complex macromolecular dynamics

EUROPEANPOLYMER

European Polymer Journal 40 (2004) 1997–2008

www.elsevier.com/locate/europolj

JOURNAL

Complex macromolecular dynamics:I. Estimation technique for time-resolved emissionanisotropy ratio of chromophores incorporated

into polymer chains

Chihiro Hashimoto a, Jaques Rouch b, Jean Lachaise c,Alain Graciaa c, Hideharu Ushiki a,*

a Laboratory of Molecular Dynamics and Complex Chemical Physics, Department of Environmental and Natural Resource Science,

Faculty of Agriculture, Tokyo University of Agriculture and Technology, 3-5-8, Saiwai-cho, Fuchu-shi, Tokyo 183-8509, Japanb Centre de Physique Moleculaire Optique et Hertzinne, UMR5798, Universite Bordeaux I,

351, Cours de la Liberation, 33405 Talence, Francec Laboratoire de Thermodynamique Etats Metastables et de Physique Moleculaire, U.F.R. Sciences et Techniques,

Universite de Pau, avenue de l’universite, 64000 Pau, France

Received 23 February 2003; received in revised form 6 April 2004; accepted 7 April 2004

Available online 9 June 2004

Abstract

Time-resolved fluorescence emission anisotropy ratios of carbazolyl groups incorporated into polystyrene chains in

polyethyleneoxide(PEO)/1,2-dichloroethane mixtures have been measured by the single photon counting method. The

fluorescence depolarization method is very excellent to clarify various dynamical modes of polymer chains, and many

theoretical and experimental researches have so far been reported in the field of polymer chain dynamics. However there

are few reports about the dynamics on the polymer side chain, because the dynamical mechanism of the polymer side

chain is very complicated. In this report we tried to analyze the dynamical modes of the polymer side chains by the

fluorescence depolarization method. Five dynamical modes of a polymer chain based on the W€oessner model wereestimated by our original analytical technique ‘v2-map method’. The value of each mode of a polymer side chain wasdiscussed above the overlap concentration (C�) of PEO and the micro-environments were clarified in the vicinity of the

chromophore attached to the polymer side chain.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: W€oessner model; Fluorescence depolarization; C*

1. Introduction

In the field of condensed matters, the twisted internal

charge transfer (TICT) probe, the fluorescence depo-

larization, and the excimer formation methods have so

far been used in order to analyze the micro-environ-

ments, the Brownian movements, and the micro-struc-

* Corresponding author. Tel./fax: +81-42-367-5616.

E-mail address: [email protected] (H. Ushiki).

0014-3057/$ - see front matter � 2004 Elsevier Ltd. All rights reserv

doi:10.1016/j.eurpolymj.2004.04.002

tures, respectively [1]. The fluorescence depolarization

method is very efficient, especially to clarify the

dynamical mechanism of polymer chains. In the field of

polymer physics, many researchers found the fact that

some dynamical modes of macromolecular chains can be

measured directly by the time-resolved fluorescence

depolarization method. Various theories for dynamical

modes of polymer chains have been reported conse-

quently. Let us explain a history of such development

of the fluorescence depolarization method as follows

[2].

ed.

mail to: [email protected]

1998 C. Hashimoto et al. / European Polymer Journal 40 (2004) 1997–2008

The first period was the establishment of the rela-

tionship between the Brownian motion of molecules and

the fluorescence depolarization principle (1940–1960).

During this period, the approximation of a rigid spher-

ical model was established [3]. The emission anisotropy

ratio, r is represented by

rðtÞ ¼ r0 expð�t=hÞ; ð1Þ

where r0 and h are the emission anisotropy ratio withoutthe Brownian motion of molecules and the rotational

correlation time, respectively. The rotational diffusion

coefficient D is represented by D ¼ 1=ð6hÞ. The meananisotropy ratio �r induced by the stationary light is

shown by the following equation using 1=h ¼ kT=vg:

1

�r¼ 1

r01

�þ sf

s

�¼ 1

r01

�þ kT

vg

�: ð2Þ

sf , v, g, T , and k are the fluorescence lifetime of the

chromophore, the volume of the spherical rotational

body, the solvent viscosity, the absolute temperature and

the Boltzmann constant, respectively. Eq. (2) is known as

the Perrin–Webber plots.

The second period was the establishment of the time-

resolved fluorescence depolarization method (1960–1980).

During this period, the approximation of a rotational

ellipsoid model was established [4], and r is expressed by

rðtÞ ¼ A1 expð�t=h1Þ þ A2 expð�t=h2Þ þ A3 expð�t=h3Þ;ð3Þ

where

A1 ¼6

5sin2 a cos2 a; A2 ¼

3

10sin4 a;

A3 ¼1

10ð3 cos2 a � 1Þ2; 1

h1¼ DL þ 5DS;

1

h2¼ 4DL þ 2DS;

1

h3¼ 6DS: ð3aÞ

DL and DS are the rotational diffusion coefficients of the

long- and short-axes of an ellipsoid, respectively. a is theangle between the long-axis and the emission transition

moment of the chromophore. Various emission aniso-

tropy ratio decay curves were measured by the time-

resolved fluorescence depolarization technique, and

some researchers had begun to notice the existence of

the non-exponential decay phenomena in nature.

The third period was the establishment of the ana-

lytical method based on the restricted rotational motions

of macromolecules (1970–1980). During this period, the

researchers discussed the theme that the non-exponential

decay curves of the fluorescence emission anisotropy

ratio were caused by various motional modes of mac-

romolecules. The complex biomolecular dynamics were

further extended by a number of investigators using ESR

and NMR techniques at that time, and the measured

emission anisotropy ratio decay curve of chromo-

phores in condensed biomolecules actually showed var-

ious non-exponential functions [5]. Researchers picked

up some important information of intramolecular mo-

tions of biomolecules by the time-resolved fluorescence

depolarization technique, that is, the restricted rotational

motions of macromolecules in membrane (Wobbling–

Cone model) and the twisted motions of super-

macromolecules (DNA) as shown in Eqs. (4) and (5),

respectively:

rðtÞrð0Þ ¼ A1 þ ð1� A1Þ exp

�� Dwt

hri

�; ð4Þ

where

A1 ¼ rð1Þ � rð0Þ ¼ fð1=2Þ cos að1þ cos aÞg2;hri ¼

Xj 6¼1

Ajrj=ð1� A1Þ ð4aÞ

and

rðtÞ ¼ A1 expð�rffiffit

p=4Þ þ A2 expð�r

ffiffit

pÞ þ A3; ð5Þ

where

r ¼ 2kTpR

ffiffiffiffiffiffigY

p : ð5aÞ

Y is a molecular elasticity.

The fourth period was the establishment of the

analytical method based on the local motions of poly-

mer chain segments (1970–now). During this period, the

micro-environmental modes of polymer main chains

were discussed in details using the time-resolved fluo-

rescence depolarization technique [6], and various model

functions of the local motions of polymer chain seg-

ments have been proposed [7]. However, the theoretical

dynamical models of the polymer side chains are little

reported except for the W€oessner model [8]. In this

model, the internal anisotropic rotations in the vicinity

of a polymer side chain are separated into five dynam-

ical modes. The discussion about the relationship be-

tween the W€oessner model and the experimental data

based on the time-resolved fluorescence depolarization

technique would be very complicated. So few research-

ers tried to estimate the dynamical modes of polymer

chains based on the fluorescence emission anisotropy

decay curves of the chromophores attached to the poly-

mer main chains. In this report, we estimate the appro-

priateness of the W€oessner model in our experiments

using our original analytical method. Our original esti-

mation technique based on the W€oessner model is veryexcellent to clarify various dynamical modes of a flexible

polymer in the region of semi-concentrated polymer

concentration.

C. Hashimoto et al. / European Polymer Journal 40 (2004) 1997–2008 1999

2. Experimental

2.1. Materials

N -(2-Naphthylmethyl)carbazole (NMC) was synthe-sized by a reaction of potassium carbazolide with

2-chloromethylnaphthalene as shown in Fig. 1 [9]. Co-

poly(styrene-vinylcarbazole) (P(St–Cz)) was prepared

using an usual radical polymerization of styrene and

vinylcarbazole as shown in Fig. 1. The number-average

molecular weight and the molecular-weight distribu-

tion of P(St–Cz) were Mn¼ 2.5· 104 and Mw=Mn¼ 1.84,respectively. The content of carbazolyl group in P(St–

Cz) was about 7.0% unit/polymer.

Poly(ethylene oxide) with Mn ¼ 300 (PEO300) and

1,2-dichloroethane were purchased from Wako Chemi-

cal Co., and used as received.

2.2. Measurements

NMC and P(St–Cz) were respectively dissolved in a

mixture solvent of PEO300 and 1,2-dichloroethane at a

carbazolyl group concentration of 1.0 · 10�7 mol dm�3.

The solvent viscosity changed by the fraction vol-

ume of PEO300 from 50% to 80% was in the range

from 0.006 to 0.033 N sm�2. The viscosities of the

mixed solvent were measured using an Ubbelohde vis-

cometer at 30 �C. The samples were deaerated by

several freeze-pump-thaw cycles under a high vacuum

system, and sealed in a cylindrical cell of 1 cm diam-

eter.

The fluorescence depolarization spectroscopy mea-

surement of NMC or P(St–Cz) in a PEO300/1,2-di-

chloromethane mixture at 25 �C was carried out as a

part of perfective laboratory automation system for

macromolecular analysis (PLASMA) which is a new

measurement system that consists of various spectro-

Fig. 1. Chemical structures o

scopic apparatus supported by a personal computer net-

work with many original electric circuits and analyti-

cal programs [2,10]. The flowchart of the fluorescence

depolarization measurement of PLASMA is shown in

Fig. 2 [2,11]. The stationary fluorescence depolariza-

tion spectroscopy measurements were carried out by

the fluorescence spectrophotometer (Hitachi: 650-60).

The excited wavelength for the fluorescence depolariza-

tion spectroscopy was selected at 340 nm according to

the absorption, excitation and emission spectra and the

nano-second time-resolved and consequently the data

were analyzed by our programs. Then the fluorescence

decay curve and the polarized fluorescence decay curve

of NMC or P(St–Cz) excited at 340 nm were measured

by a nano-second time-resolved fluorescence spectro-

scopy using a time-correlated single-photon counting

method (Horiba: NAES-1100).

Observed fluorescence decay curve IobsðtÞ and emis-sion anisotropy ratio decay curve robsðtÞ were calculatedby

IobsðtÞ ¼ IVVðtÞ þ 2GIVHðtÞ ð6Þ

and

robsðtÞ ¼IVVðtÞ � GIVHðtÞIVVðtÞ þ 2GIVHðtÞ

; ð7Þ

respectively. G is the compensating factor for anisotropicsensitivity of the photomultiplier of NAES-1100 and

defined by

G ¼Z

IHVðtÞdtZ

IHHðtÞdt:�

ð8Þ

IVVðtÞ, IVHðtÞ, IHHðtÞ and IHVðtÞ indicate the decay curvesof the polarized fluorescence intensities measured

through a sharp cut filter (Toshiba: UV35). Subscripts

f NMC and P(St–Cz).

Fig. 2. Flow chart of fluorescence depolarization measurement of PLASMA. The stationary anisotropy spectra are measured by the

fluorescence spectrophotometer (Hitachi: 650-60) and analyzed together with the absorption, excitation and emission spectra. On the

other hand, nano-second time-resolved fluorescence depolarization spectroscopy measurements were carried out by a time-correlated

single-photon counting method (Horiba: NAES-1100). Nano-second time-resolved fluorescence depolarization spectroscopy mea-

surements were carried out and consequently the data were analyzed by our programs. Integral transformation method is a new

analytical approach for the fluorescence decay curve (16). In this report we applied a v2-map method to the emission anisotropy ratioanalysis.


‘V’ and ‘H’ represent the parallel and the perpendicular

directions to the vertical line. The first suffix indicates

the direction of an incident light and the second to an

emitted light.


2.3. Calculations

In general, the fluorescence decay curve IcalðtÞ and theemission anisotropy ratio decay curve rcalðtÞ are repre-sented by

IcalðtÞ /Z t

0

P ðT ÞSðt � T ÞdT ð9Þ

and

rcalðtÞ /R t0P ðT ÞDðt � T ÞSðt � T ÞdTR t

0P ðT ÞSðt � T ÞdT

; ð10Þ

respectively. P ðtÞ, SðtÞ and DðtÞ represent a response

function of the apparatus, a fluorescence decay curve

corresponding to an infinitely short excitation, and an

orientational auto-correlation function, respectively.

P ðtÞ was assumed as an exciting light pulse. SðtÞ isusually a sum of exponential fluorescence decay curves

and was used in this report as a double exponential

function

SðtÞ ¼ a1 expð�t=s1Þ þ a2 expð�t=s2Þ; ð11Þ

where the sis are the fluorescence decay constants. DðtÞwas assumed as an exponential function for NMC, while

two-type trial functions of DðtÞ were applied for P(St–Cz), that is, a double exponential function (Eq. (12)) or a

function based on the W€oessner model (Eq. (16)):

DðtÞ ¼ A1 expð�t=h1Þ þ A2 expð�t=h2Þ: ð12Þ

We made a deconvolution and a curve fitting programs

on the basis of the Wahl’s method [12] and the quasi-

Marquardt algorithm [13], respectively. The goodness of

the fitting calculation was evaluated by a value of the

residual sum of squares v2. The program computed

adequate values of variable parameters of a fitting

function until the variation of v2 become less than

1· 10�6. On determining the value of parameters ai andsi in SðtÞ for the curve fitting of IcalðtÞ, v2 was calculatedby

v2 ¼ 1

t2 � t1 � 4

Z t2

t1

wðtÞfIobsðtÞ � IcalðtÞg2 dt; ð13Þ

where wðtÞ, t1 and t2 indicate the weighting function attime t, the lower and upper cut-off times for the fittingcalculation, respectively. The weighting function adop-

ted for IcalðtÞ [12] is

wðtÞ ¼ 1

IVVðtÞ þ 4G2IVHðtÞ: ð14Þ

In contrast, on estimating the value of parameters in

DðtÞ for the curve fitting of rcalðtÞ, Eq. (13) is substitutedby

v2 ¼ 1

t2 � t1 � m

Z t2

t1

wðtÞfrobsðtÞ � rcalðtÞg2 dt; ð15Þ

where m is the number of variable parameters of DðtÞand equals to 6 in case of the W€oessner model. Theweighting function was replaced by [12]

wðtÞ ¼ 3IobsðtÞ1þ Gþ 3GrobsðtÞ � 3robsðtÞ2 � 2ð2G� 1ÞrobsðtÞ3

:

ð16Þ

2.4. v2-map method

In order to estimate the validity of various models,

we proposed a novel method called ‘v2-map method’

[2,14]. Select two parameters and calculate the value of

v2 as the value of the parameters is changed indepen-dently. We consider an orthogonal coordinate system,

O–XYZ, and the two parameters of the fitting functionare selected as X and Y coordinate axes and the calcu-

lated 1=v2 are plotted on the Z coordinate one. The

value of 1=v2 approaches 1 when a function fits well tothe measured data. We named such three-dimension

graph a ‘v2-map’. This method enables us to estimate thefitting properties in the vicinity of a stable point in a

trial function. Fig. 3 shows a v2-map together with var-ious quantitative estimation techniques of the v2-map. Aunique peak in a v2-map means a unique fitting result forvariable parameters in a trial function. The full width at

half maximum of the peak represents the sensibility of

the parameter. The more narrow the value of the full

width at half maximum become, the more sensible the

parameter will. If selected parameters are dependent

each other, the shape of v2-map is to be a distorted

shape like a horseshoe.

3. Results and discussion

3.1. Fluorescence and emission anisotropy ratio decay

curves of carbazolyl groups incorporated into polystyrene

Fluorescence decay curves of NMC and P(St–Cz)

were fitted well by Eq. (9) and the fitting parameters

were listed in Table 1. In case of NMC, a1=a2 is almost 1at any PEO300 concentration and SðtÞ of NMC seems

like a single exponential decay. On the other hand, it is

clear that a double exponential has a good agreement

with SðtÞ of P(St–Cz) as shown in Fig. 4. SðtÞ with theseparameter values was used in Eq. (10) for the fitting of

the emission anisotropy ratio decay curve, rðtÞ. As listedin Table 1, rðtÞ of NMC was fitted well by Eq. (10) with

DðtÞ as a single exponential function. But rðtÞ of P(St–Cz) was not fitted well using DðtÞ as a single exponentialfunction. This is because the decay curve of P(St–Cz)

Fig. 3. Analytical image of v2-map. Obtained v2-map contains the information of the fitting condition and the condition can benumerized by the following images via various procedures: (a) contour graph, (b) v2 value along Y -axis of maximum, (c) maximumvalue plot of X - and Y -axis, (d) maximum v2 value of Y -axis, (e) v2 value along X -axis of maximum, (f) area over half width, (g)maximum v2 value of X -axis, (h) area over half width along X - and Y -axis, (i) distribution of v2 value within 46–255, (j) 3D graphics of

v2-map.


reflects a great amount of relaxation modes of chro-

mophores attached to a backbone polymer chain. As a

general method for the time-resolved fluorescence

depolarization, such a non-exponential decay is fitted by

a double exponential function. But, there is a problem

that the physical meaning of a trial double exponential

function is not clear. As the fitting results are listed in

Table 1, the PEO300 concentration dependence on two

characteristic times was not obvious.

On the other hand, the W€oessner model has a

clear dynamical image of polymer side chain as shown

in the left-side of Fig. 6. The orientational auto-cor-

relation function, DðtÞ of the W€oessner model is givenby

Table 1

Values of various parameters of fluorescence and emission anisotropy ratio decay curves of NMC and P(St–Cz) fitted to single and

double exponential functions, and the Woessner model

NMC P(St–Cz)

/ (PEO) 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8

Emission decay curves (double exponential function)

a1=a2 1.11 0.922 0.891 0.923 0.555 0.681 1.26 1.52

s1 (ns) 2.74 2.64 1.92 1.50 3.14 2.11 1.45 3.66

s2 (ns) 7.33 7.32 7.16 7.23 11.7 11.1 10.2 9.82

Emission anisotropy decay curves (single and double exponential functions)

A1 0.182 0.260 0.154 0.268

A1=A2 3.66 4.96 2.56 4.17

h1 (ns) 0.487 0.368 2.46 1.12 0.775 0.450 0.660 0.520

h2 (ns) 15.3 12.3 9.38 89.9

v2 1.63 1.55 1.20 1.85 0.903 1.18 0.880 0.801

Emission anisotropy decay curves (W€oessner model)A0 0.0175 0.0298 0.0415 0.0416

a (ns) 85.5 89.8 90.6 90.8

h1 (ns) 6.78 4.77 6.78 12.3

h2 (ns) 12.4 14.5 36.2 43.0

h3 (ns) 0.433 0.444 0.530 0.627

h4 (ns) 2.22· 109 1.11· 107 1.00· 104 3.32· 108v2 1.01 1.79 1.04 1.43

Fluorescence decay curves of NMC and P(St–Cz): SðtÞ ¼ a1 expð�t=s1Þ þ a2 expð�t=s2Þ.Fluorescence anisotropy decay curves of NMC: DðtÞ ¼ A1 expð�t=h1Þ.Fluorescence anisotropy decay curves of P(St–Cz): DðtÞ ¼ A1 expð�t=h1Þ þ A2 expð�t=h2Þ.W€oessner model: DðtÞ¼A0ðA1þA2 expð�t=h1ÞþA3 expð�t=h1ÞÞexpð�t=h2Þð0:578expð�t=h3Þ0:422expð�t=h4ÞÞ A1¼ð1=4Þð3cos2a�1Þ2,A2¼ð3=4Þsin2ð2aÞ, A3¼ð3=4Þsin4 a.


DðtÞ ¼ A0 A1

þ A2 exp

�� t

h1

�þ A3exp

�� t

h1

��

� exp

�� t

h2

�0:578 exp

�� t

h3

�

þ 0:422 exp

�� t

h4

��; ð17Þ

where

A1 ¼1

4ð3 cos2 a � 1Þ2; A2 ¼

3

4sin2ð2aÞ;

A3 ¼3

4sin4 a: ð17aÞ

A0 is pre-factor and the other fitting parameters are

dynamical modes of carbazolyl groups attached to poly-

mer chain, that is, a is the angle between excited tran-sition dipole moment and rotation axis of chromophore

attached to a polymer chain, h1 is the rotational relax-ation time of a polymer side chain, h2 is the rotationalrelaxation time of a polymer main chain, h3 is the

relaxation time of three-bond crankshaft and tetrahedral

lattice motions of a polymer main chain, and h4 is therelaxation time of the fluctuation of a whole polymer.

The emission anisotropy ratio decay curves of P(St–Cz)

were fitted well by Eq. (10) with DðtÞ as Eq. (16) asshown in Fig. 4 and the fitting parameters are listed in

Table 1.

In the previous paper [15], it was shown that the

rotational mode of NMC was varied from the whole to

the carbazolyl group by the addition of PEO300 mole-

cules. The crossover point of the rotational motion of

NMC were estimated by the stationary and the nano-

second fluorescence depolarization methods, and they

coincided with the overlap polymer concentration C� of

PEO300. The overlap concentration C� was evaluated

by

C� ¼ Mw

NAhs2i3=2; ð18Þ

where Mw, NA and s are the molecular weight, Avoga-dro’s constant and the end-to-end distance, respectively,

and calculated at C� ¼ ab � 53%. PEO300 molecules

are isolated from each other in the solvent at C < C�.

However PEO300 molecules begin to be packed closely

near to the overlap threshold C�, and they entangle with

each other over C�. In this report, all the fluorescence and

the emission anisotropy ratio decay curves were mea-

sured over C�. Therefore, it can be considered that only

Fig. 4. Dependence of PEO300 concentration on the fluorescence and emission anisotropy decay curves of carbazolyl groups

incorporated into polystyrene. Data are fitted by Eq. (9) (left side) and Eq. (10) (right side) with SðtÞ and DðtÞ as a double exponentialand Eq. (16). The profile of an exciting light pulse, P ðtÞ is also shown.


the carbazolyl group rotates in both cases of NMC and

P(St–Cz), and that there is no need to consider the

polymer itself in this report. The W€oessner model is

appropriate to describe such dynamical mode in the

vicinity of carbazolyl group. But the discussion about the

adequacies of the W€oessner model to the experimental

Fig. 5. v2-maps between each parameters of the W€oessner model (Eq. (16)) applied to the emission anisotropy ratio decay curve of

P(St–Cz) in PEO300/1,2-dichloroethane¼ 7/3 mixtures.


data based on the time-resolved fluorescence depolar-

ization technique would be very complicated. Because

the internal anisotropic rotations in the vicinity of a

polymer side chain are separated into five dynamical


modes according to the W€oessner model,. Therefore, weestimate the appropriateness of the W€oessner modelusing our original analytical method as follows.

3.2. v2-Maps for relationship between the W€oessner modeland the emission anisotropy ratio decay curves of carbaz-

olyl groups incorporated into polystyrene

The v2-map method was applied to the curve fittingof the anisotropy ratio decay curves by the W€oessnermodel. As mentioned above, the v2-map method en-

ables us to estimate the fitting properties in the vicinity

of a stable point in a fitting function. All v2-maps foreach fitting parameter of Eq. (17) applied to rðtÞ of thesample 70% PEO300 are shown in Fig. 5. In the v2-map for a vs. h2, for example, the large values of Z-axisexist within a very narrow width along a. It indicatesthe fact that there is no stable solution of a, while h2is identically determined in the relationship between

a and h2. In other words, h2 is independent on a. Onthe other hand, the v2-map for h1 vs. h2 represents ahorseshoe form in Fig. 5. It can be presumed that there

is a strong correlation between h1 and h2. It may bedifficult to separate h1 and/or h2 in Eq. (17) to fit theexperimental data. All estimated results of such rela-

tionship between fitting parameters in Eq. (17) are

shown in Fig. 6. ‘Narrow’, ‘wide’, and ‘horseshoe form’

in this figure mean the relationships of ‘independent’,

Fig. 6. Dynamical modes of a carbazolyl group attached to polyme

effectiveness of parameters of these six dynamical modes when the emi

(16).

Table 2

Estimation for v2-maps between each fitting parameter of the W€oessnecurves of P(St–Cz) in PEO300/1,2-dichloroethane¼ 7/3 mixtures

a h1 h

a – Very narrow V

h1 Narrow – H

h2 Wide Horseshoe form –

h3 Narrow Narrow V

h4 Very wide Very wide V

‘meaningless’, and ‘dependent’ between the parameters,

respectively.

3.3. Estimation technique for emission anisotropy ratio of

chromophores incorporated into polymer chain

The relationships between each parameter of Eq. (17)

as showed in Table 2 are summarized as a correlation

map in the right side of Fig. 6. When an arrow directs

from X to Y , X is a meaningless parameter and Y is an

independently defined one. Solid line is the horseshoe

form type and it means the dependency between these

parameters. In this schematic diagram, we find an inter-

esting fact that all arrows gather around a, h1 and h3. Itindicates the validity of these three dynamical parame-

ters, a, h1 and h3, while the discussion about parametersremaining h2 and h4 are attended by many risks in therelationship between our experimental data and Eq. (17)

(the W€oessner model). Therefore, we cannot discuss

about the rotational relaxation time of a polymer main

chain, h2 and the relaxation time of fluctuation of a

whole polymer, h4 in this report. Note that the aboveestimation result as showed in Fig. 6 was also obtained

over all PEO300 concentrations.

The angle a between the excited transition dipole

moment and the rotation axis of a chromophore at-

tached to a polymer side chain is almost constant at

85.5–90.8� and not dependent on the PEO300 concen-

r chain in the W€oessner model and a correlation map of the

ssion anisotropy ratio decay curves of P(St–Cz) are fitted by Eq.

r model (Eq. (17)) applied to the emission anisotropy ratio decay

2 h3 h4

ery narrow Very narrow Very narrow

orseshoe form Narrow Narrow

Wide Wide

ery narrow – Very narrow

ery wide Very wide –

Fig. 7. Dynamical modes for a carbazolyl group attached to polymer side chain in PEO/1,2-dichloroethane mixtures.


tration as listed in Table 1. The rotational relaxation

time of a polymer side chain, h1 is about nano-secondorder and about 10 fold of that of NMC as a monomer

model compound. The relaxation time of three-bond

crankshaft, h3 is about 100 ps order and about 1/10 foldof rotational relaxation times of a polymer side chain.

These values for dynamical parameters of a polymer

chain are reasonable [6,7]. Characteristic times of P(St–

Cz), h1 and h3 increase with increasing PEO300 con-

centration as well as the rotational correlation time of

NMC. A slight increase in the rotational correlation

time of NMC with increasing PEO300 concentration

coincides with the previous results in the higher viscosity

region of C > C�. The dynamical modes in the vicinity

of polymer side chain become slower at more concen-

trated PEO solution. It indicates that the interpene-

tration process of PEO300 into P(St–Cz) sphere will

proceed with increasing of PEO300 concentration above

C� region.

In closing, we would like to sum some new knowl-

edge in this report as follows:

(1) The fluorescence emission anisotropy ratio decay

curves of P(St–Cz) in a mixture of PEO300 and 1,2-

dichloromethane are expressed very well by the

W€oessner model.(2) Our original ‘v2-map method’ estimates the validity

of the W€oessner model. Using this analytical tech-nique, we selected the effective parameters of the

W€oessner model, that is, the angle between the ex-cited transition dipole moment and the rotation axis

of chromophore attached to a polymer side chain

a, the rotational relaxation time of a polymer sidechain h1, and the relaxation time of three-bond

crankshaft h3. The schematic diagram is shown in

Fig. 7.

(3) The values of a, h1, and h3 are 85.5–90.8�, 5–12, 0.4–0.6 ns, respectively, and they are considered to

be reasonable in the higher viscosity region of

C > C�.

(4) Our original estimation technique based on the

W€oessner model is very excellent to clarify variousdynamical modes of a flexible polymer in the region

of semi-concentrated polymer concentration.

References

[1] Horie K, Ushiki H, Winnik F. Molecular photonics.

Kodansha Ltd./Wiley-VCH Verlag GmbH; 2000;

Yamamoto M. In: NATO ASI Ser. E, vol. 327. 1996. p.

479.

[2] Ushiki H. Fluorescence probes in polymers and liquid

crystals––complex macromolecular chain dynamics [pro-

posal from far east] In: Rettig W, et al., editors. Applied

fluorescence in chemistry, biology, and medicine. Springer;

1999. p. 325–70.

[3] Perrin F. Ann Phys [Paris] 1929;12:169;

Weber G. Adv Protein Chem 1953;8:415.

[4] Tao T. Biopolymers 1969;8:60;

Wahl Ph. Biophys Cell Biol 1975;2:233.

[5] Kinoshita K, Kawamoto S, Ikegami A. Biophysics 1977;

20:289;

Barkley MD, Zimm BH. J Chem Phys 1979;70:2991;

Lipari G, Szabo A. Biophys J 1980;30:489.

[6] Valeur B, Monnerie L. J Polym Sci, Polym Phys 1976;

14:11;

Sasaki T, Yamamoto M, Nishijima Y. Macromolecules

1988;21:610;

Ediger MD. Ann Rev Phys Chem 1991;42:225.


[7] Tsunomori F, Ushiki H. Polym J 1996;28:576, 582, 588.

[8] W€oessner DE. J Chem Phys 1962;36:1.

[9] Ushiki H, Horie K, Mita I. Chem Phys Lett 1983;98:

285.

[10] Ushiki H. Rep Prog Polym Phys Jpn 1992;35:419.

[11] Ushiki H, Hashimoto C, Nozawa K, Ishii A, Rouch J. Rep

Prog Polym Phys Jpn, Rev 2000;43:633.

[12] Wahl P. Biophys Chem 1979;10:91.

[13] Nakagawa T, Oyanagi Y. Saisho-Nijou-hou Jikken-Data

Kaiseki. Tokyo: Tokyo Daigaku Syuppan-kai; 1982.

[14] Ushiki H, Tsunomori F. Rep Prog Polym Phys Jpn 1993;

36:373;

Ushiki H, Katsumoto Y, Tsunomori F, Rouch J. Rep Prog

Polym Phys Jpn 1996;39:131;

Hashimoto C, Ushiki H. Polym J 2000;32(10):807.

[15] Tsunomori F, Ushiki H. Bull Chem Soc Jpn 1996;69:1849.

complex macromolecular dynamics

Documents