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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. 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