1

Substituting k = 1 and r₀ = 0 in Eqs. (12) and (13), one obtains:

l=∫01sds1−s2=1

and

v=π∫01s3ds1−s2=32π

Appendix 2

For m → 0, k_min → 0.5, and the value of k corresponding to the maximum of L/R₀ is close to k_min: k → k_min, and thus r0−(k)≈1. Eqs. (12) and (13) can be re-written in the following form:

l=∫r01(ks2−k+1s+ks2−k+1)dss−ks2+k−1

Since the value of s in the interval of integration is close to one, we can assume s = 1 in the parenthesis:

l=12∫r01dss−ks2+k−1=12k[π2−arcsin(2kr0−12k−1)]≈π2k

where we employed the fact that according to Eq. (10) the expression in parenthesis is close to –1 for m → 0. The integral for v differs from integral for l by a factor s² ≈ 1 [see Eqs. (12), (13)]. Thus,v≈π2/2k.

Appendix 3

For the numerical solution of Eq. (5), we used the implicit scheme with linearization of the equation on each step. Let us sub-divide the X-axis into the small sections of the length Δ, and denote X_n = nΔ and R_n ≡ R(X_n, t). The mean curvature K_n ≡ K(X_n, t) in Eq. (4) can be approximated as

(A3.1) Kn=UnRn−Rn+1+Rn-1−2RnΔ2Un3 whereUn=(1+(Rn+1−Rn−1)24Δ2)−1/2

The partial differential Equation (5) is then transformed into the finite-difference equation

(A3.2) Δ2Bτ(Rn−R^n)=an+Kn+1+an−Kn-1−(an++an−)Kn

where τ is the time step, R̂_n = R(X_n, t–τ) and an± = Un±Rn+1Un+1−Rn-1Un-14Rn. After linearization of the right-hand side of Eq. (A3.2) with respect to the difference w_n = R_n – R̂_n, one obtains the set of linear equations for w_n, which can be solved by standard methods. We tested the stability of our numerical scheme for the following range of parameter values: Δ/R₀ = 10^–3 – 10^–2, Bτ/R04 = 10^–6 – 10^–5.

Received: 2005-06-30

Accepted: 2005-07-02

Published Online: 2022-02-03

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/ijmr-2005-0193

Keywords for this article

Nanowires; Bamboo microstructure; Surface diffusion; Grain boundaries