Comment on “Degenerate mobilities in phase field models are insufficient to capture
surface diffusion” [Appl. Phys. Lett. 107, 081603 (2015)]
Axel Voigt
Citation: Applied Physics Letters 108, 036101 (2016); doi: 10.1063/1.4939930
View online: http://dx.doi.org/10.1063/1.4939930
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APPLIED PHYSICS LETTERS 108, 036101 (2016)
Comment on “Degenerate mobilities in phase field models are insufficient to
capture surface diffusion” [Appl. Phys. Lett. 107, 081603 (2015)]
Axel Voigt
Institut f€
ur Wissenschaftliches Rechnen, Technische Universit€
at Dresden, 01062 Dresden, Germany
and Dresden Center for Computational Materials Science (DCMS), 01062 Dresden, Germany
(Received 14 October 2015; accepted 4 January 2016; published online 22 January 2016)
[http://dx.doi.org/10.1063/1.4939930]
In a recent paper, Lee et al.1 argue about Cahn-Hilliard
equations as approximations for motion by surface diffusion.
The considered equations read
ut ¼ r j;
j ¼ MðuÞrl;
2
2
0
l ¼ r u þ f ðuÞ;
ut ¼ r j;
(1)
j ¼ MðuÞrl;
gðuÞl ¼ 2 r2 u þ f 0 ðuÞ;
(3)
(4)
(2)
with order parameter u, quartic free energy f ðuÞ ¼ 14 ð1 u2 Þ2 ,
mobility function M(u) ¼ 1 u2, and the length scale related to the diffuse interface width. In leading order, these
equations formally converge if ! 0 to motion by surface
diffusion with an additional bulk diffusion term, which had
been ignored in several previous studies. This additional
term can alter the long time behaviour, as shown in Refs. 1
and 2 and is thus important to consider. However, several
points in Ref. 1 are at least misleading. This includes the
general statement of the title, which is already contradicted
by the authors, who admit that a double obstacle or logarithmic free energy instead of the used quartic free energy f(u)
gives rise to motion by surface diffusion without the
additional bulk diffusion term.3 But even if the authors use a
terminology where the term phase field models is only associated with equations based on a quartic free energy, their
conclusion is wrong. It has already been pointed out by
Gugenberger et al.,4 who showed using matched asymptotic
analysis, that the model in Ref. 5 converges also to motion
by surface diffusion without the additional bulk diffusion
0003-6951/2016/108(3)/036101/1/$30.00
term. With the introduced notation and scaling, the model in
Ref. 5 reads
which is obtained by defining u ¼ 2/ 1, rescaling time
by 1/2, considering an isotropic free energy and neglecting
the viscous term. There are two differences: The mobility
function is a higher order polynomial with M(u) f(u) and
the presence of a stabilizing function g(u), which is also
defined such that g(u) f(u). Such a stabilizing function is
commonly used in classical phase field models for solidification.6 Both modifications alter the analysis in Ref. 1. The
higher order polynomial in M(u) already suppresses the
presence of a normal flux in leading order and as shown
in Ref. 4 the combination with the stabilizing function
ensures also at next-to-leading order proper convergence
results.
1
A. A. Lee, A. M€
unch, and E. S€
uli, Appl. Phys. Lett. 107, 081603 (2015).
S. Dai and Q. Du, Multiscale Model. Simul. 12, 1870 (2014).
3
J. W. Cahn, C. M. Elliott, and A. Novick-Cohen, Eur. J. Appl. Math. 7, 287
(1996).
4
C. Gugenberger, R. Spatschek, and K. Kassner, Phys. Rev. E 78, 016703
(2008).
5
A. R€atz, A. Ribalta, and A. Voigt, J. Comput. Phys. 214, 187 (2006).
6
A. Karma and W.-J. Rappel, Phys. Rev. E 57, 4323 (1998).
2
108, 036101-1
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V
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