Isotropiccholesteric transition of a weakly chiral elastomer cylinder.
ABSTRACT When a chiral isotropic elastomer is brought to the lowtemperature cholesteric phase, the nematic degree of freedom tends to order and form a helix. Due to the nematoelastic coupling, this also leads to elastic deformation of the polymer network that is locally coaxial with the nematic order. However, the helical structure of nematic order is incompatible with the energetically preferred elastic deformation. The system is therefore frustrated and appropriate compromise has to be achieved between the nematic ordering and the elastic deformation. For a strongly chiral elastomer whose pitch is much smaller than the system size, this problem has been studied by Pelcovits and Meyer, as well as by Warner. In this work, we study the isotropiccholesteric transition in the weakchirality limit, where the pitch is comparable to or much larger than system size. We compare two possible solutions: a helical state as well as a doubletwist state. We find that the doubletwist state very efficiently minimizes both the elastic free energy and the chiral nematic free energy. On the other hand, the pitch of the helical state is strongly affected by the nematoelastic coupling. As a result, this state is not efficient in minimizing the chiral nematic free energy.

Article: Synthesis, structure and mesomorphism of new cholesteric monomers and smectic comblike polymers
[Show abstract] [Hide abstract]
ABSTRACT: Seven new cholesteric monomers (M1−M7) and the corresponding smectic comblike polymers containing cholesteryl groups (P1−P7) were synthesized. The chemical structures and purity were characterized by FTIR, 1H NMR, and elemental analyses. The specific optical rotations were evaluated with a polarimeter. The mesomorphism was investigated by polarizing optical microscopy, differential scanning calorimetry, thermogravimetric analysis, and Xray diffraction. The specific optical rotation values of these monomers and polymers with the same numbers of phenyl ring and terminal groups were nearly equal, however, they decreased with increasing the aryl numbers in the mesogenic core. M1−M7 showed oily streak texture and focal conic texture, or fingerprint texture, or spiral texture of cholesteric phase. P1−P7 showed the smectic A phase. The melting, clearing, and glass transition temperatures increased, and the mesophase temperature ranges widened with increasing the aryl number in the mesogenic core. Surprisingly, although the molecular structures of M6 and M7 were similar to those of M4, namely the mesogenic cores contained three phenyl rings, their phase behavior had a considerable difference, and Tm and Ti of M6 and M7 were less than those of M4. In addition, M6 and M7 also showed an obvious glass transition. TGA showed that all the polymers had good thermal stabilities.European Polymer Journal  EUR POLYM J. 01/2010; 46(3):535545.  [Show abstract] [Hide abstract]
ABSTRACT: The synthesis of two cholesteric monomers (M1 and M2), nematic crosslinking agent (C1 and C2), and the corresponding sidechain elastomers containing menthyl groups (P1 and P2 series) is described. The mesomorphism was investigated by differential scanning calorimetry, polarizing optical microscopy, Xray diffraction, and thermogravimetric analysis. The effect of the content of the different nematic crosslinking unit on the mesomorphism of the elastomers was discussed. M1 and M2 showed cholesteric and blue phases; C1 and C2 showed nematic phase. Because of the introduction of the nematic crosslinking unit, elastomers P11−P15 and P21−P25 exhibited cholesteric phase. With increasing the content of nematic crosslinking unit, T g of the obtained elastomers revealed an increased tendency, and T i of P1 series firstly increased then decreased, while T i of P2 series decreased the mesomorphism of the corresponding elastomers when the content of nematic crosslinking unit was 12mol.%. KeywordsNematic crosslinking unitMesomorphismElastomersCholestericMenthylColloid and Polymer Science 01/2010; 288(8):851858. · 2.16 Impact Factor
Page 1
arXiv:0801.3661v1 [condmat.soft] 23 Jan 2008
IsotropicCholesteric Transition of a Weakly Chiral Elastomer Cylinder
Xiangjun Xing and Aparna Baskaran
Department of Physics, Syracuse University, Syracuse, NY 13244
(Dated: February 5, 2008)
When a chiral isotropic elastomer is brought to low temperature cholesteric phase, the nematic
degree of freedom tends to order and form a helix. Due to the nematoelastic coupling, this also
leads to elastic deformation of the polymer network that is locally coaxial with the nematic order.
However, the helical structure of nematic order is incompatible with the energetically preferred
elastic deformation.The system is therefore frustrated and appropriate compromise has to be
achieved between the nematic ordering and the elastic deformation. For a strongly chiral elastomer
whose pitch is much smaller than the system size, this problem has been studied by Pelcotivs and
Meyer, as well as by Warner. In this work, we study the isotropiccholesteric transition in the weak
chirality limit, where the pitch is comparable or much larger than system size. We compare two
possible solutions: a helical state as well as a double twist state. We find that the double twist state
very efficiently minimizes both the elastic free energy and the chiral nematic free energy. On the
other hand, the pitch of the helical state is strongly affected by the nematoelastic coupling. As a
result this state is not efficient in minimizing the chiral nematic free energy.
PACS numbers: 61.30.v 61.30.Cz 61.30.Vx
I.INTRODUCTION
An isotropic chiral elastomer can be synthesized
by crosslinking a chiral nematic polymer melt in the
isotropic phase. When such a system is brought into the
low temperature cholesteric phase, the nematic degree of
freedom orders locally and tends to form a helical struc
ture. Due to the nematoelastic coupling, the polymer
network tends to stretch along the direction of the lo
cal nematic order, which continuously rotates along the
helical axis. For a system with cylindrical shape, this
leads to strain deformation which increase linearly with
the cylinder radius, as illustrated in Fig. 2B. Its elastic
energy cost is formidably high, when the cylinder radius
is much larger than the helical pitch. This frustration
due to competition between network elasticity and liquid
crystalline ordering makes it nontrivial to find the ground
state of the system in the cholesteric phase.
This problem was first studied by Pelcovits and Meyer
[1] using linear elasticity theory. In the limit of infinitely
strong chirality, it is clear that the system should first
satisfy the chirality by forming a planar helix along the
cylinder axis. On the other hand, to avoid large strain
energy, the solid can only deform uniaxially, which im
plies that the nematoelastic coupling can only be par
tially satisfied. Such a state, as illustrated in Fig. 2A and
2C, is called a planar helix state in reference [1] and a
transverse cholesteric state in reference [2]. As the chi
rality is made weaker, a conical helix state, where the
precessing director has a nonvanishing component along
the helical axis, may constitute a better solution. The
associated solid deformation as well as director pattern
for this conic state are illustrated in Fig. 2D and 2E re
spectively. Warner [2] carried out a nonlinear analysis of
the same problem using the neoclassical elasticity the
ory [3, 4]. Nontrivial dependence of the phase boundary
on the magnitude of nematic order was identified. The
multicritical point associated with the planarconic tran
sition (where the first order transition line and the second
order transition line meet) was also analyzed.
It is implicitly assumed both in reference [1] and refer
ence [2] that the pitch of the corresponding nematic liquid
crystal system (typical ≤ 0.1µm) is much smaller than
the system size, e.g. the radius of the cylinder. That
is, the elastomer is in the strong chirality limit. This
is certainly correct for many cases.
cholesteric pitch can be continuously tuned by changing
the concentration of chiral chemical groups during poly
merization. In particular it can be tuned to be compa
rable to macroscopic length scales, e.g. the system size.
This is especially true if the system has the shape of a
thin cylinder or wire. It is therefore interesting and rel
evant to study the isotropiccholesteric transition in the
weak chirality limit. In this work, we carry out a nonlin
ear elasticity analysis of this problem using variational
methods. We find that in this regime, a double twist
state has lower free energy than the usual helix director
pattern. The results obtained by Pelcovits and Meyer
[1], Warner [2], as well as the authors in this work are
summarized by the “phase diagram” of a chiral nematic
cylinder in Fig. 1.
Nevertheless, the
II. MODEL
The total free energy per unit volume of a chiral liquid
crystalline elastomer crosslinked in the isotropic phase is
given by
f = fel+ fQ, (1)
where felis the neoclassical elastic free energy
fel=1
2µTrΛTl−1Λ −3
2µ,(2)
Page 2
2
8
q a
00
q R
0
0
Experimentally
not accessible
Planar helixConical helix
1
1
8
Double twist
FIG. 1: “Phase diagram” of a chiral cylinder in nematic phase.
The parameters a0, q0, and R are defined in Sec. II. The tran
sition between planar helix state and conical helix state may
be continuous or discontinuous, while the transition between
conical helix state and double twist state is expected to be
discontinuous.
with
Λia=∂ri
∂xa
(3)
the deformation gradient matrix defined relative to the
isotropic reference state ? r = ? x, which is subject to the
incompressibility constraint:
detΛ ≡ 1.
As usual, the vector ? x coincides with the position of the
mass points in the isotropic reference state and is re
ferred to as the Lagrangian coordinate. The vector ? r
on the other hand describes the position of mass points
in the chiral nematic reference state (ground state that
minimizes the total free energy), and is usually referred
to as the Eulerian coordinate. As a general property of
nonlinear elasticity theory, it is important to distinguish
these two coordinates properly. The symmetric and posi
tive definite tensor l in the neoclassical elastic free energy
Eq. (2) is called the step length tensor [4] of the current
state, or deformed state [9] and describes the statistical
conformation of polymer chains in the current state. It
is related to the nematic order parameter Q by
l = aI − bQ.(4)
where a and b are some microscopic constants. In this
work we shall always normalize l such that it has deter
minant one. In the principle coordinate system of the
nematic order parameter, the step length tensor l can be
represented as a matrix:
l =
1
ζ
0
0 0 ζ2
0
1
ζ
0
0
= (ζ2− ζ−1)ˆ nˆ n + ζ−1I, (5)
where ζ is a monotonic increasing function of the mag
nitude of the nematic order S, whose detailed functional
form is irrelevant to our study. For an achiral nematic
elastomer, ζ turns out to be the ratio of spontaneous
stretch along the direction of the nematic director when
the system enters the nematic phase from the isotropic
phase [4]. Due to the incompressibility constraint, the
system shrinks by factor of 1/√ζ in the perpendicular
directions. Finally we note that in Eq. (2) a constant
term −3µ/2 is introduced so that the elastic free energy
vanishes in the isotropic reference state where Λ = l = I.
The second part fQin Eq. (1) is the Landaude Gennes
free energy for a chiral nematic liquid crystal. Assuming
that the nematic order is well saturated with fixed mag
nitude S in the cholesteric state, the relevant nematic
free energy is the Frank free energy for chiral nematic
liquid crystals [5, 6]:
fFrank =
1
2K1(∇ · ˆ n)2+1
1
2K3(ˆ n × ∇ × ˆ n)2+ K24∇ · (ˆ n · ∇ˆ n − ˆ n∇ · ˆ n),
2K2(ˆ n · ∇ × ˆ n − q0)2
(6)
+
where K1,K2,K3, are splay, twist, and bending constants
respectively, while q−1
0
= ℓ0is the cholesteric pitch for the
corresponding chiral nematic liquid crystal. K24 is the
saddle splay constant, which plays an important role in
the physics of blue phase [5, 7, 8]. Since the saddle splay
density is a complete differential, its volume integral can
be transformed into a surface integral by Gauss’ theorem,
and therefore scales the same as the surface anchoring of
the nematic director field, which we shall not consider
in the work. Nevertheless, it is rather straightforward to
include this surface interaction. Also, it is important to
note that all the derivatives in Eq. (6), ∇i= ∂/∂ri, are
with respect to the Eulerian coordinates, i.e. Cartesian
coordinates of mass points in their deformed states. This
is required by the liquid nature of Frank free energy: at
length scales where the Frank free energy becomes im
portant, the system is essentially a liquid. The physical
quantities of a liquid should be naturally expressed in
terms Eulerian coordinates, rather than in terms of La
grangian coordinates. To avoid confusion in notation,
we shall use ∇i = ∂/∂ri for derivative with respect to
Eulerian coordinates and use ∂a= ∂/∂xato denote the
partial derivative with respect to the Lagrangian coordi
nates.
As a first step, let us discuss the total free energy
Eq. (1) qualitatively. Within the one constant approxi
mation of the Frank free energy, and ignoring the surface
saddle splay term for a moment, there are three natu
ral length scales in this problem. q−1
rality pitch, while a0 =
?K/µ is the crossover length
scale set by the competition between network elasticity
and nematic director elasticity. The third length scale
is the radius R of the cylinder. For most liquid crys
talline elastomers, we estimate K ∼ 2 − 4 × 10−12N,
while µ ∼ 104− 106Pa. Therefore a0∼ 1 − 10nm, con
stituting the shortest length scale in our problem. On
the other hand, the chirality pitch ℓ0can vary a lot, typ
ically 0.1µm or smaller for strongly chiral materials but
0
= ℓ0 is the chi
Page 3
3
θ
DACBEGF
FIG. 2:
elastic deformation of the planar helix state at weak chirality limit, studied in Sec. IV. C: The nematic director pattern of the
planar helix state. D: The elastic deformation of the conical helix state, studied in references [1, 2]. E: The director pattern of
the conical helix state, θ is the conical angle. F: The elastic deformation of the double twist state. G: The director pattern of
the double twist state.
Elastic deformation and director pattern for various states. A: Reference cylinder in the isotropic phase. B: The
may get much larger for weakly chiral materials. In par
ticular, it can even be larger than the cylinder radius
R. Also, the regime a0/R ≫ 1 is clearly experimentally
inaccessible.
Comparing these three length scales, we are naturally
lead to the following two distinct regimes:
1. Weak chirality regime a0≪ R ≪ ℓ0.
2. Strong chirality regime a0,ℓ0≪ R.
The strong chirality limit has already been analyzed by
Pelcovits and Meyer [1], as well as by Warner [2]. It is
found that as one tunes the dimensionless ratio a0/ℓ0=
a0q0to below a critical value of order of unity, the system
goes from a planar helix director pattern to a conical
helix pattern. In this work, we shall mainly focus on
the weak chirality limit. Similar to reference [1] and [2],
we shall use variational methods, proposing two kinds of
candidate states with certain variational parameters and
minimizing the total free energy over these parameters.
III.DOUBLE TWIST OF NEMATIC DIRECTOR
AND TWIST OF CYLINDER
Consider a cylindrical block of isotropic chiral elas
tomer of radius R, aligning along the z axis. We need
to find the nematic director field ˆ n(? x) as well as the elas
tic deformation ? r(? x) relative to the isotropic reference
state that minimizes the total free energy. One possible
low energy configuration for the chiral Frank free energy
is a double twist texture, as illustrated in Fig. 2G. In
chiral nematic liquid crystals, the double twist configu
ration is energetically favorable if the saddle splay mod
ulus K24is positive and large enough [5, 7, 8]. Accord
ing to the current understanding of the blue phase, these
double twist cylinders pack into a three dimensional peri
odic structure with cubic symmetry. In liquid crystalline
elastomers, due to the nematoelastic coupling, a double
twist nematic director texture necessarily induces twist
of the cylinder, together with a uniaxial stretch λ along
the cylinder axis:
? r(? x) = Oz(αz)
1
√λ
0
0
0
1
0
√λ0
0λ
· ? x,(7)
where
Oz(αz) =
cosαz −sinαz 0
sinαzcosαz
0
0
10
(8)
is a rotation about the zaxis by an angle αz. Using
the cylindrical coordinate system, the Lagrangian coor
dinates of a point ? x are given by the triplet (ρ,φ,z):
? x =
ρ cosφ
ρ sinφ
z
.(9)
In the deformed state Eq. (7), the Eulerian coordinates
? r(? x) are given by
? r(? x) =
r cosϕ
r sinϕ
rz
=
ρ
√λcos(φ + αz)
ρ
√λsin(φ + αz)
λz
,(10)
where we have used Eq. (7) and Eq. (9). Therefore we
find
(r,ϕ,rz) = (ρ
√λ,φ + αz,λz).(11)
Page 4
4
Note that the deformed cylinder has height Lλ and radius
R/√λ.
Let ˆ eϕbe the unit vector associated with the Eulerian
cylindrical coordinate ϕ:
ˆ eϕ=
????
∂? r
∂ϕ
????
−1∂? r
∂ϕ=
−sinϕ
cosϕ
0
, (12)
In terms of the Eulerian coordinates, a double twist tex
ture of nematic director is represented:
ˆ n(? r) = ˆ ezcosθ(r) + ˆ eϕsinθ(r).
Note that the twist angle θ(r) = θ(ρ/√λ) can be equally
well represented as a function of Lagrangiancoordinate ρ.
θ(r) satisfies the boundary condition θ(0) = 0, since the
nematic director is parallel to ˆ z on the center axis of the
cylinder. On the outer surface of the cylinder r = R/√λ,
θ(R/√λ) is free to vary.
Calculation of the deformation gradient using Eq. (7)
is a trivial and tedious matter. On the other hand, by
substituting Eq. (13) into Eq. (5) we can readily calculate
the step length tensor l. Substituting these results into
Eq. (2), we find that the spatially dependent elastic free
energy density for the proposed double twist solution is
given by
(13)
fel =
µ
4ζ2λ
?ζ3− 1??λ3− α2ρ2− 1?cos(2θ)
− 2α?ζ3− 1?λ3/2ρsin(2θ)
The spatially dependent Frank free energy density can
be calculated by substituting Eq. (13) into Eq. (6), care
fully noting that all derivatives are with respect to the
Eulerian coordinates ? r. The result is
?1 + λ3ζ3+ α2ρ2ζ3+ 3ζ3+ λ3+ α2ρ2
−
?
. (14)
FFrank =
1
2K2
?1
?1
2rsin2θ +dθ
2rsin2θ +dθ
dr
?2
?
+1
2K3sin4θ
− K24sin2θ
r2
− K2q0
drr
dθ
dr,
(15)
which is identical to that for a chiral nematic liquid crys
tal in a double twist cylinder [6].
In the weak chirality limit, q0R ≪ 1, we expect θ(r)
to be small and linear in r. We can therefore expand the
elastic free energy density in terms of ρ [10] and θ(r):
fel= f0+ f2+ higher order terms, (16)
where
f0 =
µλ2
2ζ2+ζµ
µ
2ζ2λ
λ
−3
2µ,(17)
f2 =
?α2ρ2ζ3+?ζ3− 1?θ2?λ3− 1?
−2α?ζ3− 1?θλ3/2ρ
?
,(18)
are terms of order of r0and r2respectively. We shall
ignore all higher order terms in the elastic free energy.
Note that f0 is exactly the free energy density for a
monodomain nematic elastomer, with anisotropy ratio ζ,
undergoing a uniaxial deformation coaxial with the step
length tensor. Minimizing f0over λ we obtain
λ = ζ −→ f0= 0, (19)
as expected. Substituting this into Eq. (18), we find
f2→
µ
2ζ3
??ζ3− 1?θ − αζ3/2ρ
?2
, (20)
which is a complete square. Since fFrankis independent
of α, and since θ is linear in ρ as will be shown below,
Eq. (20) is minimized by
α = ζ−3/2(ζ3− 1)θ
f2 = 0.
ρ,
(21)
(22)
Hence there is no elastic free energy cost for the double
twist state up to the order of (αR)2. As we shall show
below, the parameter α is of order of q0. Hence (αR)2is
indeed a small parameter in the weak chirality limit.
Similarly, we expand the Frank free energy in terms of
θ and r. To the leading order we find
fFrank= (23)
1
2K2
?θ
r+dθ
dr
?2
− K2q0
?θ
r+dθ
dr
?
− K242θ
r
dθ
dr,
which only depends on θ. Note that the bending term
is of higher order when compared to all other terms that
we have kept.
We have to minimize the total Frank free energy den
sity
FFrank= 2πLλ
?
R
√ζ
0
fFrankrdr (24)
over θ(r) in order to determine the optimal director tex
ture. Let us define Rζ = R/√ζ in order to streamline
the notation below. Calculating the first variation of
the Frank free energy, including the boundary terms at
r = Rζ, we find
δFFrank
2πLλ
+[K2(θ(Rζ) + Rζθ′(Rζ) − Rζq0) − 2K24θ(Rζ)]δθ(Rζ).
= K2
?Rζ
0
dr
?
−rθ′′(r) − θ′(r) +θ(r)
r
?
δθ(r)
(25)
Since the twist angle θ(r) is free to vary on the bound
ary r = Rζ, we have to set both the integrand and the
boundary term to zero in order to find the minimizing
solution. This leads to the following two EulerLagrange
equations
rθ′′(r) + θ′(r) +θ(r)
r
= 0,(26)
(θ(Rζ) + Rζθ′(Rζ) − Rζq0) − 2η θ(Rζ) = 0, (27)
Page 5
5
where η = K24/K2is a dimensionless ratio. Solving these
two equations we find
θ(r) =
q0
2(1 − η)r =
q0ρ
2(1 − η)√ζ,
(28)
which explicitly shows that θ(r) is indeed linear in r. The
twist angle on the boundary is given by
θ(Rζ) =
q0R
2(1 − η)√ζ,
which serves as a small parameter controlling the validity
of the perturbative analysis. Substituting Eq. (28) into
Eq. (21) we find the parameter α given by
α =(ζ3− 1)q0
2(1 − η)ζ2,(29)
which is indeed a constant, of the same order of q0, and
independent of r. Substituting Eq. (28) into Eq. (23) we
find the Frank free energy density, which is also the total
free energy density (since the elastic free energy vanishes
at the order of (αR)2), to be given by
ftot= fFrank= −
K2ηq2
2(1 − η).
0
(30)
Summarizing Eq. (28), Eq. (29) and Eq. (30), we find
that if η < 1, our perturbative calculation is quantita
tively good in the weak chirality regime where q0R/2(1−
η)√ζ ≪ 1. The double twist state is very efficient in min
imizing both the elastic free energy and the Frank free
energy. In particular, when the saddle splay constant
K24vanishes, η = 0, and therefore the total free energy
Eq. (30) also vanishes. Note that the total free energy is
positive definite if η = 0. Hence the double twist state
is clearly the ground state, at least up to the order of
(q0R)2. By contrast, for a cholesteric liquid crystal with
K24= 0, the blue phase is clearly not the lowest energy
state, compared to the usual helical state. This shows
that unlike in the blue phase of cholesteric liquid crystal,
the saddle splay constant K24 does not play an impor
tant role in the formation of the double twist pattern in
a cholesteric elastomer. When q0R/2(1−η)√ζ is compa
rable or larger than unity, the higher order terms of the
elastic free energy and the Frank free energy can not be
neglected, and one has to minimize the full free energy
Eq. (14) and Eq. (15). Finally if η > 1, a perturbative
calculation in power of αR is qualitatively incorrect, no
matter how small the parameter q0R is. We must mini
mize the full elastic free energy Eq. (14) and Eq. (15).
IV.HELICAL STATE
In the weak chirality regime that we are interested in,
q0a0≪ q0R ≪ 1, the elastic energy scale (per unit vol
ume) µ is much larger than the chiral Frank energy scale
K q2
ence [1] and [2] can never be the ground state, as it only
partially minimizes both the Frank free energy and the
elastic free energy. There is however, another potential
candidate for the ground state, which can minimize the
elastic free energy up to the leading order. Let us con
sider a planar helix director pattern along the cylinder
axis, where the nematic director remains perpendicular
to the cylinder zaxis and rotates around this axis with
pitch α [11]:
0. Therefore the conical helix state studied in refer
ˆ n(z) = ˆ excosαz + ˆ eysinαz = Oz(αz)ˆ ex.(31)
In the following, we shall use both dyadic notation and
matrix notation of tensor quantities. The corresponding
local step length tensor is given by
l(z) = Oz(αz)l(z = 0)Oz(−αz),(32)
where
l(z = 0) =
ζ20 0
0
ζ
0 0
1
0
1
ζ
(33)
is the step length tensor at the plane z = 0. This vari
ational form of nematic director field is the same as the
planar helix sate considered in reference [1].
Due to the nematoelastic coupling, the polymer net
work prefers to stretch along the local nematic director.
This however implies that the direction of local strain
deformation rotates by an angle π/2 between two cross
sections ℓ0/4 apart along the cylinder. This leads to an
additional strain energy density µ(αR)2that is quadratic
in the cylinder radius.For a fat cylinder (or in the
strong chirality limit), αR ≫ 1 and this strain energy
is prohibitively high [12]. For a thin cylinder (or in the
weak chirality limit), however, αR ≪ 1 and this addi
tional strain energy only constitutes a perturbation to
the strain energy of the corresponding uniform deforma
tion. Nevertheless, to reduce the additional strain energy
at the order of µ(αR)2, the system can globally twist in
the direction opposite to the nematic helix. The over
all nonuniform deformation, shown in Fig. 2B, is repre
sented by the Eulerian coordinates as functions of the
Lagrangian coordinates:
? r(? x) ≡˜Λ(z) · ? x(34)
= Oz(αz)
λ
0
0
0
1
0
0
1
√λ
0
√λ
Oz(−(α + β)z) · ? x,
where β measures the global twist of the solid. We also
note that due to the additional inhomogeneous strain de
formation, the inverse pitch of the director helix α is
generically different from the inverse pitch q0of the cor
responding liquid crystal system. This will be made clear
in the calculation below.
It is important to note that the matrix˜Λ defined
through Eq. (34) explicitly depends on the coordinate
Page 6
6
z and is not the deformation gradient Λia.
ter should be obtained by taking partial derivative of
Eq. (34) with respect to Lagrangian coordinates x,y,z.
Being derived in this way, the deformation gradient ma
trix naturally satisfies the following compatibility condi
tions:
The lat
∂aΛib= ∂bΛia.
Substituting the deformation gradient and nematic or
der parameter Eq. (31) into Eq. (2), integrating over the
reference volume of the cylinder, and dividing it by the
total volume πR2L, we obtain the elastic free energy den
sity for the proposed deformation gradient as
fel[α,β,λ] = f0+ f2,
?ζ
f2=µR2
8ζ2λ
(35)
f0= µ
λ+
λ2
2ζ2
?
−3
2µ,(36)
??ζ3+ λ3?β2− 2
λ3/2− 1
?
ζ3− λ3/2??
λ3/2− 1
?
αβ
+α2?ζ3+ 1??
Note that f0is independent of the cylinder radius R and
is identical to the free energy of an achiral nematic elas
tomer undergoing uniaxial and homogeneous deforma
tion. By contrast, f2is proportional to R2, and quadratic
in α and β. f2is clearly due to the inhomogeneous de
formation. The ratio between f0and f2scales as (αR)2
as discussed earlier. The dimensionless ratio αR char
acterizes the importance of chirality in this problem. In
reference [1] and [2] this ratio is implicitly taken to be
large at the very beginning. In this work, we shall as
sume it to be a small number. More precisely we shall
assume another dimensionless ratio q0R ≪ 1. Also we
shall see below that for this proposed variational solu
tion, the Frank free energy scales the same as f2, hence
it is reasonable to first minimize f0and then the sum of
f2and fFrank. Minimization of f0leads to
?2?
. (37)
λ = ζ,
f0 → 0.
(38)
(39)
That is, the local elastic deformation is identical to that
of a homogeneous achiral nematic elastomer. Inclusion
of f2and fFrankinduces small change of λ at order of αR.
The Frank free energy density for the proposed director
pattern Eq. (31) can also be easily calculated. Again we
have to be careful with the derivatives in Eq. (6) that are
with respect to the Eulerian coordinate ? r. After some
tedious calculation and replacing λ with ζ, we find
fFrank=1
2K2α2ζ − K2αq0
?
ζ +1
2K2q2
0, (40)
which is independent of elastic constants K1, K3 and
K24.
We still need to minimize the sum of f2, given in
Eq. (35) and fFrankEq. (40), over the remaining two vari
ational parameters α,β. Since fFrankdoes not depend on
β, we minimizes f2over β and find
β =
α?ζ3/2− 1?2
2ζ3/2
R2α2?ζ3− 1?2µ
16ζ3
, (41)
f2 =
. (42)
We note that as long as ζ ?= 1, and α ?= 0, the global
spontaneous twist of solid β does not vanish. More im
portantly, unlike the double twist state, the planar helix
state considered here does cost elastic free energy at the
order of (αR)2.
We can now minimize the sum of Eq. (42) and Eq. (40)
over α, which leads to
α =
8a2
0q0ζ7/2
8a2
0ζ4+ R2(ζ3− 1)2
(43)
Remembering a0/R ≤ 10−6even for R = 1mm, and
ζ ?= 1, the first term in the denominator can be safely
ignored and α can be approximated as
α =
8a2
R2(ζ3− 1)2∼
0q0ζ7/2
?a0
R
?2
q0≪ q0. (44)
This result indicates that the helix of nematic director is
strongly resisted by the nematoelastic coupling energy
and the pitch becomes much longer than the correspond
ing value ℓ0in the nematic liquid crystals. Substituting
Eq. (44) into Eq. (42) and Eq. (40), we find
ftot≈ −a2
0
R2
4K2q2
(ζ3− 1)2+1
0ζ4
2K2q2
0≈1
2K2q2
0.(45)
The total free energy is therefore positive, in strong con
trast with the double twist state we considered in the
preceding section. The planar helix state considered here
is therefore not efficient in energy minimization. This is
clearly due to the extra elastic free energy cost Eq. (42)
caused by the nematoelastic coupling.
V.DISCUSSION AND CONCLUSION
We have shown in this work that in the weak chirality
limit q0a0≪ q0R ≪ 1, the double twist state minimizes
both the Frank free energy and the elastic free energy
up to the order of (q0R)2, and is therefore a good can
didate for the real ground state. The planar helix state,
on the other hand, is strongly influenced by the nemato
elastic coupling, with the pitch much longer than the cor
responding value in cholesteric liquid crystal. As the di
mensionless parameter q0R becomes larger than one, the
elastic energy cost due to inhomogeneous strain, scaling
as µ(q0R)2, dominates all other terms. When q0R ≫ 1
and q0a0≫ 1, the ground state is likely to be the conical
helix state with θ ≈ π/2, according to th e studies in
Page 7
7
references [1, 2]. The conical state and the double twist
state are qualitatively different, and can not be mutually
accessed in a continuous fashion. Therefore the afore
mentioned two regimes must be separated by a first or
der phase transition, located around q0R ∼ 1. Study of
this transition is technically challenging and is beyond
the scope of this work.
We acknowledge financial support from the American
Chemical Society under grant PRF 44689G7.
[1] R. A. Pelcovits and R. B. Meyer, Phys. Rev. E 66, 031706
(2002).
[2] M. Warner, Phys. Rev. E 67, 011701 (2003).
[3] M. Warner and E. M. Terentjev, Liquid Crystal Elas
tomers (Oxford University Press, 2003).
[4] M. Warner and E. Terentjev, Prog. Poly. Science 21, 853
(1996).
[5] P. de Gennes and J. Prost, The Physics of Liquid Crystals
(Clarendon Press, Oxford, 1993).
[6] O.D.Lavrentovich and M. Kleman, in Chirality in Liquid
Cyrstals, edited by H.S. Kiterow and C. Bahr (Springer,
2001).
[7] S. Meiboom, J. P. Sethna, P. W. Anderson, and W. F.
Brinkman, Phys. Rev. Lett. 46, 1216 (1981).
[8] J. P. Sethna, D. C. Wright, and N. D. Mermin, Phys.
Rev. Lett. 51, 467 (1983).
[9] Strictly speaking there is also a step length tensor l0 in
the reference preparation state, which appears in the neo
classical free energy, in front of ΛT. However, since the
reference state is isotropic, l0 is proportional to identity
tensor and therefore can be eliminated by redefinition of
l.
[10] Remembering that ρ =
√λr is proportional to r.
[11] We note that α is the helical pitch measured by the La
grangian coordinate ? x. The physical value of the pitch
however, should be defined using the Eulerian coordinate
and is therefore given by α√ζ.
[12] In the fat cylinder/strong chirality limit, the system
would prefer a uniform uniaxial deformation along the
cylinder axis instead, as studied by Pelcotivs and Meyer,
as well as by Warner.
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