Lifshitz-Safran Coarsening Dynamics in a 2D Hexagonal System
Leopoldo R. Go ´mez and Enrique M. Valle ´s
Planta Piloto de Ingenierı ´a Quı ´mica (UNS-CONICET), Camino La Carrindanga, KM 7, 8000 Bahı ´a Blanca, Argentina
Daniel A. Vega
*
Department of Physics, Universidad Nacional del Sur, CONICET, Avenida L.N. Ale ´m 1253, 8000 Bahı ´a Blanca, Argentina
(Received 6 February 2006; published 31 October 2006)
The coarsening process in a two-dimensional hexagonal system in the region close to both spinodal and
order-order transitions was investigated through the Cahn-Hilliard model. We found a distinctive region of
the phase diagram where the pinning of dislocations plays only a minor role and the dynamics is led by the
triple points. In this region, we found configurations of domains with the same features as those proposed
by Lifshitz. As a consequence, different correlation lengths grow logarithmically in time, in good
agreement with the predictions of coarsening at low temperatures proposed by Safran.
DOI: 10.1103/PhysRevLett.97.188302 PACS numbers: 82.35.Jk
The mechanism of coarsening in a two-dimensional
system undergoing phase separation following a quench
from the high temperature phase to the ordered phase has
been the subject of intense investigations for more than
three decades [1,2]. Except in certain exceptional circum-
stances, it has been clearly shown through numerous stud-
ies that different systems show a coarsening process
satisfying scaling at long times [1]. In this case, the dy-
namics can be characterized by a simple length scale t
that grows in time t as a power law ( t
q
)[1,3]. This
feature has also been observed experimentally in thin films
of block copolymers in the smectic phase [4,5]. In this
case, it was shown that the orientational correlation length
grows in time as
2
t
1=4
and that the dynamics is led by
the annihilation of multipoles of disclinations. On the other
hand, it has recently been found through simulations [6]
and experiments [7] that, in sphere-forming block copoly-
mer thin films, the orientational and translational correla-
tion lengths grow according to different kinetic exponents.
The difference in kinetic exponents has been attributed to a
preferential annihilation of dislocations located along
small angle grain boundaries [6].
In the 1960s, Lifshitz predicted the possibility of for-
mation of a stable lattice of domains on a system with
p-fold degenerate equilibrium states. According to
Lifshitz, this lattice should emerge during the coarsening
process due to the dynamic frustration to reach equilibrium
[8]. Although this grain structure would not minimize the
total free energy of the system, it was shown that it could
be kinetically stable. As a consequence of the relaxation
driven by the curvature of grain boundaries, bounded re-
gions where three grains meet [triple points (TP)] can
become pinned to their positions, slowing down the dy-
namics. Once the system becomes trapped into this dy-
namically stable state, the only path to induce further
coarsening is through fluctuations or driving forces large
enough to unlock the system from the local traps. The first
step to introducing Lifshitz’s ideas in the coarsening pro-
cess quantitatively was made by Safran [9]. It was found
that the domains grow according to a power law in time for
p<d 1 (d is the spacial dimensionality) but logarithmi-
cally in time in the case p d 1.
Although a few systems have been found where the
growth of the correlation length is logarithmic [10], to
the best of our knowledge, there are no systems clearly
verifying the Lifshitz-Safran predictions at present.
The dynamics of phase separation for a diblock copoly-
mer can be described by the following time-dependent
Ginzburg-Landau equation for a conserved order parame-
ter (Cahn-Hillard model) [11]:
@
@t
Mr
2
F
: (1)
In this equation, the order parameter is defined in terms
of the local density of each block in the block copolymer,
M is a phenomenological mobility coefficient, and F is the
mean-field free energy functional for a diblock copolymer
[11]:
F
Z
dr
3
U
D
2
r
2
b
2
ZZ
dr
3
dr
03
Gr r
0
rr
0
: (2)
Here Gr is a solution of r
2
Grr, and U
1
2
a1 2f
2
2
1
3
3
1
4
4
. The parameters
a, , b, and are related to the vertex functions derived
by Leibler [12]. The parameter depends linearly on the
Flory-Huggins parameter and provides a measurement of
the depth of quench. f is the block copolymer asymmetry,
and D is a parameter related to the segment length [11].
Equation (1) leads to spinodal decomposition for >
s
2
bD
p
a1 2f
2
and to an order-order transition
(hexagonal-smectic transition) for f f
c
1=2 (
2
bD
p
).
In this work, we solve Eq. (1) in the region near both the
order-order and spinodal transitions ( *
s
, f & f
c
) for a
PRL 97, 188302 (2006)
PHYSICAL REVIEW LETTERS
week ending
3 NOVEMBER 2006
0031-9007= 06=97(18)=188302(4) 188302-1 © 2006 The American Physical Society