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Local motions of polystyrene chain in semi-concentrated polymer solutions
European Polymer Journal 37 (2001) 475±483
Local motions of polystyrene chain in semi-concentrated polymer solutions Yukiteru Katsumoto a,*, Fumiaki Tsunomori a, Hideharu Ushiki a, Louis Letamendia a,b, Jaques Rouch a,b a
Graduate School of Bio-Applications and Systems Engineering BASE, Tokyo University of Agriculture and Technology Tokyo Univ. A and T, 3-5-8, Saiwai-cho, Fuchu-Shi, Tokyo 183-8509, Japan b Centre de Physique Mol eculaire Optique et Hertzinne CPMOH, Universit e Bordeaux I, 351, Cours de la Lib eration, 33405 Talence, France Received 12 January 2000; accepted 16 March 2000
Abstract In the 1980s, the mechanism of local motions of polymer chain was discussed theoretically and experimentally by many researchers. Finally, the intermittent motion model was proposed with respect to the measurements of emission anisotropy decay curves of a chromophore incorporated into a polymer main chain in dilute polymer solutions. The relationship between orientational auto-correlation function type based on power law and measured decay curves was discussed in these reports. The local motion of anthryl groups end-capped polystyrene (APSA) was measured in various polystyrene/di-n-butyl phthalate mixtures by using the ¯uorescence depolarization method. The intermittent motion model for local motions of polymer segments is modi®ed in order to apply to both dilute and semi-concentrated polymer solutions. The model predicts that the orientational auto-correlation function will be given by the KWW function when the segment±solvent collision is eective on the relaxation process. On the other hand, the hyperbolic function will be observed when the segment±segment collision becomes eective. It was found that measured decay curves in dilute and concentrated polymer solutions were well represented by the KWW function and the hyperbolic function, respectively. The result implies the fact that modi®ed intermittent motion model is in good agreement with the experimental data. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Fluorescence depolarization method; Local motion of polymer segment; Orientational relaxation; Intermittent motion model; Power law type auto-correlation function; Concentrated polymer solution
1. Introduction The microscopic dynamics of the polymer chain essentially governs the macroscopic properties of polymers. Therefore, comprehension of the local polymer chain dynamics is an important fundamental subject in the polymer science. During the past decades, theoretical
* Corresponding author. Tel.: +81-42-367-5616; fax: +81-42367-5565. E-mail addresses: [email protected], [email protected] (Y. Katsumoto).
and experimental studies of orientational relaxation of a polymer chain in dilute solution have been reported by many authors [1±7]. Also, many experimental methods have been performed for the investigation of such local chain motions of polymer, e.g., NMR [8], ESR [9], dielectric relaxation [10], and ¯uorescence depolarization [6,7,11]. In our previous papers [12±14], the intermittent motion model was proposed and discussed. The local motion of segments of polystyrene having an anthryl group in the center of its chain (PSAPS) and that in the end of its chain (PSA) are investigated by the ¯uorescence depolarization method [12,14]. It was found that the
0014-3057/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 0 0 ) 0 0 1 3 6 - 1
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orientational relaxation of PSAPS and PSA was successfully reproducible with the hyperbolic function r t r0 t=seff ÿb . This means that the origin of hyperbolic anisotropy decay curves is an intermittent relaxation of polymer segments [13]. Although the intermittent motion model could successfully explain the experimental data for local motions of PSAPS and PSA, it has included some details that should be revised and modi®ed. In the modi®ed intermittent motion model, it was assumed that the orientational relaxation of polymer segments is ascribed to the reaction with solvent molecules or the other segments. That is, a segment loses its orientation completely and instantaneously by this reaction. Fig. 1 shows the schematic representation of the intermittent motion model (b) compared with the conventional relaxation process (a). Therefore, Fig. 5 in Ref. [13] should be revised. The number of polymer segments preserving the initial direction, nA , decreases with time caused by the segment±solvent or segment± segment collision. Note that at t 0, nA is equal to the total number of the polymer segment, n0 . When the segment±solvent collision mainly happened in a relaxation process, the following equation should be solved [15,16]: dnA ÿ4pR0 DA DB nA nB ; dt
1
where nB , DA , DB , and R0 is the number of the solvent molecules, the diusion coecients of polymer segments and solvent molecules, and the reaction radii, respectively. For the sake of simpli®cation, Eq. (1) is solved under the condition of nB nA . Thus, nB can be con-
sidered as a constant, nB nB0 . nB0 is the average number of the solvent molecule colliding with a segment per unit time. In this case, Eq. (1) is represented approximately as follows: dnA ÿknA : dt
2
The reaction rate may be governed by the diusion of solvent molecules because in general, the diusion coecient of a segment DA is much smaller than that of a solvent molecule DB . Then, k is represented by k 4pR0 nB0 DB . The orientation of a polymer segment relaxes after the segment is activated by the reaction with a solvent molecule, B , which has energy above the threshold for the activation energy. That is, the solvent molecules, which have lower energy compared with the threshold, cannot contribute to the reaction. In nB0 and DB , therefore, the only contribution of B should be taken into account. This assumption leads a time dependent value for DB : DB tcÿ1 (Eqs. (21)±(30) in Ref. [13]). Then, a time dependent value of k is derived as k k0 tcÿ1 ;
3
where k0 is a constant. Substituting Eq. (3) into Eq. (2), the following equation is given: dnA ÿk0 tcÿ1 nA : dt
4
Solving Eq. (4), the stretched exponential (KWW) function is given as the orientational relaxation of polymer segments k0 c nA t n0 exp ÿ t : 5 c Thus, the KWW function is derived when the segment±solvent collision is only taken into account. In the earlier paper, the hyperbolic function is derived from the relaxation process by the segment±solvent collision (Eq. (34) in Ref. [13]), but it should be revised. Considering only the segment±solvent collision, the hyperbolic function cannot be derived from the intermittent motion model. When the segment±segment collision for the orientational relaxation process is considered, our basic assumptions are the following:
Fig. 1. Schematic representation of (a) gradual process and (b) intermittent process of orientational relaxation of a polymer segment. In the case of (a) the orientation of a segment gradually relaxes with time. On the other hand, in the case of (b) the orientation of a segment relaxes instantaneously at a time.
(i) At the earlier stage of the orientational relaxation process, the orientation of a few segments relaxes. The number of these segments, nA , is far less than the total number of segments in a polymer. (ii) The collision takes place only when two segments are very close to each other in a polymer solution. In other words, the number of two such segments, nA , is taken into account. (iii) Before the reaction, these two segments are uniformly distributed without correlation between them.
Y. Katsumoto et al. / European Polymer Journal 37 (2001) 475±483
Under these conditions, de GennesÕs theory [16] can be applied to the modi®ed intermittent motion model, which dealt with the diusion-control reaction between two reactants attached to a polymer in concentrated solutions. Then, it is desirable to solve the following Smoluchowski equation: dnA ÿk 0 n2A : dt
6
When the memory function for Brownian motion of both ``reactants'' decreases more slowly than tÿ1 , de Gennes found a time dependent rate value for k 0 (Section III.D.1 in Ref. [16]). According to de Gennes, k 0 tbÿ1 is expected, and then, the following equation is given: dnA ÿk00 tbÿ1 n2A ; dt
7
where k00 is a constant. Solving Eq. (7) with nA nA0 at t 0, one can obtain nA as a function of time t. nA0 nA t : 8 1 k00 nA0 tb =b When b=nA0 k00 tb , the above equation can be simpli®ed as nA t
b ÿb t : k00
9
For the segment±segment collision, therefore, the hyperbolic function is derived as the orientational relaxation function. According to de Gennes, b is derived from the diusion length of A and the dimension of space. Hence, it is obvious that the exponent b in Eq. (9) is dierent from the exponent c in Eq. (5). The orientational auto-correlation function of polymer segments measured by ¯uorescence depolarization method, r t, is proportional to nA t or nA t. As a result, the following two important aspects of the orientational relaxation of polymer segments could be led from the modi®ed intermittent motion model: (i) When the orientational relaxation of a polymer segment is caused by the segment±solvent collision, the KWW function will be obtained as the orientational auto-correlation function. (ii) When the segment±segment collision causes a polymer segment to relax its orientation, the hyperbolic function will be observed. The segment±solvent collision may be eective in a dilute polymer solution with good solvents. On the other hand, the segment±segment collision becomes eective in a dilute polymer solution with poor solvents or in a concentrated polymer solution with good solvent. In the earlier papers [12±14], local motions of PSA and PSAPS were investigated in a poor solvent by using the time-resolved ¯uorescence depolarization method. It
477
was found out that the hyperbolic function was the most adequate orientational auto-correlation function. The result indicates that the segment±segment collision is the eective process on the orientational relaxation of polymer segments in a poor solvent. In this report, the local motion of anthryl groups end-capped polystyrene is investigated in a dilute polymer solution and in a concentrated one with good solvents. The crossover between the KWW and the hyperbolic function as the orientational auto-correlation function is discussed. 2. Experimental 2.1. Materials and sample preparations The chemical structure of a,x-dianthrylpolystyrene (APSA) is shown in Fig. 2. APSA was prepared in a way similar to that reported previously [17]. The number-average molecular weight and molecular weight distribution of APSA, which were measured with a gelpermeation chromatography (Toyo Soda, HLC-802UR) at 40°C in THF, were Mn 3:1 106 and Mw =Mn 1:24, respectively. Measuring absorption spectra of APSA using a spectrophotometer (Shimadzu, MPS5000), the content of anthryl group at the chain ends was determined to be 91%. APSA was dissolved in a solution of polystyrene (Aldrich, Mw 2:8 106 )/di-n-butyl phthalate (Wako, special-grade) mixture. Polystyrene (PS) was puri®ed by reprecipitation, and di-n-butyl phthalate (DBP) was used without further puri®cation. The weight fraction of polystyrene was varied from 0.00 to 0.20. 2.2. Fluorescence measurement techniques The steady-state ¯uorescence and depolarization spectra of APSA were measured by using a ¯uorescence spectrophotometer (HITACHI 650-60). The time-resolved ¯uorescence intensity decay curves and the time-resolved ¯uorescence anisotropy decay curves of APSA were measured by single photon counting techniques with a time-resolved spectro¯uorometer
Fig. 2. Chemical (APSA).
structure
of
a,x-dianthrylpolystyrene
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(HORIBA, NAES-1100). Signals of the photomultiplier were stored into the time-resolved spectro¯uorometer apparatus. The stored data were transformed to a personal computer (NEC, PC-9801VX) via RS232C. The excitation wavelength was set 385 nm. Fluorescent intensities were monitored through a color glass ®lter (Toshiba, L-42). The sample solutions were degassed by several freeze±pump±thaw cycles, and were sealed in a columnar Pylex cell. All measurements were performed at 25°C. The molar concentration of APSA in each sample was about 1:4 10ÿ6 M OD 0:01. Fluorescence intensity decay curve Fobs (t) and ¯uorescence anisotropy decay curve robs (t) are calculated by Fobs t IVV t 2GIVH t;
10
IVV t ÿ GIVH t ; IVV t 2GIVH t
11
robs t
R IHV tdt R G ; IHH tdt
12
where IVV (t), IVH (t), IHH (t), and IHV (t) are the measured ¯uorescence intensity decay related to the intensities of emitted light polarized parallel and perpendicular to the polarized exciting light and G is a correction factor, respectively. Subscripts V and H denote vertically and horizontally polarized light, where the ®rst letter in the subscripts refers to the exciting light and the second one refers to the emitted light. Note that G factor is necessary for correcting the depolarization characteristics of apparatus. The same procedure written in Ref. [12] is adapted in order to calculate the theoretical curves. Fluorescence intensity decay curves, S(t), are analyzed by a single exponential function as follows: t S t af exp ÿ ; 13 sf
Fig. 3. (a) Variation of the steady-state ¯uorescence anisotropy r with /PS , (b) variation of the averaged ¯uorescence lifetime sf with /PS , and (c) variation of the apparent ¯uorescence rotational correlation time sh with /PS by using Eq. (14).
where the sf and af are the ¯uorescence life time and a constant, respectively. The curve-®tting program with a deconvolution was made by the use of P A S C A L language (Borland: Turbo P A S C A L ) based on WahlÕs method [18] and quasi-Marquardt algorithm [19] with a non-linear least square method. These calculations were carried out on a personal computer (NEC, PC-9821Xn).
3. Results and discussion 3.1. Steady-state ¯uorescence anisotropy measurements Fig. 3(a) shows the variation of the steady-state ¯uorescence anisotropy, r, as a function of the weight fraction of PS, /PS . Measured r values increases gradually from 0.052 to 0.15 with increasing /PS . Figs. 4 and 5
Fig. 4. Fluorescence intensity decay curve of APSA at /PS 0:00. The open circles are the measured data. The solid line is the best ®t theoretical ¯uorescence intensity curve (Eq. (13)) convoluted by the measured instrumental function (exciting light pulse) represented by the broken line.
show the obtained measured ¯uorescence intensity decay curves, Fobs (t), at /PS 0:00 and /PS 0:20, respectively. The Fobs (t) obtained, was analyzed using Eq. (13).
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479
Fig. 5. Fluorescence intensity decay curve of APSA at /PS 0:20 (same representation as in Fig. 4).
It is clear that Fobs (t) for /PS 0:20 decayed more rapidly than that for /PS 0.00. sf decreases with increasing /PS as shown in Fig. 3(b). In general, this decrement of sf with /PS is ascribed to a quenching action. Plotting sf /s =sf 0 versus the molar concentration of the polystyrene segment, /s , the qualitative agreement with Stern±Volmer relation was con®rmed. Therefore, a dynamic quenching may occur between anthryl groups and polymer segments. It is therefore indicated that the environment around anthryl groups changes remarkably as /PS increases. Fig. 3(c) shows the variation of the apparent ¯uorescence rotational correlation time sh with /PS . sh calculated from sf and r with using the following equation: 1 1 sf 1 : 14 sh r r0 r0 was estimated as 0.28 by time-resolved ¯uorescence anisotropy decays. In order to derive Eq. (14), a single exponential function should be assumed as r(t): r t r0 exp ÿt=sh . However, obtained ¯uorescence anisotropy decay curves, robs (t), were not expressed by a single exponential function as shown in the next section. Using the steady-state ¯uorescence anisotropy data, therefore, the description of the eect of /PS on sh is restricted to qualitative one. sh becomes large gradually with increasing /PS . The result implies that the mobility of the chain end in a concentrated solution is smaller than that in the dilute solution. It is believed that this result simply re¯ects the dierence of the polymer concentration.
Fig. 6. Fluorescence anisotropy decay curves of APSA at /PS 0:00. The open circles are the measured data. The solid line is the best ®t KWW function (Eq. (15)) convoluted by the measured instrumental function (exciting light pulse) represented by the broken line.
weight fraction of PS, /PS , is varied from 0.00 to 0.20. At low /PS , the experimental curve decayed gradually at the initial stage of relaxation, but it seems to have a long time tail. The initial slope of the obtained decay curve became sharp with increasing /PS . If a probe molecule can rotate isotropically, ¯uorescence anisotropy decay is represented by a single exponential function as r t exp ÿt=sh . However, the observed ¯uorescence anisotropy decays suggest that a probe molecule, which is incorporated at the end of a polymer chain, moves in the wake of orientational relaxation motion of a polymer segment. Fitting the KWW (Eq. (15)) and the hyperbolic functions (Eq. (16)) to robs (t), the dierence between the relaxation processes can be discussed. ( b ) t r t reff exp ÿ ; 15 seff
t
ÿb
3.2. Time-resolved ¯uorescence anisotropy measurement
r t reff
Figs. 6±10 show the ¯uorescence anisotropy decay curves of APSA in various DBP/PS mixtures. The
where reff , seff , and b are respectively, a constant, the eective orientational relaxation time, and an exponent.
seff
;
16
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Y. Katsumoto et al. / European Polymer Journal 37 (2001) 475±483
Fig. 7. Fluorescence anisotropy decay curves of APSA at /PS 0:025 (same representation as in Fig. 6).
Fig. 9. Fluorescence anisotropy decay curves of APSA at /PS 0:10. The open circles are the measured data. The solid line is the best ®t hyperbolic function (Eq. (16)) convoluted by the measured instrumental function (exciting light pulse) represented by the broken line.
Obtained values of ®tting parameters, reff , seff , and b for the KWW and the hyperbolic function are listed in Tables 1 and 2, respectively. In general, a ®tting of a trial function to measured data is evaluated quantitatively by a residual sum of square, v2 . If a trial function is in agreement with a measured anisotropy decay curve, v2 will approach 1 in our curve ®tting procedure (Eqs. (11) and (12) in Ref. [12]). The dierence between obtained v2 values for the KWW function and that for the hyperbolic one is small as shown in Tables 1 and 2. However, the result implies that robs (t) measured at low /PS is well represented by the KWW function. Conversely, the hyperbolic function is in good agreement with robs (t) at high /PS . 3.3. Crossover from the KWW to the hyperbolic function
Fig. 8. Fluorescence anisotropy decay curves of APSA at /PS 0:05 (same representation as in Fig. 6).
Now, the crossover behavior between the KWW function and the hyperbolic one is discussed. The solid lines in Figs. 6±8 represent the ®tting result for the KWW function (Eq. (15)). In Figs. 9 and 10, the ®tting result for the hyperbolic function (Eq. (16)) were represented. v2 values indicate that the KWW function is more adequate as the orientational auto-correlation
Y. Katsumoto et al. / European Polymer Journal 37 (2001) 475±483
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observed with increasing /PS . That is, the segment± segment collision becomes eective at higher /PS . Considering the change of the segment concentration around a probe molecule, this result may be explained. The gyration radius of a polymer, RG , in a good solvent is estimated by using the following relation: R2G 13a2 Nb2 ;
17
where a is the expansion factor, N is the number of segment, and b is the length of a segment. If one segment is constituted of two monomers, then b 3:1 10ÿ10 m. The value of the expansion factor a is 0.93 [20]. Thus, RG 9:3 10ÿ9 m was calculated. Then, the molar concentration of polymer at the hexagonal closest packing for the polymer sphere, uPS , is equal to 3:65 10ÿ4 M. The averaged segment concentration in a polymer sphere, ucoil , is estimated by the following relation: ucoil
Fig. 10. Fluorescence anisotropy decay curves of APSA at /PS 0:20 (same representation as in Fig. 9). Table 1 Best ®t parameters and v2 value obtained when the KWW function is ®tted to the experimental curves /PS
reff
seff
b
v2
0.000 0.025 0.050 0.100 0.200
0.338 0.345 0.350 0.398 0.394
5.17 3.80 3.86 4.75 10.68
0.461 0.476 0.487 0.431 0.427
1.066 0.866 1.002 0.892 1.171
Table 2 Best ®t parameters and v2 value obtained when the hyperbolic function is ®tted to the experimental curves /PS
reff
seff
b
v2
0.000 0.025 0.050 0.100 0.200
0.119 0.098 0.087 0.142 0.190
4.84 5.14 6.67 4.19 3.96
0.399 0.471 0.496 0.404 0.283
1.321 1.159 1.051 0.992 1.112
function than the hyperbolic function until /PS 0:05. Then, the hyperbolic function is the adequate one at /PS 0:10 and 0.20. It is therefore clear that the crossover from the KWW to the hyperbolic function was
3N : 4pR3G
18
When the above value is adopted, ucoil can be estimated as 0.689 M. In general, it is believed that the overlap concentration, u , is nearly equal to ucoil . The calculated values for the molar concentration of PS, uPS , and the total polymer segment concentration, us , in samples are listed in Table 3. A polymer solution is considered as a dilute one when us < u , and as a semi-dilute one when us > u . Therefore, the sample solution changes from a dilute solution to a semi-dilute one nearby /PS 0:10 in this system. Estimated uPS and u may suggest that the segment concentration around a probe molecule changes dramatically at /PS 0:10 because chromophore of APSA was attached at the end of polystyrene. The ®tting result suggests the fact that the crossover between the KWW and the hyperbolic functions was observed at /PS 0:10. In previous papers [12±14], the orientational relaxations of PSAPS and PSA were successfully reproducible with the hyperbolic function. While PSAPS has a chromophore at the center of the polymer chain, PSA has a chromophore at the end of the polymer chain. Therefore, it is believed that us around a probe molecule for PSAPS may dier from that for PSA. In spite of this dierence, the orientational auto-correlation function for both PSAPS and PSA was hyperbolic function. In Table 3 The molar concentration of PS, uPS , and APSA, uPSAPS , and the segment concentration, us , in each sample /PS
uPS 10ÿ4 M
us M
uAPSA 10ÿ6 M
0.000 0.025 0.050 0.100 0.200
0.00 0.96 1.98 4.16 8.32
0.000 0.134 0.277 0.582 1.16
1.06 1.12 1.89 1.28 1.40
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order to explain these experimental results, it may be worth considering that these polymers were dissolved in poor solvents. That is, the results suggest the fact that the segment±segment collision is eective in a dilute polymer solution with poor solvents. 3.4. Application of v2 -maps method for evaluating an adequate trial function The ``v2 -map method'' was proposed as a novel method for quantitative estimations and discussions of the relationship between an experimental decay curve and a trial function [12±14,21±23]. This method has been applied when it is dicult to discuss the suitability of trial functions to measured data only by the use of v2 . In this section, the availability of v2 -maps method is discussed. The v2 -maps of the KWW and the hyperbolic functions for experimental decay curves of APSA are shown in Fig. 11. seff and b of the KWW and the hyperbolic functions were selected as the X and Y coordinate axes
Fig. 12. Variation of DShm with /PS : when DShm is positive, the KWW function is more adequate orientational auto-correlation function than the hyperbolic one.
for v2 -map. Z axis represents the value of 1=v2 . In these ®gures, seff and b are varied from 0:1 seff to 10 seff and from 0:1 b to 1.0, respectively. The asterisk indicates the obtained values for seff and b at the maximum of 1=v2 . It is believed that the full area at half maximum, Shm , represents the sensitivity of these parameters [12]. As shown in Fig. 11, v2 -maps for the hyperbolic function have broad peaks until /PS reaches 0.10, and that for the KWW function has broad peaks clearly at /PS 0:20. Calculating the dierence between Shm for the KWW and the hyperbolic functions by DShm Hyperbolic KWW Shm ÿ Shm , the pro®le of 1=v2 can be discussed quantitatively. In the v2 -map method, it is assumed that a parameter which indicates the narrow pro®le of 1=v2 is important to explain data. Fig. 12 shows the variation of DShm with /PS . KWW function is more adequate than the hyperbolic one when DShm is positive. DShm has a positive value at the region of /PS < 0:10, and it becomes negative at /PS 0:10 and 0.20. It suggests that ¯uorescence anisotropy decay curves of APSA in dilute and concentrated polystyrene solutions were shown the best ®t based on the KWW and the hyperbolic functions, respectively. The result is in agreement with the conclusion of previous sections. It is therefore presumed that v2 -map method is useful to evaluate an adequate function. 4. Conclusions Fig. 11. v2 -maps of the hyperbolic and the KWW functions for obtained ¯uorescence anisotropy ratio decay curves of APSA in various PS/DBP mixtures.
The ``intermittent motion model'' for local motions of polymer segments was modi®ed. Two important as-
Y. Katsumoto et al. / European Polymer Journal 37 (2001) 475±483
pects of the modi®ed intermittent motion model are as follows: (i) When the orientational relaxation of a polymer segment is caused by the segment±solvent collision, the KWW function will be observed as the orientational auto-correlation function. (ii) When the orientational relaxation of a polymer segment is ascribed to the segment±segment collision, the hyperbolic function will be obtained. Hence, the local motion of anthryl groups endcapped polystyrene (APSA) was investigated in various polystyrene/di-n-butyl phthalate mixtures by the use of the ¯uorescence depolarization method. The steadystate ¯uorescence anisotropy measurements suggest that the mobility of the chain end in a concentrated solution is smaller than that in the dilute solution. Fluorescence anisotropy decay curves of APSA were measured, and the modi®ed intermittent motion model was investigated in both dilute and semi-concentrated polymer solutions. As a result of analysis with a curve ®tting procedure, the experimental decay curves at low weight fraction of polystyrene were well represented by the KWW function. Conversely, the hyperbolic function was in good agreement with the measured data at high weight fraction of polystyrene. This crossover from the KWW function to the hyperbolic one was observed near the overlap concentration. It is therefore concluded that the modi®ed intermittent motion model can successfully explain the nature of local motions of polymer segments in both dilute and semi-concentrated solutions.
483
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Ediger MD. Ann Rev Phys Chem 1991;42:225. Jones AA, Stockmayer WH. J Polym Sci 1977;15:847. Bendler JT, Yaris R. Macromolecules 1978;11:650. Hall CK, Helfand E. J Polym Sci 1977;15:841. Viovy JL, Monnerie L, Brochon JC. Macromolecules 1983;16:1845. Valeur B, Monnerie L. J Polym Sci Polym Phys Ed 1976;14:11. Sasaki T, Yamamoto M, Nishijima Y. Macromolecules 1988;21:610. Gronski W, Sch afer T, Pater R. Polym Bull 1979;1:319. Friedrich C, Laupr^etre F, Noel C, Monnerie L. Macromolecules 1981;14:1119. Mashimo S. Macromolecules 1976;9:91. Waldow DA, Ediger MD, Yamaguti Y, Matsusita Y, Noda I. Macromolecules 1991;24:3147. Tsunomori F, Ushiki H. Polym J 1996;28:576. Tsunomori F, Ushiki H, Horie K. Polym J 1996;28:582. Tsunomori F, Ushiki H. Polym J 1996;28:588. von Smoluchowski M. Z Phys Chem 1917;92:192. de Gennes PG. J Chem Phys 1982;76:3316. Ushiki H, Horie K, Okamoto A, Mita I. Polym J 1979;11:691. Wahl P. Biophys Chem 1979;10:91. Nakagawa T, Oyanagi Y. Saishou nijyou-hou niyoru jikken-data kaiseki. Tokyo: Tokyo Daigaku Syuppan-kai, 1982. Ono K, Okada Y, Yokotsuka S, Ito S, Yamamoto M. Polym J 1994;26:199. Ushiki H, Mori M. Rep Prog Polym Phys Jpn 1994;37:471. Ushiki H, Tsunomori F. Rep Prog Polym Phys Jpn 1996;39:127. Ushiki H, Katsumoto Y, Tsunomori F, Rouch J. Rep Prog Polym Phys Jpn 1996;39:131.
Local motions of polystyrene chain in semi-concentrated polymer solutions Yukiteru Katsumoto a,*, Fumiaki Tsunomori a, Hideharu Ushiki a, Louis Letamendia a,b, Jaques Rouch a,b a
Graduate School of Bio-Applications and Systems Engineering BASE, Tokyo University of Agriculture and Technology Tokyo Univ. A and T, 3-5-8, Saiwai-cho, Fuchu-Shi, Tokyo 183-8509, Japan b Centre de Physique Mol eculaire Optique et Hertzinne CPMOH, Universit e Bordeaux I, 351, Cours de la Lib eration, 33405 Talence, France Received 12 January 2000; accepted 16 March 2000
Abstract In the 1980s, the mechanism of local motions of polymer chain was discussed theoretically and experimentally by many researchers. Finally, the intermittent motion model was proposed with respect to the measurements of emission anisotropy decay curves of a chromophore incorporated into a polymer main chain in dilute polymer solutions. The relationship between orientational auto-correlation function type based on power law and measured decay curves was discussed in these reports. The local motion of anthryl groups end-capped polystyrene (APSA) was measured in various polystyrene/di-n-butyl phthalate mixtures by using the ¯uorescence depolarization method. The intermittent motion model for local motions of polymer segments is modi®ed in order to apply to both dilute and semi-concentrated polymer solutions. The model predicts that the orientational auto-correlation function will be given by the KWW function when the segment±solvent collision is eective on the relaxation process. On the other hand, the hyperbolic function will be observed when the segment±segment collision becomes eective. It was found that measured decay curves in dilute and concentrated polymer solutions were well represented by the KWW function and the hyperbolic function, respectively. The result implies the fact that modi®ed intermittent motion model is in good agreement with the experimental data. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Fluorescence depolarization method; Local motion of polymer segment; Orientational relaxation; Intermittent motion model; Power law type auto-correlation function; Concentrated polymer solution
1. Introduction The microscopic dynamics of the polymer chain essentially governs the macroscopic properties of polymers. Therefore, comprehension of the local polymer chain dynamics is an important fundamental subject in the polymer science. During the past decades, theoretical
* Corresponding author. Tel.: +81-42-367-5616; fax: +81-42367-5565. E-mail addresses: [email protected], [email protected] (Y. Katsumoto).
and experimental studies of orientational relaxation of a polymer chain in dilute solution have been reported by many authors [1±7]. Also, many experimental methods have been performed for the investigation of such local chain motions of polymer, e.g., NMR [8], ESR [9], dielectric relaxation [10], and ¯uorescence depolarization [6,7,11]. In our previous papers [12±14], the intermittent motion model was proposed and discussed. The local motion of segments of polystyrene having an anthryl group in the center of its chain (PSAPS) and that in the end of its chain (PSA) are investigated by the ¯uorescence depolarization method [12,14]. It was found that the
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orientational relaxation of PSAPS and PSA was successfully reproducible with the hyperbolic function r t r0 t=seff ÿb . This means that the origin of hyperbolic anisotropy decay curves is an intermittent relaxation of polymer segments [13]. Although the intermittent motion model could successfully explain the experimental data for local motions of PSAPS and PSA, it has included some details that should be revised and modi®ed. In the modi®ed intermittent motion model, it was assumed that the orientational relaxation of polymer segments is ascribed to the reaction with solvent molecules or the other segments. That is, a segment loses its orientation completely and instantaneously by this reaction. Fig. 1 shows the schematic representation of the intermittent motion model (b) compared with the conventional relaxation process (a). Therefore, Fig. 5 in Ref. [13] should be revised. The number of polymer segments preserving the initial direction, nA , decreases with time caused by the segment±solvent or segment± segment collision. Note that at t 0, nA is equal to the total number of the polymer segment, n0 . When the segment±solvent collision mainly happened in a relaxation process, the following equation should be solved [15,16]: dnA ÿ4pR0 DA DB nA nB ; dt
1
where nB , DA , DB , and R0 is the number of the solvent molecules, the diusion coecients of polymer segments and solvent molecules, and the reaction radii, respectively. For the sake of simpli®cation, Eq. (1) is solved under the condition of nB nA . Thus, nB can be con-
sidered as a constant, nB nB0 . nB0 is the average number of the solvent molecule colliding with a segment per unit time. In this case, Eq. (1) is represented approximately as follows: dnA ÿknA : dt
2
The reaction rate may be governed by the diusion of solvent molecules because in general, the diusion coecient of a segment DA is much smaller than that of a solvent molecule DB . Then, k is represented by k 4pR0 nB0 DB . The orientation of a polymer segment relaxes after the segment is activated by the reaction with a solvent molecule, B , which has energy above the threshold for the activation energy. That is, the solvent molecules, which have lower energy compared with the threshold, cannot contribute to the reaction. In nB0 and DB , therefore, the only contribution of B should be taken into account. This assumption leads a time dependent value for DB : DB tcÿ1 (Eqs. (21)±(30) in Ref. [13]). Then, a time dependent value of k is derived as k k0 tcÿ1 ;
3
where k0 is a constant. Substituting Eq. (3) into Eq. (2), the following equation is given: dnA ÿk0 tcÿ1 nA : dt
4
Solving Eq. (4), the stretched exponential (KWW) function is given as the orientational relaxation of polymer segments k0 c nA t n0 exp ÿ t : 5 c Thus, the KWW function is derived when the segment±solvent collision is only taken into account. In the earlier paper, the hyperbolic function is derived from the relaxation process by the segment±solvent collision (Eq. (34) in Ref. [13]), but it should be revised. Considering only the segment±solvent collision, the hyperbolic function cannot be derived from the intermittent motion model. When the segment±segment collision for the orientational relaxation process is considered, our basic assumptions are the following:
Fig. 1. Schematic representation of (a) gradual process and (b) intermittent process of orientational relaxation of a polymer segment. In the case of (a) the orientation of a segment gradually relaxes with time. On the other hand, in the case of (b) the orientation of a segment relaxes instantaneously at a time.
(i) At the earlier stage of the orientational relaxation process, the orientation of a few segments relaxes. The number of these segments, nA , is far less than the total number of segments in a polymer. (ii) The collision takes place only when two segments are very close to each other in a polymer solution. In other words, the number of two such segments, nA , is taken into account. (iii) Before the reaction, these two segments are uniformly distributed without correlation between them.
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Under these conditions, de GennesÕs theory [16] can be applied to the modi®ed intermittent motion model, which dealt with the diusion-control reaction between two reactants attached to a polymer in concentrated solutions. Then, it is desirable to solve the following Smoluchowski equation: dnA ÿk 0 n2A : dt
6
When the memory function for Brownian motion of both ``reactants'' decreases more slowly than tÿ1 , de Gennes found a time dependent rate value for k 0 (Section III.D.1 in Ref. [16]). According to de Gennes, k 0 tbÿ1 is expected, and then, the following equation is given: dnA ÿk00 tbÿ1 n2A ; dt
7
where k00 is a constant. Solving Eq. (7) with nA nA0 at t 0, one can obtain nA as a function of time t. nA0 nA t : 8 1 k00 nA0 tb =b When b=nA0 k00 tb , the above equation can be simpli®ed as nA t
b ÿb t : k00
9
For the segment±segment collision, therefore, the hyperbolic function is derived as the orientational relaxation function. According to de Gennes, b is derived from the diusion length of A and the dimension of space. Hence, it is obvious that the exponent b in Eq. (9) is dierent from the exponent c in Eq. (5). The orientational auto-correlation function of polymer segments measured by ¯uorescence depolarization method, r t, is proportional to nA t or nA t. As a result, the following two important aspects of the orientational relaxation of polymer segments could be led from the modi®ed intermittent motion model: (i) When the orientational relaxation of a polymer segment is caused by the segment±solvent collision, the KWW function will be obtained as the orientational auto-correlation function. (ii) When the segment±segment collision causes a polymer segment to relax its orientation, the hyperbolic function will be observed. The segment±solvent collision may be eective in a dilute polymer solution with good solvents. On the other hand, the segment±segment collision becomes eective in a dilute polymer solution with poor solvents or in a concentrated polymer solution with good solvent. In the earlier papers [12±14], local motions of PSA and PSAPS were investigated in a poor solvent by using the time-resolved ¯uorescence depolarization method. It
477
was found out that the hyperbolic function was the most adequate orientational auto-correlation function. The result indicates that the segment±segment collision is the eective process on the orientational relaxation of polymer segments in a poor solvent. In this report, the local motion of anthryl groups end-capped polystyrene is investigated in a dilute polymer solution and in a concentrated one with good solvents. The crossover between the KWW and the hyperbolic function as the orientational auto-correlation function is discussed. 2. Experimental 2.1. Materials and sample preparations The chemical structure of a,x-dianthrylpolystyrene (APSA) is shown in Fig. 2. APSA was prepared in a way similar to that reported previously [17]. The number-average molecular weight and molecular weight distribution of APSA, which were measured with a gelpermeation chromatography (Toyo Soda, HLC-802UR) at 40°C in THF, were Mn 3:1 106 and Mw =Mn 1:24, respectively. Measuring absorption spectra of APSA using a spectrophotometer (Shimadzu, MPS5000), the content of anthryl group at the chain ends was determined to be 91%. APSA was dissolved in a solution of polystyrene (Aldrich, Mw 2:8 106 )/di-n-butyl phthalate (Wako, special-grade) mixture. Polystyrene (PS) was puri®ed by reprecipitation, and di-n-butyl phthalate (DBP) was used without further puri®cation. The weight fraction of polystyrene was varied from 0.00 to 0.20. 2.2. Fluorescence measurement techniques The steady-state ¯uorescence and depolarization spectra of APSA were measured by using a ¯uorescence spectrophotometer (HITACHI 650-60). The time-resolved ¯uorescence intensity decay curves and the time-resolved ¯uorescence anisotropy decay curves of APSA were measured by single photon counting techniques with a time-resolved spectro¯uorometer
Fig. 2. Chemical (APSA).
structure
of
a,x-dianthrylpolystyrene
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(HORIBA, NAES-1100). Signals of the photomultiplier were stored into the time-resolved spectro¯uorometer apparatus. The stored data were transformed to a personal computer (NEC, PC-9801VX) via RS232C. The excitation wavelength was set 385 nm. Fluorescent intensities were monitored through a color glass ®lter (Toshiba, L-42). The sample solutions were degassed by several freeze±pump±thaw cycles, and were sealed in a columnar Pylex cell. All measurements were performed at 25°C. The molar concentration of APSA in each sample was about 1:4 10ÿ6 M OD 0:01. Fluorescence intensity decay curve Fobs (t) and ¯uorescence anisotropy decay curve robs (t) are calculated by Fobs t IVV t 2GIVH t;
10
IVV t ÿ GIVH t ; IVV t 2GIVH t
11
robs t
R IHV tdt R G ; IHH tdt
12
where IVV (t), IVH (t), IHH (t), and IHV (t) are the measured ¯uorescence intensity decay related to the intensities of emitted light polarized parallel and perpendicular to the polarized exciting light and G is a correction factor, respectively. Subscripts V and H denote vertically and horizontally polarized light, where the ®rst letter in the subscripts refers to the exciting light and the second one refers to the emitted light. Note that G factor is necessary for correcting the depolarization characteristics of apparatus. The same procedure written in Ref. [12] is adapted in order to calculate the theoretical curves. Fluorescence intensity decay curves, S(t), are analyzed by a single exponential function as follows: t S t af exp ÿ ; 13 sf
Fig. 3. (a) Variation of the steady-state ¯uorescence anisotropy r with /PS , (b) variation of the averaged ¯uorescence lifetime sf with /PS , and (c) variation of the apparent ¯uorescence rotational correlation time sh with /PS by using Eq. (14).
where the sf and af are the ¯uorescence life time and a constant, respectively. The curve-®tting program with a deconvolution was made by the use of P A S C A L language (Borland: Turbo P A S C A L ) based on WahlÕs method [18] and quasi-Marquardt algorithm [19] with a non-linear least square method. These calculations were carried out on a personal computer (NEC, PC-9821Xn).
3. Results and discussion 3.1. Steady-state ¯uorescence anisotropy measurements Fig. 3(a) shows the variation of the steady-state ¯uorescence anisotropy, r, as a function of the weight fraction of PS, /PS . Measured r values increases gradually from 0.052 to 0.15 with increasing /PS . Figs. 4 and 5
Fig. 4. Fluorescence intensity decay curve of APSA at /PS 0:00. The open circles are the measured data. The solid line is the best ®t theoretical ¯uorescence intensity curve (Eq. (13)) convoluted by the measured instrumental function (exciting light pulse) represented by the broken line.
show the obtained measured ¯uorescence intensity decay curves, Fobs (t), at /PS 0:00 and /PS 0:20, respectively. The Fobs (t) obtained, was analyzed using Eq. (13).
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479
Fig. 5. Fluorescence intensity decay curve of APSA at /PS 0:20 (same representation as in Fig. 4).
It is clear that Fobs (t) for /PS 0:20 decayed more rapidly than that for /PS 0.00. sf decreases with increasing /PS as shown in Fig. 3(b). In general, this decrement of sf with /PS is ascribed to a quenching action. Plotting sf /s =sf 0 versus the molar concentration of the polystyrene segment, /s , the qualitative agreement with Stern±Volmer relation was con®rmed. Therefore, a dynamic quenching may occur between anthryl groups and polymer segments. It is therefore indicated that the environment around anthryl groups changes remarkably as /PS increases. Fig. 3(c) shows the variation of the apparent ¯uorescence rotational correlation time sh with /PS . sh calculated from sf and r with using the following equation: 1 1 sf 1 : 14 sh r r0 r0 was estimated as 0.28 by time-resolved ¯uorescence anisotropy decays. In order to derive Eq. (14), a single exponential function should be assumed as r(t): r t r0 exp ÿt=sh . However, obtained ¯uorescence anisotropy decay curves, robs (t), were not expressed by a single exponential function as shown in the next section. Using the steady-state ¯uorescence anisotropy data, therefore, the description of the eect of /PS on sh is restricted to qualitative one. sh becomes large gradually with increasing /PS . The result implies that the mobility of the chain end in a concentrated solution is smaller than that in the dilute solution. It is believed that this result simply re¯ects the dierence of the polymer concentration.
Fig. 6. Fluorescence anisotropy decay curves of APSA at /PS 0:00. The open circles are the measured data. The solid line is the best ®t KWW function (Eq. (15)) convoluted by the measured instrumental function (exciting light pulse) represented by the broken line.
weight fraction of PS, /PS , is varied from 0.00 to 0.20. At low /PS , the experimental curve decayed gradually at the initial stage of relaxation, but it seems to have a long time tail. The initial slope of the obtained decay curve became sharp with increasing /PS . If a probe molecule can rotate isotropically, ¯uorescence anisotropy decay is represented by a single exponential function as r t exp ÿt=sh . However, the observed ¯uorescence anisotropy decays suggest that a probe molecule, which is incorporated at the end of a polymer chain, moves in the wake of orientational relaxation motion of a polymer segment. Fitting the KWW (Eq. (15)) and the hyperbolic functions (Eq. (16)) to robs (t), the dierence between the relaxation processes can be discussed. ( b ) t r t reff exp ÿ ; 15 seff
t
ÿb
3.2. Time-resolved ¯uorescence anisotropy measurement
r t reff
Figs. 6±10 show the ¯uorescence anisotropy decay curves of APSA in various DBP/PS mixtures. The
where reff , seff , and b are respectively, a constant, the eective orientational relaxation time, and an exponent.
seff
;
16
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Fig. 7. Fluorescence anisotropy decay curves of APSA at /PS 0:025 (same representation as in Fig. 6).
Fig. 9. Fluorescence anisotropy decay curves of APSA at /PS 0:10. The open circles are the measured data. The solid line is the best ®t hyperbolic function (Eq. (16)) convoluted by the measured instrumental function (exciting light pulse) represented by the broken line.
Obtained values of ®tting parameters, reff , seff , and b for the KWW and the hyperbolic function are listed in Tables 1 and 2, respectively. In general, a ®tting of a trial function to measured data is evaluated quantitatively by a residual sum of square, v2 . If a trial function is in agreement with a measured anisotropy decay curve, v2 will approach 1 in our curve ®tting procedure (Eqs. (11) and (12) in Ref. [12]). The dierence between obtained v2 values for the KWW function and that for the hyperbolic one is small as shown in Tables 1 and 2. However, the result implies that robs (t) measured at low /PS is well represented by the KWW function. Conversely, the hyperbolic function is in good agreement with robs (t) at high /PS . 3.3. Crossover from the KWW to the hyperbolic function
Fig. 8. Fluorescence anisotropy decay curves of APSA at /PS 0:05 (same representation as in Fig. 6).
Now, the crossover behavior between the KWW function and the hyperbolic one is discussed. The solid lines in Figs. 6±8 represent the ®tting result for the KWW function (Eq. (15)). In Figs. 9 and 10, the ®tting result for the hyperbolic function (Eq. (16)) were represented. v2 values indicate that the KWW function is more adequate as the orientational auto-correlation
Y. Katsumoto et al. / European Polymer Journal 37 (2001) 475±483
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observed with increasing /PS . That is, the segment± segment collision becomes eective at higher /PS . Considering the change of the segment concentration around a probe molecule, this result may be explained. The gyration radius of a polymer, RG , in a good solvent is estimated by using the following relation: R2G 13a2 Nb2 ;
17
where a is the expansion factor, N is the number of segment, and b is the length of a segment. If one segment is constituted of two monomers, then b 3:1 10ÿ10 m. The value of the expansion factor a is 0.93 [20]. Thus, RG 9:3 10ÿ9 m was calculated. Then, the molar concentration of polymer at the hexagonal closest packing for the polymer sphere, uPS , is equal to 3:65 10ÿ4 M. The averaged segment concentration in a polymer sphere, ucoil , is estimated by the following relation: ucoil
Fig. 10. Fluorescence anisotropy decay curves of APSA at /PS 0:20 (same representation as in Fig. 9). Table 1 Best ®t parameters and v2 value obtained when the KWW function is ®tted to the experimental curves /PS
reff
seff
b
v2
0.000 0.025 0.050 0.100 0.200
0.338 0.345 0.350 0.398 0.394
5.17 3.80 3.86 4.75 10.68
0.461 0.476 0.487 0.431 0.427
1.066 0.866 1.002 0.892 1.171
Table 2 Best ®t parameters and v2 value obtained when the hyperbolic function is ®tted to the experimental curves /PS
reff
seff
b
v2
0.000 0.025 0.050 0.100 0.200
0.119 0.098 0.087 0.142 0.190
4.84 5.14 6.67 4.19 3.96
0.399 0.471 0.496 0.404 0.283
1.321 1.159 1.051 0.992 1.112
function than the hyperbolic function until /PS 0:05. Then, the hyperbolic function is the adequate one at /PS 0:10 and 0.20. It is therefore clear that the crossover from the KWW to the hyperbolic function was
3N : 4pR3G
18
When the above value is adopted, ucoil can be estimated as 0.689 M. In general, it is believed that the overlap concentration, u , is nearly equal to ucoil . The calculated values for the molar concentration of PS, uPS , and the total polymer segment concentration, us , in samples are listed in Table 3. A polymer solution is considered as a dilute one when us < u , and as a semi-dilute one when us > u . Therefore, the sample solution changes from a dilute solution to a semi-dilute one nearby /PS 0:10 in this system. Estimated uPS and u may suggest that the segment concentration around a probe molecule changes dramatically at /PS 0:10 because chromophore of APSA was attached at the end of polystyrene. The ®tting result suggests the fact that the crossover between the KWW and the hyperbolic functions was observed at /PS 0:10. In previous papers [12±14], the orientational relaxations of PSAPS and PSA were successfully reproducible with the hyperbolic function. While PSAPS has a chromophore at the center of the polymer chain, PSA has a chromophore at the end of the polymer chain. Therefore, it is believed that us around a probe molecule for PSAPS may dier from that for PSA. In spite of this dierence, the orientational auto-correlation function for both PSAPS and PSA was hyperbolic function. In Table 3 The molar concentration of PS, uPS , and APSA, uPSAPS , and the segment concentration, us , in each sample /PS
uPS 10ÿ4 M
us M
uAPSA 10ÿ6 M
0.000 0.025 0.050 0.100 0.200
0.00 0.96 1.98 4.16 8.32
0.000 0.134 0.277 0.582 1.16
1.06 1.12 1.89 1.28 1.40
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order to explain these experimental results, it may be worth considering that these polymers were dissolved in poor solvents. That is, the results suggest the fact that the segment±segment collision is eective in a dilute polymer solution with poor solvents. 3.4. Application of v2 -maps method for evaluating an adequate trial function The ``v2 -map method'' was proposed as a novel method for quantitative estimations and discussions of the relationship between an experimental decay curve and a trial function [12±14,21±23]. This method has been applied when it is dicult to discuss the suitability of trial functions to measured data only by the use of v2 . In this section, the availability of v2 -maps method is discussed. The v2 -maps of the KWW and the hyperbolic functions for experimental decay curves of APSA are shown in Fig. 11. seff and b of the KWW and the hyperbolic functions were selected as the X and Y coordinate axes
Fig. 12. Variation of DShm with /PS : when DShm is positive, the KWW function is more adequate orientational auto-correlation function than the hyperbolic one.
for v2 -map. Z axis represents the value of 1=v2 . In these ®gures, seff and b are varied from 0:1 seff to 10 seff and from 0:1 b to 1.0, respectively. The asterisk indicates the obtained values for seff and b at the maximum of 1=v2 . It is believed that the full area at half maximum, Shm , represents the sensitivity of these parameters [12]. As shown in Fig. 11, v2 -maps for the hyperbolic function have broad peaks until /PS reaches 0.10, and that for the KWW function has broad peaks clearly at /PS 0:20. Calculating the dierence between Shm for the KWW and the hyperbolic functions by DShm Hyperbolic KWW Shm ÿ Shm , the pro®le of 1=v2 can be discussed quantitatively. In the v2 -map method, it is assumed that a parameter which indicates the narrow pro®le of 1=v2 is important to explain data. Fig. 12 shows the variation of DShm with /PS . KWW function is more adequate than the hyperbolic one when DShm is positive. DShm has a positive value at the region of /PS < 0:10, and it becomes negative at /PS 0:10 and 0.20. It suggests that ¯uorescence anisotropy decay curves of APSA in dilute and concentrated polystyrene solutions were shown the best ®t based on the KWW and the hyperbolic functions, respectively. The result is in agreement with the conclusion of previous sections. It is therefore presumed that v2 -map method is useful to evaluate an adequate function. 4. Conclusions Fig. 11. v2 -maps of the hyperbolic and the KWW functions for obtained ¯uorescence anisotropy ratio decay curves of APSA in various PS/DBP mixtures.
The ``intermittent motion model'' for local motions of polymer segments was modi®ed. Two important as-
Y. Katsumoto et al. / European Polymer Journal 37 (2001) 475±483
pects of the modi®ed intermittent motion model are as follows: (i) When the orientational relaxation of a polymer segment is caused by the segment±solvent collision, the KWW function will be observed as the orientational auto-correlation function. (ii) When the orientational relaxation of a polymer segment is ascribed to the segment±segment collision, the hyperbolic function will be obtained. Hence, the local motion of anthryl groups endcapped polystyrene (APSA) was investigated in various polystyrene/di-n-butyl phthalate mixtures by the use of the ¯uorescence depolarization method. The steadystate ¯uorescence anisotropy measurements suggest that the mobility of the chain end in a concentrated solution is smaller than that in the dilute solution. Fluorescence anisotropy decay curves of APSA were measured, and the modi®ed intermittent motion model was investigated in both dilute and semi-concentrated polymer solutions. As a result of analysis with a curve ®tting procedure, the experimental decay curves at low weight fraction of polystyrene were well represented by the KWW function. Conversely, the hyperbolic function was in good agreement with the measured data at high weight fraction of polystyrene. This crossover from the KWW function to the hyperbolic one was observed near the overlap concentration. It is therefore concluded that the modi®ed intermittent motion model can successfully explain the nature of local motions of polymer segments in both dilute and semi-concentrated solutions.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Ediger MD. Ann Rev Phys Chem 1991;42:225. Jones AA, Stockmayer WH. J Polym Sci 1977;15:847. Bendler JT, Yaris R. Macromolecules 1978;11:650. Hall CK, Helfand E. J Polym Sci 1977;15:841. Viovy JL, Monnerie L, Brochon JC. Macromolecules 1983;16:1845. Valeur B, Monnerie L. J Polym Sci Polym Phys Ed 1976;14:11. Sasaki T, Yamamoto M, Nishijima Y. Macromolecules 1988;21:610. Gronski W, Sch afer T, Pater R. Polym Bull 1979;1:319. Friedrich C, Laupr^etre F, Noel C, Monnerie L. Macromolecules 1981;14:1119. Mashimo S. Macromolecules 1976;9:91. Waldow DA, Ediger MD, Yamaguti Y, Matsusita Y, Noda I. Macromolecules 1991;24:3147. Tsunomori F, Ushiki H. Polym J 1996;28:576. Tsunomori F, Ushiki H, Horie K. Polym J 1996;28:582. Tsunomori F, Ushiki H. Polym J 1996;28:588. von Smoluchowski M. Z Phys Chem 1917;92:192. de Gennes PG. J Chem Phys 1982;76:3316. Ushiki H, Horie K, Okamoto A, Mita I. Polym J 1979;11:691. Wahl P. Biophys Chem 1979;10:91. Nakagawa T, Oyanagi Y. Saishou nijyou-hou niyoru jikken-data kaiseki. Tokyo: Tokyo Daigaku Syuppan-kai, 1982. Ono K, Okada Y, Yokotsuka S, Ito S, Yamamoto M. Polym J 1994;26:199. Ushiki H, Mori M. Rep Prog Polym Phys Jpn 1994;37:471. Ushiki H, Tsunomori F. Rep Prog Polym Phys Jpn 1996;39:127. Ushiki H, Katsumoto Y, Tsunomori F, Rouch J. Rep Prog Polym Phys Jpn 1996;39:131.