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8/10/2019 Edwards Et Al. - 1990 - Generalized Constitutive Equation for Polymeric Liquid Crystals Part 2. Non-homogeneous Systems
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Journal of Non-Newtonian Fluid Mechanics, 36 (1990) 243-254
Elsevier Science Publishers B.V., Amsterdam
243
GENERALIZED CONSTITUTIVE EQUATION FOR POLYMERIC
LIQUID CRYSTALS
PART 2. NON HOMOGENEOUS SYSTEMS
BRIAN J. EDWARDS , ANTONY N. BERIS *, MIROSLAV GRMELA *
and RONALD G. LARSON 3
I Center for Composite Materials and Department of Chemical Engineering,
University of Delaware, Newark, Delaware 19716 (U.S.A.)
* Ecole Polytechnique de Montreal, Mont&al, Qu&bec H3C 3A7 (Canada)
AT&T Bell Laboratories, Murray Hill, New Jersey 07974 (U.S.A.)
(Received November 27, 1989)
Abstract
The Hamiltonian formulation of equations in continuum mechanics
through Poisson brackets was used in Ref. 1 to develop a constitutive
equation for the stress and the order parameter tensor for a polymeric liquid
crystal. These equations were shown to reduce to the homogeneous Doi
equations as well as to the Leslie-Ericksen-Parodi (LEP) constitutive equa-
tions under small deformations [l]. In this paper, these equations are fitted
against the non-homogeneous Doi equations through the simulation of the
spinodal decomposition of the isotropic state when it is suddenly brought
into a parameter region in which it is thermodynamically unstable. Linear
stability analysis reveals the wavelength of the most unstable fluctuation as
well as its initial growth rate. Results predicted from this theory compare
well with the predictions of Doi for the spinodal decomposition using an
extended molecular rigid-rod theory in terms of the distribution function. .
This completes the development of a generalized constitutive equation for
polymeric liquid crystals initiated in Part 1.
Keywords: Hamiltonian fonnulation; Poisson bracket; polymeric liquid crystals; constitutive equation;
tensorial order parameter
1. Introduction
Recently, the Hamiltonian (Poisson bracket) formulation was used to
produce a more general theory for polymeric liquid crystals [l]. The gener-
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where $( r, it, t) is the distribution function of the molecular orientation,
and r, n and
t
are the position vector, the unit vector of orientation and the
time, respectively. From the definition (2.2), both S and
m
are symmetric
and
tr(S) = 0,
(2.3a)
tr(m) = 1.
(2.3b)
The portion of the free energy (Hamiltonian) density dependent upon the
order parameter tensor for the generalized model is
H(S) = -&J&(S) dV- &J&(S) dV,
(2.4)
where a,, is the homogeneous contribution to the entropy,
WS) = - L%YSYol + $GG&3SP~ - &J,,S,pSP,SC~ - $a;( S&J*,
(2.5)
and Qe is the inhomogeneous (Frank elastic) contribution,
@e(S) = - &%/G&u - %%,s,nSvp,u - &%,S,+%,,.
(2.6)
Following Ref. 1, the evolution equation for the order parameter tensor is
(2.7)
where
6H/6Sa,,
the functional derivative of
H
with respect to Sny, is given
by Dl,
6H
- = J( a2Syu a3( Sy&3, - t~yaS2k3)
mx,
+ a4(Sy&3S& -
~s,&?&S~/3) + d ,S~aSa~
- WauJ3.P
- b2(+Spu,p,a :Spu,p,y Spc.s.c)
+b&+&,~ - S& y,c,~ dav.r
- &xuS~&3~.s)).
(2.8)
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Here and in the following, we use Einsteins summation convention (i.e.
repeated indices imply summation) and a comma to denote spatial differ-
entiation. Subsequently, the term proportional to b, will be neglected since it
is higher-order than the other non-homogeneous terms and does not add
qualitatively to the physics of the situation. The stress constitutive equation
is
(2.9)
With the exclusion of the B and bi parameters, which control the effects
of spatial inhomogeneities in the orientation, all the parameters involved in
the above three expressions have already been determined in Ref. 1 in terms
of the Doi theory for homogeneous systems of rigid rods. This was done by
comparing the molecular equations for the time evolution of the order
parameter tensor and the free energy with the equivalent expressions pro-
vided by eqns. (2.4), (2.5) and (2.7). The two expressions are identical for
spatially homogeneous distributions if
(2.10a)
and
9 u
al=- l--
i 1
3
a3= U, ai= %U,
(2.10b)
provided that the Doi closure relationship for the fourth-order average is
used [l]. The term proportional to a4 has been dropped, since tr(S . S - S. S)
= (1/2)tr(S.
S)2
[4], and since it gives identical contributions with the
aA
term to the free energy. (For a much simpler proof than [4], see Appendix
A.) As shown in [l], the viscous stress of the Doi model can also be
recovered as well, simply by letting Mayfir = ~{~~~in,,,rn~~, and the solvent
stress is obtained when r = 7,.
Thus, the parameters of the present theory are based on a consistent
averaging of the molecular Doi theory for rigid rods. As such, some of the
molecular details (but hopefully not the important ones) are lost in the final
form of the equations, (eqns. 2.7-2.10); however, a major advantage is
realized. Namely, the molecular distribution function I,L is eliminated from
the equations while the thermodynamic consistency of the equations is
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preserved because of the adherence to the Poisson bracket formalism.
Elimination of 4 = I/.J(, n , t)-which is, in general, a six-dimensional func-
tion-makes the numerical solution of the resulting equations practical for
interesting inhomogeneous problems, such as spinodal decomposition in its
early, intermediate and late stages and flow-induced texture evolution in
liquid crystals. Furthermore, the thermodynamic consistency guaranteed
from the Poisson bracket formalism lowers significantly the risk of observing
aphysical behavior introduced from the averaging of the more detailed
molecular description.
The remaining parameters, B and bj, are determined from the more
recent, inhomogeneous Doi theory for concentrated solutions of rigid rods
[2,3] as follows. The parameter B is determined from Dois translational
diffusivity term in the following paragraphs. The parameters
bi
are obtained
from the inhomogeneous excluded volume (molecular interaction) effects by
comparing the linear stability analyses of spinodal decomposition from the
isotropic to the nematic phase, as described in the next section.
The translational term in the extended Doi diffusion equation [2,3]
(neglecting the effects of the flow field and the rotational terms, which were
addressed in Ref. 1) can be rewritten as
which also can be identified as
(2.11)
(2.12)
where pA denotes the free energy density, defined from the free energy
expression in terms of the distribution function [2,3,5]
A =
1
p ,G(r , n , t )
dn dV
= ck,T
/
(In ( r , n , t ) + W(r , n , t ))$ (r , n , t ) dn dV,
(2.13)
where w is the potential of the molecular field expressed in terms of the
interaction potential W( r - r , n , n ) [2,3] as
@=
1
W(r - r , n , n ) (r , n , t ) dn dV.
(2.14)
In eqns. (2.4), (2.5) and (2.6), the Landau-de Gennes expression for the
free energy is used in lieu of the alternative free energy expression provided
by eqn. (2.13). Furthermore, the parameters a, in eqn. (2.59, listed in eqn.
(2.10), were obtained by fitting the Doi free energy expression [5]. Therefore,
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in order to compare the translational term of eqn. (2.7) with eqn. (2.11), we
need to obtain an alternative expression to eqn. (2.11) for the portion of the
free energy depending on the order parameter tensor, H(S), which will
involve eqn. (2.4) in terms of an integral over the entire space r 8 n One
way to obtain this is to assume an approximation for the free energy of the
form
1 dH
H=Ho+-2-dS:S.
(2.15)
This approximation becomes exact in the limit of small S when the free
energy reduces to a quadratic functional of S. Thus, the free energy A is
now written as
A=&+
:~[n,n,-
,S,,]~ (r, n,
t)
dn dV/,
Ya
(2.16)
where A, involves the portion of the free energy which does not depend on
S (kinetic energy). Then, pA can be approximated as
(2.17)
As a consequence, eqn. (2.11) can be rewritten in a form compatible with
our formalism as
.
(2.18)
If now eqn. (2.18) is multiplied by n,np and integrated over n, then an
equivalent expression for the evolution of the order parameter tensor is
obtained:
where the decoupling
approximation (n,npnin,n,n,) = (n,n,)
( npng)(n,n,) is used in order to preserve the symmetry of B, and
6,,( ~H/&l$,,) = 0 since the functional derivative is traceless. Equation (2.19)
does not, in general, guarantee the symmetry and unit trace characteristics
of m However, the corresponding equations for
mpa
and tr(m) can be
formulated and used together, as in eqn. (2.7), for the definition of
am,,/&.
(We have tried to make the comparison as simple as possible.) Then,
comparison of eqn. (2.7) with a modified form of eqn. (2.19) leads to the
following expression for B
B
ally = mq
[(h,,-A,)m,,+h,6,,]mgj, (2.20)
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where
Dll
D,
A,,=--- ___
2ck,T
AL= 2ck,T
(2.21)
Equation (2.20) is positive-definite, provided that A,, 2 A _L, as is always the
case for polymeric liquid crystals. Thus, this definition is acceptable from a
thermodynamic point of view (see Refs. 1 and 6).
Although the qualitative character of the sixth-order tensor B is well
represented by the above theoretical description, the exact numerical values
of it are not, due to the assumptions involved in the derivation. An
alternative approach can be constructed as follows. In the small concentra-
tion limit, w = , and at equilibrium, the distribution function is constant,
=$,=1/47r. F or cases which are perturbed slightly from equilibrium, the
distribution function can be approximated by a truncated series in terms of
the order parameter tensor S, which as a first-order approximation is
=A exp(PS:nn)=&(l+jB:nn),
(2.22)
with the parameter p determined from the consistency requirement
S = $(nn - +a) dn = &S: /nnnn dn = %S, (2.23)
or, /3 = H/2. Therefore, in this limit, the free energy expression provided by
eqn. (2.13) reduces to
A = ck,T
s
In+ 4 dn dV
= ck,T
/
(-ln(4r) + FS: nn)- l + F-S: nn) dn dV
= ck,T
/
( -
ln(4r) + YS
: S)
dV;
where use was made of the identities [7]
&
I
n,np dn = +Saa,
and
SH
6s = lSck,TS.
(2.24)
(2.25a)
(2.25b)
(2.26)
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In the initial stages of the disturbance, it is reasonable to assume no flow
(u = O), and that the parallel translational diffusion is the only relaxation
mechanism for the molecules (A = 0, A I = 0) [3]. Under these assumptions,
and retaining only terms linear with respect to the components of the order
parameter tensor, eqn. (2.7) becomes
VDll
= -
a2Sap
blKQ3,p.p
27 ( - b&&,p,s + :Spp,r,, - f%~%+,~));~,~;
(3 -2)
where we have used
(3.3)
which is the limiting (S = 0) expression for B arising from either of eqns.
(2.20) or (2.30) with the numerical (order one) constant v assuming the
values of l/2 or 3/5 respectively.
To investigate the time evolution of various modes, let Sk be the kth
Fourier component of the order parameter tensor
Sk = Real[ Ak( t) eikx3],
(3.4
in general complex, where
Ak(t)
is a traceless, symmetric tensor. Substitu-
tion of eqn. (3.4) into eqn. (3.2) leads to a system of five independent,
ordinary differential equations coupling the five independent components of
Ak( t). As already observed by Doi [3], these equations can be separated into
five independent sets of equations, each one governing the (initial) evolution
in time of five orientational modes. These are equations involving Ak12,
A
kll
- Ak22 Ak13 Ak23 an d Ak33 -
In particular, the equations have exactly the same form discovered by Doi
[3] and separate the fluctuation modes into three types.
(1) The twist mode, with similar equations followed by A,,, and A,,, -
A
k22 is
2vDll
&A -
12= -
3L2
where K = kL /2 with L being
Equation (3.5) is the same as the
8bl
-K4
Ak12,
9L2
1
(3.5)
a characteristic length of the molecule.
corresponding equation (eqn. (3.2) in Ref.
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3) with the only (minor) difference being that the numerical factor 2v/3
appears instead of 4/7, provided that the coefficient b, is defined as
h&.
(3.6)
For U < 3 the coefficient of Akr2
in eqn. (3.5) is negative, corresponding to a
negative eigenvalue, which implies a decaying fluctuation for every wave-
length k. For UP 3 however, the fluctuation will grow for small enough
wavelengths with the maximum growth rate A,
h = 6vL+(1 - u/3)*
n2
UL2 )
attained for the most unstable wavenumber
k,,,
(3.7)
k,, = ;
(3.8)
Thus U = 3 corresponds to the critical concentration beyond which the
isotropic state becomes unstable to infinitesimal perturbations, in agreement
with the free energy analysis of the static system.
(2) The bend mode, with similar equations followed by A,,, and A,,,, is
2vDll
$Ah13 = - -
3L2
K*+ 4(269'L:b2)K4]Ak,3.
(3.9)
Equation (3.9) is the same as the corresponding equation (3.3) in Ref. 3 with
the only (minor) difference that the numerical factor 2v/3 appears instead
of 12/7 provided that the coefficient h, is defined as
(3.10)
(3) The splay mode, with the following coupled equations for AA33, A,,,
and Ak22, is
A,;;= --
2vDll 1
3L2
[i
2vDll
$A,,,= - __
3L2
[i
1
8(3h, + 2&)
K4 A,
27L2
1
h33?
-K4
%
A,,,
- -
w 8bI
9L2 3L2 27L2
K4A
x33>
(3.11)
(3.12)
K4
1
2vD,, 81,,
Am - 3~2
~KA,33-
(3.13)
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3 M. Doi, J. Chem. Phys., 88 (1988) 7181-7186.
4 D.C. Wright and N.D. Mermin, Crystalline Liquids: The Blue Phases, Rev. Mod. Phys., 61
(1989) 385-432.
5 M. Doi, J. Polym. Sci., Polym. Phys. Ed., 19 (1981) 229-243.
6 A.N. Beris and B.J. Edwards, J. Rheol., 34 (1990) 55-78.
7 R.B. Bird, C.F. Curtiss, R.C. Armstrong and 0. Hassager, Dynamics of Polymeric Liquids,
Vol. 2, Kinetic Theory, 2nd Ed., John Wiley, New York, NY, 1987.
8 B.J. Edwards and A.N. Beris, J. Rheol., 33 (1989) 1189-1193.
Appendix A
For any matrix c, the Cayley-Hamilton theorem states that
c *c - l,c + I - I -l = 0,
where
I, = trc,
I,= (1/2)[(trc)*- tr(c.c)],
and
I3 = det(c).
Multiplying eqn. (A.l) by c*, and taking the trace yields
tr(c.c.c.c) -I$r(c.c.c) +I,tr(c.c) -I,trc=O.
Substitution of S for c, knowing that trS = 0, immediately yields
tr(S.S.S.S) = (1/2)[tr(S.S)]*.
(~4.1)
64.2)
(A.3)
(A.4)
(A-5)
(A@