Smectic Liquid Crystals: Materials with One-Dimensional, Periodic Order

0.19 MB·English

by  Randall D. Kamien

SmecticLiquidCrystals:MaterialswithOne-Dimensional,
Periodic Order
Randall D. Kamien and Christian D. Santangelo
rd
Department of Physics and Astronomy, 209 South 33 Street, University of
6
0 Pennsylvania, Philadelphia, PA 19104-6396, USA
0
2 6 December2005; revised 16 January 2006

n Abstract. Smectic liquid crystals are materials formed by stacking deformable,
a fluidlayers.Thoughsmecticsprefertohaveflat,uniformly-spacedlayers,boundary
J
conditionscanimposecurvatureonthelayers.Sincethelayerspacingandcurvature

0 areintertwined,theproblemoffindingminimalconfigurationsforthelayersbecomes
2
highly nontrivial. We discuss various topological and geometrical aspects of these

materials and present recent progress on finding some exact layer configurations.
]
G Wealsoexhibitconnectionstothestudyofcertainembeddedminimalsurfacesand
briefly summarize some important open problems.
D
. Keywords: mean curvatureflow, foliations, liquid crystals
h
t
a
m
[ 1. Liquid Crystals: Phases in Between Liquid and Solid

1
First characterized at the end of the 19th century, liquid crystals have
v
4 gone from chemical curiosities to important optical and electronic de-
9 vices (de Gennes and Prost, 1993). In fact, this article relied on liquid
4
crystal technology to be produced! As their odd name implies, liquid
1
0 crystalshavesomepropertiesofacrystalandsomeofaliquid.Whether
6 nematic, cholesteric, or smectic, all these phases flow (like a fluid), at
0
least in some directions, while transmitting torques and forces (like a
/
h crystal)inothers.Thephaseweconsiderhereisthesmectic-A phase.A
t
a crystalischaracterizedasaperiodicstructure(thesimplestofwhichare
m lattices).Ad-dimensionalcrystalisaperiodicstructureinddimensions
: withdindependentperiodicities.However, itispossibletohave alower
v
i numberofperiodsthanthannumberofdimensions.Focussingond = 3
X
dimensions, a one-dimensional periodic structure is a smectic, a two-
r dimensionalperiodicstructureisacolumnarphase,athree-dimensional
a
structure is a crystal, and a phase with no periodicity is a fluid.
In this article we will consider smectic phases and discuss minima
of the free energy
1
F = d3x B( Φ 1)2+K ( N)2 (1)
1
2 |∇ |− ∇·
ZV n o
where the density of material is ρ(x) cos[2πΦ(x)/a], a is the one-
∝
dimensional spacing of the layers and N = Φ/ Φ is a unit vector
∇ |∇ |
c 2008 Kluwer Academic Publishers. Printed in the Netherlands.
(cid:13)
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.1
2 RandallD.KamienandChristianD.Santangelo
field, normal to the level sets of Φ. The constants B and K are the
1
bulkandbendmodulus,respectively,andtheintegralisoverthesample
volume V. In particular we are interested in minima with non-trivial
topologyondomainswhicharenotsimply-connected.Beforediscussing
those solutions, we will review the origins of the functional in (1).
A one-dimensional periodic structure in three dimensions can be
represented as an L summable density wave in R3
2
ρ(x) = ∞ ρ eijqx (2)
j ·
j=
X−∞
where ρ = ρ , so that ρ(x) real, and q is a vector pointing along
∗j j
−
the direction of the modulation with magnitude 2π/a, where a is the
spacing between the “layers”. Each Fourier mode ρ for j = 0 is an
j
6
order parameter for the smectic: if ρ = 0 for all j = 0, then the
j
6
material is a fluid, while if some ρ is nonvanishing, it is a smectic.
j
Sinceais thespacingbetween thelayers, it is sufficientto keep track of
just ρ since the presence of the fundamental mode is enough to signal
1
the order. Truncating (1) to the first non-trivial mode and writing
ρ = ρ e 2πiu/a (Chaikin and Lubensky, 1995) we have
1 1 −
| |
2π
ρ(x) =ρ +2ρ cos (qˆ x u) (3)
0 1
| | a · −
(cid:20) (cid:21)
To study deformations and fluctuations of this periodic structure, we
promote u from a constant to a scalar field u(x). This results in an
Eulerian description of the material, i.e. the layers sit at surfaces of
constantΦ(x) qˆ x u(x)or,equivalently, atthepeaksof thedensity
≡ · −
wave. An alternative description is Lagrangian, in which we label each
fiducial plane defined by qˆ x = n and consider deformations about
n
·
these. When the deformations are small they can be represented as a
height function or graph over each plane. If we choose our coordinates
so that qˆ= zˆ, then
n = h (x,y) u[x,y,h (x,y)] (4)
n n
−
However, a unique solution for h requires that the deformed surface
n
not have any overhangs and, moreover, awkwardly mixes spatial co-
ordinates with the Lagrangian coordinate n which labels the original
surface.Despite this,as wewillsee subsequently,in restricted cases the
Lagrangian representation is a useful tool.
There are three essential actors in the free energy of smectics: the
layer compression or strain, the bending or mean curvature, and the
topology of the surfaces characterized by the intrinsic or Gaussian
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.2
SmecticLiquidCrystals 3
curvature. Using the phase field Φ, related to the density ρ via ρ
∝
cos(2πΦ/a), the unit layer normal is
Φ
N = ∇ (5)
Φ
|∇ |
At a point on a surface, the two principle radii of curvature R and R
1 2
determine the two curvatures κ 1/R . These two curvatures can be
i i
≡
combined into the mean curvature H = 1(κ +κ ) and the Gaussian
2 1 2
curvature K = κ κ . The former is a measure of the bending of the
1 2
layers in space, while the latter is an intrinsic quantity which is a local
measure of geometric distortion and can be applied globally to gather
topologicalinformation.Fortunately, thesequantitiesmaybeexpressed
in terms of the layer normal: (Kamien, 2002)
1
H = N (6)
2∇·
(cid:12)Φ(x,y,z)=c
(cid:12)
1 (cid:12)
K = [N(cid:12)( N) (N )N] (7)
2∇· ∇· − ·∇
(cid:12)Φ(x,y,z)=c
(cid:12)
(cid:12)
where the expressions are evaluated on one of t(cid:12)he level sets of Φ. By
working in this set of coordinates, it is possible to free these geometric
quantities from the surfaces and, as we will show, allows us to relate
different terms in the free energy.
Because smectics have a preferred layer spacing, a, there is nec-
essarily a strain energy associated with changing this spacing. While
the two curvatures are natural from the geometric point of view, an
expression for the compression strain is not as well defined. Returning
to the family of equally spaced layers, we have na= Φ(x +anN) and
n
so, differentiating with respect to n, find that
1 = N Φ(x +anN) (8)
n
·∇
which implies that Φ = 1. Thus the strain, when expressed in
|∇ |
terms of Φ must vanish when this condition holds (Kleman, 1983).
For instance, we can define the strain u as
zz
u 1 Φ = 1 N Φ (9)
zz
≡ −|∇ | − ·∇
or an alternate strain u˜ as
zz
1
u˜ 1 ( Φ)2 (10)
zz
≡ 2 − ∇
h i
both of which vanish when Φ = 1. Moreover, in terms of u, the
|∇ |
Eulerian displacement field, it follows that u ∂ u + u2 and
zz z
≈ O
(cid:0) (cid:1)
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.3
4 RandallD.KamienandChristianD.Santangelo
u˜ ∂ u+ u2 are identical to linear order in u and differ only
zz z
≈ O
in their nonlinear terms. Therefore, when considering the quadratic
(cid:0) (cid:1)
compression energy:
1
F = d3xBu2 (11)
c 2 zz
Z
the harmonic term is independent of the specific choice of u – the
zz
lineartermwillalwaysbe∂ u.Whileu˜ arisesnaturallyfromaLandau
z zz
theory description of the nematic to smectic transition (de Gennes,
1972), there is no compelling reason to choose it over u – both are
zz
invariantunderrotations andvanishforequallyspacedlayers.Thecon-
stant B is known as the bulk modulusand controls compression modes
of the layers. As we will show in the following, the use of u allows
zz
us to further develop the geometric theory of smectics 1. In a similar
vein, one could argue that the mean curvature H could be replaced by
1 2Φ. However, the nonlinear form H = 1 N unambiguously has
2∇ 2∇·
the Laplacian as the first term in an expansion in powers of u, while
there are many nonlinear generalizations of the Laplacian, including
the full mean curvature.
Putting this all together we have (Santangelo and Kamien, 2005a)
B
F = d3x [1 N Φ]2+λ2( N)2 , (12)
2 − ·∇ ∇·
Z n o
where the last term is the contribution of the layer curvature to the
energy, and is quadratic in the radii of curvature. The constant λ =
K /B has units of length and sets the natural lengthscale for defor-
1
mations. For the sake of consistency, one should add to this expression
p
a term of the form
F = K¯ d3x [( N)N (N )N], (13)
K
∇· ∇· − ·∇
ZV
coming from the Gaussian curvature. Since this term is a total deriva-
tive, it can be reduced to an integral on the boundary and, therefore,
cannot play a role in the minimization of the energy that determines
theconfigurationofthelayers.Theusualgroundstatewhichminimizes
(12) depends on an arbitrary unit vector n: Φ = n x, corresponding
·
to equally spaced layers along the n direction. The free energy of this
solution vanishes since N = n is constant and Φ = n =1.
|∇ | | |
It will become clear why we have chosen this nonlinear form for
u . In the following section, however, we will linearize this theory – a
zz
process which results in an unambiguous form for u .
zz
1 Indeeduzz =1 √1 2u˜zz =u˜zz+ u˜2zz and soenergies based onuzz will
differfrom those wit−h u˜zz−only by anharmOonic terms.
(cid:0) (cid:1)
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.4
SmecticLiquidCrystals 5
Onlyinveryfewcasescanwesatisfythetermsintheenergyindivid-
ually. The difficulty of this problem has to do with the nontrivial way
that the terms in equation (12) interact. It is instructive to consider a
familyof evenly spaced layers x (s ,s )parameterized by (s ,s ),with
n 1 2 1 2
x = x +naN. At any point (s ,s ) on x , we have the two principal
n 0 1 2 0
curvaturesκ andκ . Theradii of curvature of the nth layer are just
1,0 2,0
R = R +na from which it follows that
i,n i,0
κ
i,0
κ = (14)
i,n
1+naκ
i,0
From the definitions of H and K it follows that the curvatures of the
nth layer are:
K
0
K = (15)
n 1+2naH +n2a2K
0 0
H +naK
0 0
H = (16)
n 1+2naH +n2a2K
0 0
Thus if the layers are evenly spaced, the only way for the bending to
vanish for all layers (H = 0) is for K (and thus K ) to vanish (Di-
n 0 n
Donna and Kamien, 2003). This interplay between even spacing, the
Gaussiancurvature,andthemeancurvatureis arecurringmotif inthis
work and will be discussed further in the following. It is clear that in
order to have configurations with non-vanishing curvature, the layer
spacing must necessarily vary.
1.1. Topological Defects
The ground state or equilibrium configuration of the field u subject to
boundary conditions is the goal of our analysis. While the boundary
conditions at infinity can usually be taken to be u = 0 or u = 0,
∇
we may introduce additional boundary conditions via topological de-
fects (Chaikin and Lubensky,1995). Inparticular, becauseΦ is aphase
field, it can wind by multiples of 2π around defects. In the so-called
“London limit”, we simply remove a line from R3 around which the
phase winds. It is possible to instead allow the density ρ to vary in
1
space: the solution to the full equation has ρ vanishing at the core
1
of the defect rendering the energy finite. This extra complication is
tangential to our discussion and we will use the London theory in the
following.
Much of the physical understanding of defects comes from the lin-
earized form of the theory or, equivalently, the quadratic energy (Kle-
man, 1983; de Gennes and Prost, 1993):
B 2
F = d3x (∂ u)2+λ2 2u (17)
l z
2 Z (cid:20) (cid:16)∇⊥ (cid:17) (cid:21)
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.5
6 RandallD.KamienandChristianD.Santangelo
The first term is the linearized compression strain and measures the
stretching along the layer direction, taken to be zˆ. We define =
∇⊥
xˆ∂ +yˆ∂y,thegradientintheplaneperpendiculartozˆ,andseethatthe
x
second term is the Laplacian in that plane, which we recognize as the
linearized form of the mean curvature, H. It is important to note that
theoriginaltheorywasquadraticinthestrainu andthecurvatureH.
zz
There is no reason, physical or otherwise, that this truncation should
be valid – it is used primarily because of its solvability. In addition,
the ambiguity arising from the freedom to choose different forms of the
nonlinear free energy does not arise in the linear theory. In order for
the density to be single-valued, we have
u dℓ = na (18)
∇ ·
IΓ
for some n Z whenever the contour Γ goes around the defect. The
∈
combination na= b is called the Burgers vector (Chaikin and Luben-
−
sky, 1995). This can be promoted to differential form by introducing
a defect density m(x) which is a vector field pointing along the defect
line (de Gennes and Prost, 1993). The defect line is along the curve
r(s) R3 with local unit tangent t(s)= dr/ds and we write
∈
L
m(x) = na δ3[x r(s)]t(s) (19)
−
Z0
It follows that m = 0 for defect lines that either run off to infinity
∇·
or are closed loops. From Stoke’s theorem, we have the local condition:
u= m (20)
∇×∇
To find the ground state configuration of the smectic in the presence of
defects, we write u as the sum of a regular part σ and a singular part
τ so that ∂ u= ∂ σ+τ with τ = m (de Gennes and Prost, 1993).
i i i
∇×
This is most easily done in Fourier space where
iq m(q)
τ(q) = − × (21)
q2
where we have chosen τ = 0, absorbing any of the longitudinal
parts of τ into σ. Havi∇ng· enforced the boundary conditions via the
introduction of the singular part of u, we insert the expression for u
∇
into the free energy and minimize over the remaining variable, σ. This
results in
δF zˆ (q m)
0= l = B q2+λ2q4 σ q 1 λ2q2 · × (22)
δσ (cid:26)h z ⊥i − z(cid:16) − ⊥(cid:17) q2 (cid:27)
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.6
SmecticLiquidCrystals 7
so that
q 1 λ2q2 (q m q m )
z x y y x
σ(q) = − − (23)
q2 q⊥2+λ2q4
(cid:0) z(cid:1)
⊥
and (cid:0) (cid:1)
Bλ2 d3q zˆ (q m)2
F = | · × | (24)
l 2 (2π)3 q2+λ2q4
Z z
⊥
Because of the anisotropy of the smectic, we can have defects which
areparalleltothespecial,zˆaxisanddefectswhichareperpendicularto
it.Theformerarecalledscrewdislocationswhilethelatterareknownas
edge dislocations (Kleman, 1983). Defects can, of course, have a mixed
character when they are oblique to the zˆ axis. For an edge dislocation
pointing along the yˆ axis, m = nayˆδ(x)δ(z) or m(q) = (2π)nayˆδ(q ).
y
In order to calculate the energy we will need a long length cutoff L /2
y
for q and a short distance cutoff q = π/ξ for q . We have
y max z
Bλ2n2a2 q2
F = dq dq x δ(q = 0)
edge 4π x zq2+λ2q4 y
Z z x
BL √λn2a2 qmax 1
y
= dq
z
4√2 Z−qmax |qz|
BL n2a2√π λ p
y
= (25)
√2 sξ
sothattheenergyperunitlength(F/L )isfinite(deGennesandProst,
y
1993;Kleman,1983).Forascrewdislocation,m = nazˆδ(x)δ(y)andthe
free energy vanishes meaning that the smooth part of u can “screen”
the strain from the boundary at the defect. The spatially dependent
solutions for the edge and screw dislocations are
na x2 x
u = exp 1,0, (26)
edge
∇ 4 πλ|z| (−4λ|z|)(cid:20) −z(cid:21)
na
u = p [y, x,0] (27)
∇ screw 2π(x2+y2) −
respectively.
1.2. The Edge Again
It is instructive to notice that the result for edge dislocation is simply
the gradient of
x
u(y,z) = sgn(z)b 1 √2λ|z| dte t2/2 (28)
−
− 2√2π
Z−∞
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.7
8 RandallD.KamienandChristianD.Santangelo
where we will now switch to the Burgers vector b = na. For z > 0,
−
we have ∂ λ∂2 u = 0 while for z < 0, ∂ +λ∂2 u = 0. There is a
z − x z x
discontinuity between the upper and lower half planes which precisely
(cid:0) (cid:1) (cid:0) (cid:1)
enforces the condition (18). To see this, we take the clockwise path
in the yz plane, starting at (y,z) = ( ,0 ), going to (+ ,0 ), to
− −
−∞ ∞
(+ ,0+) and back to ( ,0+). We have
∞ −∞
u dℓ = u( ,0+) u( ,0+) + u( ,0+) u( ,0 )
−
∇ · −∞ − ∞ ∞ − ∞
ZΓ
+(cid:2) u( ,0 ) u( ,(cid:3)0 )(cid:2) (cid:3)
− −
∞ − −∞
b b
= b(cid:2) = b (cid:3) (29)
2 − − 2 −
Using this result, we can calculate the energy directly by noting that
B 2
F = d3x ∂ u λ 2u 2λ∂ u 2u (30)
l z z
2 ZM (cid:26)h ∓ ∇⊥ i ± ∇⊥ (cid:27)
B 2
= d3x ∂ u λ 2u
z
2 ZM (cid:26)h ∓ ∇⊥ i (cid:27)
1
Bλ d3x (∂ u u) ∂ ( u)2
z z
± ZM (cid:26)∇⊥· ∇⊥ − 2 ∇⊥ (cid:27)
We take M to be all of R3. In the upper half plane, we pick the minus
sign,whilein the lower half planethe plussign so thatthe firstintegral
over M vanishes and the second becomes a surface integral over ∂M,
the xy plane. We see then that the energy is
b2 x2
F = BλL ∞ dx exp −
l y
16πλξ Z−∞ (2λξ′)
BL b2 λ
y
= (31)
8√2π sξ′
where,again,wehaveintroducedashort-distancez cutoff,ξ .Bychoos-
′
ing ξ = ξ/(64π2) the two expressions for the energy can be made to
′
agree.Theimportantpointis thattheenergy hasthesamedependence
on the physical parameters, b, B and λ.
The insight of Bogomoln’yi, Prasad, and Sommerfield, was to turn
thiscalculation around:by rewritingthefreeenergyas aperfectsquare
plus a total derivative we can bound the energy from below simply
throughtheboundaryconditions(Bogomol’nyi, 1976;PrasadandSom-
merfield, 1975). Moreover, when there are solutions of the lower order
differential equation in the perfect square satisfying the appropriate
boundaryconditions, theenergy arisessolely from theboundarycondi-
tions. Such states are called BPS states and their energy is sometimes
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.8
SmecticLiquidCrystals 9
called “topological” because of its sole dependence on the boundary
conditions.
Inthenextsection weconsidertherotationally invariantmodelwith
this decomposition in mind.
2. Rotational Invariance: From Here to Nonlinearity
The quadratic approximation employed in the previous section has the
drawback that it does not preserve the invariance of the free energy
under rigid rotations of the layered structure. This rotation invariance
is preserved by the higher order terms that would appear in an explicit
expansion of the energy in the deformation u(x,y,z). As it happens,
these nonlinear terms are crucial for understanding a number of physi-
cal phenomena not captured by the quadratic approximation, yet they
rendertheenergytoocomplexforastraightforwardanalyticalsolution.
2.1. BPS Solutions
In the full free energy, there are also exact minima that satisfy a BPS
flow equation (Santangelo and Kamien, 2003; Santangelo and Kamien,
2005a). We start with the nonlinear theory (12):
B
F = d3x [1 N Φ]2+λ2( N)2
2 − ·∇ ∇·
Z n o
Following the discussion in the last section, we complete the square
yielding:
B
F = d3x (1 Φ λ N)2+2λ(1 Φ ) N (32)
2 −|∇ |− ∇· −|∇ | ∇·
Z h i
While the first cross term is a total derivative ( N), the second is
∇ ·
not. However, we write
Φ N = [N( N)] Φ
|∇ |∇· ∇· ·∇
= [N( N) (N )N] Φ (33)
∇· − ·∇ ·∇
where the additional term vanishes because N2 = 1 and N Φ. Re-
||∇
casting this in terms of the geometric invariants, we have
1
Φ H = Φ[N( N) (N )N] KΦ (34)
|∇ | 2∇·{ ∇· − ·∇ }−
Thus, we can convert the cross term in (32) into a total derivative plus
a purely geometric quantity (Santangelo and Kamien, 2005a):
B
F = d3x (1 Φ λ N)2+4λΦK
2 " −|∇ |− ∇·
Z
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.9
10 RandallD.KamienandChristianD.Santangelo
+2λ N Φ[N( N) (N )N] (35)
∇·{ − ∇· − ·∇ }#
If we restrict ourselves to systems for which K = 0, we again have
the BPS decomposition. Fortunately, edge dislocations introduce layer
bending in only one direction, and hence a single edge dislocation
could be a BPS configuration, as could be a collection of parallel edge
defects. Whether or not such configurations exist depends on whether
all boundary conditions of the problem can be fulfilled by the lower
order “BPS” equation:
1 Φ = λ N (36)
−|∇ | ∇·
WehavestudiedthisequationnumericallybyswitchingtoLagrangian
coordinatesforthelayers.Intermsoflayersdescribedbyanembedding
X for layers labeled by n, this can be rewritten as the flow equation
n
a
N ∂ X = (37)
n n
· 1 2λH
−
where we have used ∂Φ/∂n = a. If we start with a layer at n = 0 with
vanishing Gaussian curvature, this flow generates new surfaces with
vanishing Gaussian curvature, enabling us to find an exact solution
for smectics with multiple parallel edge dislocations and to determine
the energy due to bending the surrounding layers. The nonlinear BPS
equation precludes, on the surface, superposition. However, in terms of
u˜ = (z Φ)/λ and the new variables α= x/√λz and β = z/λ, we find
−
an expansion around β = (Santangelo and Kamien, 2005a):
∞
1 1
0 = ∂ u˜ α(1 ∂ u˜)3∂ u˜+ (1 ∂ u˜)2 2∂2u˜ (∂ u˜)2
β − β2(1 ∂βu˜)3( − β α − β − β α
− h i
1
4(1 ∂ u˜)∂ u˜∂ ∂ u˜+2(1 ∂ u˜)2∂2u˜ + (38)
− − β α α β − β α ) O(cid:18)β2(cid:19)
Thus, for large β, we may write our solutions as a power series around
β = . The lowest order, non-trivial equation for u˜ = f(α) is indepen-
∞
dent of β and satisfies
1
α∂ f +(∂ f)2 +∂2f = 0 (39)
2 α α α
h i
Through the Hopf-Cole transformation S = exp f/2 we have
{ }
α∂ S +2∂2S = 0, (40)
α α
a linear equation.
Kamien_Santangelo.tex; 2/02/2008; 3:38; p.10