# Isotropic-cholesteric transition of a weakly chiral elastomer cylinder.

**ABSTRACT** When a chiral isotropic elastomer is brought to the 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 Pelcovits and Meyer, as well as by Warner. In this work, we study the isotropic-cholesteric transition in the weak-chirality 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 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.

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**ABSTRACT:**The synthesis of two cholesteric monomers (M1 and M2), nematic crosslinking agent (C1 and C2), and the corresponding side-chain elastomers containing menthyl groups (P1 and P2 series) is described. The mesomorphism was investigated by differential scanning calorimetry, polarizing optical microscopy, X-ray 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 P1-1−P1-5 and P2-1−P2-5 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 unit-Mesomorphism-Elastomers-Cholesteric-MenthylColloid and Polymer Science 01/2010; 288(8):851-858. · 2.16 Impact Factor - SourceAvailable from: 140.138.140.197
##### Article: Synthesis, structure and mesomorphism of new cholesteric monomers and smectic comblike polymers

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**ABSTRACT:**Seven new cholesteric monomers (M-1−M-7) and the corresponding smectic comblike polymers containing cholesteryl groups (P-1−P-7) were synthesized. The chemical structures and purity were characterized by FT-IR, 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 X-ray 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. M-1−M-7 showed oily streak texture and focal conic texture, or fingerprint texture, or spiral texture of cholesteric phase. P-1−P-7 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 M-6 and M-7 were similar to those of M-4, namely the mesogenic cores contained three phenyl rings, their phase behavior had a considerable difference, and Tm and Ti of M-6 and M-7 were less than those of M-4. In addition, M-6 and M-7 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):535-545.

Page 1

arXiv:0801.3661v1 [cond-mat.soft] 23 Jan 2008

Isotropic-Cholesteric 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 nemato-elastic 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 isotropic-cholesteric 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 nemato-elastic 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 nemato-elastic 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 nemato-elastic 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 neo-classical 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 planar-conic 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 isotropic-cholesteric 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 neo-classical 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 Landau-de 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 cross-over 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 nemato-elastic 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 z-axis 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 Euler-Lagrange

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 z-axis 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 nemato-elastic 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 nemato-elastic 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 nemato-elastic 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 44689-G7.

[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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