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This is to certify that the

thesis entitled

MICROSTRUCTURE 0F LIQUID CRYSTALLINE POLYMERS
IN SIMPLE SHEAR FLOWS

presented by
Chinh T. Nguyen

has been accepted towards fulfillment
of the requirements for

M. S . Chemical Engineering

degree in

@MQS .

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJCIRCDateDuepGS-pJS

 

MICROSTRUCI‘URB OF LIQUID CRYSTALLINE POLYMERS
IN SIMPLE SHEAR FLOWS
By

Chinh T. Nguyen

A THESIS
Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemical Engineering

2001

ABSTRACT
MICROSTRUCTURE OF LIQUID CRYSRALLINE POLYMERS IN SIMPLE
SHEAR FLOWS
By

Chinh T. Nguyen

Doi’s theory for liquid crystalline polymers (LCPS) is used to predict the response of
LCPS under simple shear flows. The moment equation for the orientation dyadic is
closed by using a new class approximation for the orientation tetradic that retains the six-
fold symmetry and projection properties of the exact fourth order moment of the
orientation distribution function. The nematic tendency for LCPS to align with each other
in one direction is characterized by the Maier-Saupe potential. The dynamic response of
the microstructure depends on the Pelect number Pe and the Maier-Saupe potential
strength U. For U = 0 and Fe = 0, the microstructure relaxes to an isotropic state from all
initial conditions. For U > O and Fe = 0, multiple equilibrium states are oblate, prolate
and isotropic. These equilibrium states are dependent on the initial orientation. For U >
' 0, and Pe > 0, multiple periodic solutions and multiple steady state solutions are possible,
a flaw-transition from tumbling to wagging and from wagging to steady state occurs as
Fe increases for a fixed value of the nematic strength coefficient U. For microstructure
that are initially nematic. When the initial orientation is planar isotropic in the plane of

flow/vorticity plane, a logrolling microstructure obtains.

to Tuy-Phuong

iii

ACKNOWLEDGEMENTS

First and foremost, I would like to thank my advisor, Professor Charles A. Petty
for his guidance and continuing support throughout this entire research. Additional
thanks to Professor Andre' Be’nard and Dr. Steven M. Parks for their helpful
suggestions during this research.

I would also like to acknowledge my family for their unending support: Mom,
Dad, Hoang and Khoa. I thank you all. I

I wish to thank the Department of Chemical Engineering and the Composite
Materials and Structures Center at Michigan State University as well as the National

Science Foundation for their financial support of this research.

iv

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES
LIST OF NOTATION

CHAPTER 1- INTRODUCTION
1.1 Background
1.2 Objective
1.3 Methodology

CHAPTER 2 - THEORY

2.1 Orientation Characterization
2.1.1 Single Polymer Molecule
2.1.2 Distribution Function
2.1.3 Average Orientation Tensors
2.1.4 States Representative of the Average Orientation Tensors
2.1.5 Realizable Orientation States

2.2 Continuity Equation

2.3 Constitutive Models
2.3.1 Rotary Diffusive Flux
2.3.2 Rotary Convective Flux

2.4 The Smoluchowski Equation

CHAPTER 3 - CLOSURE MODELS FOR THE ORIENTATION TETRADIC
3.1 The Quadratic Closure Model
3.2 The Linear Model of Hand
3.3 The Bingham Distribution
3.4 The Fully Symmetric Quadratic (FSQ-) Closure

CHAPTER 4 - BROWNIAN DIFFUSION
4.1 Introduction
4.2 Relaxation of Anisotropic States by Brownian Diffusion

4.3 Conclusions

CHAPTER 5 - BROWNIAN-D01 DIFFUSION
5.1 Introduction
5.2 Relaxation of Anisotropic States by Brownian-Doi Diffusion

5.3 Conclusions

Page
vii

viii

26
26
27
30

33
33
34
45

CHAPTER 6 - FLOW INDUCED ALIGNMENT
6.1 Introduction
6.2 Flow Induced Alignment
6.3 Conclusions

CHAPTER 7 - CONCLUSIONS AND RECOMMENDATIONS

APPENDICES

A.

B.
C.
D.
E.

FSQ-Closure

A.1 The FSQ- Closure at the Isotropic State

A2 The FSQ- Closure at the Nematic State
Component Equation for Brownian-Doi Diffusion
Order Parameter for Brownian-Doi Equilibrium State
Diagonal Initial Conditions
Computational Strategy

E.1 Introduction

E.2 Variable Listing

E.3 Dimensionless Time Step Selection

E.4 Program Listing

LIST OF REFERENCES

vi

47
47
48
61

62

65

67
69
73
75
77

77
78
8O

92

LIST OF TABLES

page
Table 1 Equilibrium States Predicted by the FSQ Closure (4.1 S U S 24) 44

vii

LIST OF FIGURES

page
Figure 1.1 Focus Topics of This Thesis 4
Figure 2.1 Orientation coordinates for a Single Axisymmetric Polymer 6
Molecule
Figure 2.2 Example of Orientation States 10

Figure 2.3 Realizable Set of Anisotropic Orientation States (Parks et al., 1999) 14

Figure 2.4 Distribution Orientation Vectors on the Unit Sphere 17

Figure 4.1 Relaxation of Anisotropic Invariants by Brownian Diffusion 28
(U = O, Pe = 0)

Figure 4.2 Relaxation of Nematic States by Brownian Diffusion 29
(U = 0, Pe = 0)

Figure 4.3 Relaxation of Planar Isotropic State by Brownian Diffusion 32
(U = O, Pe = 0)

Figure 5.1 Relaxation of Anisotropic Invariants by Brownian-Doi Diffusion 37
(U = 10, Pe = 0)

Figure 5.2 Relaxation of a Nematic State by Brownian-Doi Diffusion 38
(U = 3, Fe = 0)

Figure 5.3 Relaxation of a Planar Isotropic State by Brownian-Doi Diffusion 39
(U = 3, Pe = 0)

Figure 5.4 Comparison of Equilibrium States Predicted by the FSQ-Closure 42
(lower graph, this research) and Quadratic Closure (upper graph,
Brave et al., 1993) for Pe = O and 0 S U 510

Figure 5.5 Location of Multiple FSQ-Equilibrium States on the Realizable 43
Diagram for4.5 SUS6

Figure 6.1 Tumbling Microstructure. Components of the orientation dyadic 49

and its eigenvectors
(FSQ-closure with U = 30,Pe = 25, and 2(0) = $2393)

viii

Figure 6.2

Figure 6.3

Frgure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Tumbling Microstructure. Relative alignment of the orientation
director with the flow direction

(FSQ-closure with U = 30, Fe = 25, and 2(0) = g3 g3)

Wagging Microstructure. Components of the orientation dyadic
and its eigenvectors
(FSQ-closure with U = 30, Fe = 38, and 2(0) = 9393)

Wagging Microstructure. Relative alignment of the orientation
director with the flow direction
(FSQ-closure with U = 30, Fe = 38, and 2(0) = 2393)

Nematic-Like Microstructure. Components of the orientation
dyadic and its eigenvectors
(FSQ-closure with U = 30, Pe = 80, and 2(0) = 9393)

Flow Transition Predicted by the FSQ-Closure for Nematic
Initial State

Logrolling Microstructure. Components of the orientation dyadic
and the orientation director

(FSQ closure with U = 30, Fe = 55, and 2(0) $59191 +%g3g3)

Anisotropic Invariants Associated with Logrolling

Flow Transition Predicted by FSQ-Closure for 2D Planar
Isotropic Initial States

Figure D.1 Realizable Orientation States Predicted by the FSQ-Closure for

U = 30, and Pe =50

Figure E.1 Step Size Selection. Behavior of orientation dyadic component

a33 for U = 30, and Pe = 25

ix

52

53

54

55

56

59

76

79

English Symbols

ll”

31,32,33

LIST OF NOTATION

Orientation dyadic

Eigenvalues of 2,

Anisotropic orientation dyadic
Eigenvalues of 2

Rotational diffusivity

Unit vector in the vorticity direction

Unit vector in the direction perpendicular to the plane of
flow/vorticity

Unit vector in flow direction

Boltzman constant

Unit dyadic

Unit vector direct along molecular axis
Orientation dyadic

Orientation tetradic

Pelect number

Time

Absolute temperature

Trace of 2

Maier-Saupe nematic potential coefficient

VJ It:

IIUJ

Xi

£1,529.53

Other Symbols

1;,

111,

mb

At

Fluid velocity
Degree of orientation order parameter

Rate of deformation dyadic

Vorticity tensor

Eigenvectors of 2

Three Cartesian coordinate axes

First invariant of 2

Second invariant of 2

Third invariant of 2

Angle between molecular axis and axis x2

Angle between the projection of molecular axis on the plane
[x,,x3] and x, axis

Distribution function of LCPs
Gradient
Dimensionless time

Excluded volume potential

Strain rate

Change of step time

xi

CHAPTERI

INTRODUCTION

1.1 Backgron

In 1971, Dupont produced ultrahigh strength Kevlar fibers from liquid crystalline I
polyamides (see Collyer, 1992). Unfortunately, commercial use of liquid crystalline
polymers (LCPS) has been limited to injection molding of small high precision parts. The
main difficulty in molding a complex three-dimensional part consists in controlling
molecular orientation. 2

The rheological properties of lyotr‘opicI LCPs under simple shear has been studied
extensively (see Larson, 1990; Brave et al., 1993; and Chaubal et al., 1999). Doi
developed a statistical theory for rodlike LCPs that has been used as a starting point for
most theoretical development over the past twenty years (see Doi et al., 1986).
Langelaan and Gotsis [1996] measure the rheological properties of concentrated solution
of Poly (benzyl-L-glutamate). They observed that the first normal stress difference in
simple shear flows for this system is negative, i.e., ‘

§3°T°§r§2°1°§25N1<0~ (Ll)

In the above inequality, 93 represents the flow direction, 92 is the cross flow direction
and I is the deviatoric component of the stress tensor. Larson [1990] used an

approximation of Doi’s theory for the orientation distribution function (see Doi and
Edwards, 1986) and predicted a negative first normal stress difference for concentrated

LCPs suspension in agreement with experiments. Current closure approximations for the

 

‘ Lyotropic LCPs undergo a phase transition as the concentration changes (Larson, 1999)
Thermotropic LCPS undergo a phase transition as the temperature changes (Larson, 1999)

orientation tetradic, which directly influences 2 , predict a positive first normal stress

difference in contrast to experiments and theory (see Larson, 1990).

These studies may developed microstructures in steady shear flows that have
periodic behavior for a range of the Pelect numbers depending on the strength of an
excluded volume potential (see, esp., Chaubal and Lea], 1999). Chaubal and Lea!
showed that as the Pelect number increased, the LC? microstructure based on Doi’s
theory undergoes several dynamic transitions and eventually approaches to a flow
alignment regime. Brave et al. [1993] have shown that the ubiquitously employed
quadratic closure for the orientation tetradic fails to predict the phase flow transition

phenomena anticipated by the underlying Doi’s theory.

1.2 Objectives

Three physical processes influence the behavior of concentrated LCPs: Brownian
diffusion, hydrodynamics drag, and the excluded volume effect that tends to align
macromolecules. The relative balance of these physical effects determines the
microstructure of LCPs in the presence of flow. Model predictions of low-order

statistical prOperties characteristics of the LCPs microstructure are often based on a

moment equation for the orientation dyadic 2 x p p > . The instantaneous orientation

vector 2 represents the relative alignment of a constituent of the dispersed phase. The

solution to the moment equation for < 2 2 > requires knowledge of the local flow field

as well as a closure model for the orientation tetradic < 2 2 2 2 >. Unfortunately, the

widespread use of the moment representation for the microstructure has been limited by

the absence of a practical and accurate closure model that relates the dispersed phase
orientation tetradic to the dispersed phase orientation dyadic. Therefore, the goal of this
thesis is to addresses this fundamental issue of multiphase processing for a class of
physical problems by using a fully symmetric quadratic (FSQ-) closure that retains the
six-fold symmetry and projection properties of the exact orientation tetradic (see Petty et
al., 1999). The second moment is obtained by numerically solving a set of ordinary

differential equations governing the components of 2(1). The instantaneous orientation

states are examined for realizability.

1.3 Methodology

In this thesis, an immiscible is treated as a single dumbbell with a density equal to
the continuous phase. Spatial diffusion of the polymer phase is neglected. The
continuity equation for the orientation distribution is used to develop a moment equation

for the orientation dyadic 2 = < 2 2 >. The solution of the moment equation requires a
closure model for the orientation tetradic <2 2 2 2 >. A fully symmetric quadratic
(FSQ-) closure is used as a closure for <2 2 2 2 >. Figure 1.1 illustrates the focus

topics of this master thesis. Although the development is incomplete and requires further
studies, the methodology developed as a part of this thesis provides a clear and

unambiguous framework for the present work and further improvements.

 

Brownian Diffusion
Brownian-Doi Diffusion
Flow Induced Alignment

 

 

 

l

 

 

Continuity Equation for the Orientation Distribution Function

 

 

l .

 

Moment Equation for the Orientation Dyadic

 

 

 

l

 

Closure for the Orientation Tetradic

 

 

 

l

 

Equilibrium States
Multiple Steady States
Periodic States

 

 

 

 

 

 

Figure 1.1 Focus Topics of This Thesis

CHAPTER 2

THEORY

2.1 Orientation Characterization

2.1.1 Single Polymer Molecule
Suppose each polymer molecule is an axisymmetric particle, like a long rigid

dumbbell. The orientation of such a polymer molecule can be described by the angle 0

and ¢ (or by a unit vector 2) directed along the molecular axis (see Figure 2.1). These

two descriptions are related by,

p, = sin0 cos¢ p2 = c030 p3 = sine sin¢ (2.1)
where p1, p2, p3 are Cartesian components of the vector 2. Because the ‘head’ of the
polymer molecule is assumed to be identical to its ‘tail’, the choice of direction for 2 is

arbitrary. Hence, any description of the orientation statistics must be unchanged by the

following substitution

2—> ‘2 (2.2)
or

9—; “.9 ¢——>¢+n (2.3)
For an axisymmetric polymer molecule, the length of the symmetry axis is L; the

diameter of the effective cross section is d. The aspect ratio IJd determines the

hydrodynamic coupling coefficient A, defined by

2—
k _ (L/d) 1

_ . 2.4
(L/d)2 +1 ( )

X2

I'D

 

 

 

v
35

Figure 2.1 Orientation coordinates for a Single Axisymmetric Polymer Molecule

I'U

 

 

v
35

 

Figure 2.1 Orientation coordinates for a Single Axisymmetric Polymer Molecule

2.1.2 Distribution Function

Polymers molecules in liquid crystalline polymers are unable to align in the same
direction, not even within a very small region. Instead, most polymer molecules have
different orientations from their nearby neighbors. A description of the orientation state,
which must include the orientations of many individual polymer molecules, can be

developed, by introducing the orientation distribution function ‘P(0,¢) . This function
gives the fraction of polymer molecule having orientation coordinates in the range 9 and
0+d0; «p and «bi-dd):

P(0 S 0 S 0 + d0,¢ 5. ¢ 5 d¢) = ‘I’(0,¢) sin 6d0d¢. (2.5)
Since every polymer molecule must lie at some angle, the integration of this function

over the unit sphere is unity (Tucker et al., 1999):

2: rt

j I we, ¢, t) sin 0d9d¢ = = . (2.6)
o-oo=o

Eqs. (2.2) and (2.3) imply that
‘P(2)=‘I’(-2), and (2.7)

‘I’(G,¢) = ‘I’(7t - 9, ¢ + 1:) , (2.8)

2.1.3 Average Orientation Tensors

Consider a small volume of fluid within the polymer/solvent system. The volume
is assumed large enough to contain many polymer molecules, but small enough that the
orientation distribution is spatially uniform. This requires that the average volume to be
large compared to the polymer molecules and small enough compared to the overall

dimensions of the sample. Let B represent any quantity that can be associated with a

single polymer molecule, and let Bk represent the value of B for the km polymer

molecule. The local average of B, denoted by <B>, is defined by
< B > - ii B" (2 9)
N ’ '

where N is the number of polymer molecules in the averaging volume. If pi is any

component of the orientation vector 2 , the average of pi is always zero

< pi > = 0. (2.10)
The second and fourth moments of the orientation distribution function are defined by

33 = <pipj >, (2.11)

aijki= < P1P jPiP. >. (2-12)
with i, j, l, k taking on values 1, 2, and 3 in all possible combinations. These moments

are called orientation tensors. If all the polymer molecules in the control volume are

drawn from a population with the same probability distribution function, then
a. = I‘i’(p)p.p,-dp. (2.13)
a... = I‘I’(p)p.p,-pip.dp- (2.14)

A number of properties are associated with the orientation tensors. First, they

both satisfy the symmetry conditions
aij = aji (2.15)
aijkl = ajiin = akjil = 31de = aikjl = 3mg (2-16)

Because the length of the unit vector 2 is fixed, then

2 Papa =1 (2,17)

It follows that the trace of 2 is unity
tr(a)= 2 21,: 1. (2.18)

Furthermore, the higher order tensors provide complete information about the lower order
tensors inasmuch as

3 3 3 3 3 3
aa=23mi=23m1=23ur =2aiui =2ai318 ‘23:»: (2°19)

ssl :21 pl 3:1 :21 as]

am,I = aim = etc. (thirteen projections) (2.20)

2.1.4 States Representative of the Average Orientation Tensors

Figure 2.2a shows an isotropic (or 3-D random) orientation state with equal
numbers of polymer molecules in all directions. This corresponds to an orientation tensor
where the off-diagonal entries are all zero and the diagonal entries are all equal. Eq.

(2.18) shows that this must be

1— o o
3
3i; T 0 C;- 0 . 3-D random. (2.21)
0 0 l-
- 3 -

 

 

If all the polymer molecules lie in the 2-3 plane, then p1 = 0, and an = an = an = a23 = 0.
For a random orientation in the 2-3 plane (see Figure 2.2b), the second order orientation

tensor is

 

 

 

 

 

 

 

 

 

 

 

 

 

O
l \ \
/ 3
/\|| .
1
(a)
2
\ '\—/ ‘
/ /\
’3
(b)
2
_- 3

 

 

 

(C)

Figure 2.2 Example of Orientation States

10

ii

 

 

0

0 , planar random. (2.22)
.1.

2

0 0 0
aii = 0 0 0 , perfect alignment (2.23)
0 0

When the off-diagonal elements of aij are zero, the diagonal elements describe the

distribution of the polymer molecules along the 91,92 , and g3 directions.

2.1.5 Realizable Orientation States

The diagonal components of 2 must be non-negative and the Schwartz’s
inequality must hold for the off diagonal element:

Ian]2 5 afiajj , (2.24)
The above inequality holds for fixed values of I and j. The orientation dyadic has three
non-negative eigenvalues a,,a2 , and a,. The associated eigenvectors are (xll ,xn, x13) ,

(x2,,x22,x23),and (x3l,x32,x33). The three invariants of 2 are

I. =tr(2)=al +a2 +23 =1, (2.25)
II. = tr(2-2) = (a1)2 +(a2)2 + (a3)2, and (2.26)
111, = tr(2 - 2 - 2) = (11,)3 +(a,)3 +(a,)3. (2.27)

11

The anisotropic orientation dyadic 2 is defined by the following decomposition

of the orientation dyadic

3=g+y3 (2.28)

The order parameter 8 used to characterize the degree-of-orientation is defined as
3 ll 2
S = [22 : 2) . (2.29)

The tensor 2 also has three eigenvalues b1 , b2, and b3 . Because 2 = 21 , I = IT , it

follows that 2: 2'. From Eq. (2.18), it implies that tr(2) = o. The invariants of 2 are:

I, = my) = h, + h2 + b, = o, (2.30)
11, = tr(2-2) = (1),)2 +(h,)2 +(b3)2, and (2.31)
HI. = «(2.22) = (1),)3 + (1),)3 + (h,)’. (2.32)

The eigenvectors 5i and the eigenvalues ai of the orientation dyadic 2 are

defined by the following equation:

2-_l_t_i =ai_xi. (2.33)
With 2 = 2 +1/ 3 , Eq. (2.33) can be rewritten as

(31'1/3).’£i :21“ (2.34)
Eq. (2.34) shows that the eigenvectors of 2 are the same as the eigenvectors of 2 . The

eigenvalues of 2 and 2 are related by

b. = a. ——, (2.35)

12

The two nontrivial invariants of 2, defined by Eqs. (2.31) and (2.32), can be
rewritten in terms of the eigenvalues of 2 :

II, = (a, -1/3)2 +(a2 -1/3)2 +(a3 -l/3)2, (2.36)

m, = (a1 --1/3)3 +(a2 -1/3)3 +(a, -1/3)3. (2.37)
The two invariants of 2 must fall within a so-called realizable region (see Parks et al.,

1999 and, esp., Lunrley, 1978). Figure 2.3 shows the realizability region. Orientation

states, within this region sre realizable inasmuch as the eigenvalues of 2 are real and non-
negative.

For uniaxial alignment (1-D) (see Point A of Figure 2.3)

a,=0,a2=0,and a3=l. (2.38)
From Eqs. (2.36) and (2.37), it follows that at the perfect alignment state

IIb = 2/ 3 , (2.39)

111,, = 2/9. (240}
For 2D planar isotropic state (see Point C of Figure 2.3)

a,=0,a2=l/2,and a3=1/2. (2.41)
The anisotropic invariants for this state are

IIb = 1/ 6 , (2.42)

IIIb_ = -1/ 36 (2.43)
The eigenvalues of 2 for a 3D isotropic orientation state (see Point E of Figure 2.3) are
all equal to 1/3. Therefore,

a, =a2 =a3 =1/3. (2,44)

13

 

 

 

 

 

 

 

 

 

 

 

 

 

B. Planar \
Anisotropic A. Nematic States
States
C. Planar \ . . .
. a1 Ax tn
Isotropic / F Axr lsymme c
States
Set of Realizable
Orientation States
D. Planar Axisymmetric
E. Isotropic States
/ ’
0 I111,
Invariants of Eigenvalues of the orientation
Orientation State tensor b tensor 3
Ilb 111., a. a2 a3 Notes
A. Uniaxial Aligment (1D) 2/3 2/9 0 0 1
B. Planar Anisotropic (2D) 11., = 2/9-1-2111b 0 a l-a a =[0,1/2]
C. Planar Isotropic (2D random) 1/6 - 1/36 0 l/2 1/2
D. Planar Axisymmetric 11b: «Jug/6);”3 1-2a a a a=[1/3,1/2]
E. Isotropic (3D random) 0 0 1/3 1/3 1/3
F. Axial Axisymmetric 11,, = (5(111bl(5)2’3 a a l-2a a=[0,l/3]

 

 

 

 

 

 

 

Figure 2.3 Realizable Set of Anisotropic Orientation States (Parks et al., 1999)

14

 

The anisotropic invariants for this state are

uh = 0 (2.45)

m, =0. (2.46)
For planar anisotropic states (see Line B of Figure 2.3), the eigenvalues of 2 are

a,=0,a2=a,and a3=1-a. _ (2.47)
The anisotropic invariants on Line B are related by

IIb = 2/ 9 + 2IIIb. (2.48)
For planar axisymmetric states (see Line D in Figure 2.3), the eigenvalues of 2 are

a,=a,a2=a,and a3=l-2a. (2.49)
The anisotropic invariants on Line D in Figure 2.3 is given by

IIb = [(—IIIb I 6)"3 ] - 6. (2.50)
For the axial axisymmetric states (see Line F on Figure 2.3), the eigenvalues of 2 are

al=a,a2=a,and a3=1—2a. (2.51)
The anisotropic invariants on Line F in Figure 2.3 are

11, = [(1111, /6)2’3]-6. (2.52)

2.2 Continuity Equation
Figure 2.4 shows an arbitrary area A on the unit sphere. The area is bounded by

the contour line C. The rotary flux of orientation states across the boundary C relative to

a material frame of reference is p‘I’ . The fraction of orientation states in surface A is

constrained by the following balance equation:

15

 

'T]

4*.

1922

' d
— ‘P ~nds=— ‘1' ,tdA. 2.53
{(13)- (all (g) ( )
The surface divergence theorem (see Deen, 1998) given by
j(p‘Y) ads = H— (p‘P)dA . (2.54)
c — A 82 '-

can be used to rewrite Eq. (2.53) in term of surface integrals. This implies that the

balance equation on orientation states is equivalent to the following evolution equation

for ‘1’:

DP 3

For a spatially homogeneous LCP mixture, the substantial time derivative equals the

partial time derivative (Bird et al., 1960). Therefore, the continuity equation for the

orientation states is

3‘1’ 3 '
.3. _ -31; . (2‘1!) . (2.56)

where (53—) - is a surface gradient operator and p is the angular velocity of the LCP
p -

constituent. The rotary flux can be written as the sum of a convective and diffusive flux

with the result that
aw a ' a ' '
— = -— O W _ — O - W . 2.57

In the above equation, 21 is the angular velocity due to LCP/fluid interactions (Jeffrey,

1922). For liquid crystalline polymers, the rotary diffusive flux has two important.

16

Unit Sphere

 

Contour Line C

Figure 2.4 Distribution Orientation Vectors on the Unit Sphere

17

_o--.

 

 

 

 

contributions: Brownian motion and excluded volume effects. .
2.3 Constitutive Models

Polymer molecules that form liquid-crystalline phases in solution generally have
rigid backbones and consequently have rod-like or disk-like shapes. The model is built
based on the solution of polymer molecules as an ensemble of rigid dumbbells suspended
in a Newtonian solvent (Brave et al., 1993). The rigid dumbbell, which is equivalent to a
rigid rod as a model for rod-like macromolecules (Doi, M., 1981), accounts for the ability
of the polymer molecules to orient in the flow direction and ignores the stretching and
bending motions. The analysis hereinafter assumes that the LCP density is the same as
the bulk fluid density and that the liquid crystalline polymer solution is a continuous
phase. Moreover, the inertial force acting on the LCP dumbbells is negligible and has the

fluid is isotropic.

2.3.1 Rotary Diffusive Flux

In liquid crystalline polymers, the rotary diffusive flux depends on Brownian
diffusion and the excluded volume effect that tends to align the macromolecules. The
Brownian force is caused by bombardment and jostling of dumbbells by the solvent
molecules (Brave et al., 1993). The excluded volume effect is described in terms of a

mean field Maier-Saupe (MS-) potential (Doi, 1981; see p 358 Doi and Edward, 1986):

2:23

ch 5 nkT j [13(2, 2)‘P(2) sin édédi). (2.58a)
0 0

where n is the number of LCPs per unit volume, k is the Boltzman constant, and T is the

absolute temperature. The B(2,2) represents the effective volume of an LCP constituent

18

in:

 

 

in the presence of other LCP particles. ‘P(2) sin 0d0d<l> represents the fraction of LCP

particles having orientation angles between (0; (1)) and (0 + d0,$ + dill).
The MS-potential assumes that

. (2.58b)

I'O>

3(2.§)=Bm -(B.,... -B...)g 13:2
where 8m and [3m are positive coefficients depend on the physicochemical properties
of the LCP suspension, but not the orientation state.

If the orientation vector 2 of the test particles is orthogonal to the orientation
vector of the field particle, then B = B1m . On other hand, if 2 is collinear with 2(i.e.,

nematic), then B = Bm . A combination of Eqs. (2.58a) and (2.58b) yields the following

representation for the MS-potential:
(D .. ..
E = an - n(Bm - Ban )2 2 :< 2 2 >. (2.580)

The dimensionless nematic strength of the LCP mixture is defined as
U a MB“m — [3m ). (2.58d)
U is proportional to the number concentration of the LCP constituent and is directly

proportional to the excluded volume, (Bum — Bm ). As the polymer concentration

increases, the interaction spacing between molecules decreases, and the nematic strength
U increases. In this limit, the polymer molecules have a tendency to self-align in one
direction. Brownian diffusion drives the LCP system to an isotropic state and the
excluded volume effect causes the system to approach the nematic states (see Point A of
Figure 2.3). The linear contribution of these two factors rotary yields a model for the

rotary diffusive flux:

19

' - 3‘? 3 ch
(2 - g, )‘P - 43,133+ 53%)“ (2.59)

The transport coefficient Dr is the rotary diffusivity and has unit of reciprocal time. This

parameter is constant for this study.

2.3.2 Rotary Convective Flux
The rotary convective flux is produced by the hydrodynamic drag experienced by
the dumbbell as it rotates through the solution. The following stems from a balance of

angular momentum on an axisymmetric LCP molecule (Jeffrey, 1922)
2. =E-2+Ml-2 94.3.2): 0-60)
In the above equation, 2. is defined by Eq. (2.4), 2 and ail—V are the rate-of-strain and

vorticity tensors, respectively:

1' 'T
V2+V2 ;WE(V2—Vp_

S
" 2 ‘3 2

). (2.61)

2.4 The Smoluchowski Equation

Combining Eqs. (2.56)—(2.59), the Smoluchowski equation for LCPs can be

written as:
3‘? 8 a arr 3 ch
_ _. .v _ :V ‘1’ =D —- — ‘I’— — . 2.62

20

In this thesis, Eq. (2.61) is analyzed by using the method of moments. An equation for

the orientation dyadic 2 (E< 2 2 >) follows by multiplying Eq. (2.62) by the orientation

p p and integrating the result over the unit sphere. This yields (see Chaubal et al., 1997)

=V27.2+2.V2-2VQT :<2 2 2 2>—6D-,(2--:l;l)

9h?

+6UD,(2-_—2:<2 2 2 2>). (2.63)
For simple shear flows, V2 = @293 , where 7: (V2 :V11_T)I2 , Eq. (2.63) can be made
dimensionless by dividing both sides of the equation by 6 D, :
da 1
-d_; = Pe(§3§2 '2+2'§2§3 -29392 KB 2 B E >)-(3'§D

+U(2-2-2:<2 2 2>). (2.64)

I'D

In the above equation, The Pelect number Pe is derived as
_ i
Pe — — and (2.65)

the dimensionless time t is defined as

t=6fin. com
Eq. (2.64) governs the relaxation of the LCP microstructure as characterized by the

orientation dyadic 2. The interactions of three physical effects control the behavior of

the LCP model: rotary convection, Brownian diffusion, and excluded volume. The
competition among these three physical effects yields a complex dynamic response that

includes multiple steady state and periodic behavior. Unfortunately, Eq. (2.64) contains

21

two statistical orientation moments of the microstructure. A closure model for the

orientation tetradic is needed for closure and this is the focus of the next chapter.

22

CHAPTER 3

CLOSURE MODEL FOR THE ORIENTATION TETRADIC

3.1 The Quadratic Closure Model

The simplest way to approximate the tetradic orientation is to use the quadratic
closure adopted by Doi (1981):

<2222>=<22><22>=33o (3.1)
Eq. (3.1) maintains the trace of the trace of the orientation dyadic and is exact for nematic
states. However, the quadratic model fails to produce director tumbling and wagging of
liquid crystalline polymers in simple shear flows, as predicted by the exact Doi’s theory
(Brave er al., 1993). More significantly, and perhaps related is the observation that the

quadratic closure does not fulfill the six symmetry conditions or the six projection

properties of the orientation tetradic (see Section 2.1.3).

3.2 Linear Model of Hand
In 1962, Hand proposed a linear representation for the orientation tetradic that
satisfies all the projection and symmetry properties of the exact orientation tetradic. This

closure model, which is accurate near the isotropic state (see Figure 2.3), is given by:

l
< PinPltPl > = —-3_S'(Iijlkl +Iik1ji +Iilljk)

+‘;'(ah-Iu +3:th1 +331“ +au1ij+afilik +ajklu). (3.2)

23

3.3 The Bingham Distribution
Another approach to the closure problem is to use an explicit form for the

orientation distribution function. For example, the Bingham distribution is defined by

““22“?
z

 

L113(2) = (33’)

where Z is a normalization factor, and 2 is a symmetric dyadic valued operator with

non-negative eigenvalues. This approximate approach, albeit limited, avoids the closure
problem this arises in the method of moments and the realizability problem by using an

approximate distribution function.

3.4 The Fully Symmetric Quadraa'c (FSQ) Closure
The fully symmetric quadratic (FSQ-J closure (see Parks er al., 1999) retains all

six symmetry and projection properties of the exact orientation tetradic and is defined by
<EBBB>=CI<BBBB>I+C2<BBBB>2 0-4)
where < 2 2 2 2 >1 is given by Eq. (3.2) and < 2 2 2 2 >2 is defined as follows:

3

<2 2 2 2>2= 352:20,].1” +1215, +Inlkj)+aijau +aikajl ~1-aila,‘j

‘ gm} (2 ' 2):! + 1&(2 ' 5911 + In (9. ‘ 3919'

+ (2 -2)ijIn + (2 -2)ik Ijl +(2-2)u1kj]. (3.5)
The two tetradics on the right-hand-side of Eq. (3.4) individually satisfy all the symmetry
and projection properties of the exact orientation tetradic. Therefore, the first-order and

second-order scalar coefficients in Eq. (3.4) must be related by

24

Cl +c2 =1. (3.6)

These coefficients may depend on the invariants of the orientation dyadic, but for this
thesis C2 = g . A justification for this selection is given in Appendix A.

The following symmetric fourth order tensor is used in the FSQ-closure:

i(-)j S[_A_ 2] = Aura,l +A,,Bj, Maura,j +BijAu +B,Aj, +BuAki (3.7)

Eq. (3.7) remains the same if the indices i and j are exchanged inasmuch as Aij and Bi).

are symmetric. Eq. (3.4) also remains the same for the following five operations:

in SIA 2]: Aiji, +A,,Bj, +Am13ij +Biju +BuAj, +13,,Aij (3.8)

iHl Sig: 2]: Auraik +AkBij +A,Bj, “3,11,ni 4.13,».ji +B,A,,. (3.9)

ij SD; 2] = Aan + AnBu + AuBkj +BnAn +BijA,‘l +BnA,‘j (3.10)
j<->l S[2 2] = AflBkj +Aan +Athn +BnAkj +BikAlj +BijAk1 (3.11)
k<—>l S[2_ 2] = AijB“, +AnBjk +AmBIj +BijAk +BnAjk +BikAb. (3.12)

where H indicates the exchange of indices.

An important feature of the orientation tetradic is the following six projection properties:
<pipj > = < pspspipj > = < papipspj > = < papipjps >
= <p.p.p.p,- > = <pip.p,-p. > = <p.p,p.p. > =8. (3-13)

The fourth order tensor < 2 2 2 2 > satisfies all the six symmetry properties and the six

projection conditions.

25

CHAPTER 4

BROWNIAN DIFFUSION

4.1 Introduction
With Pe = 0 and U = 0, Eq. (2.64) reduces to the following equation, which
governs the relaxation of an initial anisotropic microstructure by Brownian diffusion:

da

1
—I-a. 4d
my: ()

For this case, the steady state microstructure is characterized by an anisotropic orientation
dyadic with II,, = 0 and 111,, = 0 (see Point E on the anisotropic invariant diagram, Figure

2.3):
as 3; . ( . )

As previous noted in Chapter 3, the components of the exact 3D random orientation

tetradic at Point E is given by
I
< PinPltPl >isotropic = Baijlkl +1ilthl + Iilij) - (43)

Clearly, Eq. (4.3) reduces to Eq. (4.2) by contracting any two indices in Eq. (4.3). Eq.
(4.3) is also invariant to the exchange of any two indices. These six-fold symmetry and
projection properties are satisfied by the Hand-closure as well as by the FSQ-closure (see

Chapter 3).

26

4.2 Relaxation of Anisotropic States by Brownian Diffusion

The solution to Eq. (4.1) subject to any realizable initial condition 2(0) is given

g<r>=§1+(g(0>—§pexp<-t). (4.4)

As 1: —> on , 2 approaches the equilibrium state defined by Eq. (4.2). Eq. (4.4) shows that

the anisotropic orientation dyadic (i.e., the structure tensors) for Brownian diffusion is

given by

33(1) = 2(0) exP(-T) - (4.5)
The relaxation of the two invariants 11b and 1m, (see Section 2.1.3) depend on

the initial conditions and the dimensionless time 1::
use)atr<2sg)=ns(0>exp(—2t). (4.6)

III}, (1:) E tr(2 2 - 2) = IIIb (0)exp(-31:) . (4.7)

The following equation is obtained directly from Eqs. (4.6) and (4.7) by eliminating the

independent variable t

3/2
H (t)

m =III o b .
13(7) b( {Hb(0)]

 

(4.8)

Figure 4.1 shows that the Brownian relaxation of an initial anisotropic nricrostructure to a
3D random orientation follows the curves defined by Eq. (4.8). As expected, IIb and In,

remain in the realizability region (see Section 2.1.5).

Figure 4.2 shows how the components of the orientation dyadic relax from the
nematic state defined by

27

0.7 l I l l r 1

 

  

 

 

 

0.6 - -
0.5 - .
0.4 - -
Hb
0.3 - .
3/2
III(I) = III(0 lK—Q
0.2 " / 11(0) ..
0.1 .
Isotropic (1 == 00)
O ‘ . l M I l L
-0.05 ' 0 0.05 0.1 0.15 0.2 0.25 0.3
1111,

Figure 4.1 Relaxation of Anisotropic Invariants by Brownian Diffusion (U = 0, Pe = 0)

28

 

 

 

 

 

 

14

Figure 4.2 Relaxation of Nematic States by Brownian Diffusion (U = 0, Pe = 0)

29

3(0) = 9.393 . (4-9)

As indicated, the equilibrium state is attained on a time scale 1: E 6tDr == 7 . Figure 4.3

illustrates the temporal relaxation of an initial 2D planar isotr'Opic state defined by

g(0)=%§2§2 +£23.23. (4.10)

For this case, the equilibrium isotropic state is also attained on a time scale for which

4.3 Conclusions

For U = 0 and Po = 0, the microstructure relaxes to the isotropic equilibrium state

by Brownian diffusion. The steady state orientation dyadic and the steady state

orientation tetradic are given by Eqs. (4.2) and (4.3), respectively. The following

conclusions stem directly from an analysis of Eq. (4.1).

(i)

(ii)

(iii)

In the absence of flow alignment (Pe = 0) and nematic alignment (U = 0), all
realizable anisotropic microstructures on the line ABC of Figure 2.3 relax to
an equilibrium isotropic state on a time scale comparable to 1: a 6tD, == 7 (see
Figure 4.1).

A microstructure on Line F of Figure 2.3 remains on this boundary during the
relaxation process (see Eq. (4.8) with 111‘, (0) > 0). Thus, a prolate
microstructure remains prolate during the relaxation to equilibrium.

A microstructure on Line D of Figure 2.3 remains on this boundary during the
relaxation process (see Eq. (4.8) with 1111, (0) < 0). Thus, an oblate

microstructure remains oblate during the relaxation to equilibrium; and

30

(iv) If IIIb(0)=0, and IIb(0)>0, then Eq. (4.7) implies that IIIb(1:)=0for

OStSm.

31

 

 

    

Isotropic State

 

 

.A A A A

mo

 

 

l l l

8 10 12

Figure 4.3 Relaxation of Planar Isotropic State by Brownian Diffusion (U = 0, Pe = 0)

32

14

CHAPTER 5
BROWNIAN-DO] DIFFUSION

5.1 Introduction

With Pe = 0, Eq. (2.64) reduces to the following equation

3"18‘

=<§;-2>+U<2-g-<2 2 2 2>=s> (51>

The second term on the right-hand-side of Eq. (5.1) accounts for the natural tendency for
liquid crystalline polymers to align due to the excluded volume effect (see Section 2.3.1).
The dimensionless group U is independent of the local orientation state, but depends on
the volume concentration and physicochemical properties of the two phases polymer
mixture. The component equations associated with Eq. (5.1) are given in the Appendix
B. The isotropic state (see point B on figure 2.3) is a steady state solution to Eq. (5.1) for

all values of U inasmuch as

da 1
[—‘] =0 for a=-I (52)
d1: 13 '3 3=

This conclusion stems from the normalization condition on the orientation vector p ,

= -1, (5.3)

33

The purpose of this chapter is to develop an analysis of Eq. (5.1) for 0 < U < 30.

The FSQ-closure is employed with C2 = 1/3 for all orientation states (see Section 3.2)

5.2 Relaxation of Anisotropic States by Brownian-Doi Diffusion

An earlier analysis of Eq. (5.1) with a quadratic closure for the orientation tetradic
showed that multiple equilibrium states occurred for U > 2.7 (Brave et al., 1993).
However, a numerical solution of the Smoluchoski equation (see Eq. (2.65) with g :-=. 0)
indicates that multiple equilibrium states only occur for U > 5 (see p. 68, Larson 1999).
Eq. (5.1) subject to the FSQ-closure also has multiple equilibrium states. Although the
isotropic state is clearly a solution to Eq. (5.1) for all U .>. 0, this state is only stable for U

smaller than some critical nematic strength Uc . The critical value of the nematic
strength (Uc ), above which multiple equilibrium states occur, is sensitive to the closure

approximation used for the orientation dyadic.

It follows directly from Eq. (5.1) that the nematic state (see Point A Figure 2.3),

defined by
g = 9.3.3.3 9 and (5.4)
< 2 2 2 2 > = 9.3939393: (5.5)

a 2. (5.6)

Thus, Eqs. (5.1) and (5.6) show that Brownian diffusion controls the short time response

from an initial nematic state:

93 --1-1— -ee #9. (57)
do 3= "33 '

The Brownian diffusion term in Eq. (5.1) has a tendency to reduce the alignment caused
by the excluded volume effect.

For the FSQ-closure, it follows that (see Appendix B for the development details)

<2 2 2 2>=3=
1 l 3 4
-— —— — I+— +- 5.8
3“ 35 73 3)= 353 73 3] ( )
l 2 2 39 10 6
+— —a --a-—tr22-2I+ —a: a —— +—

Eq. (5.8) and the condition that tr(2) = 1 imply that

tr(<2'222>:2)==:2. (5.9)
This result implies that solutions to Eq. (5.1) satisfy the normalization condition
tr(2) = 1 for 1: 2 0since

52-22)]: 0, for 1:2 0. (5.10)

Eq. (5.1) with < 2 2 2 2 >: 2 defined by Eq. (5.8) was integrated numerically for

fixed values of U in the range 0 < U < 25 (see Appendix E for a discussion of the
algorithm). The following initial conditions for the microstructure were selected from the
realizability boundary Figure 2.3

2(0) = all§l§1+(l’all-a33)§2§2 +3‘339393- (511)

35

The component a 33 was selected from the range
5 a33 S 1. (512)

For a33 = 1 and a1 1 = 0 , the initial microstructure is at the aligned nematic state (Point A
on Figure 2.3); for a33 = 1/2 and 311 = 0, the initial microstructure is 2D planar
isotropic state (Point C on Figure 2.3); for a33 =a11=1/3, the initial microstructure is at
the 3D random isotropic state (Point E on Figure 2.3).

For U = 10, Figure 5.1 shows that the final steady state depends on the initial

. . l 1 .
conditions. For — < a33 = all 5 — , the steady state microstructure corresponds to an
3 2

oblate state with 11,, =0.0915 and 111,, =-0.0113. For other initial conditions that
satisfy Eq. (5.11), the equilibrium microstructure is a prolate state with HI, = 0.4131 and
111:, = 0.1084 . The isotropic state (IIb ,IIIb) = (0,0) is an unstable equilibrium state.

For U =-' 3, the isotropic state is stable. This is illustrated by Figure 5.2, which
shows how the components of 2(1) relax from an initial nematic state, 2(0) = g3 g3 .

Although the short time response is similar to Brownian diffusion (i.e., U = 0), the
relaxation to equilibrium is clearly slower for U = 3 and takes about twice as long as
Brownian diffusion. Note that equilibrium is attained 6tD, z 14 for U = 3 and Pe = 0.
The relaxation of a 2D planar isotropic microstructure to a fully 3D isotropic structure by
Brownian and Doi diffusion for U = 3 is illustrated by Figure 5.3.

For U > Uc , the equilibrium states are not isotropic. However, all the

equilibrium states occur on the line DEF of the realizability region illustrated by

36

0.7 l I I I l I

 

0.5 -

   
    

0,3 . Stable Prolate Equilibrium State

 

 

 

0.2. - -
Stable Oblate Equilibrium State
0.1 ~ ..
Unstable Isotropic State
0 I J L l l
-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
1111,

Figure 5.1 Relaxation of Anisotropic Invariants by Brownian-Doi Diffusion
(U = 10, Fe = 0)

37

 

 

 

20 25

Figure 5.2 Relaxation of a Nematic State by Brownian-Doi Diffusion (U = 3, Pe = 0)

38

30

 

 

 

Figure 5.3 Relaxation of a Planar Isotropic State by Brownian-Doi Diffusion
(U = 3, Pe = 0)

39

30

Figure 2.3. The order parameter S, defined by Eq. (2.29) is used to characterize the
degree of orientation of the equilibrium states. For an oblate and prolate states, Eq.
(2.29) reduces to the following expressions (see Appendix C). Figure 5.4 compares the

equilibrium microstructures predicted by the FSQ-closure with those predicted by the

quadratic closure.
3 l
S = —a3 -—, prolate state (5.13)
2 2
3 1
S = -2-al -§ . oblate state (5.14)

Both closures show a trifurcation of equilibrium states above a critical value of

dimensionless coefficient U. For U < Uc 5 2.7 , the quadratic closure predicts that the

isotropic state is stable to finite disturbances. On the other hand, the FSQ closure yields

stable isotropic states for U < UO 5 4.1. The two closures show significant qualitative

differences in the transition to multiple equilibrium states. For example, the quadratic
model anticipates a range of unstable prolate states (S > 0) for 2.7 < U < 3.0. The FSQ
closure does not show this subcritical trifurcation phenomena. Furthermore, the FSQ
closure predicts that the oblate states (i.e., S < 0) are stable to finite oblate disturbances.
On other hand, Brave et al., 1993 indicate that the oblate state is unstable if the quadratic
approximation is used as a closure for the orientation tetradic.

Figure 5.5 shows the position on the realizable diagram of the multiple

equilibrium states predicted by the FSQ closure for U > Uc . A prolate equilibrium state

attracts all initial states on the boundary B and the boundary F (see Figure 2.3). The
complementary oblate state attracts all initial states on the boundary D. Although the

isotropic state is a solution to Eq. (5.1), this state is unstable to any infinitesimal

4o

disturbances for U > Uc s 4.1. Table 1 gives components of the orientation dyadic and

the anisotropic for the equilibrium states corresponding to U in the range 4.15 U S 24 .

41

 

 

 

 

 

 

 

  
  

   
 

 

 

 

 

1
Stable Prolate
- U tabl ' ‘
0'5 \ Unstable Prolate ns e rsotroprc
S ‘ / /
o "
\ ‘~ - .. _ . Unstable Oblate
Stable Isotropic ' ‘ --------- . _____ _ . _
'0.5 1 T r I I I I
o 1 2 3 4 5 6 7 8 9 10
U
1
Stable Prolate
0.5 -
\ Unstable isotropic
S / Stable Oblate
0 . .. . . . . . . 7/. ......... .
Stable Isotropic
'0.5 I T I I I I I I7 I j
o 1 2 3 4 5 6 7 8 9 10

Figure 5.4. Comparison of Equilibrium States Predicted by the FSQ-Closure (lower
graph, this research) and Quadratic Closure (upper graph, Brave et al., 1993)
forPe=0andOSU510

42

2500 r

2000-

1500 -
H.3104

I

1000

500-

 

 

Prolate

Oblate

 

a3

11.310“ 8*103

a3

HIfilO

'l S"‘103

 

4.5

0.425

126

138

0.335

-17.3 6

 

U!

0.612

1159

417

0.357

 

 

 

0.722

 

 

2270

583

 

0.389

 

 

 

 

 

Prolate States 5

 

    
 

Oblate States

 

 

 

 

 

 

-lOO

200

111,3104

Figure 5.5 Location of Multiple FSQ-Equilibrium States on the Realizable Diagram
for 4.5 S U S 6

43

Table l Equilibrium States Predicted by the FSQ Closure (4.1 S U S 24)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U equilibrium state 111, 1111, an an a33
4.1 stable isotropic 0.0000 0.0000 0.3333 0.3333 0.3333
stable prolate . 0.1159 0.0161 0.1939 0.1939 0.6122
5 stable oblate 0.0033 -0.0001 0.3569 0.3569 0.2862
unstable isotropic 0.0000 0.0000 0.3333 0.3333 0.3333
Stable prolate 0.4131 0.1084 0.0709 0.0709 0.8581
10 stable oblate 0.0915 -0.0113 0.4568 0.4568 0.0863
iunstable isotropic 0.0000 0.0000 0.3333 0.3333 0.3333
stable prolate 0.4998 0.1442 0.9105 0.0447 0.0447
15 stable oblate 0.1345 -0.0201 0.4831 0.4831 0.0339
unstable isotropic 0.0000 0.0000 0.3333 0.3333 0.3333
stable prolate 0.5422 0.1630 0.9345 0.0327 0.0327
20 stable oblate 0.1569 -0.0254 0.4950 0.4950 0.0099
lunstable isotropic 0.0000 0.0000 _ 0.3333 0.3333 0.3333
stable prolate 0.5632 0.1725 0.9461 0.0270 0.0270
24 stable oblate 0.1683 —0.0282 0.4995 0.4995 0.0001
unstable isotropic 0.0000 0.0000 0.3333 0.3333 0.3333

 

 

5.2 Conclusions

In the absence of flow, the LCPS microstructure relaxes to an equilibrium state by

Brownian-Doi diffusion. The following conclusions stem directly from an analysis of

Eq. (5.1):
(i)

(ii)

(iii)

0‘!)

(V)

Multiple equilibrium states are predicted by the FSQ-closure. Stable
isotropic states are predicted for U 5 UC =4.l. For U > Uc, the isotropic
state is unstable. Complementary stable anisotropic oblate and prolate states
occur for U > 4.1 (see Figure 5.4).

Oblate equilibrium states are predicted when the disturbances are on the
prolate states.

Prolate equilibrium states are predicted when the disturbances are on the
planar anisotropic and prolate states (see Figure 5.1).

As the nematic potential coefficient U approaches to infinity, the equilibrium

state becomes nematic (i.e., S —>1 and 11,, = 2/ 3) and planar isotropic (i.e.,
8—); and 11,, =1/6) in term of the order parameter S as well as to the

nematic states (IIb = 2/ 3) (see Figure 5.4)
There are significant qualitatively differences in the transition to multiple
equilibrium states predicted by FSQ-closure and the quadratic closure. The

quadratic closure gives Uc = 3.1 (Brave et al., 1993), while the FSQ-closure
predicts that Uc= 4.1 (see Figure 5.4), and reported by Larson (see p. 68,

Larson 1999) is Uc= 5.0.

45

(vi) Figure 5.1 shows that the FSQ-closure gives a realizable model for
Brownian-Doi diffusion for realizability initial conditions defined by Eq.

(5.11).

46

CHAPTER 6

FLOW INDUCED ALIGNMENT

6.1 Introduction

In the presence of simple shear flow (i.e., V2 = 79293 , y.=. constan t), Eq. (2.64)

becomes

+U(2-2—-<2 2 2 2>:2). (6.1)

The first term on the right-hand—side of Eq. (6.1) accounts for the interaction between the
dispersed phase and the continuous phase (see Section 2.3.2). The dimensionless group
Fe and the dimensionless nematic potential coefficient U do not depend on the local
microstructure. The purpose of this chapter is to develop an analysis of Eq. (6.1) for the
following initial conditions

2(0) = 9393 , nematic state (6.2)
1 l . .
2(0) = 3219, +3993 , 2D planar isotropic state (6.3)

The scope of this exploratory study is limited to the FSQ closure for the range of
0 S U S 30 and 4 < Pe S 100 for the FSQ closure. An earlier analysis of the
Smoluchowski equation showed periodic orientation states at low values of Pe and steady
state behavior for high Pelect numbers (see Larson and Ottinger, 1991 and Chaubal et al.,

1998). An analysis of Eq. (6.1) by Brave et al., [1993] showed that a quadratic closure

47

for the tetradic orientation fails to predict the flow transition phenomena anticipated by

the Smoluchowski equation.

6.2 Flow Induced Alignment

Eq. (6.1) was integrated numerically using a second-order Runge-Kutta algorithm
(see Appendix F). For this study, the initial condition selected for the orientation dyadic
was either the nematic state or the 2D planar isotropic state (see Eq. (6.3)). For simple
shear flows, the nontrivial components of the orientation dyadic include the diagonal

components and the symmetrical off—diagonal correlations a23 (1:) and an (t):
g = 311009191 +322(T)§292 +333 (1)2393
+ 2123 (1)9293 + 2132 (”E392 ’ (6.4)
The other components of 2 remain zero as the microstructure relaxes to its asymptotic

state. The final state may be either periodic or steady, depending on the value of U and
Fe. Tire Schwartz’s inequality is monitored in the computation to ensure the model is

realizable.

Tumbling Microstructure

With U = 30, and Fe = 25, the FSQ-closure predicts that an initial nematic state
relaxes to a periodic state. The orientation director (see Section 2.1.4) rotates through an
angle of 360° and tumbles about 93, the flow direction. The angular velocity of the
director is collinear with the vorticity of the continuous phase. Figure 6.1 shows the

behavior of components of the orientation dyadic and components of eigenvector

48

 

 

 

 

1.2
l u... I... a-.- —-.- ..
W s‘r-om¥r--.\ur-
0.8 - . -
333
0.6 ' .
0.4 " ‘
ai-
J a
0.2 - 22 , ..
. \. \.. 2 ' A f ,
O i\ ‘3‘: 1i - \\. i i 3
'0-2' l i i i i l l
'0'40 5 10 15 20 25 30
T

 

 

 

 

 

Figure 6.1 Tumbling Microstructure. Components of the orientation dyadic and its
eigenvectors (FSQ-closure with U = 30, Fe = 25, and 2(0) = 9323)

49

associated with 2(1). The dyadic component an stays very close to its initial value zero.

Hence, the eigenvector component xll is approximately equals to zero. Furthermore, the

component X31 of the director 11, defined by Eq. (6.5) below, is identically equal to zero.

This indicates that the director rotates in the shear plane spanned by the flow direction
23 and the cross-flow direction 92 . The eigenvector vector corresponding the largest
eigenvalue is called director of the microstructure. This designated as a3. This
asymptotic periodic state of LCPS is referred to as a tumbling microstmcrure.

There are three eigenvectors associated with the orientation dyadic. The
eigenvector 53 corresponding to a3 (Note 0 S al S a2 S 83) has the following
presentation

n5 33 = x31.91"”‘3292 +X33E3- (6'5)
and the angle between the director and fixed flow direction 93

cos(x) = g - e3 = x 33 . (6.6)

In figure 6.2, the director component x33 decreases slowly from 1 to 0.8, rapidly

fall to —1, rises up to about 0.8 very quickly, and then approaches to unity slowly to
complete a cycle. The rotation direction of the director 11 is shown in the lower figure of
Figure 6.2. The upper figure of‘Figlne 6.2 shows that a periodic state with the period

IP = 4.125 is obtained after transition dimensionless time 6tDr -- 2 .

Wagging Mierostructure
For U = 30 and Pe = 38, the FSQ-closure predicts that an initial nematic state

relaxes to a periodic state for which the director wags about the fixed flow direction. As

50

illustrated by Figure 6.3, the magnitude of x33 and a33 are smaller for this case. The
director It oscillates within an angle smaller than 90° and about the flow direction 23 with
period 1p: 2.997 after a transition dimensionless time 6tDr z 2. This referred as

wagging microstructure. Figure 6.4 shows the behavior of the director component for U

= 30, Pe = 38.

Nematic-Like Microshucture

For U = 30 and Pe = 80, the FSQ-closure predicts that an initial pure nematic state
(i.e., Point A on Figure 2.3) relaxes to a nematic-like state very near the original nematic
state. This referred as nematic microstructure. AS previous discussed in Chapter 5 (see

Eq. (5.7)), the nematic state is not solution to Eq. (5.1) inasmuch as

d:

1
— =-I—e e $0. 6.7
[ML 3= ’3‘3 = ( )

Eq. (6.7) also holds for Eq.(6.l) for Pe > 0. Thus, Brownian diffusion causes the
relaxation of the initial nematic State to a nematic-like state exemplified by the results
portrayed in Figure 6.5. Figure 6.5 shows the behavior of the orientation dyadic and its
director components in the nematic region foe U = 30, Pe = 80. Figure 6.6 shows the

phase behavior predicted by the FSQ-closure for limited range of U and Fe examined.

51

 

 

 

 

 

 

 

 

 

 

 

1--—-’—----- ------------ 4:
‘I \p
- ‘1 1 I. ',‘-
F i F H .4
. ., l '1 ,.
_ ~ I‘ 'I I-
X33 0 . 3 : a I.
~ I l I. :
" ‘ l ~ I.
~ I i I l-
_ I '; r,=4.125 : 1.
-1 r 3 a ’I I,
0 5 IO 15
T
lL--O-—-.-__.._ _'.’__..,,--...-a
~~‘\ 1"-
0.8? 83 ‘ 2, n
< ./ / ;. i\ .
0.4" I l -
I | x
0.2' n 4 : A . e3 .
0 . 1'

-0.2" e3 : : n -
-0.4r fi 1 '. ’N a
O l
'0-6’ // i J/ \ e3 3
-0.8- n it. .I -
-lr- ‘O’ -

 

 

 

2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5

,2.

Figure 6.2 Tumbling Microstructure. Relative alignment of the orientation director with
the flow direction (FSQ-closure with U = 30, Fe = 25, and 2(0) = 9393)

52

 

 

 

 

 

- g 5 .

aij 0.4 / g E ,5.
02 - A Pg 13 . -

a“, p . , ' .:=\ ..
O ‘i\ “(j ‘\ \i 7‘ ! ‘
'0.2 '- 323 31] z o a
'0'40 5 10 15

T

 

 

 

more / 4.. --.t- s -.L m

 

 

 

 

T

Figure 6.3 Wagging Microstructure. Components of orientation dyadic and its
eigenvectors (FSQ-closure with U = 30, Fe = 38, and 2(0) = _e_3 _e_3)

53

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.1
1.05 ' ‘
l ‘—
x33 0.95 ' 3
=2.997
0.9 ’ Tp -
« >
0.85 ’ —\ 3
0.8 1 a 1 a 1 a n
0 1 2 3 4 5 6 7 8 9
1.1 I I I I 1 I I I 1
63
1.05 - e3 1' 1 '
< 1 n
1 7 n -
X33 /
0.95 r '
0.9 - n ‘
83 C W
0.85 - < i ‘33 '
n
0'8 1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64 1.66 1.68 2.0

T

Figure 6.4 Wagging Microstructure. Relative alignment of the orientation director with
the flow direction (FSQ-closure with U = 30, Fe = 38, and 2(0) = 9393)

54

1.1

1.05

x33 0.95

0.9

0.85

0.8
0

 

 

 

 

 

 

 

 

 

 

 

1.1

1.05 F

X33

0.95 ‘

0.9 ‘

0.85 '

 

:3.“

 

 

0.8

1.50 1.52 1:54 1.56 1.58 1.60

1.62 1.64 1.66 1.68

Figtue 6.4 Wagging Microstructure. Relative alignment of the orientation director with

the flow direction (FSQ-closure with U = 30, Pe = 38, and 2(0) = 9323)

1.2

0.8

0.6

0.8
0.6
0.4
0.2

0.2
-0.4
-0.6
-0.8

Figure 6.5

 

\memu

I
I

 

 

 

 

 

 

 

 

 

 

 

. \ 7
- X33 r
l. x23 4
. / .
W ”W '
. X22 -
0 5 10 15
Nematic-like Microstructure. Components of the orientation dyadic and

its eigenvectors (FSQ-closure with U = 30, Pe = 80, and 2(0) = 9393)

55

 
  
 

       
    

Tumbling Wagging
Region Region
Steady State Region

 

 

 

Steady State Region

Isotropic Equilibrium States: 0 < U < Uc = 4.1,Pe = 0 (see Chapters 5)

 

 

0 4 20 40 60 80 100 120

Figure 6.6 Flow Transition Predicted by the FSQ-Closure for N ematic Initial State

56

Logrolling Microstructure

If the initial orientation state is 2D planar isotropic and U = 30 and Fe = 55, then
only two of the three eigenvectors associated with the orientation dyadic 2(1) participate
in the dynamic response of the microstructure. The eigenvector 5, remains fixed, while
the eigenvector _x_3 tumbles and the eigenvector 22 rolls within it in the shear plane.

This referred as logrolling microstructure. Figure 6.7 shows the behavior of the
components of the orientation dyadic and its eigenvectors for U = 30 and Fe = 55. The

symmetry x22 and x33 about the axis reflects that the angular velocity of the
eigenvectors 52 and 53 are the same. The dynamic of microstructure relaxes into a

steady state after a dimensionless time 6tDr z 0.8 . Figure 6.8 shows the orientation

states for the logrolling region for U = 30 and Fe = 55.
The flow-transition behavior in LCPS also occurs for U = 25 and U = 20. Figure

6.9 shows the flow transition diagram for initial states that are planar isotropic.

57

 

 

 

 

 

 

 

 

 

 

 

 

1.2 ' ' fl
1 r A___ _
O 8 ‘ \ 1 l ..
a11 2(0) = ‘2‘9191 + 39393
0'6 ‘ .9406 0 0
ii 0.4 - \\‘ 333 2(a) = 0 .0078 .0047 .
\ 4*“ a23 ' 0 .0047 .051
0 2 ' \J’ \ .
of. ~ \ ’ ~‘ I“-
" . ’M:. a?” n
0 13.2 [I W22” {" ' S
‘02 0 0 2 0 4 0.6 0 8 1 12
l s: ‘
- xx \ .1
0 6 ' g‘.‘ X11 x33 .
I. A. ' ‘. .‘. -
0 2 ' I, \ i “ l. \\ - .
. ‘ 2 0‘. 2 \.‘. 5 , .\\."’ \‘./0~. ____ d
“0'2 _ ‘v' \. !\ 3° 1 0 0 '
~ ‘. ! .
x22 1; Xij(°°) = - 9944 1061
'0'6 x23 0 1061 9944
- / is. .
- 1 ”..fi‘..0.o"-...’. .°o.....o~.”o.h M - __

 

 

 

0 0.2 0.4 0.6 0.8 1 1.2

T

Figure 6.7 Logrolling Microstructure. Components of the orientation dyadic and the
orientation director (FSQ closure with U = 30 and Pe = 55)

58

 

0.7 l , ,

 

0.5 .

 

 

0.4 .

 

0.1 -

 

 

 

30.180

 

 

 

 

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

1111,
Figure 6.8 Anisotropic Invariants Associated with Logrolling

59

 

Logrolling Region

6 l 1
i 3(0)=E§lgl +5939:

 

 

, Figure 6.9 Flow Transition Predicted by FSQ—Closure for 2D Planar Isotropic
initial States

6.3 Conclusions

In the presence of simple shear flows, the orientation microstructure of liquid
crystalline polymers is governed by Brownian diffusion, Doi diffusion and
hydrodynamics drag. The following conclusions stem directly from an analysis of Eq.
(6.1):

(i) For a nematic initial orientation state, the FSQ-closure predicts a flow
transition microstructure from tumbling to wagging and to alignment as the
Pe increases (see Figures 6.2, 6.4 and 6.6).

(ii) The logrolling state is obtained when the initial orientation state is planar
isotropic (see Figure 6.7).

(iii) For initial orientation dyadic that satisfy Eqs. (6.2) and (6.3), the FSQ-closure

yields realizable orientation state.

61

CHAPTER 7

CONCLUSIONS AND RECONIENDATIONS

A model based on Doi’s theory for the microstructure of liquid crystalline
polymers is analyzed using a new closure approximation for the orientation tetradic. The
model equation balances two physical processes, rotary diffusion and rotary convection.
The rotary diffusive flux includes both Brownian diffusion and an excluded volume
effect. The rotary convective flux is related to hydrodynamics drag. A fully symmetric
quadratic (FSQ—) closure for the orientation tetradic C2 = 113 was used to close the
moment equation for the orientation dyadic. This closure, which retains the symmetry
and the projection properties of the exact orientation tetradic, and represents a significant
advancement in the practical theory of LCPS.
For U = 0 and Pe = 0, the LCPS microstructure relaxes to an isotropic state by
Brownian diffusion from any realizable initial state (see Figures 4.1, 4.2 and 4.3). For

0<U <Uc =4.1 and Pe = 0, the LCPS microstructure still relaxes to the equilibrium
isotropic state, but the time required to reach equilibrium increases as U increases. For U
= 0, the relaxation time to isotropic state is t = 7 / 6D, (see Figures 4.1 and 4.2). For U =

3, the relaxation time t = 14/ 6Dr (see Figure 5.1).
For U > Uc , the new model predicts the simultaneous existence of three steady

states: an unstable isotropic state; a stable prolate state; and a stable oblate state. The
critical nematic potential Uc, predicted by the FSQ-closure, is 4.1. The quadratic closure
also predict a trifurcation potential but Uc = 2.7 for this approximation (see Figure 5.4).

As U -> co, the prolate state approaches the nematic state corresponding to Point A of

62

Figure 2.3, and the oblate state approaches to the planar isotropic state corresponding to
Point C on Figure 2.3.

For U > O and Pe > O, the microstructure goes through several phases as Pe
increases from to 80. For U = 30, the microstructure passes from a tumbling periodic
state to a wagging periodic state for Pe = 36; a transition from a wagging microstructure
to a steady state microstructure occurs for Pe = 76 (see Figure 6.9). Above this value, a
steady state microstructure is obtained.

If the initial microstructure is 2Doplanar isotropic with an =1/ 2, an =0, and
a33 = l/ 2 , the FSQ-closure yields a transient logrolling microstructure for U = 30 and

O < Pe < 100 (see Figure 6.9). For larger Pelect number, the logrolling state relaxes to a
nematic like state, but the dynamic details of interesting phenomena requires additional
study. Similar behavior was observed for U = 25 and U = 20 at lower values of the
Pelect number.

Although the foregoing results are exploratory, they nevertheless illustrate the
value of the FSQ-closure inasmuch as the trifurcation of equilibrium states and the
transition from periodic to steady state behavior are all anticipated by the underlying Doi
theory. The failure of the quadratic closure to predict the foregoing flow-transition
phenomena in simple shear flows underscores the significant of the FSQ-closure. These
results provide the strong motivation for further research on the FSQ—closure and support
the conclusion that a closure for the orientation tetradic should retain the six-fold
symmetry and projection properties of the exact orientation tetradic.

A most significant conclusion deve10ped in this study was the observation that Eq

(6.62) with C2 =1/3 states yields realizable microstructures provided the initial orientation

63

 

dyadic is diagonal and the dominant eigenvector is colinear with the flow direction. This

conclusion is limited to values U and Fe selected for analysis, viz., 0 < U < 30 and

0 <Pe < 120.

The following recommendations for further research would complement this

exploratory study and would provide the needed background to support the FSQ-closure

as a practical representation of the microstructure:

The flow transition diagrams (see Figures 6.6 and 6.9) should be fully detailed for
OSUSSO and OSPeSISO.

A comprehensive analysis of the results developed herein should be compared
to previous studies and an explanation for the significant differences should be

developed

The influence of the nematic strength and the Pelect number on the frequency of
the periodic solutions obtained (i.e., periodic wagging and periodic tumbling)
should be developed and interpreted in terms of the underlying physical
phenomena included in the theory.

The influence of the initial conditions on the dynamic response of the orientation
dyadic should be examined relative to realizability (see Appendix D).

The FSQ-closure parameter was assumed to be a constant in this study. The
Influence of the local orientation state and the physical properties of the polymer
solution on the FSQ-coefficient C2 should be critically evaluated Appendix A
shows that C2 = 1/3 for all values of U and Pe if the microstructure is nematic.
However, this is not required at other anisotropic states. The trifurcation of
equilibrium states may provide a relatively simple problem to further evaluate the
efficacy of the FSQ-closure with C2 dependent on the local orientation invariant

111.. Eve-2'2).

APPENDICES

65

APPENDIX A: FSQ-CLOSURE

A.l The FSQ-Closure at the Isotmpic State: '

With 3 = (A.l)

b) Ian—a

Eq. (3.2) reduces to:

1
<2 >,=-3—5-(1,qu +ImIjl +InIk’.)

PO

2

I'd

1 1 1 1 1 1 l
+7951,qu +31%, +351”. +351... +3-Inlj. girls)

1
=Eaijlkl + ImIjl + Inlkj). (A2)
The second-order tetradic defined in Eq. (3.5) is

—2- 146,qu +IikIj, +131“)

1

I'D

2>2=

l
+30,qu + 1&1 j, + Ifllkj)

2 1 1 l l l 1
. —-7-(-9-Iijlu +3131). +31illkj +31ijlu +3195: +'9'Iillki)

1
=I§Gijlu +Ia1 J, HUI”). (A3)

The FSQ orientation tetradic defined by Eq. (3.4) reduces to the following expression at

the isotropic point

1
< P. = £0,qu 4,1,1], Hal”). (A.4)

I'D

22>isomic

The Eq. (A.4) shows that C2 has no restriction at isotropic state.

66

A2 The FSQ-Closure at the Nematic state:

At the nematic state defined by:

The Eq. (3.2) reduces to

1
<2 2 p p>,=--3;(Iijlu +1:ij1 +Iulkj)

+%(I;5§3§3 + 139,93 + 1,3e3_e_j + 233,1” + g3ekl3, + 939.1”) , (A6)
and the second-order tetradic defined in Eq. (3.5), is

2
<2 2 p B>2='3—5'(Iijlkl +131}, +Inij)

2
"7039393 + Ii3§j§3 +Ii363§j +§3§31u +Ezek131 +§3§iIk3)

+ 3939352393 - (A.7)
The FSQ orientation tetradic defined by Eq. (3.4) reduces to the following

expression at the nematic point

'"e
I'D
I'D
I'D

ll

1 A
-3_5.(1 - 3C2)(I.,1u + 1,1], + lulu)

+ % (l - 3C2)(I,j_e_3g3 +Ii3gj§3 + Ii_,,e3§j + 93931:: + 9:33:13; + 93911”)

+ 3C 223939.393 (A8)

67

The Eq. (A.8) shows that the FSQ-closure gives the exact orientation tetradic at the
nematic state provided C2=l/3. This value is used for all orientation states in this

research.

 

68

. .0-M4 _V\ — ' .4. -

APPENDIX B: COMPONENT EQUATION FOR BROWNIAN-D01 DIFFUSION

In the absence of flow, the relaxation process to equilibrium is driven by the
Brownian-Doi diffusion. The role of the Doi diffusion term depends on the nematic
potential U. It is important to understand the coupling of the diagonal components and

the nematic potential U in the moment equation. The Doi diffusion term in Eq. (2.62) is:

'2‘<EBBB>‘£° (B-l)

IID

At the isotropic state, this contribution reduces to the null dyadic inasmuch as (B.l) for

the FSQ-closure follows:
-II 1I 1 II I II II
2. E-<E B B 2>2‘3 it; ik—B( i5 u+ «111+ in xi); 11
l l
52. (B2)

At the nematic state 3 = §3_e_3, the Doi diffusion term is also trivial at the nematic state

from Eq. (A.10), the Eq. (B.l) can be rewritten

E323 'E393 '§3§3§3§3 @393 °

59. (13.3)

ll”

'2- <2 2 2 2 >=2
With C2 = 1/3, the orientation tetradic can be expressed using the FSQ-closure is:

2 l
<2222> =3<2222>n+3<2222>2~ (3.4)
where <2 2 p 12>: and <2 2 p p), aregiven by Eqs. (3.2) and (3.5)

It follows from Eq. (3.2) that

69

(222261

1
=_§[Iijlklalk + IikIjlalk + Iillkjalk]

l
+7[Iijak,alk +Iikafian‘ +Inakjak +aijluak +aiklflan +aikljlak +aqujak]
03$
<BBBB%E
=< PinPkPr >2 an:

" 2 I I +1 I I I

2
-7[Iija5anak +Ikafiauak +Ifiafiauak +akakjluak +akaulnak +akakjluak]

(Bfi)
An analysis of Eq. (5.1) subject to Eqs. (B4), (B5) and (B6) yields the interesting
conclusion that id the initial orientation dyadic is diagonal, i.e.,
3(0) = 339.21 + 21329.29; + 8332323 (3.7)
then 2 remains diagonal for 1 > O . In the absence of flow, there is no mechanics in Eq.

(5. 1) to generate cross correlation components of g except by the initial conditions.

Thus, the following component equations for au(r), a22(r) , and 333(1) determine the

relaxation of an LCP microstructure characterized by Eq. (5.1):

70

d3“

1 2
-d—T—=(§-a“)+ U{(a,21+ a; + 3:3) " {'"IFS'II + 2311]

2
+2—l[(a121+a§2 +a§3)+4a12l +a,,]
2 1
+i'65(3121 +332 +a§3)[l+zau]+'§(au '(3121 +332 +a§3)+23131)

2
-2I[(3121+a§2 +a§3)-au+2af1+2a,2, +3an-(a121+a§2 +a§3)]}}'

(13-8)

da 1 2
EE=(§-an)+ U{(a121+a§2 +333)-{-'1'6§[1+2a22]

+%[(3121 + a; +a§3)-t-4a§2 +a22]

2 l
+F)§(afl +a§2 +a§3)[1+2a22]+-§(a22 -(a,21 + 3:2 +a§3)+ 28:2)

2
mil—lflalzl +a;2 +a§,)-an +2a‘;’2 +2a;2 +3a22 '(3121 +332 +333)]}}~

(B-9)

d333
d1

 

I 2
= (3-a33)+U{(a12, +3; +a§3)-{-fi[l+ 2333]

2
+-ZT[(a,2, +322 +a§3)+4a§3 +a33]

2 l
+£0.13, +a; +a§3)[l+2a33]+§(a33 (all +21; +a§,)+2a§3)

2 2 2 2 3 2 2 2 2
_.__[(all --|~a22 +a33)-a33 +2a33 +2a33 +3333 ~(au -l~a22 +333)]}}.

21

(B.lO)

71

The above nonlinear ordinary differential equations, which have the feature that

all + 322 + a33 = l, were integrated numerically using a second-order Runge-Kutta

algorithm (see Appendix E).

72

 

APPENDIX C: ORDER PARAMETER FOR EQUILIBRIUM STATES

The scalar parameter that characterizes the microstructure of liquid crystalline

polymers is called the order parameter and is defined as

3
S: —
(2

"O"
"0‘

). (CD

I
Because 2 = 2 -§ , then it follows that

3 I I l
S = {5(g-g):(g-§)}z. (C2)
1
= {%(g:g-%g:;+%gzg)}5. (C3)
1
= {garg-y-gue-y+éutgopv. (CA)

The orientation dyadic a has three eigenvalues: a1 5 a2 5 a3, For prolate states

a1 = a2 = (1 - a3) / 2 and Eq. (C.2) becomes

3 2 1 l
S: {-2-(af+a§+a§)-§+§}2. (C5)

3 1- l
= {§[2(—2a—3-)2 +a§ ]}2 . (C.6)

This shows that the order parameter for prolate states can be written in terms of the

maximum eigenvalue a3:

8 = —a3 — l , prolate states. (Q7)

73

For oblate states, 32 = a3 and the same procedure as above shows that the order

parameter can be expressed in terms of the smallest eigenvalue al :

3:231 ...;., oblate states. (08)

2

74

APPENDIX D: DIAGONAL INITIAL CONDITIONS

If the initial orientation state is associated with a diagonal orientation dyadic with
O S an(0) _<_ a22(0) S a33(0), the FSQ-closure with C2 = ll3 yields a realizable dynamic
response for 0 S U S 30 and O 5 Fe 5 80 . This is illustrated by Figure D.l, which shows

the relaxation of the following initial state:

2“» = 029191 + 039292 + 059.32) (DJ)

for U = 30 and Pe = 50. The invariants of the anisotropic dyadic 2 clearly stay within
the realizable region.

75

 

0.7 V I I I I I I

 

 

 

 

 

 

 

 

 

 

0.6 - / -
0 5 3(0) = 029191 + 0-3E2E2 + 05239:-
0.4 ‘ "
IIb -
0.3 ' ‘
J
0.2 " "
F
0.1 " -
‘ ‘ A 0.19s
1111:
O l
- 0 0.05 0.1 0.15 0.2 0.25

mb

Figure D.l Realizable Orientation States Predicted by the FSQ-Closure for U. = 30
and Fe =50

76

0.3

APPENDIX E: COMPUTATIONAL STRATEGY

E.l Introduction
For the purpose of computations for this study were supported by MATLAB. A

second order Runge-Kutta algorithm was used to solve the nonlinear ordinary differential

equations governing the components of g . At the beginning of the code, input variables

are introduced including U, Pe, a(i,j), ID(i,j), Du(i,j), and tstop. Others inputs are
considered to be fixed for the objective of this study as dt = 0.003, C1 =2/3, C2 = 1/3, and
lam :- 1. A time step dt = 0.003 was used for all calculations. C2 = 1/3 is a parameters
used in the FSQ closure. Lam = 1 is characteristic of liquid crystalline polymers with
very large aspect ratio. These notations are defined below. Any changes in these input

parameters may have a significant impact on the results.

E.2 Variable Listing

% notation explain
t dimensionless time 1:
' dt step time A1:
tstop running time “cf-ma]
lam relative to the aspect ration of LCP (see Eq. (2.4))
U nematic potential coefficient U
Pe Pelect number Pe
a(i,j) orientation dyadic component
pppp(i,j,k,l) tetradic orientation component
ID(i,j) unit tensor component

77

Du velocity gradient component

alpha(i,j) General closure coefficient (see Petty et al., 1999)

s(i,j) symmetric velocity gradient component

w(i,j) antisymmetric velocity gradient component

aa(i,j) dot product of two tensors g - g

larnbdal eigenvalues al ,a2,a3

v(i,j) eigenvectors components, i represents for column, j
represents for row

E.3 Dimensionless Time Step Selection

In this research, FSQ model was used to predict the behavior of LCPS system. It
is important to choose a reasonable time step that will maintain the accuracy and
sufficient computation time. Let’s consider the case for Fe = 25 and U = 30, and the
initial condition is the nematic state: an(0) = 322(0) = 0, a33 = 1. Equations were
integrated for At = 0.010 , A1 = 0.003, and Ar = 0.001 . The goal is to select a step size
that is small enough to provide the computational accuracy, but still maintain the
efficiency of the computational time. The step size is selected where there is a small
change in 333 for different A'c's. Figure E.1 shows that a33 for At = 0.003 is very close to

a3; for A1: = 0.001. This shows that there is a very small change in a3; for A1: S 0.003.

Therefore, the selected time step is Ar=0.003. This small time step is used in the

second order Runge-Kutta method to insure the accuracy in the computations.

78

 

 

 

 

 

 

 

 

Figure B] Step Size Selection. Behavior of orientation dyadic component a3; for
U = 30 and Pe = 25

79

E.4 Program Listing

%SECOND ORDER RUNGE-KUTTA COMPUTATION
clf

clear all

t=0;

dt=.003;

tstop=10;

lam=l; %for liquid crystalline polymer;
U=8;

Pu6;

Cl=2l3;

C2=113;

C3=0;

C4=1-Cl-C2-C3;

%Initial input: nematic state;
a(l.l)=0;a(l.2)=0;a(l.3)=0;
8(2.l)=0;a(2.2)=0;a(2.3)=0;
8(3.l)=0;a(3.2)=0;a(3.3)=l-0;

%Unit tensor;

ID(l.l)=l;ID(l.2)==0;ID(l.3)=0;
ID(2.1)=0:ID(2.2)=1;ID(2.3)=0;
113(3.l)=0;ID(3.2)=0;lD(3.3)=1;

%Velocity gradient for simple shear flow;
DU(1.1)=0;Du(l.2)=0;Du(l.3)=0;
DU(2.1)=0;DU(2.2)=0:DU(2.3)=1;
DU(3.1)=0;DU(3.2)=0.DU(3,3)=0;

%Defined parameter;

alpha(l,l)=—l/35;

alpha(l,2)= lfl;

%alpha(2,l) = defined during integration ...;
alpha(2.2)=0;

alpha(2.3)=l;

alpha(2,4)=-2l7;

%alpha(3,l) = defined during integration ...;
aJPM(3.2)=0;

alpha(3,3)=0; .

%alpha(3,4) = defined during integration
%alpha(3,5) = defined during integration
%alpha(3,6) = defined during integration
%alpha(4,l) = defined during integration
alpha(42)=0:

alpha(4,3)=0;

%alpha(4, 4) = defined during integration ...;
alpha(4.5)=0;

alpha(4,6)=0;

alpha(4,7)=l;

alpha(4.8)=-2r7;

.0 .0 .0 .

80

%Symmetric and Antisymmetric portion of Velocity Gradient tensor;
for i=l:3
for j=l :3
s(i.i)=5*(Du(i.j)+Du(j.i»;
w(i,j)=.5*(Du(i,j)-Du(i,i));
end
end

for n=lztstop
%First 100p of the R-K method
%Determine SH 1];
for i=1:3
for j=l:3
for k=1 :3
for l=l:3
ndSIIGJ.k.l)=ID(i.j)"‘lD(k.l)+ID(iJ<)*IDGJ)+ID(i.l)*ID(k.j);
e
end
end
end

%Determine S[I a];
for i=l:3
for j=l:3
for k=1:3
for 1:! :3
SlaGJ.k.l)=lD(iJ)*a(k.l)+lD(i.k)*a(i.l)+lD(i.l)*a(kJ);
Sla(iJJc.l)=a(iJ)*ID(k.l)+a(i.k)*DGJHa(i.l)*ID(k,j)I-Sla(i,j,k,l);
end
end
end
end

%Determine S[a a];
for i=l:3
for j=l:3
for k=1 :3
for l=l:3
ndSaaGJ“Hifi‘awhafih‘afi.l)+a(i.l)*a(kJ);
e
end
end
end

%Determine S[I(a*a)];
for i=l:3
for j=l :3
a(iJHh
bb(iJ)=0;
for k=l:3
aa(iJ)=aa(i.i)+a(i.k)*a(k.j);
mWiJ)=bb(iJ)+(a(i.k)-D(i.k)l3)*(a(k.j)-1D(k.j)/3);
e
end
end

81

 

for i=1 :3
for j=l :3
for k=1 :3
for 1:1 :3
Sham.k.1)=m(i.i>*aaac.1)+n>(ix)*aa0.1)+n>(i.r)*aa(k.i);
Sha(iJ.k.l)=aa(iJ)*fl)(kJ)+aa(i.k)*ID(iJ)+aa(i.l)*fl>(k.j)+SIaa(iJ.k.l);
end
end
end
end

%Determine S[(a*a)a];
for i=1 :3
for j=l :3
aa(iJ)=0;
for k=1:3
aa(U)=88(iJHa(i.k)*a(kJ);
end
end
end

for i=1 :3
for j=l :3
for k=1 :3
for l=l :3
Saaa(iJJ<,l)=aa(i,j)*a(k,l)+aa(i,k)*a(i.l)+aa(i,l)*a(k.j);
SaudiJcl)=a(i.i)*aa(k.l)+a(i.k)*a8(i.l)+a(iJ)*aa(kJ)+Saa8(iJ.k.l);
end
end
end
end

%Determine 81883;
for i=1 :3
for j=1 :3

m(iJ)=0;
bbb(iJ)=0;
for k=l:3
aaa(i,i)=aaa(i,j)+aa(i,k)*a(k,j);
bbb(i,i)=bbb(i,j)+bb(i,k)*(a(k,j)—ID(k,j)/3);
end

end
end

for i=l:3
for j=l:3
for k=l:3
for l=l:3
SIaaa(i,j,k,l)=ID(i,j)*aaa(k,l)+ID(i,k)*aaa(i,l)+lD(i,l)*aaa(k,j);
Slaaa(iJ.kJ) = aaa<i.i)*n>(k.1)+aaa(i.k)*n>o.1)+aaa(i.l)*n>(k.j)+51aaa(i.iJen);
end
end
end
end

82

%Determine S[(a*a)(a*a)];
for i=1 :3
for j=l:3
for k=l:3
for l=l:3
Saaaa(iJ.kJ)=aa(iJ)*aa(kJ)+aa(i.k)*aa(iJ)+aa(i.l)*aa(k..i);
end
end
end
end

%Determine S[I(a*a*a*a)];
for i =1 :3
for j=l :3
aaaa(iJ)=0;
for k=1 :3
aaaa(iJ)=aaaa(iJ)+m(i.k)*a(k.j);
end
end
end

for i=lz3
for j=l :3
for k=1:3
for i=1 :3
Slam(iJ.k.l)=m(iJ)*aaaa(kJHID(i.k)*WCi.1)+H)(i.l)*am(k.j);
SIaaaaGJJcl)=aaaa(iJ)*ID(k.l)+am(i.k)*fl>(iJ)+aaaa(i.l)‘fl>(kJ)+SIaaaa(iJ.kJ);
end
end
end
end

%Determine aza (Ila) and alpha(2,l);
Ila=aa( l , l)+aa(2,2)+aa(3,3);
IIb=bb(l,1)+bb(2.2)+bb(3.3);
alpha(2,l)=2*IIal35;

%Detennine alpha(3,1);
ALPl=aaa(1,l)-I-aaa(2.2)+aaa(3.3);
alpha(3,l)=l/35+(4/35)*(ALP1IIIa);
n1a=aaa(l,l)+aaa(2,2)+aaa(3,3);
IIIb=bbb(l,l)+bbb(2,2)-I-bbb(3.3);

%Determine alpha(3.4);
a1pha(3,4)=-1/(7*na);

%Determine alpha(35);
alpha(3,5)=llIIa;

%Determine alpha(3.6);
alpha(3.6)=-4/(7 *113);

%Determine alpha(4.l);

ALPZ=aaaa( l,1)+aaaa(2.2)+aaaa(3,3);
alpha(4, l )=2/35*ALP2+( l/35)*IIa*IIa;

83

%Determine alpha(4.4);
alpha(4,4)=(- 1/7) *IIa;

%MSU contribution:
%Determine ppppl,pppp2.pppp3.pppp4;

for i=l:3
for j=l:3

for k=1 :3
for l=1 :3

ppppS(i.i.k.l)=a(i.j)*a(k.l);

pppp1(i.j.k.1)=alpha(1.1)*sn(i.i.k.1)+alpha(1.2)*SIa<i.j.k.1);
%PPPP2(iJ.k.l)=alPha(2.l)*SH(iJ.kJ)+alPha(22)*Sla(iJLlflflpha(2.3)*Saa(iJ.kJ)
+alpha(2,4)*SIaa(iJ,kJ);
PPPP2(iJ.k.l)=alpha(2.l)*SH(iJ.k.l)+alpha(2.3)*Saa(iJ.kJHalpha(2.4)*SIaa(iJ.k.l):
PPPP3(iJ.k.l)=alpha(3.l)’SII(iJ.k.l)+ alpha(3.4)*SIaa(iJ.kJHflPha(3.5)*Saaa(iJ.k.l)
+alpha(3,6)*Slaaa(i,j,k,l);

was» alpha(4.l)‘SIIGJ.hl)+alpha(4.4)*SIaa(iJ.kJ>+ alpha(4.7)*Saaaa(i.j.k.r)
+alpha(4,8)*SIaaaa(iJ,kJ);

%Casel:Michigan State University model;
pppp0(iJ.kJ)=Cl‘ppppl(iJ.k.l)+C2*pppp2(iJ.k.l)
+(l-C1-CZ)‘ppp93(iJ.kJ)+C4‘pppp4(iJ.kJ);

%Case2:Quadratic model;
%pppp0(i.i.k.l)=pppp5(i.i.k.l);

end

end
end

end

%Determine <pppp>zS
for i=1 :3
for j=l :3
M(iJ)=0;
for k=1 :3
for l=l :3
M(i.i)=M(iJ)+ppppo(iJ.kJ)*s(l.k);
end
end
end
end

%Determine <pp>:<pppp>
for i=1 :3
for j=l :3
N(i..i)=0;
for k=1 :3
for l=l:3
N(i,j)=N(i,j)+pppp0(i.j.k.l)*a(l,k);

end
end
end

%Determine S.a+a.S, W.a+a.W(T);
for i=1 :3
for j=l :3
saaS(iJ)=0;
waaw(iJ)=0;
for k=1 :3
saaS(iJ)=saas(iJ)+S(iJ<)*a(kJ)+a(i.k)*S(k.i):
ndwaaMiJ)=\IvarnN(i.i)+W(i.k)"‘a(lc.I')+aI(i.k)"‘Wth);
e
end
end

%Right handside of the equation
for i=1:3
for j=l :3
dkl(iJ)=(1D(iti)/3-a(iti)Hlam‘Pe*(saa5(iJ)-2*M(iJ))-Pc*waaW(iJ)+U*(aa(iJ)-N(iJ));
end
end

%Computation
for i=1 :3
for j=l :3
al(iJ)=a(iJ)+dkl(iJ)*dt:
end
end

%Second loop of the R—K method
%Determine SD 1];
for i=1:3
for j=l:3
for k=l:3
for 1:1 :3
sm(is.k.1)=n>(i.i)*n>(k.1)+n>(ix)*m(i.1)+nxi.1)*m(k.j);
end
end
end
end

%Determine S[I a];
for i=1:3
for j=l:3
for k=l:3
for l=l:3
Slal(iJ.kJ)=ID(iJ)*al(kl)+ID(i.k)*al(i.l)+m(i.l)*al(Ki);
Slal(iJ.k.l)=al(iJ)"‘ID(kJ)+al(i.k)*1D(i.l)+al(i.l)*fl)(kJ)+Slal(iJJ<J);
end
end
end
end
%Determine S[a a];
for i=1:3
for j=l:3
for k=lz3
for [=1 :3

85

Saal(iJkJ)=81(iJ)*al(k.l)+a1(i.k)*alG.l)+al(i.l)*al(k,i):
end -
end
end
end

%Determine S[I(a*a)];
for i=1:3
for j=l:3
331(iJ)=0;
bbl(i,i)=0;
for k=1:3
aalfiJ) = 881(iJ)+81(i.k)*81(kJ);
bbl(i.i) = bb1(iJ)+(al(i.kHD(i.k)/3)*(al(k.i)-ID(kJ)/3);
end
end
end

for i=1:3
for j=l:3
for k=1:3
for l=l:3
Slaal(i,j,k,1)=ID(i,i)*aal(k.l)+lD(i,k)*aal(i,l)+1D(i,l)*aal(k,j);
Slaal(iJJCJHal(iJ)*1D(k.l)+aal(thWDUJHaal(iJ)*1D(kJ)+Slaal(iJok.l):
end
end
end
end

%Determine S[(a*a)a];
for i=1:3
for j=l:3
aal(iJ)=0;
for k=1:3
aal(i.i)=aal(i.i)+al(i.k)*al(k.i);
end
end
end

for i=1:3
for j=l:3
for k=1:3
for i=1 :3
Sml(iJ.kJ)=aal(iJ)*al(kJ)+aal(i.k)*al(iJHaal(i.l)*al(kJ):
Sml(iJ.k.l)=al(iJ)*aal(k.l)+al(i.k)*881(j.l)+al(i.l)*aal(RJHSml(iJ.k.l);
end .
end
end
end

%Determine 81838;
for i=1 :3
for j=l:3
ml(iJ)=0;
bbbl(i,j)=0;
for k=1:3

86

ml(i.i)=anal(i.i)+aal(i.k)*al(k.i);
bbbl(iJ)=bbbl(iJ)+bbl(i.k)*(al(kJ)-1D(1<J)l3);
end
end
end

for i=1 :3
for j=l:3
for k=1:3
for 1:1 :3
Slaaal(iJJcl)=1D(i.j)*aaa1(k.l)+lD(i.k)*ml0.1)+1D(i.l)*aaal(Ki);
SIaaa1(i,j,k.l) = ml(iJ)*fl>(k.l)+aaal(i.k)*lD(.iJ)+aaal(iJ)*H>(kJ)+SIaaal(iJ.k.l);
end
end
end
end

%Determine S[(a‘a)(a*a)];
for i=1 :3
for j=l:3
for k=1:3
for l=l:3
Saaaal(i,i,k,l)=aal(i,j)*aal(kJ)-I-aa1(i,k)*aa1(j,1)+aal(i,l)*aal(k,j);
end
end
end
end

%Determine S[I(a*a*a*a)];
for i=1 :3
for j=l:3
aaaal(i.i)=0;
for k=1 :3
aaaal(iJ)=aaaal(iJ)+aaal(i.k)*al(kJ);
end
end
end

for i=1 :3
for j=l:3

for k=1:3
for l=l:3
SIaaaal(iJ,k,l)=ID(i,j)*aaaal(k,l)+ID(i.k)*aaaal(j,l)+ID(i,l)*aaaal(k,j);
Slaaaal(iJkJHaaal(i,i)*m(k,l)+aaaal(i,k)*ID(j,l)+aaaal(i,l)*fl)(kJ)+SIaaaal(iJ,k,l);
end

end

end
end

%Determine a:a (Ila) and alpha(2,l);
IIa1=aal(l,l)-I-aal(2,2)+aal(3.3):
IIb1=bbl(l,1)+bbl(2,2)+bbl(3,3);
alphal(2,1)=2*IIa ll35;

%Determine alphal(3,l);

87

ALPll=aaa1(l,l)+aaa1(2,2)-I-aaal(3,3);
alphal(3,l)=l/35+(4I35)*(ALPI lIIIal);
IIIal=aaal(l,l)+aaal(2,2)+aaa1(3,3);
IIIbl=bbbl(l,1)+bbbl(2,2)+bbbl(3.3);

%Determine alphal(3.4);
alpha1(3,4)= l/(7*IIal);

%Determine alpha(3.5);
alphal(3,5)=l/Ilal;

%Determine alphal(3.6);
alphal(3,6)=-4l(7“IIal);

%Determine alpha(4.l);
A1P21=aaaa1(l,l)+aaaal(2,2)+aaaal(3.3);
alpha1(4,l)=2l35*ALP21+( 1135)*Ilal*IIal;

%Determine alphal(4.4):
alphal(4,4)=(-lf7)*IIal;

%MSU contribution;
%Determine ppppl,pppp2,pppp3,pppp4;
for i=1:3
for j=l:3
for k=1 :3
for l=l:3
PPPP51(iJ.k.l)=al(iJ)*al(k.l);
PPPPl 1(iJ.k.l)=81Pha(l.1)*SHl(iJ.k.l)+alpha(1.2)*Slal(iJ.kJ);
%Pppp21(i.j.k.l)=alphal(2.l)*SIIl(iJ.k.l)+alpbal(2.2)*SIal(iJ.k.l)
+alphal(2,3*Saal(iJJCJHalphal(2,4)*SIaal(i,j,k,l);
PPPP21(iJJ<.l)= alpha1(2.1)*sm(i.j.k.l)+ alpha(2.3)*Saa1(i.i.k.l)+
alpha(2,4)*SIaal(i,j,k,l);
pppp31(i,j,k,l)= alphal(3,1)‘SIIl(iJkJHalphal(3,4)*SIaal(i,j.k,l)
+alphal(3,5)*Saaal(iJ,k.l)+ alphal(3,6)*SIaaal(i,j,k.l);
pppp4l(i,i,k.l)=alphal(4,1)*SIIl(i,i,k,l)+alphal(4,4)*SIaa1(iJ,k,l)
+alpha(4,7)*Saaaal(iJ,k,l)+alpha(4,8)*SIaaaal(i,j.k,l);

%Casel:FSQ model;
ppw01(iJ.k.l)=Cl‘ppppl(iJ.kJHCZ‘pppp2(iJ.k.l)
+(l-Cl- C2)’pppp3(i,j.kJ)+C4‘PPPP4(iJ.k.l);

%CaseZ:Quadratic model;
%pppPOI(iJ.k.l)=pppp51(iJ.k.l);

%Determine <pppp>zS
for i=1:3
for j=l:3
Ml(iJ)=0;
for k=1:3
for i=1 :3
Ml(iJ)=Ml(iJ)+PPPP01(iJJ<.l)*S(l.k);
end
end
end

88

end

%Determine <pp>:<pppp>
for i=1:3
for j=l:3
Nl(i,j)=0;
for k=1:3
for l=l:3
N 1(iJ)=Nl(iJ)+PPPPOI(iJ.k.l)*al(1.10;
end
end
end
end

%Determine S.a+a.S, W.a+a.W(T);
for i=1:3
for j=l:3
saasl(iJ)=0;
waawl(i.j)==0:
for k=1:3
saasl(iJ)=saasl(iJ)+S(i.k)*al(kJHal(i.k)*S(kJ);
waaw1(iJ)=waawl(i.j)+W(i.k)*al(kJ)+a1(ix)*wti.k);
end
end
end

%Right handside of the equation
for i=1:3
for j=l:3
dk2(iJ)=(H>(iJ)/3-al(ij))+lam*Pc*(saasl(iJ)-2*Ml(iJ))-P6*Waaw1(iJ)+U*(aal(iJ)-Nl(iJ));

end

%Computation using second order R-K method
for i=1:3
for j=l:3
a(i,j)=a(i,j)+(dkl(iJ)+de(i,i))*dt"‘0.5;
E=[a(l.l).8(1.2).8(1.3):80.1)A(2.2).a(2.3);a(3.l).a(3.2).a(3.3)]z
[v.lambdal]=eig(E);
if i=2
if j==3
A(n)=a(iJ);
time(n)=(n-l)*dt;
end
end
if i=2
if j=2
B(n)=a(iJ):
time(n)=(n-l)*dt;
end
end
if i=1
ifj=l
C(n)=a(i.j):
time(n)=(n-l)*dt;
end

89

end
if i=3
if j=3
D(n)=a(iJ);
timc(n)=(n-l)*dt;
end
end

%d(n)=det(E);
e3(n)=V(3.3):
time(n)=(n-l)*dt;
62(n)=V(2.2):
time(n)=(n-l)*dt;
en(n)=V(2.3);
timdn)=(n-l)*dt;
el(n)=v(l,1);
time(n)=(n-l)*dt;
l2(n)=lambdal(2,2);
time(n)=(n-l)*dt;
13(n)=lambdal(3.3);
timdn)=(n-l)*dt;
ll(n)=lambdal(1,l);
time(n)=(n-l)*dt;

if i==l
if j=1
BB(n)=bb(iJ);
BBB(n)=bbb(i,i):
end
end
if i=2
if j==2
CC(n)=bb(iJ);
CCC(n)=bbb(i,j);
end
end
if i=3
if j==3
DD(n)=bb(iJ);
DDD(n)=bbb(iJ);
end
end
end
end
for i=1 :3
for j=1 :3
II(n)=BB(n)+CC(n)+DD(n):
III(n)=BBB(n)-t-CCC(n)I-DDD(n);
end
end
end

%Realizability region
xIII=[- l/36:0.0001:2/9];
yII=2/9+2*xIII;

nyI=[0:0.0001:ll6];
xxIII=-6*(nyI/6)."(3l2);

ynyI=[0:0.0001:2/3];
xxxIII=6*(ynyI/6)."(3l2):

%plot(xm,yII,xxIII,nyI,xxxIII,ynyI,III,II,’:’,-.01 13.0.09 l5,b’,0,0,’square’),xlabel(’IIIb’),ylabel(’IIb’),
axis([-.03 .001 -0.05 .2]);
Plot(xIII.yII.xxxlILynyI.xxIII.yyll.III.II.’--3.xlabel(’IIIb’).ylabcl(’IIb’):
%plot(time,C,time,B,’-.’,time.D,’-’,time,A,’:),xlabel('t),ylabel(’aij ’),axis([0 30 0 .5 D;
%plot(time,ev22)

%plot(time,el,time,e ,’-—’,time,en,’-.’),xlabel(’t’),ylabel(’vij),axis([0 1.2 -l.2 l.2})
%plot(time,12,’-.’,tirne,l3,’-’,time,ll),xlabel(’t’),ylabel(’lambdal’),axis([0 1.2 -.2 1.2])
%plot(time,en,’+’);

%a(1,l)

%a(2,2)

%a(3.3)

%a(2.3)

%e3

E

v

%lambdal(3)

%e3

%IIb

%IIIb

91

LIST OF REFERENCES

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