Christina Papenfußa, Jozsef Verhásb and Wolfgang Muschika
a Technische Universität Berlin, Institut für Theoretische Physik, Hardenbergstraße 36,
D-10623 Berlin
b Technical University of Budapest, Institute of Physics, 1521 Budapest, Hungary
Z. Naturforsch. 50a, 795-804 (1995); received June 3, 1994
A continuum theory of a biaxial nematic phase in liquid crystals is presented. The liquid crys
talline ordering is described by a field of three unit vectors. Because these are macroscopic directors,
the present approach is a generalization of the Ericksen-Leslie theory to biaxial nematics. The
applied tools from nonequilibrium thermodynamics are classical Linear Irreversible Thermodynam
ics and Gyarmati's variational principle.
Key words: Liquid crystals; biaxial nematics; Linear Irreversible Thermodynamics; Onsager reci
procity relations; Gyarmati's variational principle.
A Sim plified T herm odynam ic T heo ry fo r B ia x ial N em atics
1. Introduction
It is known since a long time, that in certain mate
rials a uniaxial nematic phase exits at temperatures
below the clearing point. At higher temperatures the
material is an isotropic liquid. All liquid crystalline
phases exhibit no long range positional order in three
dimensions, in contrast to the crystalline phase. They
are distinguished from the isotropic phase by the fact
that there is long range order of molecular orienta
tions. At lower temperatures there may exist smectic
liquid crystalline phases, which exhibit a one dimen
sional periodicity of the mass density.
Liquid crystals consist of formanisotropic mole
cules. If the effective molecular shape is uniaxial, the
particle orientation can be described by the orienta
tion of one axis, the microscopic director (see Fig
ure 1). The different molecular orientations give rise to
an orientation distribution function, defined on the
unit sphere S2. This distribution function can be uni
axial or biaxial (see Figure 1), i.e. there may exist a
symmetry axis of rotation for the orientation distribu
tion function or not. In both cases the distribution
function can be expanded in terms of cartesian tensors
[1]. Because of the head-tail symmetry of the molecules
the orientation distribution function is symmetric un
der inversion. Consequently odd order tensors do not
enter into the expansion. The first anisotropic mo
ment of the orientation distribution function is the
second order alignment tensors a, a symmetric trace-
Reprint requests to Prof. W. Muschik.
less tensor. In the case of uniaxial symmetry of the
orientation distribution function the alignment tensor
has two equal eigenvalues and can be written as
a = fl(/i/i-1/3 5). (1.1)
The vector n is called macroscopic director and a is
the scalar order parameter. If the orientation distribu
tion function is uniaxial, the macroscopic director is
given by the symmetry axis of the orientation distribu
tion function.
In most cases the nematic phase is observed to be
uniaxial (i.e. the orientation distribution function is
uniaxial). The possibility of a biaxial nematic phase
has been predicted more than twenty years ago [2], [3].
Later a biaxial nematic phase has been found in a
mixture [4] as well as in a pure system [5]. The present
paper is concerned with the theoretical description of
such biaxial nematic phases. The deviation of the
molecular shape from rotation symmetry will not be
taken into account, because the orientation of biaxial
particles cannot be described by a single unit vector
(the microscopic director). Rotation matrices are
needed, or equivalently unit vectors in four dimen
sional space [6]. This will not be done here.
If the phase is biaxial, the second order alignment
tensor does not have the simple form of (1.1), but three
eigenvalues are different. The corresponding eigenvec
tors will be denoted by n, I and m.
The continuum theory of uniaxial nematics with a
(macroscopic) director of unit length has been devel
oped by Oseen [7], Franck [8], Ericksen [9] and Leslie
[10]. The Ericksen-Leslie-approach has been applied
0932-0784 / 95 / 0800-0737 $ 06.00 © - Verlag der Zeitschrift für Naturforschung, D-72027 Tübingen
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
796
Rotation
symmetric
molecules
A
microscopic
director
V
ii
I \
/
C. Papenfuß et al. • A Simplified Thermodynamic Theory for Biaxial Nematics
a) . b)
microscopic
directors orientation distribution function
Fig. 1. The orientation of rotation symmetrical molecules is described by a microscopic director. The microscopic directors
in an volume element give rise to an orientation distribution function which can be a) uniaxial or b) biaxial.
to biaxial nematics too [11], yielding the differential
equations governing elastic and flow behaviour. The
constitutive equations derived using the Ericksen-
Leslie-theory have been compared with the theory
developed by Saupe [12]. In [13], the linearized hydro-
dynamic equations have been used to predict the at
tenuation of low frequency sound waves. The hydro
dynamics of biaxial nematics in an external magnetic
field have been treated in [14] in terms of thermody-
namical forces and fluxes. In a recent work [15] the
phenomenological theory of the viscosity of biaxial
liquid crystals has been developed, and experimental
geometries which allow the different coefficients to be
measured have been discussed. The linear combina
tions of the viscosity coefficients which should be ac
cessible to experiments, have also been discussed in
[15]. The particles are described as ellipsoids. By the
affine transformation model all viscosity coefficients
are expressed in terms of the two viscosities of an
isotropic reference system and the axis ratios of the
particles. Applying the theory to the problem of flow
alignment, the alignment angle has been expressed in
terms of the axis ratio of the ellipsoidal particles. All
these aproaches, including the presented here, cannot
describe variations of the degree of order in the liquid
crystal.
The present work is a generalisation to biaxial ne
matics of an earlier publication [16] on uniaxial ne
matic liquid crystals. Gyarmati's variational principle
[17] is applied to deduce the equations of equilibrium
as well as those of motion. The hypothesis of local
equilibrium is accepted. For the methods of linear
Onsager theory the book of the De Groot and Mazur
[18] is referred to.
2. The Choice of the Relevant Variables and the
Entropy of Biaxial Nematic Liquid Crystals
To handle the derivatives of the constitutive func
tions it is necessary to chose the relevant variables on
which the constitutive functions may depend. In the
case of a biaxial liquid crystal the state space includes
in addition to the internal energy u, the velocity v, the
gradient of the velocity Vv and the specific polarisa
tion P some other variables connected with the align
ment tensor of second order a. We consider only rigid
rotations of the orientational distribution function
mentioned in the introduction. In this case the eigen
values of the alignment tensor are constant and the
eigenvectors /, m and n can be taken as additional
variables. With this choice of variables we have to take
into account the constraints
m m = 1, n ■ n = 1, 1 1 = 1,
m • n = 0, m 1=0, n -1=0. (2.1)
m, n, and / will be called directors in analogy to the
theory of uniaxial nematic liquid crystals. Instead of
three vectors with six constraints it is more convenient
to use a 3 x 3-matrix Q as an element of the state
space. The orthogonal mapping Q is defined by the
equation
a(r) = Q(r,r°) • a{r°) • QT(r,r°), (2.2)
where a (r°) is the alignment tensor at some reference
point r°. If the the coordinate system in the reference
point is chosen such that the eigenvectors l{r°), m(r°)
and n (r°) are along the x-, y- and z-axis, the rows of
the matrix Q(r,r°) are the eigenvectors /(r), m(r) and
n(r) at the point r with the coordinates xt, i= 1, 2, 3.
The gradient of Q is a relevant variable, too, de
scribing the inhomogeneity of the orientation. We de
fine
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
C. Papenfuß et al. • A Simplified Thermodynamic Theory for Biaxial Nematics 797
(2.3)
and the angular distortion mapping
O: = ( 0 ,0 ,0 ,). (2.4)
The vector invariant w (F) of a tensor F occuring in
(2.3) is defined by
i (F - FT) = :» (F) x 5 «-> i (Fkl - Flk) =: ekjl w,,
(2.5)
where FT denotes the transposed of F. The above
definition (2.3), cast into rectangular components,
reads
n - I
2 M dxk (2.6)
where eirl stands for the components of the total anti
symmetric tensor of third order. From here the rela
tion
W m = e o 0
0 -tisrUskUrj (2.7)
follows for the cartesian components of the gradient of
Q (see Appendix C).
Finally, the relevant variables are
Z = (u,v,Vv,P,Q,0) (2.8)
or equivalently
Z' = (u, v, Vv, P, /, m,n, V/, Vm, V/i) (2.9)
with the constraints (2.1). For the gradients of the
directors, the relations
V/ = — / x O, Vm= —m x O, V /i= -n x O
hold. <210>
The requirement that the form of the constitutive
equations should be independent of the motion of the
observer is expressed in the principle of objectivity.
Mathematically it means that a constitutive quantity
transforms like a tensor under Galilei-transforma
tions [19]. For the scalar entropy this reads
s (u, v, Vr, P,Q ,0)
= s(u,R r,R Vt> RT,R P,R 0,R O RT)
(2.11)
with any orthogonal tensor R. In addition, the form of
the constitutive equations is restricted by the material
symmetry. For the biaxial nematic liquid crystal the
three directors have the mirror symmetry for the three
planes perpendicular to the directors /, m, and n, re
spectively:
I — I, m m, /! — n (2.12)
are symmetry operations. If we use a cartesian frame
with the axes in the direction of the directors, the
transformations read
x1 —> xlt x2->x2, x3->x3,
Xi ^ Xj, X2 ^ X2, X3 ^ X3,
Xj —► Xj , x2 —► x2, x3 -*■ x3. (2.13)
For the cholesteric phase of a chiral biaxial liquid
crystal the material symmetry is lowered to the 180°
rotations around the three directors:
*1 Xj, x2—> x2, x3—> *3»
xt Xj, x2 x2, *3
*1 Xj, x2 x2, x3—*- x 3. (2.14)
One of the most important constitutive equations is
the one for the entropy. Following Oseen [7] and
Frank [5], the entropy is approximated by a quadratic
polynomial of the components of the angular distor
tion mapping O and the polarisation P, which is in
variant under the symmetry operations (2.13):
1
2S1°111 ou on —i n n
2 i2222 Umm Umm
1
253333 ®nn Onn ~
' \ Sl°122 On Omm
1
2s 7 ^ o uonn- i n n
2 2233 mm unn
1
2Sl°212 Olm Olm "_ I c°° n n
2 1221 Ulm ml
1
2S2°121 Omi Oml -_ I c°° n o
2 ^ 1313 ulnuln
1
2^ 1*3 31 Oln Oni —i n n
1
2S2°323 Omn Omn~~ 2 52332 Omn Onm
1
2S3°232 Onm Onm_ I cPP P2
2 111
1
2oPP p2 _ I ~pp
ä22 m 2 33e(2.15)
For a biaxial cholesteric phase, terms linear in Ou,
Omm, and Om also enter into the entropy function
which are not allowd by the symmetry transforma
tions (2.13) but by the transformations (2.14).
Here we have chosen the vectors /, m, and n as the
basis of the coordinate system, e.g.
Olm:= l O m, P,:=P I, (2.16)
and analogously for the other components. The abso
lut term s0 and all the coefficients depend on u, v and
V». In (2.15) terms bilinear in the angular distortion
and the polarisation are neglected, i.e. we do not take
into account flexoelectric effects. The coefficients sf£,
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
798 C. Papenfuß et al. • A Simplified Thermodynamic Theory for Biaxial Nematics 798
are proportional to 15 independent elastic constants.
In terms of Q instead of the three directors, the poly
nomial of second degree in the components of O and
P, which is invariant under the symmetry transforma
tions, is
So - K iu ( Q T OJ2°222 (QT
-2 53333 (QT OQ)i3
-K l2 2 (QT oQ)n(QT oQ)22
-K I 33(Qt oQ)u(QT oQ)33
-i 52233 (QT oQ)22(Qt oQ)33
-K212(QT oQ)I2(Qt oQ)l2
-2 SU21 (QT oQ)I2(Qt oQ)2I
-K l21 (QT oQ)2I(Qt oQ)21
-K 3I3(QT oQ)I3(Qt oQ)l3
-2S13 31 (QT oQ)I3(Qt oQ)31
-K l3 l(Q T oQ)3I(Qt oQ)31
-2 S2°323 (QT oQ)23(Qt oQ)23
-2 52332 (QT oQ)23(Qt oQ)32
-\ S3232 (QT oQ)32(Qt oQ)32
-i SPP(QT-P)2- ■P)2
2
-
It has been used that
e ^ O e 2 = e ° (Q T O Q )e ° ,
el ,e2e{l,m,n}
(2.17)
(2.18)
,oo eoo „oo
'1111 •^1122 1133
,oo coo coo
'1122 ■^2222 ^2233
,oo coo „oo
'1133 ^2233
3. Balance Equations and Dissipation Inequality
The following notations are used: q is the mass
density, v the velocity, t the stress tensor and / the
force density exerted by external fields, r the position
vector, s the internal angular momentum (spin), II the
couple stress tensor, m the couple density, Jq the heat
current density, u the internal energy, Q the angular
velocity of the directors, E the electric field and P the
polarisation density.
The conservation laws of mass,
with the eigenvectors 1°, m° and n° of the alignment
tensor a(r°) in the reference point r°, 1° is chosen
along the xt axis, m° along the x2 axis and n° along
the x3 axis. Again flexoelectric effects have been ne
glected. In equilibrium P =0 with no external electric
field, and O = 0 for a nonchiral liquid crystal if there
are no boundary conditions inducing a distortion. As
for an adiabatic system the entropy is maximal in
equilibrium, the following stability conditions hold:
s™u > 0, ie{l,2,3},
sljij > 0, i,je{ 1,2,3},
Ue{l,2,3},
>0,
do
— + g V- r = 0,
and momentum,
(3.1)
(3.2)
sftjs'gji > (s^)2, i,je {1,2,3},
s£p > 0, ie{l,2,3}. (2.19)
are the same as for a simple fluid [20]. The balance
equation of the angular momentum has the form
d
^ - ( r x » + j) = V- (rx t + II) + e rx /+ ß m ,
dt
(3.3)
and the balance equation for the total energy is as
sumed to be
d / v1 1 \
< * { u + 2 + 2 " a ) + V-J'
d P
= v-(®-t + fl-n) + e /-r + e £ -— . (3.4)
dt
The last term on the right hand side gives the energy
supply from the electric field. The nonconvective en
ergy flux is the sum of a mechanical part Jq and an
electromagnetic part. Just so the energy and the stress
tensor can be decomposed into mechanical and elec
tromagnetic parts. It is shown in Appendix A that for
an incompressible fluid in an electric field, (3.4) is cor
rect for the energy balance if the model of Grot and
Eringen [21] is accepted.
In (3.3) and (3.4) the terms with the couple stress
tensor and the couple density and the spin are in
cluded due to the internal structure of the liquid crys
tal.
These are the balance equations for a micropolar
fluid [19]. They have been derived within an align-
menttensor theory of liquid crystals, which has been
founded statistically [22]. The angular velocity ß has
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
C. Papenfuß et al. • A Simplified Thermodynamic Theory for Biaxial Nematics 799
been defined so that
d/ _ . dm _ dn _
— = ß x /, — = ß x m , — = ß x n , (3.5)
dt dt dt
which is possible with one vector ß because we con
sider only rigid rotations of the angular distribution
function. The balance equation of the angular mo
mentum is equivalent to the spin balance
ds
2w(t) + V 11 + gm = — ,
d t (3.6)
With the balance equation (3.2) of linear momentum
and (3.6) the internal energy balance is obtained from
the conservation law of total energy (3.4):
^ + V - / = t:d + V ( ß 'I I)
dt ds
+ TI:VÜ) + Q E -P - — Q',
d t (3.7)
where the notations
3 = i[V r + (Vr)T] to = - V x v,
d P
ß ' = ß -e > , P =
-
----
(OXP
dt (3.8)
have been introduced and the couple density is re
garded to the exerted by the electric field alone:
m = P xE . (3.9)
The vector ß ' is the angular velocity in a cartesian
frame corotating with the body, and d and P are the
time derivatives of d and P in this frame. The time
derivatives of the directors in this frame are
/° = ß 'x /, m = ß 'x m , » = ß'x#i, (3.10)
It follows from this equation that
Ö = ß 'x Q , (3.11)
The corotational time derivatives and ß ' are objective
quantities.
The second law of thermodynamics is expressed in
the dissipation inequality
ds
e ^ + V - / s = <rs > 0, (3.12)
where s is the entropy density, Js the entropy current
and as the entropy production. We assume the validity
of the relation
(2.19)
With the state space Z' the time derivative of the
entropy in the corotating frame is
ds 1 du 0s o 9s o 9s o
dt T dt 9/ 9m 9n
9s o 9s o
+ 90 9P (3.14)
As the entropy is a scalar, it has the same form in any
other frame. Equivalently with the choice of the rele
vant variables Z:
ds 1 du 9s o 9s o 9s o
— =
------
+ — :Q + — : 0 +
-----
P. (3.15)
dr T dr 9Q 90 9P v '
It has been used that the velocity is not an objective
variable and that therefore by the principle of objec
tivity it appears only in the corotating time derivatives
which are objective. Moreover, use has been made of
the fact that the entropy does no depend on the defor
mation rate in a local equilibrium situation. The rela
tion
9s
du
1
T ' (3.16)
known to be valid in equilibrium, has also been used.
The derivations can be carried out explicitly using the
constitutive relations (2.15) or (2.17) for the entropy.
Examples of this calculation are given in the appendix.
Substituting (3.15), (3.13) and (3.7) into the dissipation
inequality (3.12) results in
Tffs = t:d° + V-(fl'-n) + n:(Vo>)
9s o 9s o / 9s
+ qT — :Q + o T — : 0 + o T — +oE
y 9Q 90 y dP
o ds „
P
-------
ß '.
dr (3.17)
which holds always in a local equilibrium state.
In the first term of the right hand side only the sym
metric part of the stress tensor is relevant. Inserting
the spin balance for ds/dr in the last term, one sees that
also the antisymmetric part of the stress tensor con
tributes to the entropy production.
The following identities are used to simplify this
expression:
9s o 9s ( 9s
— : Q = — : (ß x Q) = 2 ß h> — • QT ,
9Q 9Q V0Q /
v (ß' n) + n:Vo> = (V ß):n + ß ' v n ,
V ß = Ö + 0 d - ß 'x O . (3.18)
The derivation of the last identity is given in the ap
pendix.
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
For the energy dissipation function we get
C. Papenfuß et al. • A Simplified Thermodynamic Theory for Biaxial Nematics 800
T o = Q '-{ 2 w
0s
+ v n
+ ö:<|n + ö T ^ | + d:{t + o T-n}
0S ] ds _ (3.19)
The energy dissipation function fulfils the inequality
T as > 0 (3.20)
with T as = 0 in and only in an equilibrium.
4. The Equilibrium Conditions
In equilibrium the objective rates of processes ß',
O, d, and P vanish. As the energy dissipation function
(3.19) is linear in these fluxes, the equilibrium condi
tions are that the corresponding forces are equal to
zero [18j. The only constraint to be taken into account
is that d is a symmetric traceless tensor as we consider
only volume preserving motions:
d : 8 = 0, d: A = 0 (4.1)
l* = T <rs + d: (p ö + a)
0S
= ß n o T + v n
+ 0:<II + g T — > -I- P- q T Vp + qE
+ d:{t + 0 T-n + p5 + a}, (4.2)
where it has already been used that ds/dt = 0 in equi
librium.
p and a are Lagrange multipliers, which have to be
eliminated from the resulting equilibrium conditions.
These equations for equilibrium are
8s T „ T
2tr Q T— QT- n Ot
ÖQ
6s
n + 5 r - = o,
t + o T-n + pö + a = o,
0s
T — + £ = 0.
6 P
+ v n = 0, (4.3)
(4.4)
(4.5)
(2.19)
The term II • Ot in (4.3) has no analogon in the case
of a uniaxial nematic liquid crystal [16].
Equation (4.6) gains the form
qP = x J E l+ x mmE m + x„nE n (4.7)
with x ;l = T sZ, = =
(4.8)
9s
when determining the derivative ^ from (2.15). This
expression for the electric polarisation is the usual
one. Equation (4.4) is rather explicit for the couple
stress, and only the large number of material coeffi
cients in the entropy function (2.15) can cause diffi
culties.
To determine the equilibrium stress, we combine
(4.3), (4.4), (3.6), (3.9) and (4.6). In the balance of angu
lar momentum (3.6) it is used that in equilibrium ds/
dt = 0. We get first
2 qT w 0s .r 0s
----
Q +
-----
Ot
0Q v 0O
0S
2»v(t) + V -n = q T P x — = 2 qTw
0P
+ V -n = 0, (4.9)
0S
0P
with any antisymmetric tensor a. The components of
ß ' and O are all independent. The auxiliary function,
taking also the constraints (4.1) into account, is
(4.10)
and then
2w{i) = 2QT w 0s ds
P +
dP 0Q
0S
Q1 + — Ot
0O
(4.11)
0S
cO
Making use of the indentity
— • QT = P — + O • f ——V + Ot •
0Q v dP \0OJ
(4.12)
which is a consequence of the fact that the entropy is
an objective quantity, we arrive at
H>[t + 0 T-II] = 0, (4.13)
from which
t = pö + gT O T- ~ (4.14)
is obtained. The scalar pressure p is available only
when integrating the differential equation. It is worth
mentioning that the right hand side of the expression
for the stress is quadratic in the components of the
angular distortion tensor, so, in linear approximation
the distortion does not contribute to Cauchy's stress.
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
C. Papenfuß et al. • A Simplified Thermodynamic Theory for Biaxial Nematics
5. The Motion of Biaxial Nematic liquid crystals
The equations of motion in the linear Onsager
theory have a very simple form if the generalized
forces and fluxes are independent and the body is of
high symmetry. This is not so in our case because the
constraint equations (4.1) hold and the symmetry of a
biaxial nematic liquid crystal is rather low. A further
difficulty is that a transformation of fluxes producing
independent variables spoils the frame independent
notation and makes the calculations clumsy. The vari
ational principles of thermodynamics worked out by
Gyarmati [17] in the 60's offer a more elegant and
easier procedure. The most convenient representation
for dealing with biaxial nematic liquid crystals is the
flux representation of the local principle in the energy
picture.
To apply the variational principle we have to look
for the actual form of the dissipation potential (p. It is
a homogenous quadratic function of the fluxes, and
the derivatives of (p with respect to the fluxes are equal
to the generalized forces. In our case the dissipation
function depends on ß ', O, d, and P and contains the
local state variables u, P, O, and Q as parameters. To
make the task a bit easier we suppose that the dielec
tric losses are negligible, i.e. (4.6) holds also for dissipa-
tive processes. In this way cp does not depend on P.
Moreover, we supose that the sample of the liquid
crystal is large enough to neglect the angular distor-
sion O and its time derivative in the dissipation poten
tial. This supposition is encouraged by the thermody
namic theory of uniaxial nematics, where the same
simplification leads to the Ericksen-Leslie-Parodi the
ory [16]. In this way the function we are looking for is
of the form
(p = <p(ß',d, Q). (5.1)
This function has to be objective, i.e. the equality
(p(R ß',R d RT,R Q) = <p(ß',d,Q) (5.2)
holds for any proper orthogonal tensor R. Moreover,
it has to be invariant under the transformations (2.13).
The special choice of R = QT leads to
<p = <p(QT ß ,QT d Q,5), (5.3)
making use of which we conclude that tp consists
of three kinds of terms, terms quadratic in QT ß ',
terms bilinear in QT • ß ' and QT • d • Q and quadratic
in QT • d • Q. As the mapping QT transforms the axial
vector ß ' and the tensor d into the abstract reference
frame, we put down the dissipation potential as a
function of the vector and tensor components in the
reference frame. The invariant form of (p is
cp = \(Q T- ß') • (QT • ß') + (QT • ß')
R™:(QT d Q) + f(QT d Q):Rdd:(QT d Q).
(5.4)
Here R ^is a second order tensor, Rßd is a third order
one (remember that Rßd is an axial tensor) and Rdd is
a fourth order one. To avoid misunderstanding, (5.4) is
repeated in cartesian coordinates:
cp = j E RlfQsi Vs Qjr V + Z R g Qsi Q Qrj Q* L
ijsr ijkrst
+ i E R$aQ*Q,jQ*Q*W *- (5-5)
ijklprst
We determine the possible form of polynomial (5.5)
similarly as it has been done with the entropy func
tion. According to the transformations (2.13) the
nonzero coefficients of the phenomenological tensors
are listed here:
n nn n nn p nn
*ni » ^22 > n 33 >
n Od TJ Od n ad n nd n Od n ad
*M23' -*M 32» ^213> ^231» ^312' ^321'
< 1 1 , ^2222' ^3333' (^-6)
ndd odd ndd ndd ndd ndd
■*M 122 ' ^1212' *M221' ^2112' ^2121' ^2211'
ndd ndd ndd ndd ndd ndd
*M 133 » ^1313' 1331' 3113 ' ^3131' ^3311'
ndd ndd ndd ndd ndd ndd
^2233' ^2323' ^2332» -^3223' ^3232' ^3322-
Next we make use of the symmetry relations
Rijki — Rk!ij> (5-7)
which are well known from the algebraic theory of
quadratic forms and are equivalent to Onsager's re
ciprocal relations [2]. Moreover, because d is a sym
metric tensor, the relations
Rdijki — Rdijik> (5-8)
can be required without hurting the generality. This
way the independent coefficients reduce to
n na n nn n nn
*M1 » 22 j ^33 »
Rnd n nd n nd
123' ^213' ^312'
ndd ndd ndd
'M ill' 2222' 3333'
ndd ndd
*M122' *M212'
Ddd ndd
^1133' *M313'
^2233» ^2323- (5-9)
The number of the independent coefficients, is further
reduced, when regarding the fact that, for volume pre-
801
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
802 C. Papenfuß et al. • A Simplified Thermodynamic Theory for Biaxial Nematics 802
serving motions
o o o
dll + d22 + d33 = 0- (5.10)
For this reason only the 6 coefficients of the 9 elements
of Rdd in (5.9) are independent, and the coefficients
R ^ 22, #2233» and #33ii can t>e chosen for zero. Now
we are able to put down the actual form of the dissipa
tion potential:
cp = \ r™ q ? + i R ™ v2 + \
+ 2R™ Q[dmn + 2R Z Qmd„, + 2R™ Q'ndlm
, 1 ndd j2 , 1 ndd j°2 , 1 ndd j°2
2 »» » 2 mmmm umm 2 ^nnnn Unn
+ 2R%lmd?m + 2R%J?n + 2RZmj L , (5.11)
where again the directors /, m, and n are chosen to be
parallel to the coordinate axes. As the dissipation po
tential is always positive except for equilibrium, when
it is zero, the following inequalities hold:
R™> 0, R™> 0, R™> 0,
#2323 ^ 0, #1*313 ^ 0, #1212 ^ 0,
n nn ndd ^(n{2d\2 una ndd ^/n{2d\2
^11 ^2323 — v-** 123/ > ^22 ^1313 — 1^23V >
n C2n ndd ^ ( n fid \2
^33 K1212 ^ 1^312/ >
#1111 + #2222 ^ #!!!! + #3333 > 0,
#2222 + #3333 ^ 0>
ndd ndd , ndd ndd 1 ndd ndd \ r\
^1111 ^2222 ^ llll ^3333 ^2222 ^-3333 — u>
(5.12)
The coefficients appearing in the dissipation potential
are material coefficients depending on the chemical
composition of the biaxial nematic liquid crystal and
on the temperature. The equations of motion are ob
tained from a partial form (flux representation) of
Gyarmati's local variational principle, which says that
L* = T as — cp is maximal if the fluxes belong to real
processes [17]. Applying the variational principle, the
constraint conditions (4.1) have to be kept in mind.
The auxiliary function of the variational problem is of
the form
L* = Tos - cp + (p5 + A):d, (5.13)
where p and A are Lagrange multipliers; the latter is
an antisymmetric tensor. The differential equations
describing the processes are obtained from here. First
of all notice that neither the components of P nor O
enter the dissipation potential; consequently, the
derivatives of the Lagrangian L* with respect to them
exhibit the same equations as in equilibrium, i.e. (4.4)
and (4.6) remain valid for nonequilibrium situations.
Making use of these, we can reduce the actual form of
the dissipation function. Combining (3.6), (3.9) and
(4.6) so as it was done in the case of equilibrium,
moreover applying the identity (4.12), we arrive at
+ d ' - . r o ^ l .
e o j (5.14)
This form of the energy dissipation function shows
that it is useful to introduce the viscous stress tensor
by
t = :eT O T~ + t vs + fvax8,
SO (5.15)
where tvs is a symmetric second order tensor and tva is
an axial vector. The energy dissipation function gets
the form
T <7S = — 2 ß ' • fva + d:tvs,
or in components
(5.16)
T<rs = -2(Q 'r t? + Q'm-C + Q'n- 0
o o o
+ dn tJi + dmm + dnn tynl
+ 2dImC + 2 < L C + 2<L^- (5.17)
The axial vector fva includes the term qds/dt of the
time derivative of the internal angular momentum.
Within a statistically founded alignment-tensor the
ory [22] this term can be identified with the expression
Qds
dt = ^ H -^ a -2 a > x a J = i W[ä], (5.18)
where a is the time derivative of the alignment-tensor
in the corotating frame. From here it can be seen that
ds/dt is an axial vector. On the other hand there is
some evidence that the relaxation of the spin is so fast
that for experimentally observable times the spin bal
ance can be assumed to be stationary, i.e. ds/dt = 0.
The constitutive equations are determined as
t\a
_
1 d Ofir)' n {2d J
Zl —~ 2 KH — Klmnar
2 II I * (mn mn'
va
_
1 ndilr)' n {2d J
m — 2 mm "m — ^mnl Unl >
f\a
_
1 n {2{2(~y n {2d i
ln — ~ 2 nn "n ~~ ^nlm alm>
vs dd °
+ P = #1/1/
Cm + P = R"
C + P = #»--- d.
tvim + Alm = R™i &'„ + 2 Rdiiim
tmn + Amn = R™„ ßj + 2 #jj?„m„ dmn,
+ Anl = R™n Qm + 2 RdJnl dnl,
TTITTITTITTI 771771 '
(5.19)
(5.20)
(5.21)
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
C. Papenfuß et al. • A Simplified Thermodynamic Theory for Biaxial Nematics 803
As all tensors in (5.22) are symmetric, the multiplicator
A turns out to be zero:
tVlm = R-nml + ^ tfilm
Cn = ß/ + 2 Rm„mndmn,
C = + 2 R^ni d„,, (5.22)
The above equations completely determine the trace-
less part of the viscous stress tensor. The scalar multi
plicator p, i.e. the scalar pressure, can be determined
during the integration of the balance equations. It can
be seen that both the deformation and the rotation of
the director have influence both on the symmetric and
on the antisymmetric parts of the viscous stress tensor.
The number of independent viscosity coefficients is 12,
the same number as in the ansatz used by Carlsson,
Leslie and Laverty [23]. It is very interesting that the
structure of the equations is more transparent than in
the theory of uniaxial liquid crystals. This may be due
to the fact that neither the direction of the angular
velocity of the director nor the couple stress tensor are
restricted. In the light of the present theory it seems
worth reformulating the thermodynamic theory of
uniaxial liquid crystals.
The assumptions we have made in the derivation of
these constitutive equations can be summarized as
follows:
Gyarmati's variational principle has been used, the
validity of which can be proved only in the case of
linear relations between the thermodynamical forces
and the fluxes [14, 24]. The relations
1 ds 1
J = —Ja and — = —
s T du T
have been assumed, which are valid in a situation of
local equilibrium. It has been shown that in some
examples the first of these relations is not valid [25].
But this equation does not mean that we have ne
glected the contribution of the pointing vector to the
nonconvective energy flux, as Jq is only the mechani
cal heat flux. The model of Grot and Eringen [21] has
been accepted for the interaction of the electromag
netic field with matter. It is shown in appendix A how
the form of the energy balance used in the present
work follows. The couple stress exerted by the electro
magnetic field has been assumed to be
m = P xE .
The eigenvalues of the alignment-tensor are taken to
be constant, i.e. we consider only rigid rotations of the
angular distribution function.
The entropy has been approximated by a polyno
mial quadratic in the components of O and P, and
flexoelectric terms have been neglected. These simplifi
cations are only made for the sake of convenience.
o o
O and P have been neglected in the dissipation
potential with the argumentation given above.
Acknowledgements
Many helpful discussions with H. Ehrentraut und
Dr. S. Sellers are greatfully acknowledged. We thank
the Technical University of Budapest and the Techni
cal University of Berlin for financial support of the
cooperation. Financial support by the DAAD spon
soring the co-operation between both our depart
ments is gratefully acknowledged.
Appendix
A) The Balance of Energy in the Electric Field
The energy e, the nonconvective energy flux q and
the stress tensor T can be decomposed into mechani
cal parts u, Jq, t and electromagnetic parts eem, qem,
^em •
e = u + eem, q = Jq + qem, T = t + tem,
Within the model of Grot and Eringen [21] the electro
magnetic parts are in a nonrelativistic approximation
2 n0 2
with
& = E + v x B , = H -v x D .
With these equations the electromagnetic terms in the
energy balance are
d
^"em + V-?em- t em:Vt>
= B B + - + +
d t\2 n 0 2 J
- l-^—B B - - ß B)b + $D
LV2^o 2 J
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
Using the Maxwell equations in the formulation [26]
with the objective quantities <£, and the objective
time derivatives
B = B + B V v - B Vr,
D = D + D V v - D Vr,
and
f = J~ Q eV
we arrive at
d+ t em:Vr
1 . . *
= — B B + E0S £ + ß P + ß P - B 3 t f
Ho,
- D $ - tem:Vr.
C. Papenfuß et al. •
If we assume that the fluid is incompressible and that
the constitutive equations
M = - B. y' = 0 and ge = 0
Ho
hold and that B = 0, this results in
j t uem + V q em- t em:Vv = £ P = £ - ( E - 8 06 )E -P ,
which is the last term in the energy balance equation
(3.4).
A Simplified Thermodynamic Theory for Biaxial Nematics 804
[1] I. Pardowitz and S. Hess, Physica 100 A, 540 (1980).
[2] M. J. Freiser, Mol. Cryst. Liq. Cryst. 14, 165 (1971).
[3] R. Alben, J. Chem. Phys. 59, 4299 (1973).
[4] L. S. Yu and A. Saupe, Phys. Rev. Lett. 45, 1000 (1980).
[5] B. Chandrasekhar, B. Sadasniva, R. Ratna, and V. N.
Raja, J. Phys. 30, 491 (1988).
[6] H. Ehrentraut and W. Muschik, Mol. Cryst. Liq. Cryst.
262, 561 (1995).
[7] C. W. Oseen, Trans. Faraday Soc. 29, 883 (1933).
[8] F. C. Frank, Discussions of Faraday Soc. 25, 19 (1958).
[9] J. L. Ericksen, Arch. Rat. Mech. Anal. 9, 371 (1962).
[10] F. M. Leslie, Arch. Rat. Mech. Anal. 28, 265 (1968).
[11] U. D. Kini, Mol. Cryst. Liq. Cryst. 112, 265 (1984).
[12] A. Saupe, J. Chem. Phys. 75, 5118 (1981).
[13] E. A. Jacobsen and J. Swift, Mol. Cryst. Liq. Cryst. 78,
311 (1981).
[14] W. M. Saslow, Phys. Rev. A 25, 3350 (1982).
[15] D. Baalss, Z. Naturforsch. 45 a, 7 (1990).
[16] J. Verhäs, Acta Physica Hungarica 55, 181 (1984).
[17] J. Gyarmati, Non-equilibrium Thermodynamics,
Springer-Verlag, Berlin 1970.
[18] S. R. Groot and P. Mazur, Non-equilibrium Thermody
namics, North Holland Publ. Co, Amsterdam 1962.
[19] See for example I. Müller, Pitman Publishing (1985).
[20] See for example A. C. Eringen, Continuum Physics, Vol.
I-IV, Academic Press (1975).
[21] R. A. Grot and A. C. Eringen, Int. J. Engng. Sei. 4, 611
and 639 (1966).
[22] S. Blenk, H. Ehrentraut, and W. Muschik, Mol. Cryst.
Liq. Cryst. 204, 133 (1991); Physica A 174, 119 (1991).
[23] T. Carlsson and F. M. Leslie, Liquid Crystals 10(3), 325
(1991).
[24] I. Gyarmati, Ann. Phys. 7, 353 (1969).
[25] I. Müller, Arch. Rat. Mech. Anal. 26, 118 (1967).
[26] R. Ellinghaus, Kovariante Auswertung der Dissipations-
gleichung nach Liu für Flüssigkeiten und Gase im elek
tromagnetischen Feld, Thesis TU, Berlin 1989.
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 14.11.18 14:18
