edwards et al. - 1990 - generalized constitutive equation for polymeric liquid crystals part 2....

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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245

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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251

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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252

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@

edwards et al. - 1990 - generalized constitutive equation for polymeric liquid crystals part 2....

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