Ordem e formação de padrões em sistemas com interações
competitivas
Daniel A. Stariolo, Porto Alegre
Sergio A. Cannas, Córdoba
Francisco (Pancho) Tamarit, Córdoba
Mateus Michelon, Campinas
Daniel G. Barci, Rio de Janeiro
Ana Carolina Ribeiro-Teixeira, Porto Alegre
Lucas Nicolao, Porto Alegre
Orlando V. Billoni, Córdoba
Santiago Pighín, Córdoba
Marianela Carubelli, Córdoba
Roberto Mulet, Habana
Danilo Pescia’s group at ETH, Zurich
Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de 2008
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Ultrathin ferromagnetic films
Fe/Cu(001) ultrathin films, imaged
with SEMPA. In a) the effective temperature increases from left to right.
Image width=155 µm, thickness varies
from 2.06ML to 1.98ML, T=294K. In
b)-d) the temperature is increased
from left to right for the same region.
width=36µm,
thickness≈2.19ML,
Tb ≈298K, Tc ≈304K and Td ≈305K.
e)The labyrinthine pattern. In the inset
a triangular V disclination and the
structure factor. Thickness≈1.97ML,
T≈293K f) The labyrinthine pattern on
a stepped substrate. The disclination
has a square symmetry, as does the
structure factor. Parameters as in e),
thickness ≈ 2.15ML.
From O. Portmann et al., Nature 422, 701 (2003)
Diblock copolymers
R. Ruiz et el.
Phys. Rev. B 77, 054204 (2008)
Microemulsions
From P. M. Chaikin and T. C. Lubensky
Principles of Condensed Matter Physics
●
Physics ruled by competition between short range and long range interactions.
●
Long range interactions frustrate the tendency for long range order coming from
s.r.i. and then complex structures appear as stripes, bubbles, labyrinth, tetragonal,
nematic, columnar phases.
●
There is also competition between positional and orientational order giving rise to
phases similar to liquid crystals.
●
Examples go from classical to quantum systems like ultrathin ferromagnetic films,
diblock copolymers, low dimensional electronic systems, superconductors,
microemulsions, Raleigh-Bénard convection, etc.
Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de 2008
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Modulated phases in condensed matter
M. Seul and D. Andelman
Science 267, 476 (1995)
The Brazovskii model in two dimensions: stripe solutions
H=
1
2
Z
Z
Z
h
i
u0
g
d2 ~x κ(∇φ(~x))2 − r0 φ2 (~x) + φ4 (~x) +
d2 ~x d2 x~′ φ(~x)G(|~x − x~′ |)φ(x~′ )
2
2
In reciprocal space H = H0 + Hi , with:
Z
d2 k
2
~
~
φ(k) r0 + A2 (k − k0 ) + . . . φ(−k)
H0 =
2
(2π)
Z 2
d k1 d2 k2 d2 k3 d2 k4 ~
Hi = u0
φ(k1 )φ(~k2 )φ(~k3 )φ(~k4 ) δ(~k1 + ~k2 + ~k3 + ~k4 )
2
2
2
2
(2π) (2π) (2π) (2π)
●
●
●
First order isotropic-stripe transition
induced by flucutations
φ(~x) = A cos (~k0 · ~x)
Anglular fluctuations of ~k0 do not cost
energy.
Ky
K2
Kx
K1
K3
t
K4
Brazovskii, JETP 41, 85 (1975)
Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de 2008
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Beyond Brazovskii: nematic-like order
Consider a generic quartic interaction term of the form:
Z 2
d k1 d2 k2 d2 k3 d2 k4 ~ ~ ~ ~
u(k1 , k2 , k3 , k4 , ) ×
Hi =
2
2
2
2
(2π) (2π) (2π) (2π)
φ(~k1 )φ(~k2 )φ(~k3 )φ(~k4 ) δ(~k1 + ~k2 + ~k3 + ~k4 )
Ky
K2
Kx
K1
K3
t
K4
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Nematic symmetry
●
●
●
●
Isotropy force the wave vectors to lay on the circle and hence
u(~k1 , ~k2 , ~k3 , ~k4 ) = u(θ1 , θ2 , θ3 , θ4 ).
Conservation of total momentum allows to eliminate one of the four k′ s.
Fixing two of them, the other two are automatically fixed by the requirement that
the four vectors lay on the circle.
Because of rotational invariance, the interaction can only depend on the difference
between the two independent angles:
u(θ1 , θ2 , θ1 + π, θ2 + π) = u(θ1 − θ2 ) = u(θ)
●
Finally, index permutation invariance in u(~k1 , ~k2 , ~k3 , ~k4 ) implies u(θ) = u(θ + π),
which is a signature of nematic symmetry. Then u can be expanded in Fourier
series:
u(θ) = u0 + u2 cos (2θ) + u4 cos (4θ) + . . .
Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de 2008
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Nematic energy
Up to the first two terms in the expansion, the interaction energy with nematic
symmetry can be written as:
Z
n
o
H4 = d2 x u0 φ4 (~x) + u2 tr Q̂2
with
1 2
Q̂ij (~x) = φ(~x) ∇i ∇j − ∇ φ(~x)
2
a traceless symmetric tensor which can also be represented in two dimensions as a
complex nematic order parameter
Q = α ei2θ
The order parameter is a quadrupolar moment.
Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de 2008
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Mean field results
●
●
●
If u2 = 0 =⇒ Brazovskii “smectic” solution. Nevertheless, this solution is unstable
to fluctuations in d = 2 and TBr → 0.
u2 > 0 corresponds to repulsive quadrupole interactions. In this case the only
solution is α(T ) = 0.
u2 < 0 corresponds to attractive quadrupole interactions.
In this case mean field gives a secondp
order isotropic-nematic transition, with
α(T ) ≃ |Tc − T |1/2 and Tc = 2/(mk02 ) u0 /|u2 |.
Smectic
T
Nematic
Isotropic
D.A.Barci and D.A.S., PRL 98, 200604 (2007)
Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Março de 2008
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Quasi-long-range order and experimental scales
The continuous symmetry implies angular fluctuations in the director cost little energy
and can destroy long range nematic order. Evaluating local fluctuations:
Z
2
~
KT energy
δF ≡ F (Q̂(x)) − F (< Q̂ >) = ρs (T )
d2 x |∇ϕ(x)|
Now, the mean field nematic order parameter is given by
< Q̂ > = α exp(−W )
T
L
W =
ln
4πρs
a
Debye − W aller f actor
with TKT = (π/8)ρs (TKT ).
Thermodynamic limit: L/a → ∞ ⇒< Q̂ >→ 0, no nematic long range order.
For the Fe/Cu(001) films in O. Portmann et al., Nature 422, 701 (2003)
L ≈ 10−3 m
< Q̂ >≈ 0.8 α
a ≈ 10−10 m
W ≈ 7/32
Long range nematic ordershould be observable !
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