DIFFUSION IN LIQUIDS

DIFFUSION IN LIQUIDS

Uwe K. A. Klein Chemistry Department, University of Petroleum and Mi'nerals,

Dhahran, Saudi Arabia

:""",~I

~)\:JI

'-)#

~.lSj ~jl J~~ o~ 4;)\40 ~ ",'IA};,~I J#I 4J"'~ o~\i loa. d'~

~)

,ro

'-""¥!" h:-J" ~.tp..1 ^o.,ij

lJi

~ ~y' ~ ~ ~..ul ~~I 'i~I.)I~~1 ~~

~ ~ ",~~I J.,bJ1 4J"'~ j~ ~\,JI ~~~..uIJ 'J.I' ~ , ;; JJ.J! . ~ . .)

r

^J\1z:J1 d'fJ

.J.I'~ £n - 'Y,O

~I ~l..S'.rlI JI)J"u1 )u.:.;~ 4J"'l:Al ",~~I J#I 4J"'l:Al j~ll~ ~~;; ~..bI:i ~i

~I.)J) ~l:i

lJi

^4S) ~..bdl ~J ~J . h:-J)JI Ul$ ~4J., ~ ~J.J ~, ~I.)J)J ~\i.,a... 4S~ ~J.~

~

^IJ

JJ.o ~~I ~

^t).-I

~ ~)I 4J"'~1 ~ ~)u.:.;~1

1J1)J"u1

~l:i

; ;

~~ .r;

l.

~ J40i ~...r.JIJ ~

^,

. .tt\:J1 ~IJ

JJ.o

^IJ~I t).-IJ .tt\:J1

ABSTRACT

The principle of phasefluorometry is presented, together with a brief comparison with pulse methods and the historical development. A new instrumental setup is described, including a specially designed 50 Q Pockels cell. This cell has a half-wave voltage of only 135 V and a 3dB bandwidth of more than 1 GHz. The operating frequencies of the phasefluorometer are between 12.5 and 400 MHz. Finally, appli­

cations of the phasefluorometer to the measurement of picosecond rotational diffusion of oxypyrenetrisulfonate, rhodamine 6G, and perylene in various low­

viscosity solvents are presented. In particular it is shown that the results for irhodamine 6G and perylene agree well with recently published data for rotational

diffusion on the nano- and picosecond time scales.

0377-9211/84/040327-18$01.80

© 1984 by the University of Petroleum and Minerals

The Arabian Journal for Science and Engineering, Volume 9, Number 4. 327

PICOSECOND FLUORFSCENCE DECAY STUDIED BY PHASEFLUOROMETRY AND ITS APPLICATION TO

THE MEASUREMENT OF ROTATIONAL DIFFUSION IN LIQUIDS

1. INTRODUCfION TO THE PRINCIPLES OF PHASEFLUOROMETRY

For each system to be investigated having Iight­

emitting states, excitation with a light flash of infinitely short duration, IO(t) (&-excitation)

I'(t) =0 for t '" 0; jt'(t) dt = I, (1) guarantees a maximum of kinetic information if the measuring system is likewise capable of responding by a o-shaped emission, I~(t).

This ideal case is physically not realizable; instead the real case of a time dependent excitation, I a(t), is always dealt with. If the time-dependence of the system to be investigated is described by linear differential equations, the time-dependence of the emission, I dt), is given by a convolution integral

I.(t)=

I 1.(t-O)I~(O)dO_

⁽²⁾

Whereas with pulse methods one tries to shorten the excitation flash, i.e. to make the flash more and more nearly an ideal o-flash, with phasefluorometry one excites with light that is amplitude modulated [.1-3]:

I a(t) =const.(l + excoswt). (3) In Equation (3), ex is the modulation depth: O<ex< 1, and w = 21[/, where / is the modulation frequency.

Using complex notation, Equation (3) can be replaced by

Ia(t)=A [1 exexp( -iwt)]. (4) By insertion of Equation (4) into Equation (2) with an upper limit t = 00, one may obtain

I E(t,w)=B[1-pexp{ -i(wt-qJ)}]. (5) This means the phasefluorometric response is also a periodically amplitude modulated emission, having the same modulation frequency, w, a smaller modulation depth,

p,

and a phase shift, qJ, with respect to the exciting light.

In the case where the &-response I t(t) may be

expressed as a sum of exponentials

It(t)=IAiexp( - ,lit), (6)

i

the modulation depth

p

(as a function of w) is given by

A.W)2 ( A.A.

)2Jl/2( A.)-l

P

ex [(

~w2~,lf + ~w2~Ji ~ A; ,

⁽⁷⁾

and the phase shift qJ (as function of w) is given by

(8)

In the simplest monoexponential case (where only one exponential function is important in It(t»), Equations (7) and (8) simplify to

p=

---;:=====:::: = ex cos qJ (9)

and

w

qJ arctan,lF (10)

where l'F= 1jAF is the lifetime of the emitting state, usually a fluorescing one.

For a general shape of the o-response It(t), the modulation depth,

p,

and phase shift, qJ, may be calculated from Equations (11) and (12):

p={

(f

sinwt-/t(tjdt)'

+

( Jo ^roo

coswt'It(t)dt

)2J

^1/2 _. ^[

_Jo ^roo

_I~(t)dt

J-1

(11)

The Arabian Journal for Science and Engineering, Volume 9, Number 4.

328

and

oOO sin wt

cp = arctan

f

₀₀ ^-I

^~(t)d

^{t )}. (12)

( 1

^coswt-

^I~(t)dt

Excitation with modulated light consequently corresponds to a Fourier analysis of the b-response, i.e.

for each frequency, a modulation depth and a phase shift is received. Pulse methods and phasefluorometry are physically equivalent, but the latter is an indirect method of getting the same information as the former gets directly.

2. WHY PHASEFLUOROMETRY?

A COMPARISON WITH PULSE METHODS The main advantage of pulse methods may be the direct determination of the decay function; e.g. it may be possible to notice at a glance if the decay is mono­

or biexponential. It therefore satisfies the scientist's need to have an intuitive grasp of the system kinetics.

So why should an indirect method like phasefluoro­

metry be used?

Up to the 1950s, before short pulses were experi­

mentally available, phasefluorometry was the only method for determining fluorescence lifetimes. Light modulation up to about 10 MHz enabled measure­

ment of lifetimes in the 1-10 ns region or even shorter (cf Section 3, 'Development of Phasefluorometry').

Since then, the development of short excitation pulse production methods (and pulse characterization) has enabled pulse methods to overtake phasefluorometric ones. Nowadays, commercially available apparatus for the measurement of decay curves (with a few excep­

tions) is consequently based on pulse methods. Mean­

while Shank's research group [4, 5] has succeeded in cutting the half-width of pulses to about 30 fs by means of a colliding pulse mode-locked laser.

Another advantage of pulse methods is that a reference measurement is not needed. With phase­

fluorometry, a reference material is used in order to determine what the phase angle would be without the fluorescing sample. With pulse methods, one need only switch off the light source to establish the base line.

In view of the power of pulse methods, phasefluoro­

metry seems at first to be an unsatisfactory alternative.

It is uncontested that until now the author's phase­

fluorometer with a time-resolution of 1 ps (which enabled measurement of a rotational relaxation time of 9 ps for perylene in methanol at a modulation frequency off=400 MHz [6]), is not quite as good as the subpicosecond pulse apparatus described above. In spite of this, there are two essential advantages of phasefluorometry, as follows.

(1) In contrast to pulse methods, in phasefluoro­

metry it is not necessary to have ideal condi­

tions for transmittance fidelity. (In pulse methods, 'ideal' means that all Fourier com­

ponents are transmitted at the same time with the same amplitude factor and frequency-depen­

dent phase angle.) With phasefluorometry, each frequency is treated separately, yielding the appropriate modulation depth and phase shift.

In these parameters, the transmittance ampli­

tude ratio and the characteristic phase angle do not appear. This is the physically decisive reason, because the above mentioned ideal conditions for transmittance fidelity may never be fulfilled completely and therefore results of measure­

ments made by pulse methods may sometimes be invalid.

(2) As only very low intensities of illumination (in the region of 1~I00mW) are used in phase­

fluorometry, the interference with the investi­

gated system is correspondingly low. Two­

quantum processes may be excluded from con­

sideration. The low intensity may also be a great advantage with biological systems, especially for in vivo measurements.

Furthermore, in many cases it is not even necessary to concede superiority in time-resolution to pulse methods; the results of the Shank group are an excep­

tion. Other research groups applying pulse methods, particularly to rotational diffusion of molecules in solution [7-19], have a somewhat lower time resolu­

tion than can be achieved with the author's phase­

fluorometer. Moreover, the currently achievable cut-off frequency of 400 MHz is not a physical barrier. There are already examples where light modulation in the gigahertz region has been realized [20, 21].

3. DEVELOPMENT OF PHASEFLUOROMETRY The historical development of phasefluorometry is strongly coupled with the development of devices for the generation of amplitude modulated light of the proper frequency, and photomultipliers in connection with measurement devices allowing the detection of

The Arabian Journal for Science and Engineering, Volume 9, Number 4. 329

phase differences between excitation and fluorescence light.

In addition to the direct modulation of the light source via the power supply, the following methods are used for the generation of amplitude modulated light.

(1) Kerr effect. Here, an alternating field applied perpendicularly to the direction of the light beam causes birefringence in liquids like nitro­

benzene. The reason for this is an anisotropy of the polarizability of the nitrobenzene molecules.

If the Kerr cell is placed between crossed Nicol prisms, amplitude modulated light is obtained.

Besides this transverse Kerr effect there is also the corresponding longitudinal Kerr effect.

(2) Debye-Sears effect. Here, standing or propagat­

ing ultrasonic waves can be generated in a liquid filled vessel by means of a quartz oscillator. The fluctuations thus induced in the density cause changes in the refractive index. To see how this effect might be used, imagine a light beam emerging from a liquid and incident upon the liquid-air interface at the critical angle for total reflection, ex; sin ex = n

1/

n₂ • If there are periodic changes in the refractive index of the liquid,

there will also be a periodic change in the ratio of the reflected and the transmitted beams.

(3) Electrooptic effect (Pockels effect). Here, as in the Kerr effect, one applies a periodic field to an optically anisotropic crystal, e.g. KD2 P04 , be­

tween crossed polarizers. This method is used in the author's apparatus and will be discussed later. The corresponding magnetooptic effect has hardly any practical significance.

After changing the light signal into an electric signal with a photomultiplier of short enough rise-time (N B a 1 ns rise-time corresponds to about a 1 GHz bandwidth), the phase detection may be carried out with a lock-in amplifier or with an RC phase shifter in connection with a null-detector. Instead of an electro­

nic phase shifter, a phase shift may be produced by varying the length of the light path or of the electrical signal path.

Instead of the phase shift, use could be made of the modulation depth for the determination of lifetimes.

However in view of numerous sources of error (e.g., those caused by fluctuations in the irradiance) this alternative is not used.

In Table 1, the historical development of phasefluorometry is demonstrated by some selected

Table 1. Development of Phase8uorometry

Year Research group Frequency Light modulation Signal detection Results/Comments (MHz)

1929 Hupfeld [1] 10 Kerr cell Wollaston prism

polarizer

Fluorescence lifetime of I ₂

,r

^IOns

1953 Bailey and Rollefson [22]

5.2 Ultrasonic wave RC phase shifter Lifetime of acrid one, r 15.9 ns;

quinine sulfate, T

=

22.8 ns;

fluoresceine, T = 4.5 ns 1953 Schmillen [23] 10.7 Ultrasonic wave Brightness modulation

of an oscilloscope and stroboscopic

measurement of the

Conversion of the signal into a 50 Hz signal by aid of a rotating spool

phase shift by the angular position of the beam

1953 Birks and Little [24]

15 Modulation of an air filled discharge tube

RC phase shifter Measurement of fluorescence lifetimes in crystals in the region of 5 ns also with modulation depth measurements

1956 Bonch- Bruevich [25]

12 Ultrasonic wave Light path phase compensation

Resolution of 0.1 ⁰ corresponding to 20ps

1959 Venetta [26] 5.2 Ultrasonic wave RC phase shifter Microscope phasefluorometer, measurement of proflavine bonded to tumor cell nuclei with a resolu­

tion of 0.4 ns

continued ...

330

The Arabian Journal for Science and Engineering, Volume 9, Number 4.

Table 1. Development of Phasefluorometry (continued)

Year Research group Frequency Light modulation Signal detection Results/Comments (MHz)

1961 Ware [27] 5.2,

10.6

Ultrasonic wave RC phase shifter Investigation of energy transfer.

Measurement of rate constants to be 12 times faster than predicted by diffusion theory

1961 B. Brehm and others [28]

18 Ultrasonic wave Light path phase compensation

Measurement of the emission lifetimes of Ga, AI, TI, Mg, and Na in region 9 to 15 ns

1962 Brewer and others [29]

5.2 Ultrasonic wave RC phase shifter Fluorescence lifetimes of perylene, 1: = 4.8 ns; rhodamine B, 1: = 6.2 ns.

Conversion into 1 kHz signal by heterodyne technique

1963 Hauser [30] 10 Ultrasonic wave Difference voltmeter Fluorescence lifetimes of pyrene and its derivatives in region 1.7 to 130 ns. Conversion into 1 kHz signal by heterodyne technique 1963 Brewer and

others [31]

5.2 Ultrasonic wave RC phase shifter Fluorescence lifetime of 1₂ ,

1:=720ns 1965 Muller and

others [32]

14 21 27

Pockels cell Cable path-length phase compensation

Organic dyes like fluoresceine, 1: = 3.8 ns; safranine, 1: = 2.3 ns.

Commercial components, discussion of photomultiplier properties

1970 Hauser and Heidt [33]

0.5 1.5 4.5

Pockels cell Vector voltmeter Excimer formation of pyrene. First discussion of biexponential decays 1975 Hauser and 0.5 to Pockels cell Vector voltmeter Determination of the rate constants

Heidt [34] 72 of the system

2-naphthol/naphtholate 1977 Mantulin and

Weber [35]

10 30

Ultrasonic wave Null-detection by mixing in the multiplier

Anthracene, perylene, and chrysene in propyleneglycol. Anisotropic rotational diffusion found.

Conversion into a 1 kHz signal by heterodyne technique

1977 Popovic and Menzel [36,37]

1.6 27 9 to 29

Standing and propagating ultrasonic waves

Light path phase compensation by aid of a laser and a radio

Organic dyes like rhodamine B, 1: = 3.5 ns and rhodamine 6G, 1: =6.4 ns

1978 Haar and Hauser [38]

5 to 250 Pockels cell Vector voltmeter Frequency doubling by using quadratic operating point on the characteristic curve for the Pockels cell

1980 Gugger and Calzaferri [39,40]

28.8 36.3

Pockels cell Spectrum analyzer Calibration by light path phase differences; time resolution 15 ps 1981 Klein and

others [41-44]

12.5, 25, 50, 100, 200, and 400

Pockels cell in 50 Q waveguide technique

Lock-in amplifier Conversion into a 5 kHz signal by heterod yne technique

Note. The author's investigations have been made since 1977 [6,41-44] using the new phasefluorometer constructed at the Institute of Physical Chemistry at the University of Stuttgart, West Germany.

The Arabian Journal for Science and Engineering, Volume 9, Number 4.

331

examples. From the modulation frequency, the time resolution may be calculated immediately, The optimum time resolution occurs when '[' = l/ro = 1/21Cf, which produces a phase shift of 45°. For example, when

f=

1 MHz one has ^'['= 150ns, and when

f=

100 MHz one has ^'['= 1.5 ns. The resolution of the apparatus (at a phase resolution of 1°) is 3 ns at 1 MHz and 30 ps at 100 MHz.

With apparatus having just one frequency, only systems with monoexponential behavior may be investigated. Otherwise, a mean lifetime is detected depending upon the frequency applied.

4. THE NEW PHASEFLUOROMETER

4.1 Operation

The setup of the new phasefluorometer is shown schematically in Figure 1.

The vertically polarized light beam of the krypton ion laser Kr-L (Spectra Physics 164) undergoes a high­

frequency rotation of the polarization plane by passing through a home-made 50

n

Pockels cell. The modulation is produced by a high frequency alternating signal between 12.5 and 400 MHz. It is generated with the quartz-stabilized signal generator SG 1 (Ortec 462 and home-made second harmonic generators from 100 to 200 MHz and 200 to 400 MHz) and amplified with the power amplifier PA (EN I 510 L) to an effective voltage of 20 V. This is a very low voltage for driving a Pockels cell. The electric wave passes through the Pockels cell and is attenuated with an attenuator chain A (46 dB General Radio) and damped without reflections by the terminator T. In order to operate the Pockels cell at its optimum working voltage, a DC voltage is added. The light beam passes through a Nicol prism, PO I, with its polarization direction also vertical producing high

Figure I. Block Diagram of New Phasejluorometer

332 The Arabian Journal for Science and Engineering, Volume 9, Number 4.

frequency, amplitUde modulated, vertically polarized light. Via mirror Sl' lens Ll' and mirror S2 (diameter =3 mm), the beam is directed vertically on the thermostatted sample P (1 mm quartz cell).

This then excites the fluorescence. If the excitation beam is autocollimated, the associated reflected beam (at 4% of the incident intensity) goes back to the mirror S2 and is negligible. The lens L2 (f= 5 cm, diameter = 3 cm) has its focus at the excitation point within the cell and hence makes the rays of the fluorescent light parallel. The fluorescence passes through the Nicol prism PO II and is separated into parallel and perpendicular components. If it is required to obtain fluorescence free from contributions from rotational diffusion, it is necessary to fix the prism PO II at an angle of 54.7° with respect to the vertical plane [15]. Filters F between L2 and PO II will allow the selection of the interesting wavelength region of the fluorescence. Finally the fluorescent light arrives at the photocathode of the multiplier (RCA C 31024 A) via a light-diffusing plate SP and is changed into DC and AC signals. In order to eliminate the influence of the wavelength of the light upon the phase of the AC signal, the photomultiplier is connected so that about 1 k V of the 3 k V high voltage drops between the photocathode and the first dynode.

The DC voltage signal (proportional to the continuous portion of the fluorescence) is measured with a voltmeter VM and stored by a data registration unit DS (Ziegler RS 24-500). The AC signal is mixed in the mixer MS with a second high frequency AC signal (offset 5 kHz with respect to the primary frequency of SG 1 and generated by the signal generator SG2, HP 8654). The phase angle of the high frequency signal is kept the same in the 5 kHz signal (heterodyne technique). The resulting 5 kHz i.f. signal is then analyzed by the lock-in amplifier LIA (PAR 129) to determine its phase and amplitude. For this purpose, it is compared with a reference signal, also of 5 kHz, generated in the mixer MR by pick-off signals ST of SGI and SG2. Phase and amplitUde are stored in the data registration unit DS. This unit also controls the position of the prism PO II (via a flip-flop in a time of 15 s), insuring that its direction of polar­

ization is either parallel or vertical. Thus, for each position, the DC signal, the amplitude of the AC signal, and the phase signal are registered simul­

taneously. The ratio of the AC amplitude to the DC amplitude depends upon the modulation depth.

In order to prevent errors by electrical noise, all high frequency leading parts are screened thoroughly

with a copper pipe. This is absolutely necessary because, at these high radar-like frequencies, noise (each connection generates noise) may influence the phase signal dramatically, depending on the position of the operator.

Figure 2 shows a photograph of the new phase­

fluorometer.

It is evident that the new phasefluorometer may be improved by using commercially available waveform synthesizers SG 1 and SG2, sweeping them at a constant difference of, e.g., 5 kHz between 1 MHz and 1 GHz to get a whole Fourier spectrum for the fluor­

escence decay. This is possible owing to the author's new 500 waveguide Pockels cell described in Subsection 4.2.

Figure 2. Photograph of New Phasejluorometer

4.2 The New 500 Pockels Cell

The theory of optically anisotropic crystals is described by Nye [45] and Chen [46]. The author uses a KD* P-crystal. The rotation of the plane of polarization, ~a., of linearly polarized light with frequency v (which passes through a crystal of length L with a longitudinally applied electric field of strength E₃ ) is given by [21]

nn r 4

~a. = _ _^6_3 LE (13)

c 3

where

n is the refractive index of the crystal;

r⁶³ is the electrooptical constant; and c is the speed of light in vacuo.

The half-wave voltage U 1/2' necessary for a rotation of the plane of polarization through an angle of n/2

(in the case of a transverse applied field) is given by

d

A

U 1 / 2 =-L'-24 ^' (14)

n r63

where

d is the width of the crystal; and Ais the wavelength of the light.

From Equation (14) it can be seen that if a low half­

wave voltage is required, a long crystal may have to be chosen, possibly also having a small cross section, d²•

For example, in the case of KD*P, U1/2 =90V for a ratio d/L= 1/50.

Besides the possible requirement of a low half-wave voltage, two other important points had to be considered when constructing the new Pockels cell:

(a) the temperature-dependence of the static birefringence; and

(b) the operability at high frequencies.

As the temperature-dependence of the static birefringence is of the same order of magnitude as the change of the refractive index owing to an electric field (for 20 V, ~n= 8.4 x 10-⁷; for a change in temperature of 0.1 °C, ~n= 7 x 10- 7), the tern perature effect must be compensated for. This may be possible using two crystals instead of one. These crystals have to be of the same length but rotated 90° with respect to each other and positioned one after the other [47]. In this way the field-dependent effect will be added, but the temperature-dependent effect is eliminated.

c

~ ^fR

Crystal

Figure 3. Pockels Cell Made Part of Oscillating Circuit If the Pockels cell is made part of an oscillating circuit as shown in Figure 3, the cut-off frequency is determined by the capacitance of the modulator. In this case, frequencies higher than about 100 MHz will not be achieved with, for example, the commercially available Lasermetrics modulator, M 3078 FW. In order to achieve higher frequencies, the Pockels cell must be made part of a waveguide as shown in Figure 4. For this purpose the impedance of the cell must be

Effeclivesignal20V .. ~-- - - --~---~----r-

Coaxial coble 50 Q 50 Q Coaxial coble 50 Q

Figure 4. Pockels Cell Made Part of Transmission Line

The Arabian Journal for Science and Engineering, Volume 9, Number 4.

333

matched to that of the coaxial cable, ^Zo = 50

n.

For a waveguide of the form shown in Figure 5, the impedance may be calculated from Equation (15) [21]

Crystal

,

a

..

.... 0 ­ +-w-+

Figure 5. Geometry of Transmission Line

as follows

Zo = (15)

where

Jlo is the magnetic permeability of free space;

is the electric permittivity of free space; and.

80

8 is the dielectric constant of the crystal.

With KD* P, ^Zo = 50

n

for a guide width w= 1.4 mm and a crystal width a 1 mm. A crystal width of 1 mm seems to be reasonable, as it exceeds thy beam diameter of the laser.

Finally, the length of the crystals must be determined. By lengthening the crystals on the one hand the required half-wave voltage may be reduced, but on the other hand the bandwidth will be decreased. This is due to the difference in the transit time of the electrical wave and the light wave through the crystals. For an overall length, L, of the crystals, the rise-time, ^{L M} of the modulator is given by [20]

(16)

where Vs and _{V L} are the speeds of the electric and the light wave respectively.

Using Fourier transforms, the 3dS' bandwidth, iJdB' of the modulator may be calculated as [20]

(17)

The bandwidth and rise-time of the modulator as a function of the modulator length is shown in Figure 6 for various crystals (from [20]).

The Arabian Journal for Science and Engineering, Volume 9, Number 4.

14

...---., f3dB 12

N

J: 10

~ .s:::.

8 800

:0

5 10 15 20 25 30

iii _c..

~ ~

c 6 600

Cl ..:

ro

III oj

~ 4 400

.... E

,

<I>

200 .~

0::

35 40 4S 50 mm Modulator length

Figure 6. Bandwidth and Rise-Time of Modulator as a Function of Modulator Length (from [20])

The preceding considerations led to the following crystal arrangement: two KD* P crystals 1 x 1 x 25 mm, rotated at 90° with respect to each other between two silvered brass plates of 1.4 mm width and 5 mm thickness as shown in Figure 7. The crystals

~

Figure 7. Crystal Arrangement of New Pockets Cell

were supplied by Gsanger, Gdifelfing. The electrical connection of the silvered brass plates to the SMA con­

nectors at the housing was done with due regard for the high frequencies to be employed and may be seen from the photograph of the new Pockels cell shown in Figure 8.

The plates are imbedded in a cylindrical piece of Plexiglas held together by a copper case. In order to avoid losses due to reflections of the light beam, the Plexiglas part is completely immersed in decaline.

Decaline has almost the same refractive index as the crystals and does not attack the materials used. After purification of the decaline via an Alz 0₃ column, its transmittance is 100% down to a wavelength of 250nm.

Theoretically, the new Pockels cell so constructed should have the following properties:

334

-100

Figure 8. Photograph of New Pockels Cell

(1) a half-wave voltage of about 90 V;

(2) a 3dB bandwidth of about 1 GHz; and (3) thermomechanical stability.

The experimental results are shown in Figures 9 and 10.

-27V

100 200 300 400

U(V)

Figure 9. Light Intensity of Laser Beam After Passing Through New Pockels Cell and a Nicol Prism as a Function

of Applied DC Voltage

Uout U. _n IdB) 0

-1

- 0 - 0 _ 0 _ 0 - 0 ­

0 _ _

0 " ' 0 ~ 0

'­'­

o 0,­

-3

o ~o

"

-4

-5 ^o \

-6 0\

~ -7

0.5 1.0 1.5 2.0 'IGHz)

Figure 10. Electrical Damping of Incoming Signal After Passing Through New Pockels Cell as a Function of

Frequency

The curve in Figure 9 was obtained by means of a laser beam, wavelength 413 nm, while applying a DC voltage between -100 and +400 V. It obeys the expected relationship (I,...., sin² U) quite well, except at higher voltages where a slight deviation can be detected. Owing to diffraction, an intensity of zero may not be achieved. From this experimental curve, the half-wave voltage is found to be U 1/2 Exp= 135 V, not far from the theoretical value of 90 V. The modulation depth of the light, when the peak-to-peak high frequency signal is 20 V, is about 40% when operating the cell at the linear working point of - 27 V.

Figure 10 shows the damping of the incoming electrical signal. The electrical 3dB bandwidth is about 1.6 GHz. Up to 400 MHz, the modulation depth of the light follows that curve closely. At the moment, the light modulation cannot be checked at frequencies higher than 400 MHz. But there is no doubt that this new Pockels cell will also work at 1 G Hz.

The new cell is also thermally stable.

4.3 Test of Phase Detection by Variation of Light Path Length

Lengthening the light path between Polarizer PO I and sample P (cf Figure 1) by L\x results in a phase shift, L\qJ, for L\x = 5 cm.

(18)

The wavelength, A, of the amplitude modulated light at various modulation frequencies is presented in Table 2 together with the expected values of the phase shift, ~qJ, for ~x= 5 cm.

Table 2. Dependence of Wavelength, Frequency, and Phase Shift of Amplitude Modulated Light

f(MHz) 12.5 25 50 100 200 400

A(m) 24 12 6 3 1.5 0.75

~q>C) 0.75 1.5 3.0 6.0 12.0 24.0

Figure 11 shows the experimentally determined values for the phase shift. These values could only be achieved when the photocathode of the photo­

multiplier was totally illuminated by aid of a diffusing plate. With single-spot illumination the values were nonuniform.

The phase measurements may therefore be considered to be calibrated. The light wavelength has

The Arabian Journal for Science and Engineering, Volume 9, Number 4. 335

20 _./ . /

./

, / , / ,I , /

10 _{. /} , /

. /

100 200 300 400

f(MHz)

Figure 11. Experimental Values of Phase Shift, ll<p, by Light Path Lengthening of llx = 5 em as a Function of Modula­

tion Frequency

practically no influence on the phase shift. Since, with rotational diffusion, parallel and perpendicular components have the same wavelength, there is no need to consider the wavelength-dependent transit time of the electrons [32]. As already mentioned, this wavelength dependence is greatly reduced by the high voltage drop of about 1 k V between photocathode and first dynode.

5. APPLICATION OF NEW PHASE FLUOROMETER TO PICOSECOND ROTATIONAL DIFFUSION

5.1 Theory

In the introduction it has been shown that it is necessary to know the 8-response of a system in order to determine its phasefluorometric response. A general theory of the fluorescence depolarisation by aniso­

tropic rotational diffusion was derived by Chuang and Eisenthal [48]. They solved the problem using a differential equation, analogous to the SchrOdinger equation of an asymmetric rotator, proposed by Favro [49J. They found that the 8-response of the fluorescence involves up to six different time constants.

The time-dependence of the parallel component is given by

I~(t)=exp(

^{- tiT}

^F)U+ l~qxq,

^YxY,exp[ - 3(D,

+

D)t]

4

+ 15qyqz yy yz exp[ - 3(D x + D)tJ

+ 15qxqz Yx yz exp[ - 3(Dy+ D)tJ 4

The Arabian Journal for Science and Engineering, Volume 9, Number 4.

+T5(P +a)exp[ - (6D + 2d)tJ 1

+~(P

- a)exp[ - (6D - 2d)t J}. (19) 15

The time-dependence of the perpendicular component is given by

(20)

The parameters are as follows:

D D and D are the diffusion coefficients in the

x' y' z . . .

X-, y-, and z-dlrectlOns, respectively;

D is the mean diffusion coefficient, given by

are the projections of the unit vector of the absorption transition dipole along the main rotation axes of the molecule;

are the projections of the unit vector of the emission transition dipole along the main rotation axes of the molecule;

d=(D2+D2+D2_D D -D D -D D )1/2; x y z xy yz xz (22) a= (D) d)(q; y; +q; y; - 2q; y; +y; +q;)

+ (Dy/d)(q;y; +q;y; -2q;y; +y; +q;) + (D z/ d)(q; y; + q; y; - 2q; y; + y; + q;)

-2D/d; and (23)

R 2 2 2 2+ 2 2 1

p=qxyx+qyyy qz yz-"3' (24)

These equations contain all possible cases and are partly discussed by Chuang and Eisenthal [48J. In the simplest case of isotropic rotational diffusion, i.e.

Dx=Dy=D z and yx=qx=l; yy=yz=qy=qz=O, Equations (19) and (20) reduce to

and

336

where '! Rot IS the rotational relaxation time

('t _{Rot =} l/A_{Rot )} derived by Debye, Stokes, and

Einstein [50] to be

1 ",V

1" _Rot-- 6D - kT ­ (27)

where

", is the viscosity of the solvent;

k is Boltzmann's constant;

V is the hydrodynamic rotational volume; and T is the absolute temperature.

Transformation of Equations (25) and (26) into the phasefluorometric responses yield for the modulation depths

f3"

and

f3

^1-[41]

(28)

(29)

and for the phase shifts [41]

(30)

- arctan(3A

3~A

^{). (31)}

F+ Rot

As quantities that are experimentally easy to detect, one may measure the ratio of the modulation depth

f3

^1-

I

f31! and the difference in phase shifts, ~CfJ, where

~CfJ = CfJ 1­ - CfJII[41]:

(32)

and

licp = arctan

(9AF~ARJ -

^arctan

CAF:SARoJ

⁽³³⁾

Nevertheless

f3 1-1 f3"

is essentially less sensitive to changes in A_{Rol ,} so that only the phase shift difference,

~CfJ, is of practical use for the determination of A_Rot or the rotational relaxation time'! Rot.

Figures 12 to 14 show together:

(1) the 8-responses I~(t) and Ii(t);

(2) the corresponding transformations

f3

₁₁(f),

f3

^1-(f);

CfJII(f), and CfJ 1-(f) as functions of the modu­

lation frequency, f; and (3) the magnitudes

f31-1

f3 ₁₁ and ~CfJ.

All curves are calculated for a fluorescence lifetime of 1" F= 5.7 ns and a rotational relaxation time of

1"Rot= 50 ps.

0.2 r - - - t - ­ - - - - t - - - t - - - I

200 400 600 ---+ t(ps)

Figure 12. 8-Responses I~(t) and Ii(t) (from [41])

~i

100 r -- - - - = == -- , - - - . . . , - - -- --;n---,100· JIP 07')

OSO -- ­ -­

_ _ _ __ __ _ - - ' - -__ __ _ ---1-_ _ _ _- - '

10 100 tOOO

~f(t"Hz)

Figure 13. Transformations (J11(f), (J1-(f), ({J11(f), and

({J 1-(f) Corresponding to Figure 12 as a Function of

Modulation Frequency (from [41])

The Arabian Journal for Science and Engineering, Volume 9, Number 4. 337

095

0.90 t-- ...-.~---~---!---+---I----\I

0.8S r - - - -..----+---+---.,L---IS·

~.l-~II

10 100 1000

---III-fIto! HzI Figure 14. The Magnitudes

P.dP"

and A<p as a Function of

Modulation Frequency (from [41])

It can be seen that long times in the I versus t diagram correspond to low frequencies in the phase­

flu orometric diagrams. Short times correspond to high frequencies; for III(t) and 11.(t) there is also a considerable difference between q>l,(f) and q> 1. (f). The modulation depths are in practice only determined by the long fluorescence lifetime t" F' Only at frequencies above 100 MHz does P1./ PI! become less than unity, and even at 1 GHz the ratio is still 0.9 whereas the phase shift difference, Ilq>, is already 18°.

5.2 Rotational Diffusion of Oxypyrenetrisulfonate in Water

Figure 15 shows the experimental values

P

_II, ^{/3 1.' q>1l'}

and q> 1. for oxypyrenetrisulfonate in slightly basic aqueous solution at 20°C as a function of the modula­

tion frequency.

100~r

7S~

oso I---.---+---~I - - - 4 - - - lso'

o2S i---+--#---~---l2S·

;, 338

10 100 1000

----too f(MHz)

Figure 15. Experimental Values of PIl' Pl.. <Pll' and <P.L of Oxypyrenetrisulfonate in Water at 20°C as a Function of

Modulation Frequency (from [41])

The Arabian Journal for Science and Engineering, Volume 9, Number 4.

With a fluorescence lifetime of t"F= 5.7 ns [51,52] the best fit was with t"Rot=104ps. It can clearly be seen that the phase shifts q>1I and q> 1. differ considerably with increasing frequency, whereas the modulation depths PI! and /31. are almost identical over the whole range. In the following, the phase shift differences Ilq> = q> 1. - q>"

are determined in order to calculate values for the rotational relaxation time.

Figure 16 shows experimental values of Ilq> (f) for various temperatures.

10 100 1000

- ' f ( M H z )

Figure 16. Experimental Values of A<p(f) for Various Temperatures (from [41] )

There was no indication of anisotropic rotational diffusion. The best fits for t" Rot together with water viscosities [53] are presented in Table 3.

Table 3. Temperature Dependence of Rotational Relaxation Time and Water

Viscosity

T (OC) 'Y/ (cP) TRot (ps)

10 1.307 136

20 1.002 104

30 0.7975 79

51 0.5378 47

70 0.4042 33

95 0.2975 26

As predicted from Equation (27), the experi­

mental values of t"Rot plotted versus l1/kT fall on a line passing through the origin as shown in Figure 17.

From the slope, the hydrodynamic volume was determined to be 412

A3.

The resulting molecular diameter is in good agreement with the diameter along the major axis of the molecule as shown in Figure 18.

t

¹⁵⁰

'tRo1(ps) I

100 ~----t---+--"7"''---T---j

50 - - - ­-+--ct"---+­- ­- - -+___

Figure 17. Rotational Relaxation Times LRot as a Function of 1J / k T (from [41J )

Figure 18. Rotating Sphere of OxypyrenetrisulJonate (from [41J)

As Fleming, Morris, and Robinson [15J have found for Rose-Bengal and Eosin, some water molecules seem to attach to the dye molecule to form an approximate sphere.

5.3 Rotational Diffusion of Rhodamine 6G in Water

Figure 19 shows the experimental values of dq> of rhodamine 6G at 20⁰ and 80°C in water.

The values may be well-fitted assuming isotropic rotational diffusion with a fluorescence lifetime of

't F = 5.0 ns. Values of 'tRot' together with literature values of the water viscosity at various temperatures [53J, are presented in Table 4.

On plotting values of 'tRot as a function of y/ /kT as shown in Figure 20, only the values between 30°C and 80°C fall on a line passing through the origin. The slope is V= 724

A

^3, corresponding to a molecular radius of 5.6

A.

At lower temperatures, the values

i

²⁵

l\<ll l

20

15

10

-.- /

/

20 /

1

/ '

/

^10· / ⁽

/

^/

/ /'~

, /

--====: ~ ,--- ^{/ '}

10 ^{100 1000}

- - - . f (MHz)

Figure 19. Experimental Values of ~<fJ(f) of Rhodamine 6G at 20° and 80⁰ C (from [42J)

measured here are higher than those predicted from the Debye-Stokes-Einstein theory [50J. The hydro­

dynamical radius increases to 6

A

at 20°C and 6.4

A

at 10°C. Probably there is an additional solvent attach­

ment caused by hydrogen bridge bondings which is not considered in the macro viscosity of the water.

Table 4. Temperature Dependence of Rotational Relaxation Time and Water

Viscosity

T (DC) 1J (cP) L Rot (ps)

10 1.307 280

20 1.002 196

30 0.7975 137

40 0.6529 109

50 0.5468 88

60 0.4665 76

70 0.4042 61

80 0.3547 49

200

'00

" " ^"

I ^/

/ /

" ^"

" " "

/ "

/ '

/ '"

/ ,/

/ ,/

/

01 02 03 0.4

Figure 20. Rotational Relaxation Times of Rhodamine 6G as a Function of1J/kT(from [42J)

The Arabian Journal for Science and Engineering, Volume 9, Number 4. 339

10000

trot Ips)

1000

1

100 - -

=== =7..

... ^:r

t;/

I

~k

.

^'6."

.

^{6 1}

.

6

'if oil

Debye

...

line

- Stokes- Einstein­

for r= 6l

10

0.1 t

It ' t

¹⁰ t 100 1000

CHflH HPI C:f\PH1 Cll H23 0H "Glycerol. "fl(e P) Cl\OH CsH"OH

t::,. Chuang and Eisenthal [7]; x Porter and others [17]; Y' Eichler and others [19];.0 van Resandt and De Mayer [57]; I Heiss and others [56]; 0 von Jena and Lessing [13] ; • Rice and Kenney-Wallace [58] ; 0 the present author.

Figure 21. Rotational Relaxation Times of Rhodamine 6G versus Viscosity as Measured by Various Groups

The rotational diffusion of rhodamine 6G has probably been more thoroughly investigated than that of any other molecule. In Figure 21 rotational relaxa­

tion times, ^T Rot' of rhodamine 6G at 20°C as functions of the viscosity are presented as measured by various groups. For comparison, measurements at other tem­

peratures were fitted by a simple extrapolation, TRol (20°C)= [Tx(K)/293]TRol(Tx). Owing to the large range in viscosity and rotational relaxation time, a log-log scale was used.

It can be seen that for viscosities below 10 cP the values fit very well to a Debye-Stokes-Einstein line with a hydrodynamic radius r= 6

A.

Also it is shown that there is a very good agreement between the measured values of T Rot found by several different methods of the various groups. Above 10 cP the values seem to become almost independent of viscosity. This might be due to the inhomogeneous microviscosity of these solvents. A similar effect was also found for translational diffusion, e.g. for excimer formation of pyrene in paraffin oil [54] or in ethylcellulose dis­

solved in toluene [55].

The dependence of the rotational relaxation time on the applied light excitation frequency has so far been investigated only by Heiss and others [56]. The mean value fits weB within the other experimental values.

The present author's measurements also fit well within the other values. So far, they are the shortest relaxa­

tion times detected for rhodamine 6G, indicating the good time resolution of the new phasefluorometer.

5.4 Rotational Diffusion of Perylene in Methanol, Dodecane, and Hexadecane

The rotational diffusion of perylene is also well established in the literature [56, 58-61], but the present investigations are the first in the picosecond region. The reason might be that, except for the Krypton laser, there were only a few weak dye laser lines available in the 400 nm region where perylene has its absorption. From molecular considerations (cf Figure 22), anisotropic rotational diffusion would be expected for perylene, with the rotational diffusion constants:

D _x =217x10_• ¹0s-1. _, D y =027x10• ¹0s-1.,

These values were calculated for pery lene in ethanol at 20°C (17 = 1.2 cP) by the method of Labhart and Pantke [59, 60]. The principal rotational axis, x, coin­

cides with the direction of either absorption or emis­

sion transition dipoles:

~-...- ...+-~~ 50 - 52 - transition

50 -51 -transition - y

Figure 22. Transition Dipoles of Perylene

From the theory of anisotropic rotational diffusion [48], this leads to 8-responses I~(t) and Ii(t):

I~(t)=exp( -tITF){~+ 115 (i+C()exp( - tITRo!,,) +

/5

(~-C()exp( -tITRol.2)} ⁽³⁴⁾

and

I i

^(t) = exp( - tITF) {~-310(~+C()exp( - tiT Rot.l) - 310(~-C()exp( -tITio!.2)}' (35)

The Arabian Jot/rnal for Science and Engineering, Volume 9, Number 4.

340

where

TRot,! = (6D

+

2L\) -1; and

T Rot,2= (6D - 2L\) -1.

It is seen that the effects of rotational diffusion are biexponential with time. The calculated parameters are:

a ~0; TRot,! = 5.0 ps; and T Rot,2 = 15.4 ps.

The expected rotational relaxation times are in the neighborhood of 10 ps. It follows from these con­

siderations that with the author's apparatus, even at 400 MHz a phase shift difference of less than 2⁰ will be expected. This means that the results are near the limit of the time resolution and, although it would be possible to fit experimental values to Equations (34) and (35), it seems more reasonable to fit the values to Equation (33), i.e. to one mean rotational relaxation time. Owing to the very short rotational relaxation time, ^TRot' compared to the long fluorescence lifetime, 5.5 ns, and the low frequencies with respect to the rotational relaxation time, Equation (33) reduces to

(36) Figure 23 shows the experimental values of L\cp as functions of the applied modulation frequency.

100 200 300 1.00 f( M Hz)

Figure 23. Experimental Values of 11<p of Perylene as a Function of the Modulation Frequency (from [6])

Except for hexadecane where there seems to be a slight curvature, the phase shift differences plotted versus modulation frequency fit well to the predicted line, Equation (36), passing through the origin [6].

Table 5 represents the parameters best fitted to Equation (36), together with viscosities [53J at 20°C.

Fitting the values of TRot versus viscosity suggests an almost linear dependence as expected from the Debye­

Stokes-Einstein Equation (27). This is shown in

Table 5. Rotational Relaxation Times Best Fitted to Equation (36)

Solvent Yf (cP) LRot(PS)

methanol 0.597 9

dodecane 1.508 26

hexadecane 3.484 49

Figure 24. From the slope, a hydrodynamical volume of 31

A

³ maybe calculated. The corresponding isotropic hydrodynamical radius is r = 2.5

A,

which is low in comparison to the radii measured along the two principal axes (r x=3.7

A

and ry=5.0A). But if the fact that the hydrogen atoms contribute only slightly to the hydrodynamical radius is taken into account (since they are relatively mobile with respect to the carbon skeleton) the values of _, rand x r y are found to be 2.2

A

and 3.5

A

respectively. These values are very close to the experimental value. The low value of ^{T Rot} could also be explained assuming that the solute slips a little as it rotates in the solvent cage rather than binding to the solvent nearest-neighbours, as described by Kivelson [62J and Hu and Zwanzig [63].

'tRot (ps) 50

40

30

/

t:, 20

10 _t:,

3 'Tl(cP)

Figure 24. Rotational Relaxation Times of Perylene versus Viscosity (from [6])

Calculating r _z from the volume of a hydrodynamical ellipsoid gives r_z = 2.0

A.

From these considerations, it may be said that D ~D , and according to Equation (21) Dy=nDx with ;=3D/Dx-2. For perylene in met­

hanol, with a mean rotational diffusion constant of D = 1.85 X 1010 s -l,Dx' Dy' ^and D'Z (with the assumption that Dx~D'Z=2rD/(rx+r'Z))can be estimated to be:

D _x ~D ~ 1.2D~2.2 x 10IOS-l and

'Z

D _y ~0.5D~0.9 X 10IOs- 1.

The experimental verification of these values may only be accomplished by evaluating two rotational relaxation time constants from the phase shift dif­

ferences at higher frequencies. As the absolute values

The Arabian Journal Jor Science and Engineering, Volume 9, Number 4. 341