PHYSICAL REVIEW E 74, 061802 共2006兲
Nonlocal model for nematic liquid-crystal elastomers
R. Ennis,1,2 L. C. Malacarne,1,3 P. Palffy-Muhoray,1 and M. Shelley4
1
Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA
2
Pressco Technology, Solon, Ohio 44139, USA
3
Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil
4
Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
共Received 15 December 2004; revised manuscript received 25 April 2006; published 8 December 2006兲
We have developed a fully nonlocal model to describe the dynamic behavior of nematic liquid-crystal
elastomers. The free energy, incorporating both elastic and nematic contributions, is a function of the material
displacement vector and the orientational order parameter tensor. The free energy cost of spatial variations of
these order parameters is taken into account through nonlocal interactions rather than through the use of
gradient expansions. We also give an expression for the Rayleigh dissipation function. The equations of motion
for displacement and orientational order are obtained from the free energy and the dissipation function by the
use of a Lagrangian approach. We examine the free energy and the equations of motion in the limit of
long-wavelength and small-amplitude variations of the displacement and the orientational order parameter. We
compare our results with those in the literature. If the scalar order parameter is held fixed, we recover the usual
viscoelastic theory for nematic liquid crystals.
DOI: 10.1103/PhysRevE.74.061802
PACS number共s兲: 61.41.⫹e, 61.30.⫺v
INTRODUCTION
Liquid-crystal elastomers 共LCEs兲, first proposed by de
Gennes 关1兴 and synthesized by Finkelmann et al. 关2兴, are
solid orientationally ordered rubbers. They consist of weakly
cross-linked liquid-crystal polymers with orientationally ordered side- or main-chain mesogenic units. The salient feature of LCEs is the coupling between mechanical deformation and orientational order 关3兴. As a result of this coupling,
these materials can exhibit exceptionally large responses to
external stimuli, suggesting a variety of potential applications. These range from artificial muscles 关4兴 to mechanically
tunable photonic band gap materials 关5兴 and bifocal contact
lenses 关6兴.
Much of the experimental and theoretical work on LCEs
has been carried out during the past decade. Equilibrium
properties are relatively well understood, but dynamic phenomena have not yet been thoroughly characterized. Since
these often involve large deformations and complex viscoelastic behavior, they are incompletely understood. In
nematic LCEs, both the scalar order parameter and director
orientation are coupled to the mechanical deformation. At
temperatures far from the nematic-isotropic transition, straininduced changes of the scalar order parameter are typically
small, and frequently assumed to be spatially uniform 关7兴.
However, some of the most interesting aspects of nematic
elastomers, such as changes in shape due to excitations, are
associated with nonuniform changes in the order parameter
tensor; see, e.g., Refs. 关8–10兴.
We present a nonlocal continuum description of nematic
LCEs. Our fundamental variables are the displacement vector of cross-links and the orientational order parameter tensor. We ignore chain entanglements, and allow the chains to
cross freely. We also ignore heat flow, and keep the temperature fixed. Since gradient expansions of the free energy can
lead to ill-posedness of the problem of its minimization 关11兴,
we use a nonlocal formalism, where the effects of spatial
variations of the displacement and order parameter on the
free energy are taken into account via the fully nonlocal in1539-3755/2006/74共6兲/061802共8兲
teractions. We also propose a Rayleigh dissipation function,
which is local due to the short range of the dissipative interactions. Using a Lagrangian approach, we obtain the equations of motion for the material displacement and the orientational order parameter. Finally, we examine the free energy
and the equations of motion in the limit of long-wavelength
and small-amplitude variations of displacement and orientational order parameter, and compare our results with those
existing in the literature.
I. FREE ENERGY
A. The elastic free energy
The elastic part of the free energy of either an isotropic or
a liquid-crystal elastomer in a continuum description can be
written in terms of nonlocal interactions between connected
cross-links. In our formalism,
Fel =
1
2
冕
␳o Po共r,r⬘兲g共r + R,r⬘ + R⬘兲d3r d3r⬘
共1兲
where ␳o is the cross-link density, Po is the probability density for finding the ends of a polymer chain at r and r⬘ in the
undeformed sample, and g is the interaction kernel. R共r兲
denotes the displacement of a material point from its original
position r in the undistorted sample, and the vector X = r
+ R is the Eulerian, while r is the Lagrangian coordinate. The
integrals are taken over the sample volume. In an isotropic
system, the probability density of finding the ends of a
polymer chain of length L at r and r⬘ is given by
Piso共r,r⬘兲 =
冉 冊 冉
3
2␲La
3/2
exp −
3共r␣ − r␣⬘ 兲Lo−1␣␤共r␤ − r␤⬘ 兲
2La
冊
共2兲
L−1
o
where a is the persistence or step length, and
is the
dimensionless inverse step-length tensor. Since the system is
isotropic, L−1
o is just the identity.
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©2006 The American Physical Society
PHYSICAL REVIEW E 74, 061802 共2006兲
ENNIS et al.
In the anisotropic LCE system, following Warner and
Terentjev 关3兴, we take
Po共r,r⬘兲 =
冉 冊
冉
3
2␲La
3/2
1
共det Lo兲1/2
3共r␣ − r␣⬘ 兲Lo−1␣␤共r␤ − r␤⬘ 兲
⫻exp −
2La
冊
共4兲
where k is Boltzmann’s constant, T is the temperature, and
P共r , r⬘兲 is the probability density for the coil ends, given by
P共r,r⬘兲 =
冉 冊
冉
3
2␲La
⫻exp −
3/2
1
共det L兲1/2
−1
3共r␣ − r␣⬘ 兲L␣␤
共r␤ − r␤⬘ 兲
2La
冊
共5兲
where L␣␤ is the effective dimensionless step-length tensor in
the deformed sample. In general, L␣␤ depends on the degree
of orientational order, and hence on the order parameter
tensor. The free energy then takes the form
1
Fcel = kT
2
冕
␳o Po共r,r⬘兲
冉
冉
J = det ␦␣␤ +
共3兲
where Lo␣␤ is the effective dimensionless step-length tensor
in the undeformed sample, and Lo appears in the denominator as a result of normalization. The interaction kernel is the
free energy of a Gaussian coil, given by
g共r + R,r⬘ + R⬘兲 = − kT ln P共r + R,r⬘ + R⬘兲
conserving. Rather than constructing a more detailed free
energy, capable of producing an equation of state for the
system, we therefore propose two approaches for enforcing
volume conservation. Volume conservation requires that
1
B共J − 1兲2
2
⫻exp −
3
共r⬘ + R␣⬘ − r␣ − R␣兲
2La ␣
冊
冉 冊 冕
冉
⫻exp −
⫻
冉
1
共det Lo兲1/2
3共r␣ − r␣⬘ 兲Lo−1␣␤共r␤ − r␤⬘ 兲
2La
冊
3/2
1
共det Lo兲1/2
3共r␣ − r␣⬘ 兲Lo−1␣␤共r␤ − r␤⬘ 兲
2La
⫻
冉
+
1
ln det L
2
冊冊冋
3
−1
共r⬘ + R␣⬘ − r␣ − R␣兲L␣␤
共r␤⬘ + R␤⬘ − r␤ − R␤兲
2La ␣
冊册
1
d3r⬘ + B共J − 1兲2 .
2
共10兲
B. The nematic free energy
The free energy associated with liquid-crystalline order of
the mesogenic constituents of the elastomer can be written in
terms of the nonlocal dispersion interactions using an inhomogeneous mean field approach. We assume the mesogens
are effectively cylindrically symmetric, with the symmetry
axis along the unit vector l̂. The orientation of the mesogen is
specified by the symmetric, traceless tensor
冊
3
−1
共r⬘ + R␣⬘ − r␣ − R␣兲L␣␤
共r␤⬘ + R␤⬘ − r␤ − R␤兲
2La ␣
1
+ ln det L d3r⬘ .
2
冉 冊 冕冋
冉
1
3
Fel共r兲 = kT␳o
2
2␲La
omitting an additive constant. By writing the free energy as
F = 兰F d3r, we have, explicitly for the elastic free energy
density,
3/2
共9兲
which penalizes volume change. The quantity B corresponds
to the bulk modulus and approaches infinity in the limit that
the elastomer becomes incompressible. Alternately, one may
include a potential energy density function P共R兲 in the free
energy which is determined by the requirement that Eq. 共8兲
remain satisfied. P may be regarded as an internal pressure.
Both approaches accomplish the desired result; here we
choose the first, and write for the elastic contribution to the
free energy density
共6兲
1
3
= kT␳o
2
2␲La
共8兲
The first approach to enforce this is include in the free
energy density the term
1
−1
⫻L␣␤
共r␤⬘ + R␤⬘ − r␤ − R␤兲 + ln det L d3r d3r⬘ ,
2
Fcel共r兲
冊
⳵ R␣
= 1.
⳵ r␤
1
␴␣␤ ⬅ 共3l␣l␤ − ␦␣␤兲
2
共7兲
Compressibility
The expression for the elastic free energy above describes
the entropic contributions associated with changing the
distance between the end points of polymer chains. It does
not describe van der Waals or steric interactions responsible
for the condensed phase of the system and for determining
bulk compressibility. Experimentally, it is found that most
rubbers and liquid-crystal elastomers are nearly volume
共11兲
where l␣ is the ␣th cartesian component of l̂ and ␦␣␤ the
Kronecker delta. This description occurs naturally when considering anisotropic dispersion forces 关12兴 and retains the
inversion symmetry of the nematic phase 关14兴. The
orientational order parameter Q␣␤ is
Q␣␤ ⬅ 具␴␣␤典
共12兲
where the angular brackets denote the ensemble average.
Q␣␤ is real, symmetric, and traceless; it can be written as
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PHYSICAL REVIEW E 74, 061802 共2006兲
NONLOCAL MODEL FOR NEMATIC LIQUID-CRYSTAL…
1
1
Q = − 共s − p兲L̂L̂ − 共s + p兲M̂M̂ + sN̂N̂
2
2
共13兲
Uo共X兲 ⬅ 3C
where the eigenvector N̂ associated with the largest
eigenvalue s is the nematic director;
s=
and
p=
冓
冓
1
共3共l̂ · N̂兲2 − 1兲
2
冔
冔
共15兲
J␣␤␥␦共X − X⬘兲关␦␣␤ + ␣a␴␣␤共X兲兴关␦␥␦ + ␣a␴␥␦共X⬘兲兴
E12 = − C
兩X − X⬘兩6
共16兲
where C is the interaction strength, and ␣a is the relative
polarizability anisotropy of the mesogen, given by
2共␣储 − ␣⬜兲
共2␣⬜ + ␣储兲
Z ␣Z ␥
− ␦␣␥
Z2
3
Z ␤Z ␦
− ␦ ␤␦
Z2
冊
共18兲
where Z␣ is a component of the vector Z = X⬘ − X. In
the mean field approximation, at constant mesogen density
␳m, the single-particle pseudopotential for a particle with
orientation ␴␣␤ in the bulk becomes 关12兴
共21兲
冕 冉
exp −
E共X, ␴␣␤兲
kT
冊
共22兲
d2l̂.
冕 冉
exp −
E2共r, ␴␣␤兲
kT
+ ␳mEnonlocal共r兲,
冊
d2l̂
共23兲
where the single-particle pseudopotential, as a function of
Lagrangian coordinates, is E共r , ␴␣␤兲 = E1共r兲 + E2共r , ␴␣␤兲
+ Enonlocal共r兲 and
1
E1共r兲 ⬅ − ␳mUo共r兲 + ␳mU␣␤␥␦共r兲Q␣␤共r兲Q␥␦共r兲, 共24兲
2
共25兲
3
Enonlocal共r兲 ⬅ − ␳m␣aC
2
关Q␣␤共X⬘兲 − Q␣␤共X兲兴Z␣Z␤ 3
⫻
d X⬘
兩X − X⬘兩8
冊
1
− ␳m␣2a C
2
1
− ␳m ␴␣␤共X兲 − Q␣␤共X兲 Q␥␦共X兲U␣␤␥␦共X兲
2
⫻
兩X−X⬘兩⬎d
E2共r, ␴␣␤兲 ⬅ − ␳m␴␣␤共r兲U␣␤ − ␳mU␣␤␥␦共r兲␴␣␤共r兲Q␥␦共r兲,
兩X−X⬘兩⬎d
1
− ␳m␣2a CQ␣␤共X兲
2
J␣␤␥␦共Ẑ兲 2
d X⬘ .
兩X − X⬘兩6
Fnem共r兲 = ␳mE1共r兲 − ␳mkT ln
冕
冉
冕
兩X−X⬘兩⬎d
For continuous deformations, X is a single-valued function
of r, and the free energy density can be expressed in terms of
Lagrangian coordinates. Assuming incompressibility,
兰F共r兲d3r = 兰F共X兲d3X, and the nematic free energy density is
given, explicitly, by
E共X, ␴␣␤兲 = − ␳mUo共X兲 − ␳m␴␣␤共X兲U␣␤共X兲
3
− ␳ m␣ aC
2
Z ␣Z ␤ 3
d X⬘ ,
兩X − X⬘兩8
Fnem共X兲 = − ␳mkT ln
共17兲
冊冉
共20兲
The distance of closest approach d for anisometric mesogens is a function of their relative orientation, and hence,
on the average, a function of their the nematic order parameters Q␣␤. It is well known that these steric interactions are
required for the correct description of the nematic phase,
including its deformations 关15,16兴. The nematic free energy
density at X is given, to within an additive constant, by
where ␣储 and ␣⬜ are the polarizabilities parallel and
perpendicular to l̂. The directional coupling tensor J␣␤␥␳ is
冉
1
d 3X ⬘ ,
兩X − X⬘兩6
冕
U␣␤␥␦共X兲 ⬅ ␣2a C
are the uniaxial and biaxial scalar order parameters. To retain
conventional notation, we write the nematic director N̂ = n̂.
The biaxial order parameter p can be significant in strongly
deformed systems.
The interaction energy of two mesogens located at
X = r + R and X⬘ = r⬘ + R⬘ due to London–van der Waals
dispersion interactions is of the form
J␣␤␥␦共X − X⬘兲 ⬅ 3
兩X−X⬘兩⬎d
U␣␤共X兲 ⬅ 3␣aC
共14兲
3
关共l̂ · L̂兲2 − 共l̂ · M̂兲2兴
2
␣a =
冕
冕
兩X−X⬘兩⬎d
关Q␥␦共X⬘兲 − Q␥␦共X兲兴 3
d X⬘
兩X − X⬘兩6
⫻
J␣␤␥␦共Ẑ兲
共19兲
where d = d共Q共X兲 , Q共X⬘兲兲 is the distance of closest
approach, and
冕
冕
兩Z兩⬎d
兩Z兩⬎d
关Q␣␤共r⬘兲 − Q␣␤共r兲兴Z␣Z␤ 3
d r⬘
兩Z共r,r⬘兲兩8
J␣␤␥␦共Ẑ兲
Q␣␤共r兲关Q␥␦共r⬘兲 − Q␥␦共r兲兴 3
d r⬘
兩Z共r,r⬘兲兩6
共26兲
where Z共r , r⬘兲 = X⬘ − X = r⬘ + R共r⬘兲 − r − R共r兲 as before.
For simplicity, it may be assumed that the dependence of
the distance of closest approach d does not depend on
nematic order. In this case, the nematic free energy density
simplifies to give
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PHYSICAL REVIEW E 74, 061802 共2006兲
ENNIS et al.
1
2
Fnem共r兲 = ␳2mUQ␣␤
共r兲 − ␳mkT ln
2
⫻
⫻
冕 冉
冕
冊
1
␳ mU
␴␣␤共r兲Q␣␤共r兲 d2l̂ + ␳mU
2
kT
exp
兩Z兩⬎dJ␣␤␥␦共Ẑ兲
Q␣␤共r兲关Q␥␦共r兲 − Q␥␦共r⬘兲兴 3
d r⬘
兩Z共r,r⬘兲兩6
共27兲
cous coefficients ␯␣␤␥␦ are constructed from ␦ij and Q0ij with
the appropriate symmetry. The dissipation is quadratic in the
generalized velocities. The associated change in the free energy, obtained by expanding the free energy about an equilibrium state, is quadratic in changes of the order parameters.
In analogy to the usual practice in nematodynamics, the viscous coefficients are taken to have the same structure as the
coefficients of the corresponding quadratic terms in the free
energy expansion.
The equations of motion are obtained by solving
冕冉
where
4␲ ␣2a C
U=
5 d3
共28兲
or
and we have neglected the constant term
−
␳2mUo
=−
4␲C
␳2m 3 .
d
共29兲
The first two terms in Eq. 共27兲 constitute the tensor version
of the local Maier-Saupe free energy, while the nonlocal
third term describes the free energy cost of inhomogeneities
in the order parameter field 关12兴. The two contributions Eqs.
共10兲 and 共23兲 or 共27兲 give
F共r兲 = Fel共r兲 + Fnem共r兲,
共30兲
冊
␦R
␦F
+
d3r = 0.
␦Q␣␤ ␦Q̇␣␤
␦F
共3兲
共4兲
e
e e
+ ␯␣␤␥
␦⵱␤Q̇␥␦ + ␯␣␤␥␦共⵱␤⵱␦Ṙ␥兲
␦R␣
共31兲
共32兲
共2兲
␯␣␤
␴␰Q̇␴␰ = −
共35兲
共36兲
共37兲
␦F
共3兲
e
− ␯␴␰
␣␤共⵱␰Ṙ␴兲.
␦Q␣␤
共38兲
In the above, we have relied on the Lagrangian approach
to obtain the equations of motion, which implicitly conserve
linear and angular momentum. A more direct alternative approach would have been to consider the equations of
continuity for momentum density explicitly, similar to 关13兴.
The equations of motion 共37兲 and 共38兲, together with the
free energy densities Eqs. 共10兲 and 共23兲 or 共27兲 are our main
results. They form the basis for our subsequent studies of
deformations in nematic elastomers.
III. SMALL-AMPLITUDE LONG-WAVELENGTH
APPROXIMATION
where ␳ is the mass density.
We write the local Rayleigh dissipation density function
as an expansion in terms of Q̇␣␤ and the velocity gradient
⵱␣e Ṙ␤, and have, to lowest order,
A. Free energy
1. The elastic free energy
The free energy in our model is
1 共2兲
共3兲
e
R = TṠ = ␯␣␤␥
␦Q̇␣␤Q̇␥␦ + ␯␣␤␥␦共⵱␤Ṙ␣兲Q̇␥␦
2
1 共4兲
e
e
+ ␯␣␤␥
␦共⵱␤Ṙ␣兲共⵱␦Ṙ␥兲
2
d 3r = 0
and for the nematic order parameter
where Ekin is the kinetic energy density, and F = Fel + Fnem
is the free energy density, and the Rayleigh dissipation
function R. We ignore contributions from the nematic order
parameter to the kinetic energy, and so have
1
Ekin共r兲 = ␳Ṙ2共r兲
2
␥ ␬
冊
共34兲
Evaluating derivatives gives the equations of motion for the
material points
The dynamical equations can be obtained from the generalized Lagrange’s equations. Two scalar functions must be
specified, the Lagrangian
冕
␬
冕冉
␳R̈␣ = −
II. EQUATIONS OF MOTION
共Ekin − F兲d3r
⳵R
d ⳵ Ekin ␦F
+
− ⵱␥e
dt ⳵ R˙
␦R␬
⳵ ⵜe Ṙ
and
the total free energy density for the elastomer.
L=
冕冉
冊
d ⳵ Ekin ␦F ␦R 3
+
+
d r=0
dt ⳵ R˙
␦R␬ ␦Ṙ␬
␬
F共r兲 = Fel共r兲 + Fnem共r兲
共33兲
where the spatial derivatives are with respect to Eulerian
coordinates. We assume that terms that depend on the
gradient of the order parameter tensor ⵱␬e Q̇␣␤ are small compared to the other terms present, and are ignored. The vis-
共39兲
where Fel共r兲 and Fnem共r兲 are given by Eqs. 共10兲 and 共23兲
in terms of the step-length tensor L␣␤共Q兲 and the distance of
closest approach d(Q共X兲 , Q共X⬘兲). This is the free energy
density in its most general, nonlocal form. We now consider
approximations to this free energy density to examine its
behavior and to make contact with existing results in the
literature.
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PHYSICAL REVIEW E 74, 061802 共2006兲
NONLOCAL MODEL FOR NEMATIC LIQUID-CRYSTAL…
First, we assume that the nematic order parameter Q␣␤ in
the deformed system differs only slightly from the order
0
in the undeformed system, and write
parameter Q␣␤
0
Q␣␤共r兲 = Q␣␤
共r兲 + ⌬␣␤共r兲
共40兲
where ⌬␣␤ is small compared to unity. We shall refer to ⌬␣␤
0
as the deviation of the order parameter from its value Q␣␤
共r兲
in the undeformed system. Second, we assume that the wavelengths of deformations are long compared to the relevant
interaction lengths a and 冑La, and that the integrand can be
adequately approximated by a gradient expansion to second
order. The deviation ⌬␣␤共r⬘兲 of the nematic order parameter
at location r⬘ can then be expressed in terms of its value at r;
1
⌬␣␤共r⬘兲 = ⌬␣␤共r兲 + ⌬␣␤,␥共r兲z␥ + ⌬␣␤,␥␳共r兲z␥z␳ + ¯ ,
2
共41兲
冉 冊
1
3
␮ ⬅ kT␳o
2
2␲La
We expand the free energy density to second order in ⌬ and
R, with the highest-order terms ⌬2, R2, and R⌬, respectively
and permit at most two gradients in the expression. It is
straightforward to carry out the expansion. We define ␧␣␤
⬅ R␣,␤, Q␣␤ ⬅ Q␣␤关 21 共r + r⬘兲兴, assume that L␣␤ = L␣␤共Q兲, and
take Qo␣␤ to be spatially uniform. The elastic part of the free
energy density is then given by
1
具z␣z␤典 = Lo␣␤La.
3
L␣␤ = ␦␣␤ + bQ␣␤ ,
where b is the dimensionless step-length anisotropy. L−1 and
detL can be evaluated at once using the Cayley-Hamilton
theorem which gives
冏
冉冏
冏 冏 冊
冉冏 冏 冊
3
=␮
2La ⳵ Q̄
+
␣␤ ⳵
1 ⳵2 ln det L
2 ⳵ Q̄ ⳵ Q̄
␣␤
共3兲
␮␣␤␥
␳=␮
3 ⳵ L␣−1␴
La ⳵ Q̄
具z␴z␤典 ,
冊
具z␣ ¯ z␤典 ⬅
冕
1
1
1
tr共L3兲 + 共tr L兲3 − tr共L2兲tr L,
3
6
2
共52兲
det L =
where tr L ⬅ L␣␣ is the trace of L and I is the identity.
共4兲
The term ␮␣␤␥
␳␧␣␤␧␥␳ has the expected form of the
elastic energy of an anisotropic solid; in our model, the
anisotropy arises through the nematic order parameter Q0 of
the undeformed sample.
We expand both the local and nonlocal parts of the
nematic part of the free energy density in Eq. 共23兲 similarly,
and obtain
共45兲
共53兲
Terms from Eq. 共23兲 containing ␧␣␤ are of higher order in
共2,nem兲
wave number and do not appear. The coefficients ␮␣␤␥
␳ and
共nem兲
K␣␤␬␥␳␶ are functions of Qo; they can be determined for any
particular choice of the distance of closest approach d. For
example, if d does not depend on nematic order, the nematic
free energy density is given by Eq. 共27兲, and it follows that
共2,nem兲
␮␣␤␥
␳ ⌬␣␤⌬␥␳ = A⌬␣␤⌬␤␣
¯ =Q0
Q
and
3 −1
=␮
L 具z␤z␳典 ,
La o␣␥
where
and
冉
共44兲
¯ =Q0
␥␳ Q
␥␳
共4兲
␮␣␤␥
␳
¯ =Q0
Q
,
共51兲
1 共2,nem兲
1 共nem兲
Fnem = ␮␣␤␥
␳ ⌬␣␤⌬␥␳ + K␣␤␬␥␳␶⌬␣␤,␬⌬␥␳,␶ .
2
2
具z␴z␰典
Q̄␥␳
1
L2 − L tr L + 关共tr L兲2 − tr共L2兲兴I
2
L−1 =
1
1
1
− tr共L2兲tr L + tr共L3兲 + 共tr L兲3
2
3
6
and
where the coefficients are
共2,el兲
␮␣␤␥
␳
共50兲
2. The nematic free energy
共43兲
−1
⳵2L␴␰
共49兲
Next, we assume a simple dependence of the step-length
tensor on the nematic order parameter,
1 共2,el兲
共3兲
Fel共r兲 = ␮␣␤␥
␳⌬␣␤共r兲⌬␥␳共r兲 + ␮␣␤␥␳␧␣␤⌬␥␳
2
1 共4兲
+ ␮␣␤␥
␳␧␣␤␧␥␳
2
共48兲
and we have omitted an additive constant. Incompressibility
requires that ␧␣␣ ⬅ 0. It is straightforward to show that,
sufficiently far from any surface,
where z␣ ⬅ r␣⬘ − r␣ is the distance in Lagrangian coordinates,
and the comma denotes differentiation so that
f ␣,␤ ⬅ ⳵ f ␣ / ⳵x␤. The displacement R共r⬘兲 can be similarly
expanded in terms of its value at r to give
1
R␣⬘ 共r⬘兲 = R␣共r兲 + R␣,␥共r兲z␥ + R␣,␥␳共r兲z␥z␳ + ¯ . 共42兲
2
3/2
共46兲
共nem兲
K␣␤␬␥
␳␶⌬␣␤,␬⌬␥␳,␶ = B
冉
12
11
⌬␣␤,␥⌬␣␤,␥ − ⌬␣␤,␤⌬␣␥,␥
14
14
共54兲
冊
共55兲
Po共z兲z␣ ¯ z␤d z
3
共47兲
where A and B are positive constants 关12兴.
The total free energy density for the nematic elastomer,
expanded to second order, is
061802-5
PHYSICAL REVIEW E 74, 061802 共2006兲
ENNIS et al.
1 共2兲
1 共4兲
共3兲
F = ␮␣␤␥
␳⌬␣␤⌬␥␳ + ␮␣␤␥␳␧␣␤⌬␥␳ + ␮␣␤␥␳␧␣␤␧␥␳
2
2
1
1
1
+ K␣␤␬␥␳␶⌬␣␤,␬⌬␥␳,␶ + B共␧␣␣兲2 + C共⌬␣␣兲2 共56兲
2
2
2
共2,nem兲
共nem兲
共2,el兲
共2兲
where ␮␣␤␥
␳ ⬅ ␮␣␤␥␳ + ␮␣␤␥␳ and K␣␤␬␥␳␶ ⬅ K␣␤␬␥␳␶. Since
there are bilinear terms, imposing a strain would cause the
system to relax, via the coupling described by the second
term on the right-hand side, to a nonvanishing order parameter deviation. The quantities B and C may be regarded as
Lagrange multipliers enforcing the constraints of constant
volume and traceless nematic order parameter. The free
energy density in Eq. 共56兲 is positive definite.
Equation 共56兲 shows that the presence of the cross-linked
elastomer matrix changes the nematic-isotropic transition
temperature, but does not, to lowest order, contribute to the
elastic constants of the nematic. A change in the transition
temperature due to contributions from the elastomer has been
pointed out in Ref. 关17兴, where only a scalar nematic order
parameter was considered.
In our second-order gradient approximation, we have
omitted surface terms. It is well known that surface terms
that arise from transforming second gradient terms to
squared first gradient terms can lead to a free energy that is
unbounded from below, and to an ill-posed minimization
problem 关11兴. This can be avoided by using the fully nonlocal free energy to obtain the dynamical equations, and only
then perform the gradient expansion 关12兴.
3. Limiting cases
If the reference state is isotropic, on enforcing incompressibility, we have for the uniform contribution of the
elastic free energy
␮
共b⌬␥␳ − 2␧␥S␳兲2 ,
Fun
el =
4
共57兲
omitting a surface term. This leads to a soft 共spontaneous兲
deformation when ␧␥S␳ = 21 b⌬␥␳, where ␧␥S␳ is the symmetric
part of the deformation tensor ␧␥␳.
It is straightforward to show that Eq. 共43兲 contains the
usual continuum elastic free energy density in terms of the
nematic director 关3兴. We take the scalar order parameter s to
be constant, and writing n = no + ␦n obtain
3
⌬␣␤ = s0共n␣0 ␦n␤ + n␤␦n␣ + ␦n␣␦n␤兲.
2
共58兲
We arrive at the usual free energy by considering terms in
Eq. 共43兲 to second order in the director deviation ␦n̂ and
enforcing ⌬␣␣ = 0.
lengths of deformations are long compared to the relevant
interaction lengths a and 冑La, and that the deviation of the
order parameter is much smaller than unity. We also assume
here that the displacement R is small compared to r, and we
therefore do not distinguish between gradients in the Euler
and Lagrangian coordinates. The equations of motion 共37兲
and 共38兲 become
␳R̈␣ =
冉
⳵F
⳵
共3兲 ˙
共4兲
+ ␯␣␤␥
␭⌬␥␭ + ␯␣␤␥␭Ṙ␥,␭
⳵ x ␤ ⳵ R ␣,␤
冊
共59兲
and
共2兲 ˙
␯␣␤␥
␭⌬ ␥␭ = −
⳵F
⳵ ⌬␣␤
+
⳵ ⳵F
− ␯␥共3兲␭␣␤Ṙ␥,␭
⳵ x␥ ⳵ ⌬␣␤,␥
共60兲
with F given by Eq. 共56兲. We assume the viscous coefficients
共k兲
共k兲
␯␣␤␥
␭ to be proportional to the coupling tensors ␮␣␤␥␭ in Eq.
共56兲, as is commonly done in nematodynamics. These can be
written in terms of s0 and n̂0, the scalar nematic order parameter and director in the undeformed sample, as shown in
the Appendix.
1. Constant scalar nematic order parameter
For the case of constant scalar nematic order parameter,
we write the order parameter deviation as
3
⌬␣␤ = s0共n␣0 ␦n␤ + n␤0 ␦n␣ + ␦n␣␦n␤兲
2
共61兲
since ⌬␣␣ = 0 and n␣n␣ = 1. From the expressions for the viscosity tensors in the Appendix , we obtain the viscous stress
␴␥⬘ ␳ =
⳵R
⳵ ␧˙ ␳␥
S 0
= ␯1n␥0 n␳0共n␣0 ␧˙ ␣␤
n␤兲 + ␯2n␥0 N␳ + ␯3N␥n␳0 + ␯4␧˙ ␳S␥
S 0
+ ␯5n␥0 ␧˙ ␳S␣n␣0 + ␯6n␣0 ␧˙ ␣␥
n␳
共62兲
and viscous director field
g␥⬘ = −
⳵R
⳵ ␦ṅ␥
S
= ␭1N␥ + ␭2n␣0 ␧˙ ␣␥
共63兲
A 0
S
A
where N␥ ⬅ 共␦ṅ␥ − ␧˙ ␥␣
n␣兲, and ␧␣␤
and ␧␣␤
are the symmetric
and antisymmetric parts of ␧␣␤. The viscous stress is in the
Ericksen-Leslie form 关18,19兴, and the coefficients are given
by ␯1 = −␯共r − 1兲2 / r, ␯2 = ␯共1 − r兲, ␯3 = ␯共1 − r兲 / r, ␯4 = 2␯,
␯5 = ␯共r − 1兲, and ␯6 = ␯共1 − r兲 / r. The parameter r ⬅ l储 / l⬜is a
measure of the anisotropy of the polymer chain. The
coefficients obey the Parodi relations ␯2 + ␯3 = ␯6 − ␯5,
␭1 = ␯2 − ␯3 and ␭2 = ␯5 − ␯6. The dissipation function is positive definite 关19兴 because for ␯ ⬎ 0 the following inequalities
are satisfied: ␯4 ⱖ 0, 2␯4 + ␯5 + ␯6 ⱖ 0, 2␯1 + 3␯4 + 2␯5 + 2␯6
ⱖ 0, and −4␭1共2␯4 + ␯5 + ␯6兲 ⱖ 共␯2 + ␯3 − ␭2兲2.
2. Small Q0
B. Equations of motion
The general equations of motion are given, in terms of the
dissipation and the nonlocal free energy, in Eqs. 共37兲 and
共38兲. We now consider approximations to these to examine
the behavior of the system and to make contact with existing
results in the literature. As before, we assume that the wave-
We next consider the case when the nematic order parameter Q0 in the undeformed sample is small, and expand to
second order in Q0 and ⌬. The viscosity coefficients become
061802-6
␯
共2兲
␯␣␤
␴␰ = 共␦␣␰␦␴␤ + ␦␣␴␦␤␰兲,
2
共64兲
PHYSICAL REVIEW E 74, 061802 共2006兲
NONLOCAL MODEL FOR NEMATIC LIQUID-CRYSTAL…
␯
共3兲
0
0
␯␣␤␥
␳ = − 共␦␣␥␦␤␳ + ␦␤␥␦␣␳兲 + ␯共Q␣␥␦␤␳ + ␦␤␥Q␣␳兲,
2
共65兲
which gives the dynamical equation for the nematic order
parameter
冉
冊
⳵F
1 ⳵F
A
0
0 A
S
− ⵱␬
+ ␧˙ ␣␤
+ 共␧˙ ␣␬
Q␬␤
− Q␣␬
␧˙ ␬␤兲
⌬˙ ␣␤ = −
⳵ ⵜ␬⌬␣␤
␯ ⳵ ⌬␣␤
0 S
S
0
␧˙ ␬␤ + ␧˙ ␣␬
Q␬␤
兲.
− 共Q␣␬
共66兲
This is in agreement with the results of Olmsted and Goldbart 关20兴 for pure nematics, except for the last term in Eq.
共66兲, which they did not explicitly give since they regarded it
as a higher-order reactive term. This type of term, and
higher-order terms in Q0, appear in a nonlinear expansion in
their formalism. A similar result in pure nematic theory is
obtained using the Poisson-bracket approach by Stark and
Lubensky 关21兴 and by Pleiner et al. 关22兴 in a polymeric
system.
3. Spatially uniform strain
Finally, we consider the case when a uniform strain is
applied to the sample, and the nematic tensor order parameter is spatially uniform. The equilibrium configuration can
be obtained from the stationary solution of Eq. 共60兲,
⳵F / ⳵⌬␣␤ = 0, which gives
共2兲
共3兲
␮␣␤␥
␳⌬␥␳ + ␮␥␳␣␤␧␥␳ = 0
共67兲
where ⌬␣␣ = 0. This expression, together with the explicit
expressions for the coefficients, shows how the order parameter tensor responds to applied strain. If the strain is along
the initial director direction, a change in the scalar order
parameter results. Strain applied perpendicular to the director
leads to biaxiality, but in this small deformation approximation the director remains constant. Shearing the sample gives
rise to off-diagonal terms in the order parameter tensor,
implying director rotation.
We have examined the behavior in the limit of longwavelength and small-amplitude variations of the displacement and the orientational order parameter. In this limit, the
predictions of our model are in agreement with existing results in the literature. To facilitate the comparison, we have
carried out a gradient expansion of the free energy density
instead of the dynamical equations in the long-wavelength
limit, and ignored surface contributions when converting
second gradient to squared gradient terms. Incorporating
these surface terms would lead to an ill-posed minimization
problem 关11,12兴. The formally correct approach is to first
derive the dynamical equations in fully nonlocal form, as we
have done, and only then take the long-wavelength limit
关12兴. More detailed work, including numerical simulations,
is currently under way.
ACKNOWLEDGMENTS
Support for this work was provided by the National Science Foundation 共NSF兲 and the Brazilian National Research
Council 共CNPq兲. We are grateful to M. Warner for his critical
comments.
APPENDIX
共k兲
The viscous coefficients ␯␣␤␥
␭ are proportional to the cou共k兲
pling tensors ␮␣␤␥␭ in Eq. 共56兲. These can be written in terms
of s0 and n̂0, the scalar nematic order parameter and director
in the undeformed sample. They become, to second order in
Qo, assuming Lij = ␦ij + bQo and ignoring terms which do not
contribute,
0 0
0 0
共2,el兲
共2兲
共2兲
␮␣␤␥
␭ = ␣ 共␦␣␥␦␤␭ + ␦␣␭␦␤␥兲 + ␤ 共␦␣␥n␤n␭ + ␦␤␥n␣n␭
+ ␦␣␭n␤0 n␥0 + ␦␤␭n␣0 n␥0 兲 + ␥共2兲n␣0 n␤0 n␥0 n␭0 ,
共3兲
0 0
共3兲
共3兲 0 0
␮␣␤␥
␭ = ␣ 共␦␣␥␦␤␭ + ␦␣␭␦␤␥兲 + ␤ 共n␣n␥␦␤␭ + n␣n␭␦␤␥兲
+ ␥共3兲共␦␣␤ − n␣0 n␤0 兲␦␥␭ ,
冋
In this paper, we propose a nonlocal continuum model to
describe the dynamics of nonuniform deformations in nematic elastomers in terms of the material displacement vector
and the nematic order parameter tensor. We have derived
expressions for the nonlocal free energy including contributions from both elastic and nematic interactions. Our formalism for elastic interactions is based on the work of Warner
and Terentjev 关3兴, but note that other general approaches
have been developed 关23,24兴. We have obtained equations of
motion for the material displacement and the order parameter
from the free energy and the Rayleigh dissipation function
via a Lagrangian formalism. The equations of motion are
expected to be valid in the case of large and inhomogeneous
deformations and strains. We point out nonetheless that
discrepancies can exist between results obtained via
Lagrangian approaches and other methods 关25,26兴.
+␮
共A2兲
冉 冊
冉 冊冉 冊
共4兲
␮␣␤␥
␭ = ␮ ␦␣␥␦␤␭ +
IV. SUMMARY
共A1兲
l⬜
−1
l储
冉 冊
l⬜
l储
− 1 n␣0 n␥0 ␦␤␭ +
− 1 n␤0 n␭0 ␦␣␥
l储
l⬜
l储
− 1 n␣0 n␤0 n␥0 n␭0 ,
l⬜
册
共A3兲
where we define
l储 ⬅ 1 + bs0 ,
共A4兲
l⬜ ⬅ 1 − bs0/2,
共A5兲
and the scalar coefficients in Eqs. 共A1兲 and 共A2兲 are given by
061802-7
␣共2兲 ⬅
␤共2兲 ⬅ −
␮ b2共1 + bs0兲 ␮b2 1
=
2 ,
4
4 l⬜
det Lo
冉
共A6兲
冊
1
␮ 3 3 0 ␮b2 1
bs =
− 2 ,
4 l ⬜l 储 l ⬜
det Lo 8
共A7兲
PHYSICAL REVIEW E 74, 061802 共2006兲
ENNIS et al.
␥共2兲 ⬅
冉 冊
冊
␮ 9 2 共bs0兲2 ␮b2 1 1
b
=
−
2 l⬜ l储
det Lo 8 1 + bs0
冉
2
,
␤共3兲 ⬅
共A8兲
冉
冊 冉 冊
␮ 3 2 0
␮b 1 1
bs0
b s 1−
=
−
. 共A10兲
2
det Lo 4
2 l⬜ l储
共bs0兲2
bs0
␮
␮b
b
+b
− −b
=−
, 共A9兲
4
4
det Lo
2
2l⬜
The corresponding viscosity tensors are given by replacing ␮
with ␯ ⬅ c␮ in the above equations, where c is a constant.
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