Analysis of Progression and LAM Bands Observed in Infrared and

Infrared and Raman spectra observed for a series of vinylidene fluoride oligomers with TGTḠ conformation have been analyzed focusing on the progress...
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Macromolecules 2002, 35, 10210-10215

Analysis of Progression and LAM Bands Observed in Infrared and Raman Spectra of a Series of Vinylidene Fluoride Oligomers Makoto Hanesaka and Kohji Tashiro* Department of Macromolecular Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan Received July 15, 2002; Revised Manuscript Received September 24, 2002

ABSTRACT: Infrared and Raman spectra observed for a series of vinylidene fluoride oligomers with TGTG h conformation have been analyzed focusing on the progression bands in the fingerprint region and the Raman bands of longitudinal acoustic mode in the low-frequency region. By the quantitative analysis of progression bands, the vibrational frequency-phase angle dispersion curves could be obtained on the basis of normal-coordinates treatment. From the analysis of longitudinal acoustic mode, the limiting Young’s modulus of poly(vinylidene fluoride) crystal form II was evaluated to be 95.8 GPa. This value was found to be consistent with the modulus evaluated from the slope of acoustic branch of the abovementioned dispersion curves.

Introduction To investigate the chain conformation and crystal structure of poly(vinylidene fluoride) [PVDF] in detail, we employed a series of oligomers with the chemical formula CF(CF3)2-(CH2-CF2)n-I.1 Each component of the oligomers was successfully separated by utilizing a supercritical fluid chromatography with a mixture of carbon dioxide and ethanol as mobile phase. The purified components were crystallized from the ethanol solutions, and the crystal structures were analyzed by the X-ray diffraction technique. As shown in Figure 1, the conformation of VDF chain is a repetition of the TGTG h sequence (T: trans; G: gauche) and is essentially the same with that of PVDF form II. The similar situation is seen also for the chain packing mode: the CH2CF2 sequential parts are packed in the same manner with the parent PVDF form II.2,3 The crystal structures of these oligomers can be changed by changing the preparation conditions such as solvent, cooling rate from the melt, etc. Among them, however, the form II can be obtained relatively easily from the ethanol solution. Therefore, in the present study we will focus our attention on the oligomers crystallized from ethanol solutions, which can be treated as typical model compounds of PVDF form II. When we discuss the characteristic features of this conformation in more detail, the vibrational spectra may play an important role because of their high sensitivity to the local structure and intermolecular interactions. In a previous paper,4 we analyzed the infrared and Raman spectra of PVDF form II and assigned the observed bands to the various normal modes on the basis of lattice dynamical calculation. However, the analysis cannot be said to be perfect but still contains some ambiguous points. Since the oligomers have limited chain length, the infrared and Raman spectra give more additional information. The first point to be focused on is the progression bands. When we consider the phase relation between the neighboring monomeric units along the chain axis, the infrared and Raman modes with the phase difference of 0 and π can be observed because of symmetrical constraint. But, this symmetrical constraint is broken for the oligomers with finite lengths, and a series of the so-called progression

Figure 1. (a) Crystal structure of VDF oligomer with n ) 71 and (b) subcell structure. The as, bs and cs are the unit cell axes of the subcell lattice.

bands can be detected in the infrared and Raman spectra. These progression bands are predicted to shift their positions depending on the chain length of the oligomer. By analyzing these progression bands quantitatively, we may estimate the vibrational frequencyphase angle dispersion curves, which can be compared with the theoretically calculated curves to check the reasonableness of the force constraints used in the calculation. The dispersion curves, in particular, the branch of acoustic phonon propagating along the chain axis, are important for the estimation of Young’s modulus of a chain. The refined force constants will give an accurate value of this modulus. Another important point to be focused in the spectra is about the longitudinal acoustic mode (LAM) or the accordion-like mode, which should be observed in the

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low-frequency region of the Raman spectra of the oligomers. Several examples of LAM bands were reported for the oligomers of polyethylene (n-alkane),5-8 the oligomers of poly(tetrafluoroethylene) (n-perfluoroalkane),9,10 the oligomers of poly(oxymethylene),11,12 etc. The vibrational frequency of the LAM band shifts depending on the chain length of the oligomer and follows approximately the dispersion curve of the acoustic branch along the chain axis. Therefore, by analyzing the observed LAM bands, the Young’s modulus along the chain axis may be estimated. This crystalline modulus of a polymer chain is important as an index of the limiting value of the modulus to be exhibited by this polymer. The crystalline modulus has been estimated in various ways including the LAM method. For example, the X-ray diffraction method is one of the useful techniques but must assume a homogeneous stress distribution in the bulk sample subjected to the tensile stress.13-18 The LAM band analysis is more useful in such a point that we do not need to introduce any assumption about the stress distribution in the sample. Of course, the LAM band analysis method has also many problems to be solved, about which many trials have been reported as will be mentioned in a later section.5-8,19-23 Anyway, we have succeeded in measuring the LAM bands in the low-frequency region of the Raman spectra of a series of VDF oligomers for the first time, from which the Young’s modulus along the chain axis of PVDF form II could be evaluated. This value can be checked by comparing with the modulus obtained from the slope of the dispersion curve of the acoustic branch along the chain axis, which can be calculated theoretically as already mentioned in the previous paragraph. In the present paper we will report the infrared and Raman spectra of a series of VDF oligomers with TGTG h conformation by focusing on two important points. One is to analyze the progression bands in the fingerprint regions of the infrared and Raman spectra, from which the refined force constants can be evaluated, and the accurate estimation is made for the Young’s modulus of a chain from the calculated dispersion curve of acoustic phonons. The second point is to analyze the LAM band in the low-frequency region of the Raman spectra and to estimate the Young’s modulus which can be compared with the value calculated from the dispersion curve. The effect of end groups on the LAM frequency is also discussed. Experimental Section Samples. A mixture of VDF oligomers X-(CH2-CF2)n-Y was supplied by Daikin Co. Ltd., Japan. This sample was separated into components by filtering through the supercritical fluid chromatography (Super-201 SFE/SFC system of Japan Spectroscopic Co. (JASCO)) using a mixture of carbon dioxide and ethanol as a mobile phase. The thus purified oligomer components (n ) 6-10) were crystallized from the ethanol solutions, and the crystal structures were analyzed by the X-ray diffraction method for the components of n ) 6, 7, and 8 as reported in a previous paper.1 From the analysis of crystal structure, the end groups X and Y were found to be CF(CF3)2 and I, respectively. Measurements. Infrared spectra of the oligomer components n ) 6-10 were measured for the KBr disks with a BIORAD FTS-60A/896 and JASCO FT/IR-510 Fourier transform infrared spectrometers. The Raman spectra were measured for the pressed powder samples with a BIO-RAD FTS Fourier transform Raman spectrometer in the frequency region 4000100 cm-1 and with a JASCO NR-1800 Raman spectrometer

Vinylidene Fluoride Oligomers 10211

Figure 2. Infrared spectra of PVDF form II and a series of VDF oligomers (n ) 6-10) in the frequency region of 5501350 cm-1. in the frequency region 750-25 cm-1. In the former case a beam with 1064 nm wavelength from a YAG:Nd laser was used as an excitation light source, and the scattered Raman signals were collected by a Ge detector. In the latter case the beam with 514.5 nm wavelength from an Ar+ laser was incident on the samples, and the Raman signals were detected by a CCD detector at a resolution power of 2 cm-1. Calculation of Vibrational Frequency-Phase Angle Dispersion Curves. The vibrational frequency-phase angle dispersion curves were calculated for a TGTG h single chain on the basis of the GF matrix method by taking into account the phase angle between the neighboring unit cell along the chain axis. The atomic coordinates were quoted from the X-ray analyzed results.1 The force constants of valence-force-field type were used, the numerical values of which were transferred from ref 4 with some modifications to get as good agreement between the observed and calculated vibrational frequencies as possible.

Results and Discussion Progression Bands. Figure 2 shows the infrared spectra measured for a series of oligomers. As already pointed out in the previous paper,1 the main bands observed at 1210, 758, and 610 cm-1 indicate that these oligomers take the chain conformation of TGTG h , essentially the same with PVDF form II. The infrared spectra in the frequency region of 825-600 cm-1 are enlarged and are given in Figure 3. Many weak bands, which cannot be detected in the infrared spectra of PVDF itself, are observed to change their positions systematically depending on the chain length of the oligomer. These bands are identified as a series of progression bands originated from finite chain length of the oligomer. By referring to the band assignment made for PVDF form II,4 the observed progression bands are assigned to the scissoring modes of CF2 groups. A similar situation can be seen also in the Raman spectra (600-800 cm-1) as shown in Figure 4. According to the simply coupled oscillator model,24 the phase difference φk between the neighboring unit cells may be given as follows:

φk ) kπ/(m + 1) (k ) 1, 2, ..., m)

(1)

where m is the number of translational units included in the chain: one translational unit contains two h ) conformation. The CH2CF2 monomeric units of TG (TG dispersion curves of an infinitely long PVDF single chain of TGTG h calculated under the assumption of periodic boundary condition are shown in Figures 5, 6, and 9. Table 1 lists the force constants used in this calculation.

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Macromolecules, Vol. 35, No. 27, 2002 Table 1. Force Constants Used in the Calculation of Frequency-Dispersion Curves of PVDF Form II Chain

Figure 3. Infrared spectra of PVDF form II and a series of VDF oligomers (n ) 6-10) in the frequency region of 600825 cm-1. A fraction indicated for each band is a phase angle φk in eq 1 of the text in a unit of π.

no.

force constants

coordinates involved

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Kd KR FR1 FR FRγ FR$ FRφ Fd K1 F11 F1ζ Hγ Fγ F′$ Hδ H$ f$t Hφ Hζ T F1φ Fφ F′φ f$g f$φg fγφt fγφg F$γt fγφg fγφt

CH CC CC, CF CC, CC CC, CCH CC, CCC CC, CCF CH, CH CF CF, CF CF, CFF CCH CCH, CCH CCH, CCH CHH CCC CCC, CCC(t) CCF CFF CCCC CF, CCF CCF, CCF CCF, CCF CCC, CCH(g) CCC, CCF(g) CCH, CCF(t) CCH, CCF(g) CCH, CCH(t) CCH, CCH(g) CCF, CCF(t)

common atoms

C C CC CC CC C C CF CC CH CC

CF CC CF CC CC CC CC CC CC CC

valuesa 4.876 4.875 0.510 -0.162 0.225 0.008 0.642 0.051 5.169 0.395 1.142 0.620 0.040 0.071 0.491 1.388 0.499 1.151 1.371 0.201 0.514 0.412 0.130 -0.042 0.097 0.176 -0.188 0.168 0.038 0.013

a The stretch constants have units of mdyn/Å, the stretch-bend intreactions have units of mdyn/rad, and the bending constants have units of (mdyn Å)/rad2.

Figure 4. Raman spectra of VDF oligomers (n ) 6-10) in the frequency region of 1450-250 cm-1.

As mentioned in the Experimental Section, the numerical values of these force constants were transferred from ref 4 with some small modification. This modification was needed because the geometry of a chain was slightly different from that used in the ref 4. This change is frequently seen when the valence-force-fieldtype force constants are used in the normal modes calculation, because the vibrational frequency is affected more or less by a small change in molecular geometry. In Figures 5, 6, and 9, the vibrational modes at phase angle 0° or Γ point in the reciprocal space are classified into A′ and A′′ species or the in-plane and out-plane modes with respect to the glide plane of the chain since the line group of TGTG h chain is isomorphous to the point group Cs. This can be adapted also for the modes of general phase angle φ. Therefore, the dispersion curves are classified into two types (A′ and A′′) indicated by solid and broken lines. Figures 5 and 6 show the comparison of the observed infrared and Raman frequencies with the calculated dispersion curves, respectively. Originally the observed progression bands do not

Figure 5. Comparison of the observed infrared progression bands with the calculated dispersion curves.

give us any information about the phase angle φk. As being indicated in a textbook,24 the progression bands were assigned to the possible phase angles so that the resultant dispersion curves were as smooth as possible and fitted the calculated curves as well as possible. In Figure 3, the thus determined phase angles are indicated for main bands as an example. The agreement between the observed and calculated curves is relatively good. LAM Bands. Figure 7 shows the Raman spectra in the low-frequency region of 25-250 cm-1. The band

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Vinylidene Fluoride Oligomers 10213

Figure 8. Plot of observed LAM frequency against the inversed chain length. Broken and solid lines correspond to eqs 2 and 3, respectively. Table 2. Parameters Used for the Analysis of LAM Frequency of VDF Oligomers n

Figure 6. Comparison of the observed Raman progression bands with the calculated dispersion curves.

obsd LAM L /Å

/cm-1

6

7

8

9

10

63.0 21.4

56.9 23.8

53.6 26.1

52.0 28.4a

45.6 30.6a

a The chain lengths L for n ) 9 and 10 were not known and so were estimated by using a Ceirus2 software on the basis of molecular geometry of n ) 6-8 components revealed by the X-ray analysis.

a continuous elastic body. In the actual case a finite chain is connected by weak interchain interactions as understood from the stacked layer structure of the oligomers (see Figure 1). Following the model developed by Strobl and Eckel for the stacked lamellar structure of n-alkane crystal,6 we try to use the following equation to analyze the LAM bands observed for the VDF oligomers: Figure 7. Raman spectra observed for a series of VDF oligomers (n ) 6-10) in the low-frequency region of 25-250 cm-1. The bold lines indicate the LAM bands.

indicated by bold line shifts its position depending on the chain length of the oligomer: the lower position for the longer oligomer. Judging from the behavior, it is speculated that these bands may be assigned to the LAM mode of the oligomer chains. As reported for the LAM bands of n-alkane chains,5-8 the vibrational frequency of LAM band is predicted to be inversely proportional to the length L of the extended chain:

ν ) (1/2cL)(Ec/F)1/2

(2)

where ν is a wavenumber, Ec is the Young’s modulus along the chain axis, F is the density of the crystal, and c is the velocity of light. Strictly speaking, the density F of the crystal changes depending on the oligomer component. But, in the present rough analysis, the F was assumed to be constant, 2.09 g/cm3. The chain lengths L for n ) 9 and 10 were not known by the X-ray structure analysis and were estimated by using a Ceirus2 software on the basis of molecular geometry of n ) 6-8 components revealed by the X-ray analysis. Figure 8 shows a plot of LAM frequency against 1/L. Table 2 lists the various parameters needed for this plot. A broken line in Figure 8 corresponds to eq 2 and passes through the origin. From the slope of this straight line, the Young’s modulus Ec is evaluated as 144.1 GPa. Equation 2 is useful for the evaluation of Young’s modulus but gives an overestimated value in general. This is because eq 2 is derived for an acoustic mode of

ν ) βπ/L + 2βf/(πEc)

(3)

β ) (1/2πc)(Ec/F)1/2

(4)

where f is a parameter to represent the interchain interaction. The solid straight line in Figure 8 corresponds to eq 3. From the slope and intercept of this line, the following values are obtained: Ec ) 95.8 GPa and f ) 1.4 × 1010 GPa/m. It is a little difficult to judge which straight line shown in Figure 8 agrees better with the observed LAM data because of the limited number of observed data point. At the present stage we will employ the Ec and f values obtained from eq 3 in the subsequent discussion. Of course, it is more ideal to have additional data of LAM bands for the oligomer components with longer chain length. But it is a quite hard work to actually get these oligomer components. The Young’s modulus can be evaluated also from the slope of the acoustic branch in the vibrational frequencyphase difference dispersion curve. The dispersion curves in the low-frequency region calculated for a single chain of PVDF form II are plotted as shown in Figure 9. The slope corresponds to (Ec/F)1/2. The thus estimated Ec is 102.3 GPa when the density of PVDF form II crystal is assumed to be 1.93 g/cm3. The open circles in this figure are the observed LAM frequencies, fitting relatively well with the calculated dispersion curve. The phase angle was approximately estimated by the equation φ ) 2π/ (n + 2) with the end groups assumed to participate into the LAM mode.5 The crystalline modulus can be evaluated also by the X-ray diffraction method. For PVDF form II the Young’s

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Figure 10. Plot of LAM frequency against 1/L: (b) observed data of oligomers; (O) calculated for crystal structures revealed by X-ray analysis; (4) calculated for single chains picked up from the crystal structures.

Figure 9. Vibrational frequency-phase difference dispersion curves in the low-frequency region calculated for PVDF form II single chain. The open circles are the observed LAM frequencies, fitting relatively well the calculated dispersion curve. The phase angle was approximately estimated by the equation φ ) 2π/(n + 2) with the end groups assumed to participate to the LAM mode.5

modulus was reported to be 59 GPa.15 This value is too small when compared with the modulus estimated by LAM method. There might be several reasons for this large discrepancy. For example, the X-ray experiment is made under static conditions by increasing the tensile stress step by step. On the other hand, the LAM band originates from the vibrational mode of a time scale of ca. 1 ps. The difference in modulus between these two methods might come from such a difference between static and dynamic conditions.25 But this difference or the difference between isothermal and adiabatic conditions is too small to explain the above-mentioned large difference in Young’s modulus.26,27 Another important origin might come from the stress distribution in the bulk sample. For example, the analysis of LAM bands of n-alkane crystals gives 280 GPa for polyethylene single chain in the crystalline state,6-8 which is much higher than the value 235 GPa13-15 estimated by the X-ray diffraction method. By taking into account a heterogeneous distribution of tensile stress applied to the bulk polyethylene samples, the Young’s modulus is corrected to 260 GP,16 much closer to the LAM value, 280 GPa. The molecular dynamics calculation gave the value 280 GPa at room temperature and 310 GPa at 0 K.28 The large discrepancy of Young’s modulus in the case of PVDF form II seems to come from the similar reasons with those proposed for n-alkane and polyethylene. Another important influence might come from the large end groups of the oligomers CF(CF3)2 and I. In the above estimation there is no explicit contribution of the end groups to the LAM frequency. But the observed frequency of the LAM band should contain the effect of these end groups more or less. For example, since these groups are heavy, the LAM frequency is predicted to be lower than the case without such heavy groups. Besides the effective chain length might be also affected more or less by the influence of these large groups. Interchain interactions are also affected by these groups. To check these effects, the LAM modes were calculated actually for the oligomer molecules with these

end groups. The calculation was made about the isolated chain and the crystal structure of the oligomers with n ) 6, 7, and 8 (Figure 1). The molecular chain conformation and the packing structure were transferred from the X-ray analysis. For the calculation of normal-mode frequencies the force constants listed in Table 1 might be used, but we do not have any reliable force constant values for the end groups. Besides, the preparation of the input data for the numerical calculation is quite troublesome even for a finite oligomer chain as well as for the crystal structure including many these oligomer chains. To check the effect of end groups on the LAM frequency more conveniently, therefore, we employed here the commercially available software Cerius2 (version 4.2, Accerlys Inc.). The accuracy of the calculated normal-mode frequencies is dependent on the force field parameters used in the calculation. In the present study the COMPASS was used as one of the best force fields.29 The results are shown in Figure 10. When CF(CF3)2 and I are introduced as the end groups, the tendency of LAM frequency to be lower for the oligomer with longer chain length can be seen reasonably for the calculated results. As shown in Figure 10, the values calculated for the crystal structures are higher than those for the isolated chains due to an effect of interlayer interaction. The Young’s modulus of a chain was evaluated to be 162.2 GPa for the crystal model and 141.6 GPa for an isolated chain; this difference comes from the effect of intermolecular interactions. In principle, the result calculated for the crystal structure should correspond to the observed value. But, when we see Figure 10, the chain length dependence of LAM frequency calculated for isolated chains are rather closer to the observed one than those calculated for the crystal structures with the intermolecular interactions taken into account. The Young’s modulus calculated for an isolated chain seems an overestimation. This situation might come from an employment of COMPASS force field which is said to give higher vibrational frequencies in general, though only slightly. In fact, the Young’s modulus of PVDF form II chain calculated by this force field was 154.1 GPa, higher than the value 102.3 GPa, which was estimated from the dispersion curve of the acoustic phonon calculated by using the force constants listed in Table 1. Anyway, as seen in Figure 10, LAM frequencies calculated for the realistic oligomer models with end groups of CF(CF3)2 and I are in a relatively good agreement with the observed data. When the LAM frequency was calculated for an oligomer chain without any such heavy end groups, it was appreciably higher than the value

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shown in Figure 10. From these considerations we may say that the Young’s modulus evaluated by using LAM data of the oligomers with bulky end groups might be different more or less from the modulus predicted for pure PVDF form II chain. To estimate the Young’s modulus of PVDF form II chain exactly, we have to employ the model compounds with more realistic chemical formulas, for example, X-(CH2CF2)n-Y with X ) H and Y ) F. Of course, the value n is desired to be long enough to erase the end group effect as much as possible. Conclusions In this paper we analyzed the infrared and Raman spectra measured for a series of oligomers with the chemical formula CF(CF3)2-(CH2CF2)n-I, which were assumed as model compounds of PVDF form II crystal. The progression bands could be detected and interpreted reasonably by carrying out the calculation of vibrational frequency-phase angle dispersion curves. This analysis gave us a set of force constants useful for the estimation of the Young’s modulus of the chain. The Young’s modulus calculated from the acoustic phonon dispersion curve was 102.3 GPa, which was found to be in a good agreement with the value 95.8 GPa evaluated from the analysis of the LAM bands detected in the low-frequency Raman spectra. The thus estimated modulus is much higher than the value 59 GPa obtained for PVDF form II by the X-ray diffraction method. At the present stage, the definite reasons for this discrepancy cannot be revealed clearly. Acknowledgment. The authors thank Daikin Co. Ltd., Japan, for kindly supplying VDF oligomer samples. They also thank Accerlys Inc. for giving us a chance to use the software Cerius2. References and Notes (1) Tashiro, K.; Hanesaka, M. Macromolecules 2002, 35, 714. (2) Hasegwa, R.; Takahashi, Y.; Chatani, Y.; Tadokoro, H Polym. J. 1972, 3, 600.

Vinylidene Fluoride Oligomers 10215 (3) Takahashi, Y.; Matsubara, Y.; Tadokoro, H. Macromolecules 1983, 16, 1588. (4) Kobayashi, M.; Tashiro, K.; Tadokoro, H. Macromolecules 1975, 8, 158. (5) Schaufele, R. F.; Shimanouchi, T. J. Chem. Soc. 1967, 71, 1320. (6) Strobl, G. R.; Eckel, R. J. Polym. Sci., Polym. Phys. Ed. 1976, 14, 913. (7) Kobayashi, M.; Sakagami, K.; Tadokoro, H. J. Chem. Phys. 1983, 78, 6391. (8) Snyder, R. G.; Strauss, H. L.; Alamo, R.; Mandelkern, L. J. Chem. Phys. 1994, 100, 5422. (9) Rabolt, J. F.; Fanconi, B. Polymer 1977, 18, 1258. (10) Rabolt, J. F.; Fanconi, B. Macromolecules 1978, 11, 740. (11) Rabolt, J. F.; Fanconi, B. J. Polym. Sci., Polym. Lett. Ed. 1977, 15, 121. (12) Runt, J.; Wagner, R. F.; Zimmer, M. Macromolecules 1987, 20, 2531. (13) Sakurada, I.; Nukushina, K.; Ito, T. J. Polym. Sci. 1962, 57, 651. (14) Sakurada, I.; Ito, T.; Nakamae, K. J. Polym. Sci., Part C 1966, 15, 75. (15) Sakurada, I.; Kaji, K. J. Polym. Sci., Part C 1970, 31, 57. (16) Tashiro, K.; Wu, G.; Kobayashi, M. Polymer 1988, 29, 1768. (17) Kitagawa, T.; Tashiro, K.; Yabuki, K. J. Polym. Sci., Part B: Polym. Phys. 2002, 40, 1269. (18) Kitagawa, T.; Tashiro, K.; Yabuki, K. J. Polym. Sci., Part B: Polym. Phys. 2002, 40, 1281. (19) Hsu, S. L.; Krimm, S.; Krause, S.; Yeh, G. S. Y. J. Polym. Sci., Polym. Phys. Ed. 1976, 14, 195. (20) Hsu, S. L.; Ford, G. W.; Krimm, S. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 1769. (21) Hsu, S. L.; Krimm, S. J. Appl. Phys. 1978, 48, 4013. (22) Krimm, S.; Hsu, S. L. J. Polym. Sci., Polym. Phys. Ed. 1978, 16, 2105. (23) Chang, C.; Krimm, S. J. Polym. Sci., Polym. Phys. Ed. 1979, 17, 2163. (24) Zbinden, R. Infrared Spectroscopy of High Polymer; Academic Press: NewYork, 1964. (25) Mason, W. P. Piezoelectric Crystals and Their Application to Ultrasonic; Van Nostrand: Princeton, NJ, 1950. (26) Wu, G.; Tashiro, K.; Kobayashi, M. Macromolecules 1989, 22, 188. (27) Tashiro, K. Prog. Polym. Sci. 1993, 18, 377. (28) Tashiro, K. Comput. Theor. Polym. Sci. 2001, 11, 357. (29) Sun, H. J. Phys. Chem. B 1998, 102, 7338.

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