A Study of the Polarization of Fluorescence of Ordered Systems With Application to Ordered Liquid Crystals

The fluorescence polarization of uniaxial molecules dissolved in an ordered medium is studied. A theoretical model is developed which relates the polarization of the fluorescence emission to molecular structure, orientation of absorption and emission dipole oscillators and the degree of ordering. This theory was tested experimentally using all trans 1,6-diphenyl-1,3,5-hexatriene dissolved in an ordered liquid crystal.


Introduction
In continuation of previous polarization studies of ordered liquid crystal [1, 2]1 and isotropic liquid sys tems [2,3,4], thi s work is concerned with the polarization of fluorescence by molec ules di ssolved in an ordered liquid crystal. A general theoretical model is developed that can be used with various lin early dichroic systems, such as streaming solutions or stretched polymer films, molecular crystals or membranes. The model is verified experimentally for the case of a fluorescen t dye dissolved in an ordered nematic liquid crystal.
The liquid crys tal solvent used here was a mixture of cholesteryl chioride and cholesteryl laurate [5 , 2, 1]. These mixtures are normally c holesteric with a helical arrangement of the molecules. However, at a definite co mposition and temperature, T nem , a nematic state results [5].
In the " unordered" nematic s tate of the mesophase the molecules are parallel within domaines.
These domain es are oriented randomly with respect to each other thus giving the characteristic cloudiness and opacity of ne matic liquid crystals. When an electric field is applied the domaines orient in such a way that the solvent molecules are parallel to the electric field [5].
The liquid crystal mixture (L.C.M.) c hanges from an opaque to a clear state within a fe w seconds, thi s c hange being reversible once the field is turned off. This ordering phenomenon can be us ed to order solute molecules which are dissolvAd in the L.C.M. at low concentrations. The L.C.M. used was a 1.95/1 by weight ratio of cholesteryl chloride to cholesteryllaurate, which is nematic at Tnem = 30°C [5, 2, 1]. Ordering was achieved by application of an electric field of the order of 40 kV/cm. The solute molecule used here as well as previously [1,2] is 1,6·diphenyl·l,3,5·hexatriene (DPH), which is a long "rod like" molecule that absorbs light in the ultraviolet imd fluo~esces with a blue emission. DPH is fairly soluble in this L.C.M. and is aligned with its long molecular axis parallel to the direction of orientation of the ordered liquid crystal [1, 2].

Definitions and Outline of Approach
As a theoretical model for the fluorescence polarization of an ordered sample, we consider a collection of uniaxial solute molecules with absorption and emission dipole oscillators that have a fixed orientation with respect to the molecular axis as indicated in figure lao This fixed orientation is defined by three angles: p and 0, the angles between the unit vector in the direction of the molecular axis, I, and the unit vectors in the directions of the absorption and emission dipole oscillators, Po and qo, respectively; and v, the angle enclosed by the planes (I, Po) and (I, qo). Thus we have, relative to a molecular reference frame consisting ofl and two arbitrarily chosen orthogonal unit vectors m and n in the plane perpendicular to I, qo= [cos 0, sin 0 cos (u+v), sin 0 sin (u+v)], where u is defined in figure lao A further angle defined by the molecular structure is 8, the angle between the absorption and emission dipole oscillators. According to eqs (la, b), it is related to p, 0, and v by cos 8 = Po . qo = cos p cos 0 + sin p sin 0 cos (v).

(lc)
It will be assumed that each molecule can rotate about its molecular axis. Then, different molecules have rotated through different angles u at the time of excitation so that, for the sample as a whole, u is a random variable. The angle v that enters into the theory depends on the relative orientation of the absorption dipole oscillator Po at the time of absorption with respect to the emission dipole oscillator qo at the time of emission. Therefore, v becomes equal 10 the intramolec· ular angle shown in figure la in the borderline case of molecules that do not rotate appreciably during the time interval between absorption and emission, but becomes a random variable for a large number of molecules that rotate fast and have long fluorescence lifetimes. The theory pre· sented here treats the general case in which v falls anywhere within this range.
The ordering of the molecules is produced by an external force, such as an electric or magnetic field, that is applied to the sample. It is assumed that this force hinders the tumbling motion of the individual molecular axes so that, for each molecule, the angle X between the molecular axis and the direction of the applied force does not change appreciably between the times of absorption and emission. The applied force produces a preferential orientation of the molecular axes with respect to the x axis of the laboratory reference frame in figure lb. This alinement of the molecules is defined by a distribution function f(x), so thatf(x) sin XdXis proportional to the number of molecules with molecular axes that enclose angles between X and X + dX with the x axis. The angular distribution of the molecular axes is assumed to be symmetric about the x axis, so that f( -X) = f(x); and it is also assumed that the probabilities for "upside·up" and "upside·down" alinement of a molecule are the same, so that f(x + 7T) = f(x). A suitable form for f(x) is the quasi·gaussian distribution function used in an earlier paper [1], where e K = /(0) //( 1T) defines the proportion of molecules oriented parallel and perpendicular to th e x axis, respectively. This function f(x) leads to the following closed expressions for the averages of cos 2 X and cos 4 X that will appear in the equations to be developed:  tion). The corres pondin g limits for Mare 1/5 and 1.0 r es pectively. The transition between these two borderlin e cases depends on the specific form off (X).
As also indi cate d in figure lb, the exciting radiation is assumed to be incident upon the sample in the y-direction of the laboratory reference fram e, and is assumed to be plane polarized with its electric fi eld vector, EV or EH, parallel to the x or z axes. The corresponding photon flux densities of the exciting radiation are denoted by SV and SH, l\es pectively. The squared magnitudes of the absorption and emis sion dipole vectors p and q an~ taken to be e qual to the probabilities for absorption and e mission by a molecule, so that for excitation with vertically polarized radiation, (3a) and for excitation with horizontally polarized radiation, (3 b) where the supe rscripts V and H denote th e polarization of the exciting radiation. H er e, (j>V and 4>H are the r es pective angles between the absorption oscillator and the exciting electric fi eld , (To is the absorption cr oss secti on for absorption oscillators that are parallel to the exciting fi eld, and Q is the fluorescence quantum yie ld. The co mpone nts of th absorption and emi ssion dipole vectors with r es pect to th e laboratory axes can now be calcula te d as follows. According to fi gure lb we have 1= (cos X, sin X cos ' T/ , sin X sin 'T/), where'T/ is a random angle because of the rotation of the molecule about th e molecular axis. The n, the two unit vec tors m = (-sin X, cos X cos ' T/ , cos X sin ' T/ ), (4b) n = (0, -sin 'T/, cos 'T/) (not shown in figure lb) may be take n to represent the other two axes of the molec ular reference frame, so that is a suitable matrix to transform molecular coordinates into laboratory coordinates. Hence. e qs (la) and (4d) show that the x and z co mponents of the unit vector in the direction of the absorption dipole oscillator p in figure Ib are Similarly, eqs (lb) and (4d) yield the unit vector Aqo in the direction of the e mission dipole oscillator q, so that according to eqs (3a, b) and (6a, b). Here, In order to describe the sample as a whole, these expressions must be replaced by their rms averages for a large number of molecules.
The fluorescence intensity I emitted by the sample into a given viewing direction is proportional to the squared rms components of qV and qH in the plane perpendicular to the viewing direction, and is polarized in the directions of these components. Therefore, the four intensity readings that are obtained for the four possible combinations of vertical and horizontal setting of a pair of excitation and emission polarizers are, for 90 0 viewing in the direction of the z axis, In this work, the averaging of these expressions was done in three steps. First the terms were multi· plied through and averaged with respect to all possible angles u using du 0 u being a random variable. Next they were averaged over all possible angles Tj , using analogous eq uations to eqs (10) since Tj was also assumed to be a random variable. Finally the terms were collected and expressed as functions of Nand M as given by eqs (2 b) and (2c). The resultant expressions were found to be (3 cos 2 0 -1) (3 cos 2 p -1) The total in tensities emitted by the sample for the two types of excitation, vertical and horizontal, are thus, Since the total intensity e mitte d by the sa mple is directly proportional to th e total inte nsity absorbed the di chroi c ratio for a bsorption can be given by This equation is th e sa me as the one de rived in ll] for the absorption of polarized light by molec ules ori e nted in a n orde red liquid crys tal. At thi s point it is desirable to tes t eqs (l2a-f) for two limitin g cases of M a nd N to c heck the correctn ess of th e de rivation. where eq (Ic) has been used to express the results in terms of e. These equations are the same as the ones obtained in a previous paper on the polarization of fluorescence of a random but frozen distribution of absorbing molecules [3]. Notice that information on e, the angle between the absorption and emission dipole oscillators, is easily obtained when polarizers are placed in the beams. When eqs (I6a, b) are used to calculate the total flux emitted by the sample for vertically and horizontally polarized excitation, one obtains (17a) This result shows that the sample is an isotropic absorber and that there is no dependence flU p, as there was in the case of perfect ordering shown by eqs (lSa, b).

Application to the DPH-liquid crystal system
Previous studies have shown [1, 2] that the longest wavelength absorption band of DPH is due to a transition involving the fully conjugated molecule and that the absorption. dipole oscillator lies along the long molecular axis; i.e., p = 0°. Also, the fluorescence emission dipole oscillator is almost parallel to the absorption dipole oscillator [1,2], and therefore it also lies along the long molecular axis; i.e., a = 8 = v = 0°. Finally it has been shown that for DPH dissolved in the ordered liquid crystal mixture used here, the excited DPH molecules do not rotate at all during the time interval between absorption and emission [2]. With these conditions, the six expressions given by eqs (12a-f) reduce to The fluorescence intensity readings which one obtains for the two viewing positions, 90° (rt angle) and lSO° (straight through), of an ordered DPH sample can now be written with reference to eqs (Sa-d) as where F == SV/SII is the polarization ratio of the exciting flux reaching the sample and G == Tv/Til is the polarization ratio of the emission detection system. As written, F and G are instrumental paramo eters and, with this liquid crystal system, can be determined easily since the unordered liquid crystal (i.e., no electric field applied to the sample) is cloudy and opaque, and acts as a light scatterer and depolarizer. Thus randomized fluorescence intensity readings , analogous to those given by eqs (19a-e), can be taken with the electric field off, and can be expressed as R~(900)r = R~(lSOO)r = k,SIITIIFGu-oQ, 25 where kr differs from k due to the scattering of the unordered crystal.
Rt (900) with one another but also the individual readings, such as R~ (900),. and R~(lBOO)r, should agree. These two conditions, as well as the assumption that the solvent does not absorb any of the exciting light are inherent in the derivation of the equations and, as will be shown, must be modified if they do not apply. Nevertheless, corresponding readings of the two sets of data (field on and field off) can always be divided to give where k' = k/kr and VV(900) = R~(900)/R~(900)r, etc. The first letter of this new notation refers to the mode of the excitation polarizer and the second to the emission polarizer. Ratios of these equations will give information on the values of Nand M that the sample exhibits.

Experimental Data [6]
Experimental data were taken using two different spectrofiuorimeters. Preliminary data were taken using the Nottingham spectrofiuorimeter [7], which is a right·angle viewing instrument. Polacoat 105 UV -visible polarizers were used in both the excitation and emission beams. The sample used was a 7.2 X 10 -5 g DPH/1.055g L.C.M. mixture where the L.C.M. was also a 1.95/1 by weight mixture of cholesterol chloride to cholesterollaurate [5]. The sample cell consisted of a piece of 7.5 mm diameter Spectrosil tubing with a stationary bottom electrode and an adjustable top electrode with the liquid crystal mixture sandwiched between. A detailed description of this cell is 26 given in [2]. The cylindrical cell was enclosed in a three-windowed aluminum heating block. The nominal electrode gap used was ~ 4 mm and the temperature of the system was kept at Tnem = 30 ± 1 0c.
Additional data, allowing a test of the above theory, were taken using the new NBS goniospectrofluorimeter [8]. This versatile instrument allows viewing of a cylindrical sample from any arbitrary viewing angle from about 20° to 180°. However, the three-windowed aluminum heating block used in study limited the viewing of the liquid crystal sample cell to the 90° and 180° viewing positions. The sample cell used here was similar to the one above except that 3 mm diameter tubing and electrodes were used. Also, a lower concentration of DPH in the L.C.M. was used-1.2 X 10 -7 DPH/l.I04g L.C.M. This was done to approximate a point source sample by making the cell physically small, and also by keeping the light intensity constant as it propagated through the cell. Clan-Taylor polarizers were used with this instrument with a Corning 7-54 filter placed in the excitation beam and a 1 cm path length cut-off filter (saturated solution of NaN02 in water) in the emission beam. The filter combination allowed viewing of the sample cell at 180° (straight through) by reducing the amount of scattered and/or nonabsorbed exciting light which reached the detector. Again, the electrode gap used was 4 mm and the temperature of the cell was kept at 30 ± 1 dc.

.1. Degree of Polarization as a Function of Applied Electric Field Strength
Using the Nottingham spectrofluorimeter and sample, the excitation wavelength was set at '-ex = 370 nm and the emission wavelength at '-em = 425 nm. Without the electric field applied, the four readings R~(900)r, R~(900)r, R~(900)r and R~(900)r were taken. Then the electric field was applied in steps up to 60 kV/cm and the four readings Rn900) , R~(900), R~ (90°) and R~~ (90°) ., were taken at each voltage. These readings were then divided by the ones with the field off to give VV(900) , VH (90°), HV(900) and HH (90°) for each voltage setting. These data are plotted in figure  3 against applied voltage and show that the readings reach steady state values after 30 kV/cm, thus showing that single domain conditions are achieved, and that as long as the voltage is above 30 k V / cm the signal level will remain constant and will be unaffected by small drifts in the electric field strength. Therefore, all further experiments were performed with voltages of 40 kV/cm.

Polarization as a Function of Wavelength
Again using the Nottingham spectrofiuorimeter and sample, the emission wavelength was set at hem = 425 nm and the four readings with the electric field on-Rn900)1 , R~(900)" R~(900) and R~(900)-were taken as a function of excitation wavelength from 245 to 400 nm. Ratios of these readings were then taken with reference to eqs (19a-e) to give .7 were then calculated and are also shown in figure 4. Th e nearly horizontal straight lin es s how that the degree of polarization of the flu orescence emission is also ind e pe nde nt of e mi ssion wavelen gth, as one would expect, and that fluorescence occurs from only one e mi ssion dipole oscillator.

Quantitative Test of Theoretical Model
Using the NBS spectrofluorimeter and sample with the excitation wavelength se t at Aex=370 nm and with the e mi ssion wavelength set at Aem = 465 nm, fluoresce nce intensity readings were taken with and without th e applied electric field at the two vi ewing positions, a = 90° and 180°. These data are s hown in table 1. Th e readings with the electric fi eld off were divided according to eqs (21a, b) to give the es timates of F and G shown in table 2. Th e agreement of the individual fi eld off Experimental readin gs ta ke n with and without the applied e lec tri c field for th e determin ation of th e degree of polariza· tion of the flu orescence emission a nd for the determination of the co rrection factor due to the birefringence of th e ordered liquid crystal solvent.  T a ble of in strume ntal polarization correc tion factors obtained from the data ta ke n with the electric field off. going from the 90° to the 180° viewing position. This is not surprising since the emission is being detected through two different windows in the heating block. When the electric fi eld is turned on, the individual polarization re adings change from the values they had with the electric field off. However , the data in table 1 indicate that the two viewing positions are no longer equivalent sin ce according to eqs ~19a-e) only the RH readin g should have chan ged in going from 90° to 180° viewing, and then only by a factor of 3 instead of 3.43. Even if the assumed model for the DPH dipoles is in error the R ~ (90°) and R ~(900 ) readings should still equal the R ~ (l800) and R ~(l800) readings respectively_ Therefore, this dis cre pancy is attributed to the fact that, with the field on, the sample is no longer a point source. There are two reasons for this. First, the ordered liquid crystal mixture is strongly birefrigent and, therefore, not equivalent to the unordered sample_ Thus, the G correction factor estimated from the latter is not quite correct. Secondly, the liquid crystal solvent has been shown to have its own dichroic ratio for absorption when the electric fi eld is on and, the refore, will modify the polarization ratio F of the exciting radiation in wavelength regions of solvent absorption. Thus, the ratio of the field on to field off data as given by eqs (22a-e) should really be written as,

R" V
where a = 90° or 180°, A (V) and A (H) are the absorbances for vertically and horizontally polarized exciting light, respectively, Tv' is another responsivity factor for vertically polarized emission due to the birefrigence of the sample, and Tf/' is the horizontal responsitivity. The proportionality cons tants k90° and kl800 diff~r because the sample is no longer a point source.
The best way to eliminate any effect due to the solvent absorption is to obtain ratios only from the readings which have constant excitation polarization and constant viewing angle. Thus, we define the ratios The polarization ratio, G', due to the birefringence of the ordered liquid crystal solvent was determined· experimentally by removing the filter in the excitation beam and setting the excitation monochromator wavelength to A ex = 465 nm to match the fluorescence emission wavelength.
With the emission detection system set at the 180° viewing position the 465 nm flux was monitored as it propagated through the liquid crystal cell. The four intensity readings -R ~' (180°), R:;' (180°) , W?' (180°) and R ~~' (180°) -were also taken with and without the applied electric field (see table 1). Since neither the liquid crystal solvent nor the DPH absorb at this wavelength the effect of the birefringence can now be determined directly. The readings with the electric field off were then divided according to eqs (21a) and (21b) to give estimates of G and F (465 nm) which are then given in Equations (27a) and (27b) can now be solved simultaneously, using the 90 0 data of the "Ratio" column in table 1 to obtain estimates of M and N. Likewise, eqs (27c) and (27d) with the 180 0 data in the same column will give estimates of M and N. These are tabulated in table 3a along with a value of K for each estimate from the curves in figure 2.
In order to check the precision of these results, the measurements were repeated. The electric field was turned off and it was found that after a few minutes the sample returned reversibly to its initial unordered state, as the polarization readings were the same as those obtained before the electric field had been applied the first time. Then a second set of data was taken and processed in the same manner as before.
[6 ] In order to descri be materi als and expe rim ental procedures adequ ately, it is occasionall y necessa ry to ide ntify co mmercial products by man ufac tu rer's na me or label. In no in stances does such identifi cation impl y endorse ment by the Na tional Burea u of Sta ndard s, nor does it impl y that th e pa rticul ar product or equip ment is necessa ril y th e bes t avail able fo r th at purpose.
[8] Miele nz , K. D., to be published , see refs. 3 and 4 for a short desc ri ption of th e in strum ent.