Download Detennining the Phase Diagra1n and Aggregate Size of a Chro1nonic Liquid Crystal J

Detennining the Phase Diagra1n
and Aggregate Size of a
Chro1nonic Liquid Crystal
Jessica Gersh
Honors Thesis
Advisor: Peter J Collings
March 3) 2006
Swarthmore College
Department qf Physics and Astronomy
2006
Abstract
Although the most recent studies of Sunset Yellow FCF, dis odium chromoglycate,
and several other chromonic liquid crystals suggest that chromonic liquid crystals form
rod-shaped aggregates with a distribution of sizes that shifts towards larger aggregates as
the concentration increases, limited studies of another chromonic liquid crystal,
Benzopurpurin 4B (BPP 4B), suggest that the aggregation process is very different for BPP
4B. These studies found that unlike other chromonic liquid crystals, BPP 4B solutions
scatter visible light and form a liquid crystal phase at significantly lower concentrations,
which implies that their aggregates are much larger. To extend this research, both the
phase diagram in water and the aggregate size of BPP 4B were investigated. To
determine the phase diagram, the temperatures marking the beginning and end of the
coexistence region between the liquid crystal and isotropic liquid phases were measured
optically. The results suggest that BPP 4B forms aggregates with a distribution of sizes
and has a liquid crystal phase at significantly lower concentrations and volume fractions
than other chromonic liquid crystals. Additionally, measurements of the hydrodynamic
and optical radii, the relative scattering intensity, and the absorption coefficient suggest
that the size distribution does not change with concentration. One possible explanation is
that BPP 4B forms micelles with a distribution of sizes. However, the presence of an
impurity much larger than the aggregates might also explain these results. Although the
exact aggregate structure of BPP 4B remains largely uncertain, the results of these
experiments suggest that it is very different from the simple, rod-like structures of other
chromonic liquid crystals.
2
Table of Contents
Chapter 1
Illtr()(illcti()Il •••••••••••••••••••••••••••••••••••••••••••••••••••• 6
1.1 An Additional Phase of Matter ...................................... 6
1.2 Prior Research............................................................. 8
1.3 The ExperilD.ents....................................................... 10
Chapter 2
1lhe()r)' •••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 11
2.1 Aggregation ............................................................... 11
2.1.1 Micelles .......................................................................... 11
2.1.2 Chrorrtonic Aggregates....................................................... 15
2.2 The Optics of Polarized Ught...................................... 21
2.2.1Jones
~ctors
andMatrices............................................ ..... 22
2.2.2 Birefringence .......................................... ......................... 24
2.3 Light Scattering......................................................... 26
2.3.1 Static Light Scattering........................................................ 26
2.3.2 Dynarrtic Light Scattering... ................................................ 31
2.4 Absorption, Scattering, and Extinction .......................... 36
2.4.1 Absorption ............................................ .......................... 36
2.4.2 Scattering........................................................................ 40
2.4.3 Extinction............................................. .......................... 41
3
Chapter 3
Experim.ental Methods .................................... 42
3.1 Sanlple Preparation .................................................... 42
3.2 DeterDlining the Phase Diagranl in Water................... 42
3.3 Light Scattering......................................................... 47
3.4 Absorption Measurenlents .......................................... 48
Chapter 4
Results .......................................................... 49
4.1 The Phase Diagrant. ................................................... 49
4.2 The Optical and H ydrodynanlic Radii ........................... 51
4.3 The Relative Scattering Intensity and the Absorption
Coefficient...................................................................... 54
Chapter 5
Discussion ..................................................... 57
5.1 The Phase Diagrant. ................................................... 57
5.2 The Radii, the Relative Scattering Intensity, and the
Absorption Coefficient..................................................... 59
5.3 The Possible ForDlation of Micelles with a Distribution of
Sizes .............................................................................. 60
5.4 The Possible Presence of a Large IDlpurity .................... 61
Chapter 6
Conclusion .................................................... 62
4
Acknowledglllents
References
5
Chapter 1
Introduction
1.1 An Additional Phase of Matter
As nearly every elementary school student knows, there are three fundamental
phases of matter: the solid, liquid, and gas phases. Transitions between these phases are
governed by the temperatures and pressures to which a material is subjected, and the
majority of compounds change from solids to liquids to gases as the temperature increases
under constant pressure. For one out of every several hundred randomly synthesized
organic compounds, however, there exists an additional phase known as the liquid crystal
phase [1]. As its name suggests, the liquid crystal phase falls in between the solid and
liquid phases, and the three phases are differentiated by the amount of order possessed by
molecules in each state. In a crystalline solid, the molecules are highly ordered, for they
are confined to a lattice structure. Since there is a tendency for the molecules to be
located at specific positions, a crystalline solid is said to possess positional order.
Additionally, since the molecules tend to be aligned along particular directions, a
crystalline solid possesses orientational order as well. Macroscopically, a crystalline solid is
characterized by its ability to maintain its shape. In an isotropic liquid, by comparison,
the molecules possess no order since they are randomly arranged, and macroscopically, a
liquid is characterized by its ability to flow to take the shape of its container. In between
these two extremes, a liquid crystal flows like a liquid while maintaining a small amount of
the orientational order possessed by crystalline solids. Depending upon the specific type
of liquid crystal, some positional order may be present as well (Figure 1.1.1) [1].
6
o
Crystalline Solid
Isotropic Liquid
Liquid Crystal
Increasing Temperature
Figure 1.1.1 Microscopic View of the Solid, Liquid Crystal , and Liquid Phases
In a crystalline solid, the molecules are highly ordered since they are confined to a lattice structure, and
macroscopically, the solid is characterized by its ability to maintain its shape. In an isotropic liquid, the
molecules possess no order since they are randomly arranged, and macroscopically, a liquid is characterized
by its ability to flow to take the shape of its container. In between these two extremes, a liquid crystal flows
like a liquid while maintaining a small amount of the orientational order possessed by a crystalline solid.
Depending upon the specific type of liquid crystal, some degree of positional order may be present as well.
In general, there are two ways in which a compound can form a liquid crystal
phase. As described above, temperature changes cause some pure compounds to move
into or out of a liquid crystal phase. Liquid crystals formed in this manner are known as
thermotropic liquid crystals. Alternatively, for some compounds in solution, changes in the
concentration cause the molecules to aggregate, forming larger structures. These
aggregates flow like the molecules in a liquid, and as they diffuse throughout the solution,
they partially align, creating a liquid crystal phase. Liquid crystals produced in this
manner are known as f;yotropic liquid crystals, and they are further classified by aggregate
shape into micelles, whose aggregates are closed structures of a particular size, and
chromonic liquid crystals, whose aggregates are shaped like rods and have a distribution of
sizes (Figure l.l.2) [1].
7
Various Micelles
and a
Representative Molecule
Various Chromonic Aggregates
and a
Representative Molecule
(Side View of Disk)
Figure 1.1.2 Micelles vs. Chromonic Aggregates
A closed structure, micelles require a specific number of molecules to form , and this restriction
fixes the size of the aggregates. Chromonic aggregates, on the other hand, are rod-shaped
structures formed by any number of molecules, and as a result, there is a distribution of aggregate
sizes. These differences in aggregate structure stem from differences in the properties of the
component molecules.
1.2 Prior Research
The majority of the research on chromonic liquid crystals has centered on
disodium chromoglycate, an asthma medication that was one of the first compounds
identified to form a chromonic liquid crystal phase [2]. Disodium chromoglycate is liquid
crystalline at room temperature at concentrations of 10 wt.% or higher [3,4,5], and the
most recent x-ray measurements suggest that the aggregates are columns of single
molecules in cross-section [6,7]. Very recent light scattering and viscosity experiments in
the liquid phase place the diameter and the average length of the columns at the liquid
crystal-liquid phase transition at 2 nm and 20 nm, respectively [2]. However, the exact
structure of the aggregates is uncertain; prior experiments suggest that the aggregates
form hollow columns of four molecules in cross-section [7,8,9].
8
In addition to disodium chromoglycate, several other liquid crystals have been
studied , albeit less extensively. Notably, Horowitz et al investigated Sunset Yellow FCF, a
food dye considered to be representative of chromonic liquid crystals in general [10].
The results of this study suggest that, like disodium chromoglycate, Sunset Yellow FCF is
liquid crystalline at room temperature at high concentrations, with no lower limit on the
concentration at which aggregates form. (The evidence that this is the case for Sunset
Yellow FCF is much stronger than the corresponding evidence for disodium
X-ray measurements provide strong evidence that the
chromoglycate, however.)
aggregates are columns of stacked single molecules, and the way the absorption
coefficient decreases with increasing concentration suggests that the size of these
aggregates changes with concentration. In fact, the outstanding agreement between the
absorption measurements and a simple theoretical model based on the law of mass action
offers convincing evidence that Sunset Yellow FCF forms aggregates with a distribution of
sizes that shifts towards larger aggregates as the concentration increases. Additional
evidence that chromonic liquid crystals form aggregates of a distribution of sizes, with the
average aggregate size increasing with concentration, comes from many less direct studies
of several other chromonic liquid crystals, including xanthone derivatives [11] , acid red
266 [12], phthalocyanine and porphyrin derivatives [13], Levafix Goldgelb [14], Violet
20 [15,16], direct blue 67 [17,18], and Blue 27 [16].
Another little-studied chromonic liquid crystal is Benzopurpurin 4B (BPP 4B), a
red textile dye used to color cotton, wool, silk, and nylon (Figure 1.2.1 ) [19].
(f)
Na
o
II
S
II " 0 8
o
Na
(f)
o=S=o
I
08
Figure 1.2.1 The Molecular Structure of Benzopurpurin 4B
To date, only one paper has been published on BPP 4B, presenting several
thermodynamic measurements and a general aggregation model [20]. In particular,
9
Bykov et al present a phase diagram of BPP 4B in water that suggests that BPP 4B forms a
liquid crystal phase at room temperature at concentrations as low as 1 wt.% and possibly
even lower. Disodium chromoglycate and Sunset Yellow FCF, by comparison, do not
form liquid crystal phases at room temperature until concentrations of approximately 10
wt.% and 29 wt.%, respectively [3,4,5,10]. To explain the low concentration, Bykov et al
offer a variety of thermodynamic measurements that suggest that the BPP 4B aggregates
incorporate water, with the amount of incorporated water decreasing with concentration.
This incorporation of water should increase the size of the aggregates and allow the
aggregates to interact at lower concentrations [20].
1.3 The Experhnents
To extend this research, both the phase diagram in water and the aggregate size of
BPP 4B were investigated. To determine the phase diagram, the temperatures marking
the beginning and end of the liquid crystal-liquid coexistence region were measured for a
range of concentrations. Since the liquid crystal phase is birefringent, these temperatures
were measured optically by relating the intensity of the light passing through a pair of
crossed polarizers that sandwiched a BPP 4B sample of known concentration to the
temperature of the sample. To determine the aggregate size, both static and dynamic
light scattering techniques were used, with the aggregates modelled as spheres for
simplicity. Unlike disodium chromoglycate and Sunset Yellow FCF, BPP 4B strongly
scatters visible light, which suggests that its aggregates are significantly larger and must
have an entirely different structure. Finally, to examine further how the aggregation
depends on concentration, both the relative scattering intensity and the absorption
coefficient were measured for a wide range of concentrations. Since BPP 4B scatters
visible light and appears to form a liquid crystal phase at much lower concentrations than
disodium chromoglycate and Sunset Yellow FCF, it was hoped that these measurements
would reveal details of an aggregate structure completely different from the simple, rodlike structure of the other dyes.
10
Chapter 2
Theory
2.1 Aggregation
An equilibrium process, aggregation occurs when it is energetically more favorable
for some of the molecules to form larger structures than for all of the molecules to remain
dissociated. These larger structures are known as aggregates, and they may have several
different shapes and sizes, depending upon the nature of the molecules. Several simple
mathematical models have been developed to describe the formation of some types of
aggregates.
2.1.1 Micelles
One type of aggregate, a micelle, is a closed structure formed by a specific
number of amphiphilic molecules. An amphiphilic molecule consists of two distinct ends, a
polar head that is soluble in water and a nonpolar tail that is insoluble in water. One
example is sodium laurate, a molecule typically used in soap (Figure 2.l.1).
Na EE>
Figure 2.1.1 A Typical Amphiphilic Molecule
The chemical structure for sodium laurate is shown, with the amphiphilic
molecule representation drawn below. The dark circle represents the polar
head, and the zig-zag line represents the nonpolar tail.
11
When amphiphilic molecules are mixed in water, they tend to arrange in a way that
minimizes the amount of contact between the water and the nonpolar tails and that
maximizes the amount of contact between the water and the polar heads. Strongly polar
amphiphilic molecules achieve this by clustering into spheres, with the polar heads in
contact with the water and the nonpolar tails sheltered in the interior of the sphere
(Figure 2.l.2).
Figure 2.1.2 A Micelle in Water
When strongly polar amphiphilic molecules are mixed in water, they cluster into spheres, with
their polar heads in contact with the water and their nonpolar tails sheltered in the center of the
sphere.
Alternatively, weakly polar amphiphilic molecules form vesicles, spherical shells
incorporating water (Figure 2.l.3).
Figure 2.1.3 A Vesicle in Water
When weakly polar amphiphilic molecules are mixed in water, they cluster into vesicles, spherical
shells that incorporate water. These molecules align so that their polar heads are in contact with
12
the water either outside the vesicle or in its center, leaving the nonpolar tails sheltered in the
shell.
In both cases, the aggregates require a specific number of molecules to form, so all of the
aggregates are approximately the same size [1].
For micelles, the mathematical model describing the aggregation process is
particularly simple [21]. Since aggregation is an equilibrium process, the rate at which
aggregates of any size form is the same as the rate at which they dissociate. For the case
of micelles, when only one size of aggregate forms, this condition may be written in terms
of the chemical equation
(2.1.1)
where N is the number of monomers (dissociated molecules) Al contained m an
aggregate AN. This reaction is characterized by an equilibrium constant K, defined by
(2.1.2)
where XN is the volume fraction of aggregates containing N molecules and Xl is the
volume fraction of all the dissociated molecules. (The volume fraction for a particular
entity is defined as the volume occupied by that entity divided by the total volume.) The
volume fractions XN and Xl are related by
(2.1.3)
where 1J is the total volume fraction of all the molecules, so the equilibrium condition
may be rewritten as
(2.1.4)
In terms of the molal concentration
eM , 1J can also be written as
(2.1.5)
where MW is the molecular weight of the sample in grams per mole and P is the density
in grams per liter [10].
13
Given N , K , and f), it is possible to investigate how Xl and XN depend on f}
by using Equation 2.1.3 to plot Xl and XN as functions of f).
The most distinctive
characteristic of such a plot is the existence of a critical volume ftaction, a volume fraction
below which micelles do not form. Above the critical volume fraction, the number of
aggregates increases linearly with f), while the number of monomers increases at a much
slower rate (Figure 2.1.4) [21]. When given in terms of the concentration, the critical
volume fraction is called the critical micelle concentration.
--e-- X1 (Volume Fraction of Monomers)
--a - XN (Volume Fraction of Aggregates)
The Volume Fraction of Monomers and Aggregates
as a Function of the Total Volume Fraction
0.00015
i
/
*
/
0>
i!'
0>
0>
«
/
0.0001
"0
c
'"
'"
/
!!?
E
0
c
0
:2
/
'0
c
/
0
~
§'"
LL
"0
>
510"5
/
o
r-
/
510"5
/ '
/
0.0001
0.00015
0.0002
Total Volume Fraction
Figure 2.1.4 The Volume Fractions of Monomers and Micelles as Functions of the Total Volume Fraction
Using Equation 2.1.3 and the parameters K = 1080 and N =20, the volume fractions of monomers (single
molecules) and micelles are plotted as a function of the total volume fraction 1J. For small values of 1J ,
only monomers exist, and the total number of monomers increases linearly with 1J. (Although the above
plot shows the volume fraction of monomers increasing linearly with 1J, the total number of monomers
also increases linearly with 1J since the volume fraction is proportional to the total number.) Once some
critical value of 1J is reached (about 5xlO-5 in the figure), micelles form, and as 1J increases, the number of
micelles increases linearly, while the number of monomers increases at a much slower rate.
14
2.1.2 Chr01nonic Aggregates
Another type of aggregate, a chromonic aggregate, is a rod-like structure formed by a
number of molecules. Unlike the molecules that form micelles, the molecules that form
chromonic aggregates possess weak polar and nonpolar regions. As a result, their
aggregation is driven as much by intermolecular attraction as by an attempt to minimize
the amount of contact between the water and the weak nonpolar regions. Disodium
chromoglycate is one example of a compound that forms chromonic aggregates (Figure
2.1.5).
o
o~o
o
OH
EE>
Na
e
e
eoo
ooe
EE>
Na
Figure 2.1.5 The Molecular Structure of Disodium Chromoglycate
For chromonic aggregates, the aggregation model is slightly more complex than it
is for micelles. Rods form as molecules link together in one-dimensional chains, which
can be of any length, and the formation of these rods serves to reduce the free energy.
For two non-interacting molecules, the free energy is simply twice the mean free energy
f.l~ of a monomer, or 2 f.l~ . However, if these two molecules interact to form a chain, the
interaction decreases the free energy by an amount aksT , where a
constant, kB is Boltzmann's constant, and T
is a positive
is the temperature. Similarly, for every
additional molecule added to the end of the chain, the free energy decreases by an
additional amount akBT (Figure 2.1.6).
15
Monomer (Free Energy f.1~)
Aggregate of 2 Molecules (Free Energy -akBT + 2f.1~ )
o
o
Monomer
Cross-section
(Single molecule)
Aggregate of 3 Molecules (Free Energy -2ak BT + 3f.1~ )
Aggregate
Aggregate of N Molecules (Free Energy -(N -l)akBT + N f.1~ )
o
Aggregate
Cross-section
(Single molecule)
Cross-section
(Single molecule)
Aggregate
o
Cross-section
(Single molecule)
Figure 2.1.6 Rod Formation
Rod-shaped aggregates form as molecules link together in one-dimensional chains, which may be
of any length, and the formation of these rods serves to reduce the free energy. For N noninteracting molecules, the free energy is simply N times the mean free energy J1~ of a monomer,
or N J1~. However, if these molecules interact to form chains, the interactions reduce the free
energy by an amount akBT for every molecule added to the end of a chain, where a
is a
positive constant, kB is Boltzmann's constant, and T is the temperature.
The free energy of an aggregate of N molecules, then, is given by
N J1~=- (N -l)aksT + N J1~,
where J1~ is the mean free energy per molecule in an aggregate of size N
(2.1.6)
[21]. Solving
for J1~, Equation 2.1.6 can be rewritten as
(2.1. 7)
As before, the rate at which aggregates form must be equal to the rate at which
they dissociate. If the rate at which aggregates of N molecules form is given by
16
rate of formation=KlX~,
(2.1.8)
and if the rate at which aggregates of N molecules dissociate is given by
rate of dissociation =KN (
~) ,
(2.1.9)
where Kl and KN are constants, then the equilibrium condition can be written as
(2. 1. lO)
or
(2.1.11)
where K is defined as the equilibrium constant.
Additionally, the law of mass action states that
This relationship is true for all N .
-M'0J
K=exp [ - ,
(2.1.12)
kBT
where M'° is the standard free energy of the reaction. In this case,
M'o = N(tI~ -
tin,
(2.1.13)
so
K = ~ = exp[ -N (Il' _/1,")
NX~
kBT
N
J
(2.1.14)
or
XN=N( x.exp [ Il~~:~
17
Jr
(2.1.15)
Using Equation 2.1. 7 to rewrite 11~ and simplifying,
(2.1.16)
Since a chromonic liquid crystal contains aggregates of a distribution of sizes instead of
aggregates of a single size, the total volume fraction i} for chromonic aggregates can be
defined in exactly the same manner as it was for micelles, with the volume fraction of
aggregates of one size, X N ,replaced by the sum of the volume fractions of aggregates of
=
all sizes in the distribution,
LX
N •
Mathematically,
N=l
f( N[
i}= fXN=
N=l
N=l
Xlear e- a ),
(2.1.17)
which simplifies to
(2.1.18)
Solving for Xl,
(2.1.19)
The negative sign is chosen because it restricts the values of Xl to being less than or
equal to i} ; in other words, it requires that the volume fraction of monomers be less than
or equal to the total volume fraction, as must be the case [21].
Given
a
and i}, it is possible to plot the distribution of XN as a function of N
using Equation 2.1.16 and Equation 2.1.18. As i} increases, the values of XN mcrease,
and the distribution broadens (Figure 2.1.7).
18
o
o
l
Vo lu me Fraction (theta;() .0 1
Vo lu me Fraction (theta;().25
The Volume Fraction of an Aggregate of Size N
0.006 r - - - - - . - - - - - - - - - r - - - - - - , r - - - - . - - - - - - - - - ,
0.005 r ············E1···············!·······y,.················ ...... , ..................................., .................................. ~ .. .
o
o
o
0.004 r································;····················.."R ........ ; ....................................; ...................................;
o
o
0.003 r ································!····································;'+r····························,; ...... ·............................ ;
o
o
0.002
r .. ·........·................·. ·!....·................ ·........ ·....·,·........·........ lh: .........; ...................................;
o
0.001
o
20
40
60
80
100
Number of Molecules in an Aggregate
Figure 2.1. 7 The Distribution of Volume Fractions for Aggregates of N Molecules
The volume fraction for aggregates of N molecules, X N
,
is plotted as a function of N for
a=7.0 and two values of 1J. As 1J increases, the individual X N increase, and the distribution
broadens.
Similarly, a plot of the distribution of the fraction of aggregates of N molecules may be
XN
obtained by dividing Equation 2.1.16 by N and normalizing. The quantity
Ii
is the
volume fraction of all aggregates of N molecules divided by N , so it is proportional to
XN
the number of aggregates of size N. The fraction of aggregates of size N is then
=
divided by
X
~;
(Figure 2.1.8).
19
N
o
o
NLllm'ber of Agg regates !theta;().01)
Num'ber of Agg regates lheta=<l.25)
The Fraction of Aggregates of Size N
0.3
0.25
---·-----'---- r ----·---
0.2
o---r---T---r---r--o----r--- T-- --r- -- -r- --
(/)
2
'"~
0>
0>
0>
<t:
'0
0.15
c
0
U
~
LL
o
l
l
l
l
:--r--"]"-- r ---1--
0.1
-- T- -r- -r - r--
0.05
o
o
20
40
60
80
100
Number of Molecules in an Aggregate
Figure 2.1.8 The Fraction of Aggregates of N Molecules
The fraction of aggregates of N molecules is plotted for two values of tJ. For higher
tJ , the distribution of sizes is broader than it is for lower tJ.
One other quantity that may be determined is the average size of the aggregates,
Mathematically,
(N)
is defined as
fN(XN)
(N) = N=!
N
fX N
N=! N
Evaluating the sum,
(N) .
(N)
fx N
= N=!
f
XN
N=! N
may be rewritten as
20
(2.1.20)
fX N
N=! N
Using Equation 2.1.21,
(N)
may be plotted as a function of
19
for a given value of
a
(Figure 2.1.9).
r=- <Nl
The Average Number of Molecules in an Aggregate
as a Function of the Total Volume Fraction
16
14
Q)
15
'"
i!?
'"
'"
""c:
'"
.s
12
U>
Q)
3
&l
-0
::;:
'0
8
Q;
.0
E
6
:J
: : :-7
Z
Q)
'"
>
""
ill
Q)
4
2
V
/
10
/
L
/
.L
/
/
o
o
0.05
0.1
0.15
0.2
Total Volume Fraction
Figure 2.1.9 The Average Aggregate Size as a Function of the Total Volume Fraction
Using Equation 2.1. 21 and the parameter a= 7 .0, the average aggregate size is plotted as
a function of the total volume fraction. As the total volume fraction increases, so does the
average aggregate size.
2.2 The Optics of Polarized Light
An electromagnetic wave, light is composed of an electric field and a magnetic
field that oscillate at right angles to each other and to the direction in which the wave
propagates. The polarization describes how the directions of oscillation change over time;
by convention, it describes how the direction of oscillation changes for the electric field.
(Since the electric and magnetic fields oscillate at right angles to each other and to the
21
direction of propagation, this simultaneously describes how the direction of oscillation
changes for the magnetic field.) Ordinarily, the direction in which the electric field
oscillates changes randomly, and the light is said to be unpolarized. When the electric field
oscillates along a fixed line, the light is linearf)! polarized. More generally, the direction of
oscillation of polarized light rotates, and the light is ellipticalf)! polarized.
2.2.1Jones Vectors and Matrices
Mathematically, the polarization can be described using Jones vectors, column
vectors in which each element corresponds to a component of the electric field along a
particular spatial direction [22]. For light travelling in the z-direction, the electric field
vector, E, can be written as
(2.2.1)
where
Ex
field, and
is the x-component of the electric field,
Ey
is the y-component of the electric
X and yare the unit vectors in the x- and y-directions, respectively. To allow
for the time and space dependence of a travelling wave explicitly,
Ex
and
Ey
can be
rewritten as
Ex
= Eo,xei(kz-rot+¢x)
(2.2.2)
and
- E
i(kz-rot+¢y)
E yo,ye
,
where
Eo,x
is the amplitude of
Ex, Eo,y
equals 21t divided by the wavelength),
Ex
and
Ey,
(2.2.3)
is the amplitude of
E y,
k is the wavenumber (and
is the frequency, and ¢Jx and ¢Jy are the phases of
0)
respectively. By substituting Equations 2.2.2 and 2.2.3 into Equation 2.2.1
and factoring, E can be rewritten as
(2.2.4)
or, in column vector form,
~
_
E-
i¢X]
[E0, xe 'n. e i(kz-rot)
Eo,ye''f'Y
.
(2.2.5)
The column vector in Equation 2.2.5 is the general form of the Jones vector, containing
the relative phases and magnitudes of the various components of the electric field. The
22
exponential factor
ei(kz-rol)
is not included since it is a property of the wave as a whole
and does not affect the polarization.
For linearly polarized light, the general form of the Jones vector is
[:~:;],
where
Y is the angle of polarization measured relative to the x-direction. SeveralJones vectors
for linear polarization at specific angles are summarized below:
Horizontal Polarization (polarized in the x-direction):
[~l
Vertical Polarization (polarized in the y-direction):
[~l
Polarized 45° relative to the x-direction:
Polarized -45° relative to the x-direction:
Just as column vectors are used to represent the polarization state of the light,
square matrices are used to represent various optical elements, and the polarization state
of the light after passing through one of these elements is given by the product of the
Jones matrix for that element and the Jones vector for the input light [22]. For example,
the Jones matrix
Mhoriz
for a horizontal polarizer is
(2.2.6)
and if arbitrarily linearly polarized light of amplitude
Eo
is passed through this polarizer,
the output light will be in the state
.
output llght
[COSY] = Eo cos Y
= MhorizEo.
SIllY
23
[1]
0
.
(2.2.7)
The resulting Jones vector corresponds to the polarization state of the output light, and
the modulus-squared of its constant prefactor corresponds to the intensity. In this case,
the output light is horizontally polarized since its Jones matrix is
[~], and its intensity I
is
given by
(2.2.8)
where 10
= lEo 12
is the intensity of the light before passing through the polarizer. Several
usefulJones matrices are listed below:
Horizontal Polarizer (transmission axis along x-direction):
M honz.
=
[1 0]
0
0
Vertical Polarizer (transmission axis along y-direction):
45° Polarizer (transmission axis 45° relative to x-direction): M" =
~[:
:]
1[1 -1]
_45 0 Polarizer (transmission axis _45 0 relative to x-direction): M -45 =-
2 -1
1
ei¢x
General Phase Retarder:
Mphase
=
[
0
2.2.2 Birefringence
As described in Section 1.1, a liquid crystal is characterized by its ability to flow
like an isotropic liquid while maintaining a small amount of orientational order. For a
chromonic liquid crystal, this orientational order is due to an anisotropy in the shape of
the aggregates. Typically, these aggregates are shaped like rods, and they tend to align
their long axes along a unique direction denoted by a line called the director (Figure 2.2.1).
24
Director
Figure 2.2.1 Aggregate Alignment and the Director in a Chromonic Liquid Crystal
A liquid crystal is characterized by its ability to flow like an isotropic liquid while maintaining some
of the orientational order possessed by crystalline solids. In a chromonic liquid crystal, this
orientational order is due to the alignment of the aggregates along a unique direction. This
direction is denoted by a line called the director, which is vertical in the figure above.
This alignment creates an anisotropy within the liquid crystal, and as a result, the index of
refraction depends upon the direction in which the light is polarized. Light polarized
parallel to the director experiences one index of refraction,
n il'
while light polarized
perpendicular to the director experiences another, n.l . This phenomenon is known as
linear birifringence, and it is often expressed as the difference between the two indices of
refraction:
birefringence = /::"n = n il
- n.l .
(2.2.9)
Since the index of refraction depends upon the direction of polarization, the
speed at which light propagates through the liquid crystal also depends upon the direction
of polarization. As a result, light propagating through a liquid crystal will accumulate a
25
relative phase difference between the components polarized parallel and perpendicular to
the director.
Assuming that light of vacuum wavelength
Ao propagates a distance d
through the liquid crystal, the parallel component will accumulate a phase
(2.2.lO)
and the perpendicular component will accumulate a phase
(2.2.11)
The relative phase difference, then, is
(2.2.12)
Since the liquid crystal introduces a relative phase difference, it may be considered a phase
retarder. As a result, its Jones matrix is given by M phase' where ¢JII and ¢J.l correspond to
¢Jx and ¢Jy when the director is along the x-axis [22].
2.3 Light Scattering
2.3.1 Static Light Scattering
In a static light scattering experiment, a beam of light is directed into a sample,
and the intensity of the scattered light is measured as a function of the scattering angle (),
the angle from the incident light. (A scattered ray pointing along the same direction as
the incident light would be oriented at zero degrees.) This scattered light arises from
interactions between the incident light and the scatterers (aggregates, molecules, or
particles) in the sample; it corresponds to the electromagnetic field induced by the
oscillating electromagnetic field of the incident light. For scatterers much smaller than
the wavelength of the incident light, this induced field may be modelled as that of an
oscillating electric dipole, with the intensity given by
26
(2.3.1 )
where Po is the induced dipole moment,
(j)
is the frequency of oscillation, r is the
distance from the dipole, and cp is the polar angle [23]. One important feature of this
e;
in other
intensity pattern is that it is completely independent of the azimuthal angle
words, the intensity is uniform in the scattering plane, the plane defined by the incident and
scattered light (Figure 2.3.1 ).
y
x
Incident Beam
z
Scattered Beam
Sample
Figure 2.3.1 Static Light Scattering
A beam of light travelling in the z-direction is directed into a sample, and a scattered beam
emerges at an angle f}. For the incident and scattered light shown above, the scattering plane is
the xz-plane. The polar angle cP is also shown.
For scatterers that are comparable in size to the wavelength of the incident light,
the dipole approximation is not applicable because light scattered from different points on
the scatterer can destructively interfere. Instead, the intensity must be calculated by
summing the contributions from the light scattered from each point. In general, the
contributions for any two points A and B on the scatterer will be out of phase, having
travelled different distances from the light source to the detector (Figure 2.3.2).
27
Figure 2.3.2 Light Scattering From Two Arbitrary Points A and B
-
-
Light is incident on A and B, with an incident wave vector k, and a scattered wave vector k, .
Rays of light scattered from points A and B have a path length difference given by AD - Be ,
which varies with
e; this path length difference is responsible for the interference of the two rays.
This phase difference is given by
L1phase
=
AD-Be
(2.3.2)
A
m
where Am is the wavelength of the light in the sample. Since the incident wave vector
-
_
and the scattered wave vector ks are given by ki
A
= km . ki
_
and ks
A
= km. ks ,where
ki
2n
km = Am
is the magnitude of the wave vector in the sample, the phase difference may be rewritten
as
by using simple geometry. The quantity
(ks-kJ·rAB
q·rAB
q·rAB
kmAm
kmAm
2n
q = ks -
(2.3.3)
k i that appears in Equation 2.3.3 is
known as the scattering wave vector, and its magnitude may be calculated in terms of the
scattering angle () as follows:
28
(2.3.4)
So, adding the electric field contributions from A and B,
EA (t) + EB(t) = Eocos(rot) + Eocos(rot + -q. rAE) = [2Eocos(t-q· rAE )Jcos( rot + t-q· rAE)
and the corresponding intensity is
Since the intensity contribution due to light scattered from A and B depends upon -q ,
which depends upon the scattering angle, it too depends upon the scattering angle. As a
result, the total intensity, which is the sum of all such contributions, must also depend
upon the scattering angle. The exact angular dependence is determined by the size and
shape of the scatterer, and it is described by the structure flctor S ( q) , which is defined for
randomly oriented scatterers as
S ( ) = scattered intensity at ()
q
scattered intensity at (}=o
or, mathematically,
S(q) = /
~ i>iq.~ 2),
\ N
(2.3.5)
j=l
where N is the number of contributions and where the averaging is done over all possible
orientations of the scatterer [24]. For a continuous scatterer, the discrete sum is replaced
by an integral. For the case of a uniform sphere, averaging over all orientations is
unnecessary due to symmetry, so Equation 2.3.5 becomes
29
R
ff
2n
S(q)
2
1C
eiqrcosa sinadar 2dr
= __~r=~o~a~=o~~R~_________
f
4n
f sin(qr) r 2dr
R
-3
1
R3
r 2dr
2
(2.3.6)
qr
r=O
r=O
Integrating by parts, this reduces to
S(q) =
where x
= qR
I:' (sin x - XCOSX)I' ,
(2.3.7)
(Figure 2.3.3).
S!ql
for R=1 microns
S q for R=O.4 microns
o
o
<>
S q for R=O.1 microns
The Structure Factor as a Function of Angle
ffi<><><><> 1
~ol::b
<>~
o
o
i<><>
~
·........·o........gdJ~....·. ·2~o~~·_ . . . ·. . . . ·. ·. . ·. . . . . . . . . . . . . . . . ~. . . . . . . . . . . . . .+. . . . . . . . . . . .
0.8
0
1
o
.
i O
o
t .......r.J.
............ 0. ...
0.6
o
0
o
1
o
o
. ,
<>
<>
<>
v<>
<>
<>
<>
<>
.... 0 ......
0.4
cp
"0'.........
<><>
<>
<><>
o
io
0
i
. . . ·. ·. . . . ·. · . ·r. ·g ..·....·......·....·· .... ·~r.Jt]' ..·....·..· ......·....·......··..~~~~o;. ·. . . . ·. ·. . !. ·. . ·. ·. . . . ·. ·. ·
0.2
00
00
o
o
10
i
0q..,
o
'i:i:
I'
30
40
i~
20
<><>
i
<><><><>+<><><>
, o<><><>A.
50
60
Angle (Degrees)
Figure 2.3.3 The Structure Factor for a Sphere as a Function of Angle
The structure factor for a spherical scatterer is plotted as a function of angle for various
radii. As the radius decreases, the plot broadens.
30
2.3.2 Dynamic Light Scattering
In a dynamic light scattering experiment, a beam of light is directed into a
sample, and the intensity of the scattered light is measured as a function of time at some
fixed scattering angle. The choice of scattering angle affects the measured intensity, since
it determines the size of the scattering volume (the part of the sample from which a
scattered photon may reach the detector) and the region of the structure factor being
examined. The measured time dependence of the scattered light is also affected by how
quickly the scatterers diffuse through the scattering volume and, if the scatterers are
anisotropic, how the orientations of the scatterers change as a function of time. Since the
scatterers diffuse and tumble erratically through the sample, the intensity might be
expected to fluctuate randomly over time. However, since there are upper limits on the
rates at which the orientations and positions of the scatterers can change, the intensity
cannot fluctuate completely randomly; this may be understood by considering the
analogous case of noise in an electric circuit.
By definition, white noise in an electrical circuit is a signal whose Fourier
transform has a constant amplitude for all frequencies. Since there are contributions at
very high frequencies, the plot of the white noise as a function of time oscillates extremely
rapidly and seems random. One way to verify that white noise is a random function of
time is to plot its autocorrelation function I(t) as a function of time t, where I(t) is
defined to be
I(t) = S:f(r)f(t+r)dr.
(2.3.8)
If a function f(t) is random, it oscillates so quickly that the function at time r
completely unrelated to the function at any other time r
function at t
= O.
+ t. As a result, I (t) is a delta
This is indeed the case for white noise (Figure 2.3.4) [25].
31
is
I- F(W
)I
1-
The Fourier Transform as a Function of Frequency
For White Noise
White Noise as a Function of Time
180
1.5
[
[
:
:
1(1)1
,------r------,------r------,-----,
100
•
•
• .............. .
................ • .......................................................
-50
-100
-180 L -_ _ _ _--'-_ _ _ _ _ _L -_ _ _ _--'-_ _ _ _ _ _L -_ _ _ _--'
20
40
60
100
80
200
400
Frequency w
600
800
1000
Time
1-
1(1)1
The Autocorrelation Function as a Function of Time
For White Noise
0.8
--------------------r--------------------- ---------------------r--------------------
0.6
~
~
--------------------r----------------------------------------r-------------------
0.4
····················f····················· ·····················f····················
0.2
------ •.•• •• ••..••.. , •••••..•• •• •• •• ••..••••••••..•• •• •• •• ••..• , ••••..••..•• ••••..•.
:
:
-0.2 L -_ _ _ _ _ _-'--_ _ _ _ _ _---'-_ _ _ _ _ _ _ _L -______--'
-5
-10
10
Figure 2.3.4 White Noise and Its Fourier T ransform
By definition , white noise is a signal whose Fourier transform has a constant amplitude at all
frequencies, as shown in the top left graph (in arbitrary units). Since the signal is completely
random , as shown in the top right graph (in arbitrary units), the autocorrelation function is a delta
function centered at the origin, as shown in the bottom graph (in arbitrary units).
However, if the white noise is directed through a low-pass RC filter, the higher frequency
components are removed, and
some correlation between
j( r)
j(t)
and
cannot oscillate as quickly. As a result, there will be
j( r + t)
when t is small.
Correspondingly, let) is
not a delta function, but
(2.3.9)
32
where K is a constant [25]. This autocorrelation function decays exponentially with a
time constant RC, which is the characteristic response time of the circuit (Figure 2.3.5).
o
modulus 01 F(w)
I
1-
The Modulus of the Fourier Transform
Of Low-pass Filtered White Noise
1(1)1
Low-pass Filtered White Noise as a Function of Time
40
30
0.8
0.6
04
.....
·············1-···············-1················-1-················1···············
••.
ii
i
i
l
l
l
l
1
1
l
l
l
l
l
······· ·· ········ff·········· · ··,·····
....... .. ., .... .
··········r················j·················r················r ··············
••••• ••
~
10
--
·····"]""··············"]""··············r··············r···..........
···········j················T···············r········.....
0.2
-10
._ .
-
.20 L -_ _---'-_ _-----L_ _ _-'--_ _---"-_ _----.J
100
200
400
300
500
20
40
60
80
100
Time
o
1(1)
I
The Autocorrelation Function for Low-Pass Filtered
White Noise
0.8
~
0
g
0.6
"'§
"
~
~
"
04
0.2
-··
..
.:
.:
.
·::
.:
.:
··
..
..
·
.,
...
..
.............. ..,..··.......................................................................
.
.:
:
.
·:
.:
.::
·:
.:
.:
·
,,
,,
..
,
,
............... ......................................................
_............... .
iii:
0.2
04
0.6
0.8
Time t
Figure 2.3.5 Low-pass Filtered White Noise
When white noise is directed through a low-pass fiiter, the higher frequency components are
removed. As a result, the amplitude of the Fourier transform decreases with increasing frequency,
approaching zero for high frequencies (top left). Since the higher frequency contributions have
been removed, f(t) does not oscillate as quickly (top right), so there is some correlation between
function values separated by short times (bottom).
Similarly, since the scatterers in a dynamic light scattering experiment cannot
translate and rotate infinitely quickly, there is an upper limit on how quickly the intensity
can fluctuate.
So, by analogy to the case of low-pass filtered white noise, the
autocorrelation function is not a delta function, but
33
let)
= I B + Ie-trw
0
,
(2.3.10)
where IBis a constant representing the background that arises from the fact that the
scattering volume is large enough to contain uncorrelated scatterers and where Tw is the
relaxation time, assuming that both the incident and scattered light are vertically
polarized [26]. The value of Tw is directly related to the size of the scatterers. As the
scatterers increase in size, they diffuse more slowly through the sample, so the intensity
does not fluctuate as rapidly. As a result, the autocorrelation function does not decrease
as quickly, so the value of Tw must be larger.
In other words, larger values of Tw
correspond to larger scatterers. The exact relationship may be found by using Fick's first
law of diffusion [27]:
lei,t) = -DT VCer,t) ,
(2.3.11)
where l(r,t) is the flux of the scatterers, C(r,t) is the mass concentration of scatterers,
and DT is the translational diffusion constant, which is related to Tw by
D
1
=--,--T
(2.3.12)
2 q 2Tw
By definition, l(r,t) is the mass passing through a unit of area in a unit of time, and it
may be expressed in terms of the mass concentration as
l(r,t) = v(r,t)C(r,t) ,
(2.3.13)
where v(r,t) is the average velocity of the scatterers [24]. Combining Equation 2.3.11
and Equation 2.3.13,
v(r,t)C(r,t) = -DT VC(r,t).
(2.3.14)
Assuming steady-state conditions, the scatterers move under the influence of two forces: a
driving force F and a translational frictional force
-iT v , where iT
is the translational
friction coefficient. Using Newton's second law, these forces are related by
(2.3.15)
34
so
F
v=-
(2.3.16)
iT .
Substituting Equation 2.3.16 into Equation 2.3.14,
~ C(r,t) = -DT VC(r,t).
(2.3.17)
Additionally, since the driving force for translational diffusion is due to a gradient in the
chemical potential, F is defined as
-V/1
F=-NA '
(2.3.18)
where V/1 is the chemical potential of the solute. Since V/1 is given by
(2.3.19)
F can be written in terms of the mass concentration as
F
=
kBT nc()
v
r,t.
(2.3.20)
C(r,t)
Substituting this expression into Equation 2.3.17,
- k;: VC(r,t)
= -DT VC(r,t).
(2.3.21 )
Matching the coefficients of VC(r,t),
D
= kBT
(2.3.22)
T iT
This result is known as the Stokes-Einstein equation. For a sphere,
35
iT
is given by
iT = 6n17R,
(2.3.23)
where R is the radius of the sphere [24], so the Stokes-Einstein equation becomes
(2.3.24)
Substituting this result into Equation 2.3.12,
(2.3.25)
So, for spherical scatterers,
(2.3.26)
2.4 Absorption, Scattering, and Extinction
As a beam of light passes through a sample, it interacts with the various molecules
in the sample. This interaction takes one of two forms: the molecules either scatter some
of the light, redirecting it along a new trajectory, or they absorb some of the light,
preventing it from travelling any farther through the sample. Typically, when a photon is
absorbed, the energy is converted into heat, increasing the temperature of the sample.
Alternatively, part of the absorbed energy can be transferred into a photon of longer
wavelength, a process known as florescence. Whether absorption or scattering occurs, these
interactions reduce the intensity of the beam, and they can be described very simply by
modelling the molecules as disks that absorb or scatter photons.
2.4.1 Absorption
In the case where only absorption occurs, this description is particularly simple. A
beam of light of intensity 10 and cross-sectional area A is directed into a sample of
thickness z and cross-sectional area greater than or equal to A. As the beam passes
through the sample, it interacts with the various molecules, and to describe these
interactions, each molecule is represented by a disk of area
Cabs'
where a photon is
absorbed if it enters one of these disks and is unaffected otherwise. Experimentally,
36
Cabs
is known as the absorption cross-section of the molecule, and its value is empirically chosen so
that the experimental fraction of photons is absorbed. This absorption reduces the
intensity of the beam, and the beam exits the sample with an intensity fez) (Figure
2.4.1).
Intensity 10
Intensity I(z)
Cross-sectional
Area A
Thickness z
Intensity I
Intensity I +dI
Cross-sectional
Area A
Thickness dz
Figure 2.4.1 Absorption in a Sample
A beam of light of intensity 10 and cross-sectional area A travels through a sample of thickness
z
and exits with intensity I(z). The sample can be divided into slices of thickness dz, where the
light incident on a particular slice has intensity I. The illuminated portion of each slice contains
N molecules, which are represented by disks of area Cabs' and a photon passing through a slice
is absorbed if it enters one of these disks and is unaffected otherwise. As a result, the intensity of
the light leaving the slice is 1 + dl ,where dl is negative since the intensity is reduced.
37
To find an expression for I(z) , the sample is divided into slices of thickness dz .
The light entering a particular slice has intensity I , and after interacting with the N
molecules in the beam, the beam exits with an intensity I + dI , where dI is negative
since the intensity is reduced (Figure 2.4.1).
Since each of these N
molecules is
represented by an absorption cross-section of area Cabs' the total area of the sample that
absorbs light is NCabs , so NCabsi A is the fraction of the beam that is absorbed. As a
result, dI can be written as
dI
= change in intensity
= -(fraction absorbed) ( initial intensity)
_ -NCabJ
A
(2.4.1 )
Rearranging this expression,
dI
I
-NCabs
A
(2.4.2)
Since the area of the slice can be written as
A= Adz =~,
dz
dz
where V
= Adz
(2.4.3)
is the volume of the slice irradiated by the beam, Equation 2.4.2 can be
written as
dI
I
NCabsdz
V
(2.4.4)
This is a separable differential equation that can be solved to find the expression for I(z).
Integrating over the entire sample,
C
f 2d = - Nabs
f dz
I(z)
10
z
I
V
so
38
0
(2.4.5)
'
(2.4.6)
Rewriting this expression,
-NCahs Z
l(z) = 10 e-v- .
(2.4.7)
By definition, the absorbance c9l of a sample is given by
c9l=_IOg(I(Z)]= _ _
l In(I(Z)I,
10
2.3
(2.4.8)
10 )
and it is related to the amount that the intensity decreases.
rewrite this definition as
Using Equation 2.4.6 to
(2.4.9)
it is apparent that the absorbance depends upon the thickness, or path length, Z of the
sample. Although it is not immediately apparent from Equation 2.4.9, the absorbance
also depends upon the concentration of the sample. To make this dependence explicit,
N can be rewritten in terms of Avogadro's number N A as
(2.4.lO)
where n is the number of moles, and V can be rewritten
concentration c as
III
terms of the molar
n
c
V=-
(2.4.11)
Then, using these two relationships, the absorbance can be written as
c9l= nNACabsZ = NACabsCZ .
2.3n/c
2.3
(2.4.12)
Since the absorption depends upon both the concentration and the path length of
the particular sample, it cannot be compared directly from one sample to another. For
39
these comparisons, another, related quantity is used instead. This quantity is known as
the absorption codJicient a, and it is defined as
1
1
a=-Bl=-NAC b
cz
2.3
as.
(2.4.13)
Unlike the absorbance, the absorption coefficient only depends upon Cabs' which is a
molecular property. As a result, it is ideal for comparisons between samples.
2.4.2 Scattering
The case where only scattering occurs is very similar to the case where only
absorption occurs. Just as absorption can be described by modelling the molecules as
disks of area Cabs' where a photon is absorbed if it enters one of these disks and is
unaffected otherwise, scattering can be described by modelling the molecules as disks of
area C scat , where a photon is scattered if it enters one of these disks and is unaffected
otherwise. Experimentally, C scat is known as the scattering cross-section of the molecule, and
like Cabs' its value is chosen empirically. Aside from replacing Cabs with C scat , the case
where only scattering occurs can be treated identically to the case where only absorption
occurs. This is due the fact that when a photon is scattered, it is redirected along any path
except its initial trajectory, so it would miss a detector placed directly behind the
scattering molecule. As a result, the beam intensity is reduced in much the same way as it
is reduced by absorption. So, by analogy, the amount of scattering S and the scattering
coefficient s can be defined as
S = -10 [f(Z)]
g
fo
= NCscatz
2.3V
(2.4.14)
and
S
s =-
CZ
1
= -NACscat.
2.3
(2.4.15)
Like a ,s only depends upon molecular properties, so it can be compared between
samples.
40
2.4.3 Extinction
When both absorption and scattering occur, the quantity of interest is the amount
of extinction E, which is the sum of the amounts of absorption and scattering:
E=A+S.
(2.4.16)
Defining the extinction coefficient in the same way as a and s are defined and using
Equations 2.4.13, 2.4.15, and 2.4.16,
E A+S
e=-=--=a+s.
cz
cz
(2.4.17)
Since the extinction coefficient is simply the sum of the absorption and scattering
coefficients, it must also depend upon molecular properties alone, making it suitable for
comparisons between samples. The extinction coefficient is particularly useful because it
is easily measured using a spectrophotometer, and it is often assumed to be approximately
equal to the absorption coefficient. This assumption is valid for the majority of the
samples analyzed in a spectrophotometer, since they are solutions of molecules and
therefore scatter very little. However, for solutions containing entities larger then
molecules, the amount of scattering can contribute significantly to the amount of
extinction, so it is no longer valid to assume that the extinction and absorption coefficients
are the same. In this case, scattering experiments must be performed to determine the
value of the scattering coefficient in order to find the value of the absorption coefficient.
41
Chapter 3
Experimental Methods
3.1 Sample Preparation
For every experiment, the solutions were prepared by mixing BPP 4B powder with
the appropriate amount of Millipore water and vortexing until the powder appeared to
dissolve completely. This powder was obtained from Sigma-Aldrich, and although it
contained sodium salt, it was left unpurified after a variety of purification attempts
suggested that the formation of a liquid crystal phase strongly depended upon the
presence of some unknown salt or other impurity. For scattering experiments, the
Millipore water was filtered by hand using a syringe and a O.2-micron nylon filter; the size
of the BPP 4B aggregates precluded any filtering of a prepared solution. Before use, all
solutions were heated in an oven to a temperature of approximately 75°C and allowed to
cool to room temperature. This initial heating was necessary to achieve reproducible
results; although the solutions scattered a significant amount of light prior to the initial
heating, indicating the presence of very large aggregates, they scattered far less
afterwards, indicating that the initial heating process changed the solution and greatly
reduced the size of the aggregates. However, the amount of scattering was unaffected by
any subsequent heating.
3.2 Detennining the Phase Diagra1n in Water
For a chromonic liquid crystal, the phase transition of interest is the liquid crystalliquid transition. Since a chromonic liquid crystal is composed of aggregates of various
sizes, each of which changes phase at a slightly different temperature if pure, the liquid
crystal-liquid transition does not occur at one well-defined temperature. Instead, this
transition takes place over a range of temperatures, and as a result, there are some
temperatures for which part of the sample is in the liquid crystal phase while another part
is in the liquid phase. The range of temperatures for which this occurs is called the
coexistence region, and determining the phase diagram consists of determining the
temperatures marking the beginning and end of the coexistence region as a function of
concentration (Figure 3.2.1).
42
Liquid
Crystal
Isotropic
Liquid
Part Liquid Crystal
Part Isotropic Liquid
(Coexistence Region)
Aggregates
of one size
melt
Aggregates
of another
size melt
!
t
t
t
Lower Trans. Temp.
(Transition starts)
Temperature
Upper Trans. Temp.
(Transition ends)
Figure 3.2.1 The Liquid Crystal-Liquid Transition
A chromonic liquid crystal is composed of aggregates of various sizes, each of which changes phase at a
slightly different temperature if pure. As a result, the liquid crystal-liquid transition does not take place at
one well-defined temperature, but occurs over a range of temperatures instead. This range is known as the
coexistence region, and in the coexistence region, part of the sample is in the liquid crystal phase, while
another part is in the liquid phase.
Since the liquid crystal phase is birefringent, while the isotropic liquid phase is not,
the temperatures marking the beginning and end of the coexistence region were
determined optically by measuring the amount of light passing through a pair of crossed
polarizers that sandwiched a BPP 4B sample of known concentration. As described in
Section 2.2.2, the birefringence of a liquid crystal causes it to behave like a phase retarder,
introducing a relative phase difference between the components of the light polarized
parallel and perpendicular to the director and altering the polarization state. As a result,
a liquid crystal placed between a pair of crossed polarizers can allow some light to pass
through these polarizers, and the exact amount can be determined using the Jones matrix
formalism of Section 2.2.1. Assuming that the first polarizer is oriented at 45°, the
second polarizer is oriented at _45°, and that the director lies along the x-axis, the state of
the output light is given by
-1]
1
[e-i¢x
0
0] V"2[[I]E
e1¢y
1
0
(3.2.1)
where Eo is the amplitude ofthe light entering the BPP 4B sample (Figure 3.2.2).
43
-45 0 Polarizer
45 0 Polarizer
45 0 polarized light
Amplitude Eo
BPP4B
Sample
(Phase
retarder)
-45 0 polarized light
._----------------------------------------------------~
Direction of Propagation
Figure 3.2.2 The Path of the Light
Light passing through a 45 polarizer is directed into a BPP 4B sample of known concentration,
which introduces a relative phase difference between the components of the light polarized parallel
and perpendicular to the director. The light then passes through a _45 polarizer, and the intensity
is measured.
0
0
As expected, the output light is polarized at -45 0 , and its intensity is given by
(3.2.2)
Using Equation 2.2.12 to simplify the exponential factor and carrying out the calculation,
this expression becomes
(3.2.3)
Since the phase retardation cfJ and the birefringence are linearly dependent,
Equation 3.2.3 states that the output intensity depends upon the birefringence of the
sample. The birefringence, in turn, depends upon both the phase and the temperature of
the sample. For a liquid crystalline sample, the birefringence is due to the orientational
order of the aggregates, and as the temperature increases, the amount of orientational
order slightly decreases. As a result, the birefringence slowly decreases with temperature
until the sample enters the coexistence region. As the sample passes through the
coexistence region, the birefringence drops rapidly in a roughly linear fashion. This rapid
decrease is due to the fact that the only contributions to the birefringence come from the
areas of the sample that are still liquid crystalline, and as the temperature increases, more
and more of these areas change to the liquid phase. Finally, as the sample becomes an
isotropic liquid, the birefringence disappears (Figure 3.2.3).
44
Birefringence
Liquid Crystal
-----------------------~,
,,
,,
,,
,,
,,
\ Coexistence
\Region
,,
,,
,,
,,
,
,,
,,
,
,,
,,
,,
L..-_ _ _ _ _ _ _ _ _ _ _ _ _\._
}~?~~~12~':. ~!9~!~
________ __ _
Temperature
Figure 3.2.3 The Birefringence as a Function of Temperature
For a liquid crystal, the birefringence is due to the amount of orientational order among the
aggregates, which slightly decreases with temperature. As a result, the birefringence of a liquid
crystal slowly decreases as the temperature increases. For a sample in the coexistence region, the
only contributions to the birefringence come from the parts of the sample that are still liquid
crystalline, and as the temperature increases, more of these areas change into the isotropic liquid
phase. As a result, the birefringence decreases rapidly in a roughly linear fashion. Finally, as the
sample becomes an isotropic liquid, the remaining birefringence disappears.
The temperature dependence of the birefringence for each phase is very distinctive, and
since a plot of the intensity as a function of temperature possesses a similar shape, it is
possible to determine the temperatures marking the beginning and end of the coexistence
region by measuring the intensity.
To measure the intensity as a function of temperature, a BPP 4B solution of
known concentration was drawn into a homemade cell constructed by pressing a pair of
parallel double-layer Parafilm® M strips onto a microscope slide, placing a coverslip on
top, cutting off the excess Parafilm®, and applying a minimal amount of heat to melt
everything into place. Once the sample had been drawn into the cell, the openings were
sealed with five-minute epoxy. These cells were approximately 0.2 mm thick (Figure
3.2.4).
45
Microscope Slide
Parafilm® M
(Two Layers)
Coverslip
Figure 3.2.4 Schematic of a Homemade Cell
A pair of double-layer Parafilm® M strips was pressed onto a clean microscope slide and topped
with a coverslip. The excess Parafilm® was removed, and a minimal amount of heat was applied
to melt everything into place. Once the cell cooled, it was filled with a sample, and the openings
were sealed with five-minute epoxy.
The prepared cell was then placed in a heating stage, which was taped onto a rotating
microscope stage between two crossed polarizers. A detector taped over one of the
eyepieces on the microscope measured the intensity of the light passing through the
polarizers, and a 630-nm filter was used to select light outside the absorption band (Figure
3.2.5).
Detector taped over
eyepiece
Crossed poiarizers
~
Heating stage
containing sampk_
~
Rotating microscopestage
630-nm filter
-+
~~===::::=:::::
-----+
C::;:::=:::::;:J
Lightsource~
Figure 3.2.5 The Microscope Set-up
A heating stage containing the sample was taped onto a rotating microscope stage between two
crossed polarizers. A detector taped over one of the eyepieces on the microscope measured the
intensity of the light passing through the polarizers, and a 630-nm filter was used to select light
outside the absorption band.
Before any measurements were taken, the microscope stage was rotated to
maximize the intensity. Then, the stage was fixed into place, and the sample was heated
at a rate of O.5°C/minute, with the intensity measured every degree. The exact heating
46
procedure depended upon the concentration. For concentrations higher than 15 mM, the
solutions were heated from room temperature to 90°C at the rate specified above. For
concentrations less than or equal to 15 mM, the solutions were cooled to 15°C and held
at that temperature until the intensity readings stabilized. Then, the solutions heated
naturally to room temperature before being ramped to a temperature of 70°C at 0.5°C/
minute. No measurements were taken as the solutions cooled, for BPP 4B aggregates
slowly reach equilibrium upon cooling. After each set of measurements, the intensity was
plotted as a function of concentration, and the temperatures marking the beginning and
end of the coexistence region were determined. Finally, these transition temperatures
were plotted as a function of concentration to generate the phase diagram.
3.3 Light Scattering
For the light scattering experiments, the Brookhaven laser light scattering system
was used. In this system, laser light with a wavelength of 647.1 nm was directed into a
glass vial containing a BPP 4B sample of a known concentration between 0.0 1 mM and
10 mM. This vial was held in a chamber filled with an index-matching fluid that
minimized the amount of reflection off the glass, and an aperture controlled the amount
of light reaching the detector, which was mounted on a computer-controlled goniometer.
Depending upon the type of scattering experiment, the detector measured either the
intensity or the correlation function of the scattered light at a particular angle (Figure
3.3.1).
Index-Matching
Fluid
Ion Laser 647.1 nm
Figure 3.3.1 The Brookhaven Laser Light Scattering System
47
A glass vial filled with a BPP 4B sample of known concentration was placed in a chamber filled
with an index-matching fluid that minimized the amount of reflection off the glass. Laser light
with a wavelength of 647.1 nm was directed into the sample, and a detector mounted on a
computer-controlled goniometer measured either the intensity or the correlation function of the
scattered light, depending upon whether a static or dynamic scattering experiment was being
performed. An aperture controlled the amount of light reaching the detector, with the aperture
size adjusted at the beginning of each experiment.
For the static light scattering experiments, the intensity of the scattered light was
measured at angles ranging from 15° to 155°. Then, the intensity data were plotted as a
function of q and fit to the theoretical function for spherical scatterers (Equation 2.3.7) to
find the optical aggregate radius. For the dynamic light scattering experiments, the
detector was parked at 90° and left to measure both the average intensity of the scattered
light and the correlation function. Then, the correlation function was plotted as a
function of time and fit to Equation 2.3.10 to find the value of Tw. Finally, using
Equation 2.3.26, the hydrodynamic aggregate radius was calculated.
3.4 Absorytion Measurements
To examine further how the aggregation changed with concentration, aJasco UVvis spectrophotometer was used to measure the extinction of solutions ranging from 0.1
mM to 3 mM in concentration. Although some scattering most likely occurred, it did not
contribute significantly to the extinction, so these extinction measurements were equated
with the absorption. Then, the absorption coefficients at 400, 500, and 600 nm were
calculated using Equation 2.4.13. (As a technical note, the solution concentrations were
given in molality, while Equation 2.4.13 required that the concentrations be given in
molarity. However, since there was no more than about a 0.3% difference between the
two types of concentration for these solutions, this distinction was ignored.) Finally, the
results were summarized in a plot of the absorption coefficient as a function of
concentration for those wavelengths.
48
Chapter 4
Results
4.1 The Phase Diagram
To determine the phase diagram, the intensity of the light passing through a pair
of crossed polarizers that sandwiched a heated BPP 4B sample of known concentration
was measured. Then, the intensity was plotted as a function of temperature, and three
lines were used to represent the behavior of the intensity at low, intermediate, and high
temperatures. The intersections corresponded to the temperatures marking the beginning
and end of the coexistence region at that concentration (Figure 4.1.1).
49
o
Intensity
The Intensity as a Function of Temperature
8
GB-r+~~~0'T-r.;:~·············· ;. ·
7
~
.
.
20
30
................... :..................... .:. ..................... :.................. .
. 0
ii
i
:
:
:
50
60
70
6
5
4
3
2
10
40
80
Temperature (Degrees C)
Figure 4.1.1 A Typical Plot of the Intensity as a Function of Temperature
At relatively low temperatures, the sample is liquid crystalline, and the intensity slowly decreases as
the temperature increases. At relatively high temperatures, the intensity is at a minimum. At
intermediate temperatures, the sample is partly liquid crystalline and partly liquid, and the
intensity falls rapidly as the temperature decreases. The three lines on the plot approximate the
behavior of the intensity at low, intermediate, and high temperatures, and the intersections
correspond to the temperatures marking the beginning and end of the coexistence region.
Finally, these temperatures were plotted as a function of concentration to generate the
phase diagram (Figure 4.1.2).
50
---+-- Upper Trans. Temp
-
- Lower Trans. Temp
The Phase Diagrant of BPP 4B in Water
80
70
II)
;...,
:
i
l
........
1. ........................
i
l
,-
/ ........................
~~~i~~·~·~~ ····
... . . . . . . .... .
~~.~.~~~¥
/"
:
l
j
:
. ... . . . . . . . . . . . . . . . . .+: . ... . . . . . . . . . . . .... . . . :!. ···········1~····················j-·~·~9~i.
? ..qD:'.S.~~!.....
/:
·········································t············.............................. -l-.......................................-l-.......................................
40
l
~
;...,
0..
.
:
. ...... . . . . . . . .. .. .J. ...... . . . . . . . .... .
;:;
.....
II)
;:
:
50
II)
~
;:
·······································l·········································l···························
Cfl
;...,
bJ)
.
60
U
II)
II)
.
·· · ···· · ···· ···· ·· ·· ···· ··· ····· ·· ·· ··· ·r · · ···· · ···· ···· ·· ·· ·"I~~t~~pi·q·"L;·q·;:;id·· ·· ···· ··· ··· · ····· ··· ···
l
:
30
S
II)
:
f-<
20
V'
:
:r:":1:
10
0
0
5
10
15
20
Concentration (mM)
Figure 4.1.2 The Phase Diagram of BPP 4B in Water
The temperatures marking the beginning and end of the liquid crystal-liquid transition are plotted
as a function of concentration. The area above the curves corresponds to the set of conditions for
which the sample is in the isotropic liquid phase; the area below the curves corresponds to the set
of conditions for which the sample is in the liquid crystal phase; and the area between the curves
corresponds to the coexistence region, the set of conditions for which the sample is in a mixture of
the isotropic liquid and liquid crystal phases. On this diagram, the liquid crystal phase appears at
concentrations as low as 10 mM, which corresponds to a volume fraction of approximately 0.005.
4.2 The Optical and Hydrodyna1nic Radii
To determine the size of the aggregates, both static and dynamic light scattering
techniques were used on samples of known concentration. For the static light scattering
experiments, the intensity of the scattered light was measured as a function of angle and
plotted as a function of q, the magnitude of the scattering wave vector. Then, to find the
optical aggregate radius, the data were fit to the theoretical functions for spherical
scatterers and Gaussian coils. Although neither fit was particularly accurate, this was to
51
be expected; since the aggregates are thought to have a distribution of sizes, it would be
unreasonable to expect a perfect fit from any model designed to describe a single
structure. However, for simplicity, the fit to Equation 2.3.7 for spherical scatterers was
used since it was the more accurate fit (Figure 4.2.1).
I ---+- Intensity (0.5 mM , heated) I
Intensity vs. q
~
-
!
l--
y=
_. . . . . . . . . . r-.._.. . . . . . . .
m1'((3/((~ZI~:2)A 3W(Si. Error
ll--c-=-S:-: l-+-:--:,c-~:- -:-:-~:-,- -~_~c-~:- -+-_2_._~0_.~-~-:--;N:-~-:-A~-II:R
NA
0.99182
_..............................+...................................................................+................................. ................................~
I
~
~
~
I
I
l
l
l
l
I
l
l
-
r--r --r --r
-
~
. ::~] ::]
~
~
j
o
o
5
1
-~
__
10
~
L.
15
::]
1
_ L
20
-
ill
25
q (inverse micrometers)
Figure 4.2.1 Sample I vs. q Plot
To determine the optical size of the aggregates, the intensity was plotted as a function of q and fit
to the theoretical model for spherical scatterers for simplicity (Equation 2.3.7). The parameter m2
above corresponded to the optical radius.
For the dynamic light scattering experiments, the correlation function was measured and
plotted as a function of time. Modelling the aggregates as spheres, the data were fit to
Equation 2.3.10 to find the value of T vv, and the hydrodynamic radius was calculated
using Equation 2.3.26 (Figure 4.2.2).
52
I ---e--- Correlation Function I
Correlation Function vs. Time
y
=m1+(m2-m1)*exp(-MO/m3)
m1
- l --r--r- eEl
j
5.7106
j
j
R
Value
5.4067e+06
Error
1591 .7
0.9991
NA
::~:~::~ ~ !;!
c
0
:g
C
:::l
LL
c
0
5.6106
~
~
0
u
5.5106
1 1II
o
1000
2000
3000
4000
5000
6000
7000
Time (microseconds)
Figure 4.2.2 A Sample Correlation Function
To determine the hydrodynamic radius, the correlation function was plotted as a function of time
and fit to the theoretical model for spherical scatterers (Equation 2.3.10). In the fit above, the
parameter m3 corresponded to the value of T w, and by substituting this value into Equation
2.3.26, it was possible to determine the hydrodynamic radius.
To determine how the aggregation changed with concentration, both the optical and
hydrodynamic radii were plotted as a function of concentration (Figure 4.2.3).
53
I---+-
1 _______ Static Radius (microns) I
Dynamic Radius (microns)
I
Aggregate Radius as a Function of Concentration
3
0.3
.
2.5
.
0.25
2
0.2
(j)
c
eu
I
(/)
:::J
'6
co
1.5
0.15
a:
. . . . . . . -. . . . . . ·.··. ·-·-·.·. . ·. · . · .·. . T.·. . ·. · . · .·. . ·-.-. . .
u
~
(fJ
0.1
--......r . . --........ . T........ . -- .
0.5
0.05
o
0.01
o
0.1
10
Concentration (mM)
Figure 4.2.3 The Optical and Hydrodynamic Radii as a Function of Concentration
The optical and hydrodynamic radii were both plotted as a function of concentration, and
although the concentration range covered three orders of magnitude, the radii did not change
significantly, showing only a slight downward trend.
4.3 The Relative Scattering Intensity and the Absorption Coefficient
To examine further how the aggregation depends on concentration, changes in
the relative scattering intensity and the absorption coefficient were also tracked. Although
the average intensity of the scattered light was measured as part of each dynamic light
scattering experiment, this value could not be compared directly from one sample to
another since it depended upon the size of the aperture in front of the detector, with
larger apertures corresponding to higher averages. To correct for this, the relative scattering
intensity, or the average intensity divided by the aperture area, was considered instead. By
plotting this relative scattering intensity as a function of concentration, it was possible to
investigate how the amount of scattering varied with concentration (Figure 4.3.1).
54
1 _ _ _ Average Intensity/Aperture Area I
§J
3000
The Relative Scattering Intensity
As a Function of Concentration
E
Rr·98581
.§
u
Q)
f:Q
(J)
c
2500
.. j.t • •••
.8
o
.J::
D..
'0 2000
(J)
"0
C
co
!
(J)
1500
C
>-
' (j)
c
2c
!
r·r· r
::J
~
1000
OJ
c
·c
Q)
~
u
(J)
Q)
>
~
ID
0:
500
·· ··r· r ·r· r
0 ~------~------~--------~------~------~------~
o
2
4
6
8
10
12
Concentration (mM)
Figure 4.3.1 The Relative Scattering Intensity as a Function of Concentration
To investigate how the amount of scattering varied with concentration, the relative scattering
intensity was plotted as a function of concentration. Although slight differences in experimental
parameters like the laser power and the absorbance of the sample can affect the relative scattering
intensity, these effects are not significant enough to alter the clearly increasing trend of the data.
Also, while the relative scattering intensity might be expected to increase linearly with
concentration, slight increases in the aggregate sizes could account for the nonlinearity.
To investigate how the absorption coefficient changed with concentration, the absorbance
measurements were converted into absorption coefficients using Equation 2.4.13. Then,
these absorption coefficients were plotted as a function of concentration (Figure 4.3.2).
55
---+- Absorption Coefficient at 600 nm ~M"-1 cm"-1 ~
____ Absorption Coefficient at 500 nm M"-1 cm"-1
---+- Absorption Coefficient at 400 nm M"-1 cm"-1
The Absorption Coefficient as a Function of Concentration
2.5104
,....
<-
E
(.)
c
Q)
.(3
~
o
()
1 104
C
o
li
o
(/)
.D
«
5000
o
o
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
Concentration (M)
Figure 4 .3.2 T he Absorption Coefficien t as a Function of Concen tration
T he absorp tion coefficien ts at 400, 500, a nd 600 nm we re plotted as a function of concen tration.
None of them cha nged significantly as the concen tration increased from 0.1 m M to 3 m M .
56
Chapter 5
Discussion
5.1 The Phase Diagra1n
In several respects, the shape of the phase diagram for BPP 4B in water suggests
that the aggregation process of BPP 4B is very different than that of Sunset Yellow FCF
[10], a food dye considered to be representative of chromonic liquid crystals in general.
First, the liquid crystal phase for BPP 4B occurs within a concentration range two orders
of magnitude below the range for the liquid crystal phase of Sunset Yellow FCF;
although the liquid crystal phase for Sunset Yellow FCF occurs for concentrations on the
order of 1 M, the liquid crystal phase for BPP 4B occurs for concentrations as low as 10
mM and possibly even lower. In terms of the volume fractions, Sunset Yellow FCF forms
a liquid crystal phase for volume fractions of at least approximately 0.2S, while BPP 4B
forms a liquid crystal phase for volume fractions as low as approximately O.OOS. (This
approximate volume fraction for BPP 4B was obtained by estimating the density of pure
BPP 4B as 1400 grams per liter--the approximate density of other chromonic liquid
crystals--and applying Equation 2.1.S.) At such a low volume fraction, relatively little
BPP 4B is present in the solution, and in order for the aggregates to interact sufficiently to
form a liquid crystal phase, they must incorporate a significant amount of water into their
structure. This incorporation of water should increase the volume fraction of the
aggregates substantially, allowing them to interact even though there is relatively little BPP
4B present in the solution. If this were the case, then the aggregates would have a very
different structure than the simple, rod-like aggregates of Sunset Yellow FCF. Additional
evidence that the aggregation process for BPP 4B is very different than that of Sunset
Yellow FCF comes from differences in the shapes of the phase boundaries in the two
diagrams. Although the phase boundaries for Sunset Yellow FCF are linear and have
slopes of approximately 200 C/M, the phase boundaries for BPP 4B are nonlinear and
have slopes in the range of approximately SOO--SOOODC/M; these differences most likely
stem from differences in the aggregate structures.
D
Although the differences in the phase diagrams for BPP 4B and Sunset Yellow
FCF indicate that there are differences in the size and structure of the two types of
aggregates, the presence of a wide coexistence region in each diagram suggests that like
Sunset Yellow FCF, BPP 4B forms aggregates with a distribution of sizes. As described in
Section 3.2, a coexistence region occurs in a chromonic liquid crystal whose aggregates
57
have a distribution of sizes since the aggregates of a particular size, if pure, melt at a
slightly different temperature than aggregates of another size. Since the size distribution
tends to be significant, this coexistence region is relatively wide. Although coexistence
regions can occur for pure compounds containing some impurities as well, the widths of
these coexistence regions depend upon the specific types of impurities, and they are often
much narrower. As a result, the presence of such a wide coexistence region in the phase
diagram for BPP 4B is more likely due to the presence of a distribution of aggregate sizes
than to the presence of impurities in an otherwise pure sample. The presence of a
similar coexistence region in the one published phase diagram for BPP 4B [20] also
suggests that this coexistence region is due to the presence of a distribution of aggregate
sizes, since it is more likely that the two forms of BPP 4B possess similar distributions of
aggregate sizes than similar impurities (Figure 5.1.1 ).
Upper Trans. Temp
-- . Lower Trans. Temp
The Phase Diagram of BPP 4B in Water
70
80
.-------r-----r----...,-------,
Sunset Yellow FCF
60
G
en
Q)
G
Q)
....
isotropic
(II
50
O'l
Q)
"""
0
0)
.a
~
!'!.
40
Q)
....
-
70
. . . . . . . . . . ..'~tr~:I'I"g~~~~~I~ •
60
50
u.u...u., u.u...u. ! .u. l
40
E
ctl 30
....
Q)
{!'.
30
20
E
Q)
(N)
I-
20
0 .6
0.7
0 .8
0.9
1.1
10
0
1.2
Concentra1ion (M I
.
. .... . . . . . .... . ... .¥. . .
0..
nematic
u
... . . ... . . . . . ........,. . ... . . . . . .... . ... . :i .L.,iql,J.igiG.rY.§1~L .
:::J
... ... :....................... ........... !..
0
........ :.....
10
15
5
Concentration (mM)
20
Figure 5.1.1 A Comparison of the Phase Diagrams of Sunset Yellow FCF [10] and BPP 4B
The phase diagrams of Sunset Yellow FCF and BPP 4B contain some key differences that suggest that the
aggregation process of BPP 4B is very different than that of Sunset Yellow FCF. First, the concentration
scale on the phase diagram for Sunset Yellow FCF is in molals, while the concentration scale on the phase
diagram for BPP 4B is in millimolals; if the two phase diagrams were plotted on the same graph, the liquid
crystal phase for BPP 4B would occur within a concentration range more than an order of magnitude
below the concentration range for the liquid crystal phase of Sunset Yellow FCF. In addition, the phase
boundaries for BPP 4B are nonlinear and have slopes of approximately 500--5000°C/M, while the phase
boundaries for Sunset Yellow are linear and have slopes of approximately 200°C/M. These differences
most likely stem from differences in the size and structure of the aggregates, although the presence of a
wide coexistence region in each diagram suggests that like Sunset Yellow FCF, BPP 4B forms aggregates
with a distribution of sizes.
The differences in the phase diagrams are also reflected in a comparison between
the phase diagram for Sunset Yellow FCF and the one published phase diagram for BPP
4B [20]. Although Bykov et al used a slightly different form of BPP 4B that contained
58
cesium salt instead of sodium salt, their results fall roughly in the same range as those
given above. Their phase diagram shows a liquid crystal phase forming at concentrations
as low as approximately 6 mM, possibly even lower, and estimating the density of their
form of BPP 4B as 1400 grams per liter and using Equation 2.1.5, this corresponds to a
volume fraction of approximately 0.004. The phase boundaries are also nonlinear and
have slopes of approximately 400--1,200DC/M.
Additionally, the phase diagram presented by Bykov et al possesses one particularly
interesting feature. Although both their phase diagram and the phase diagram for Sunset
Yellow FCF possess a concentration below which a liquid crystal phase cannot occur, the
reasons for this are very different. On the phase diagram for Sunset Yellow FCF, this
concentration occurs at the point where the lower phase boundary and the concentration
axis intersect. Physically, this point corresponds to the concentration below which the
liquid crystal phase cannot occur because the water freezes before the temperature can be
lowered sufficiently. (Once the water freezes, the sample is no longer in the liquid crystal
phase.) This freezing is the only thing that prevents the liquid crystal phase from forming;
if freezing did not occur, the slope of the lower phase boundary suggests that the liquid
crystal phase could form at lower concentrations [10]. On the published phase diagram
for BPP 4B, however, this critical concentration occurs because the lower phase boundary
becomes vertical; even if freezing did not occur, the liquid crystal phase could not occur
at lower concentrations. As a result, Bykov et al argue that this critical concentration must
be due to an entirely different effect, which again suggests that Sunset Yellow FCF and
BPP 4B aggregate very differently [20].
Additional evidence that the aggregation process of BPP 4B differs from that of
Sunset Yellow FCF came from visual observations of the liquid crystal-liquid phase
transition. These observations were taken using the same microscope set-up described in
Section 3.2, with the intensity detector removed. For Sunset Yellow FCF, this transition
produced a marked change in the appearance of the sample; as the sample passed
through the coexistence region, droplets of the isotropic liquid phase noticeably formed
and grew within the liquid crystal phase. For BPP 4B, however, the transition produced
no visible change in the sample except for a gradual dimming of its brightness. As a
result, the change from the liquid crystal phase to the isotropic liquid phase may occur on
a much smaller length scale for BPP 4B.
5.2 The Radii, the Relative Scattering Intensity, and the Absorption
Coefficient
In addition to the phase diagram, the measurements of the hydrodynamic and
optical radii, the relative scattering intensity, and the absorption coefficient suggest that
BPP 4B forms aggregates of a very different structure from Sunset Yellow FCF. For
Sunset Yellow FCF, the aggregates are shaped like rods and have a distribution of sizes
59
that shifts towards larger aggregates as the concentration increases [10]. For BPP 4B,
increasing the concentration results in a very clear increase in the relative scattering
intensity, despite the fact that small variations in experimental parameters like the laser
power and the absorbance of the sample could alter the measured values slightly. This
increase in the relative scattering intensity could be due to any combination of an
increase in the size and number of aggregates. However, changing the concentration by
three orders of magnitude did not produce any significant change in the hydrodynamic
and optical radii or the absorption coefficient, which implies that the aggregate size was
unaffected by the concentration as well. This suggests that while BPP 4B may form
aggregates of a distribution of sizes, only the number of aggregates, and not their size
distribution, changes with concentration. As a result, the aggregates must have a different
structure than the simple rod-shaped aggregates of Sunset Yellow FCF.
5.3 The Possible Fonnation of Micelles with a Distribution of Sizes
One possible explanation for these results is that BPP 4B forms micelles with a
distribution of sizes, where the micelles might incorporate various salts into their
structures. If this were the case, it would account for both the salt dependence noted
during the purification attempts and the invariance of the size distribution. As described
in Section 2.l.1, compounds that form micelles possess a critical volume fraction, or a
volume fraction below which micelles cannot form. Above this critical volume fraction,
the number of micelles increases linearly while the number of monomers increases at a
much slower rate. As a result, if the experiments were performed well above the critical
volume fraction, the samples would seem to be composed of micelles alone. Since
micelles have a specific size, the size distribution would not change with concentration,
and as a result, both the radii and the absorption coefficient would be unaffected by
increases in the concentration. However, since the number of micelles increases linearly
with concentration, the relative scattering intensity would increase linearly with
concentration as well. Although the relative scattering intensity did not increase linearly
with concentration, a fairly linear trend is clearly visible, and the nonlinearity might be
due to slight increases in the aggregate sizes.
Since no significant changes were observed in either the radii or the absorption
coefficient over the entire concentration range that was examined, this explanation would
require that the lowest tested concentration fall well above the concentration range at
which the various micelles start to form. Approximating the density of pure BPP 4B as
1400 grams per liter and using Equation 2.l.5, this concentration, 0.01 mM, corresponds
to a volume fraction of approximately 10-6 , which means that micelles would be forming
at volume fractions far less than one in one million! For this to occur, the micelles would
have to incorporate a significant amount of water into their structure, making them
extremely large. This extremely large size could account for how BPP 4B scatters visible
light and forms a liquid crystal phase at very low concentrations, unlike both Sunset
Yellow FCF and disodium chromoglycate.
60
5.4 The Possible Presence of a Large Impurity
Another explanation is that the solutions contain an impurity that is much larger
than the size of the BPP 4B aggregates. If this were the case, then the static and dynamic
light scattering experiments would be measuring the size of the impurity instead of the
size of the aggregates. As a result, since the size of the impurity would be constant, the
hydrodynamic and optical radii and the absorption coefficient would be unaffected by
changes in the concentration. Additionally, since the number of impurity molecules
would increase linearly with concentration, the relative scattering intensity would also
increase linearly. Again, while the relative scattering intensity did not increase linearly
with concentration, an increasing trend is clearly visible, and slight increases in the
aggregate sizes could explain the nonlinearity.
However, while the presence of a large impurity could also account for how BPP
4B scatters visible light, unlike Sunset Yellow FCF and disodium chromoglycate, it is less
likely to account for how BPP 4B forms a liquid crystal phase at very low concentrations.
Assuming that the impurity does not interact with the BPP 4B, it does not affect the
concentrations at which a liquid crystal phase forms. Therefore, the fact that BPP 4B
forms a liquid crystal phase at significantly lower concentrations than Sunset Yellow FCF
must be due to the size and structure of the BPP 4B aggregates. Since BPP 4B forms a
liquid crystal phase within a concentration range two orders of magnitude below the
range for the liquid crystal phase of Sunset Yellow FCF, its aggregates should be roughly
one to two orders of magnitude larger. As a result, since Sunset Yellow FCF forms
aggregates roughly 1--10 nm in size, the BPP 4B aggregates should be roughly 100--1000
nm in size. Within this range of sizes, the BPP 4B aggregates are sufficiently large to
scatter light, so if they are to be undetected by the scattering experiments, they must be
significantly smaller than the impurity. However, the scattering results would place the
size of this impurity at 500--2500 nm. Although it is not impossible that the sizes could
work out in such a way that the BPP 4B aggregates were significantly larger than the
Sunset Yellow FCF aggregates and significantly smaller than the impurity, it is not
particularly likely.
Alternatively, the impurity might be incorporated into the aggregate structure,
acting as a sort of scaffold to which the BPP 4B molecules stick. Given the low volume
fraction at which a liquid crystal phase occurs, these aggregates would have to incorporate
water as well, regardless of the size of the impurity; since the impurity would be
contained in the BPP 4B powder, its volume fraction would be at most the volume
fraction of the powder used to form the solution, which would not be high enough to
allow the aggregates to interact necessarily. If the aggregation were to occur in this
manner, then, again, the aggregation process for BPP 4B would be very different than the
simple process for Sunset Yellow FCF.
61
Chapter 6
Conclusion
A particularly little-studied chromonic liquid crystal, BPP 4B seems to aggregate
very differently from Sunset Yellow FCF and disodium chromoglycate, two previously
Unlike Sunset Yellow FCF and dis odium
studied chromonic liquid crystals.
chromoglycate, BPP 4B scatters visible light, and it forms a liquid crystal phase at
significantly lower concentrations. Additionally, although both Sunset Yellow FCF and
disodium chromoglycate probably form rod-shaped aggregates with a distribution of sizes
that shifts towards larger aggregates as the concentration increases, measurements of the
hydrodynamic and optical radii, the relative scattering intensity, and the absorption
coefficient suggest that BPP 4B forms aggregates with a distribution of sizes that is
unaffected by changes in the concentration. One possible explanation is that BPP 4B
forms micelles with a distribution of sizes in response to the various salts in the solution,
with the micelles possibly incorporating some of the salts into their structures. If this
were the case, then as the concentration increased, the number of aggregates would
increase while the size distribution remained constant. However, the presence of an
impurity much larger than the aggregates might also account for these results. While the
exact aggregate structure remains largely uncertain at this time, the results of these
experiments seem to suggest that BPP 4B forms aggregates that are very different from
the simple, rod-like structures of Sunset Yellow FCF and disodium chromoglycate.
62
Acknowledgments
First and foremost, I would like to deeply thank my advisor, Professor Peter
Collings, for taking the time to teach me about a truly fascinating phase of matter, for
correcting my liquid crystals thoughts as they almost invariably went astray, and especially
for answering the million questions that always followed. I would also like to thank
Swarthmore College, the University of Pennsylvania Laboratory for Research on the
Structure of Matter, and the National Science Foundation Research Experience for
Undergraduates program for making my amazing research experience possible. I would
like to thank Viva Horowitz too for suggesting that I apply to study liquid crystals and for
leaving behind the work off of which so many of my studies were based.
In addition, I would like to warmly thank the faculty of the Swarthmore Physics
Department for reminding me that there is a life after thesis, and I would also like to
thank all of my late-night comrades-in-arms for making the long hours all the more
bearable for a morning person. And, I would especially like to thank my brother, Brad
Gersh, for his beautiful picture of disodium chromoglycate.
Finally, last but by no means anywhere close to least, I would like to
enthusiastically thank my lab partner in crime, Michelle Tomasik. Without her chemistry
and butterfly knowledge, her sense of adventure, and her great sense of humor, the
summer would have been far less enlightening, interesting, and, above all, far less
entertaining. To her more than anyone, lowe the memories.
63
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