Mobility of Methyl Groups in Polycarbonate and in Poly--methylstyrene

39 Mobility of Methyl Groups in Polycarbonate

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and in Poly-α-methylstyrene Analyzed by Nuclear Magnetic Resonance R. KOSFELD, U. VON MYLIUS, and G. VOSSKÖTTER Institute of Physical Chemistry of the RWTH, Aachen, Germany The high polymers polycarbonate and poly-α-methylstyrene were investigated by wide line NMR spectroscopy. The activation energies and correlation frequencies of the rota tional motion of the methyl groups in the polymers are determined from the measured line-widths and second moments. These are too small for a classical leap process. With the model of the quantum mechanical rotator there is sufficient conformity at low temperatures only. At higher temperatures the results seem to point to a correlation spectrum. From the measured absorption lines, the line -widths and intensities of the components are determined by iteration. This makes it possible to state the number of the rotating and non-rotating methyl groups which are tempera ture dependent.

TDreezing of internal degrees of freedom has been observed frequently *·" in high molecular substances below the glass transition temperature by NMR spectroscopy. Measurements have been carried out on poly carbonate (PC) by Kovarskaja et al. (7), Murakani et al. (9), and Slonin et al. (19), and on poly-a-methylstyrene (PMST) by Odajima et ai. (10). Most investigators studying freezing processes determine only linewidths and second moments of the experimentally determined resonance lines and compare them with the theoretically calculated values. Using these results they try to derive information about correlation times of molecules or molecule groups and about activation energies for thawing certain degrees of freedom of such microscopic systems. Certainly, more information can be obtained from NMR measurements, especially from 592 Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

39.

KOSFELD E T A L .

593

Mobility of Methyl Groups

complicated absorption lines as in PC and PMST. Our studies were directed toward obtaining a better understanding of the freezing process. [The polycarbonate was supplied by Bayer, Uerdingen, Ger many and the poly-a-methylstyrene by BASF, Ludwigshafen, Germany.] Experimental Results Since the distance between two extremes of the measured absorption lines is defined as linewidth, Figure 1 shows the temperature dependence of the linewidths of PC. Figure 2 shows the course of the linewidths of PMST. The slight dip in the linewidth curve of the small component observed around 200 °K. can be reduced to superposition by another absorption line with a larger linewidth.

(AH)I/

2

20

15

10

100

Figure 1.

200

150

Linewidth (AH)

1/2

250

300

of polycarbonate as a function of temperature

It is well known that gaussian curves undergo a shift of their maxima by superposition in such a direction that their maxima move toward each other. The results for both linewdiths below 150 °K. show that minor linewidths are shifted to major values and major linewidths to minor

Platzer; Addition and Condensation Polymerization Processes Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

594

ADDITION A N D CONDENSATION POLYMERIZATION

PROCESSES

values. This is valid also for temperatures around 200°K., but here it is impossible to distinguish the two absorption lines because of the small intensity of the absorption line which has the major linewidth. However, around 200°K. a distinct decrease in linewidth of the minor component caused by the influence of the major component can be observed. Mea surements with PMST give essentially the same results.

ALL G

15

10

5

•Κ Figure 2.

Linewidth (AH)

1/2

of PMST as a function of temperature

The temperature dependence of the second moment ΔΗ gives even more evidence of structural changes in the two substances. The second moment is defined by: 2

+ 00

f

g* ( Η ) ' (H — H ) dH 0

3

* " * = 4 ^ f

(υ g*

(H)'(H-H )dH 0


39.

KOSFELD E T A L .


595

where g* (ff)' is the differentiated, field dependent absorption curve. Corresponding to Equation 1 values of the second moment were deter mined from the experimentally obtained curves by a Stiltjes planimeter. Figure 3 shows the temperature dependence of the second moment of PC, and Figure 4 gives the temperature dependence of PMST. The observed change of the second moment with temperature corre sponds to a change of 13.6 G for PC and of 10.4 G for PMST. 2

2

Discussion To determine the molecular process which causes a change in the second moment of 13.6 G for PC and 10.4 G for PMST, it is necessary to calculate the second moment. According to Van Vleck (22) the second 2

ι

25

2

Î

μ


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Theoretischer Wert

ΔΗ "G*

1

20

15

10

200

100

300

°K Figure 4. Second moment AH vs. temperature for PMST 2

moment of a substance which contains only one type of nucleus capable of resonance and which is isotropic can be calculated from Equation 2. 6 AH =-^I (1+1)

(2) Σ r j>k Here r is the distance between the nuclei j and k, g is the Landé-factor of the nuclei, I is the nuclear spin, and Ν is the number of nuclei which are summarized. The following distances and angles are taken from published data (β, 21): 2

g μ N2

2

1

jk

(1) In the aromatic ring: a C — C distance of 1.39 Α., a C—Η dis tance of 1.08 Α., and the C — C — C and C—C—Η angles of 120°. (2) In the aliphatic chain: a C — C distance of 1.54 Α., a C—Η dis tance of 1.09 Α., and C — C — C and C—C—Η angles of 109°28'. (3) In the junction between aromatic and aliphatic part of the molecules: a C — C distance of 1.50 A. Considering these values, Equation 2 gives ΔΗ = 18.7 G for the second moment of PC for internal molecular interactions. To calculate 2

2


39.

KOSFELD E T A L .

597


the contribution of the interactions between the molecules, the results from x-ray measurements of Prietzschk (15) have been used. From his values for the unit cell of PC in the crystalline state, a ΔΗ value of 6.7 G is calculated. A value of ΔΗ = 25.4 G for the second moment for 100% crystalline polycarbonate was obtained. This value is marked by a dashed line in Figure 3. For PMST a second moment for internal molecular interactions of 17.6 G is calculated. The contribution of the interaction between the molecules was estimated only because no structural data of PMST are known. Compared with literature data for similar substances, a value of about 6.0 G seems adequate. Thus, one obtains 23.6 G for the second moment of PMST. This value is marked by a dashed line in Figure 4. In calculating the second moment it was obvious that in PC as well as in PMST the largest contributions to this moment occur always when at least one of the interacting partnei s is a proton of a methyl group. Calculating the moment for this case a value of 17.4 G for the internal molecular inter actions of PC and a value of 13.5 G for PMST were obtained. According to Gutowsky and Pake (5) this moment decreases to one-fourth its value in a rigid methyl group if a rotational motion occurs in the molecule. For an initial methyl rotation 17.4 G for PC should decrease to 4.4 G and that from 13.5 G should decrease to 3.4 G for PMST respectively. This causes an increase of 13.0 G for PC and of 10.1 G for PMST. Considering further that for an initial C H rotation the contributions of the interactions between the molecules decrease to the second moment, good conformity is obtained with the experimentally found values of 13.6 and 10.4 G respectively. These results with PC and PMST show a freezing of methyl group rotation. Comparing these ex perimentally determined values with corresponding values found by Slonim et al. (19), r clear difference beyond experimental error is notice able. Slonim et al. (19) corrected their values by considering the ampli tude of the wobbling field. A correction like this does not reduce the difference found here since a modulation amplitude of 0.8 G, as used in our experiment, influences only insignificantly the size of the second moment compared with the values of Perlmann and Bloom (11), Andrew (1), and Visweswaramurthy (23). Besides a miscalibration of the magnitude (H-H ) the main reason for the often large differences in the experimentally found second mo ments seems to be that the outer dropping sides of the resonance curves which do not emerge clearly from the noise level, make an important contribution to the second moment. It is difficult to decide at which point the integration according Equation 1 can be stopped without great errors. 2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

3

2

0



598

PROCESSES

Despite the relatively large differences between the values found here and those given by Slonim (19) there is a good agreement in the amount of increase of the second moment [13.6 G from our experiments and with 13.0 G from Slonim (19)J. Good agreement is found between our measurements on PMST and those of Odajima et al. (10); therefore, no discussion of these values is necessary. The freezing of methyl-group rotation is always observed in the temperature range 60°-300°K. if the methyl groups are hindered by neighboring groups. This has been reported by Powles and Gutowsky (13) for low molecular weight substances. Gutowsky and Meyer (4) and Slichter (17) confirmed such results for different rubbers. Similar observations have been made by Sinnot (16) and Slichter and Mandell (18). Therefore, as shown by Odajima (10) with PMST, the hindering potential, which must be overcome by the methyl groups during rotation, must be estimated. For this purpose an equation derived by Powles and Gutowsky (14) is used: 2

2

log a = 0.4343

AT? - log v*

(3)

kT

where: (ΔΗ - AH ) 2

2

γ (AH ) 2

1/2

' (AH y

2

t

h

(

4

)

- AH ) 2 2

h

v* is a dimensionless number. It is determined by v«, — v* · v , with v = 1 sec." . Between the activation energy ΔΕ, the correlation frequency v and the correlation frequency v for infinite temperatures a dependence corre sponding to the Arrhenius equation is assumed: 0

0

1

c

00

c — x exp

v

v

(-3)

(5)

AH represents the second moment at T and AH the value at T (see Figure 3). Log a is calculated as function of the reciprocal of absolute temperature. The values of the second moment were taken from Figures 3 and 4. From the slopes of the resulting straight lines and from the abscissa intersection an activation energy AE of 1.2 kcal./mole and a limiting frequency of 1.26 MHz. for PC and 1.47 kcal./mole and 0.63 MHz. for PMST, respectively, are obtained. These values are too small. According to the Transition State Theory (3) a limiting frequency of 2.7 Χ 10 MHz. at very high temperatures for a classical leap process is obtained which is assumed from previous calculation. With Powles (12) and Odajima (JO) the frequently observed low activation energies in polymers can be explained by a correlation spectrum. That means it is 2

h

B

2

t

Vao

6


8

KOSFELD E T A L .

39.

599


necessary to assume not only one correlation frequency v but a correla tion spectrum because of the many degrees of freedom in complicated high molecular weight substances. Steiskal and Gutowsky (20) give another explanation for low molecu lar weight solid matter. They consider the methyl groups as quantummechanical rotators which overcome the hindering potential by a tunnel ing effect. The hindering potential which counteracts rotation is given by Equation 6. c

V=Y±

[1 +cos (3 φ)]

(6)

V is the potential height, and φ describes the rotation of the methyl groups. Steiskal and Gutowsky (20) give the dependence of the tunneling frequency v of the reciprocal absolute temperature as parameter for V . Figure 5 gives these results. 0

0

T

10Q0-°K Τ

Figure 5. Comparison of the correlation frequencies calculated according Equation 7 with the averaged tunneling frequencies v calculated by Steiskal and Gutowsky for the methyl groups assumed as quantum-mechanical rotators as a function of the temperature for PC (x) and PMST (o). The potential height in units of kcal./mole is the parameter T

To confirm whether or not the results reported here can be described by a tunneling effect, an equation is derived for the correlation frequency v from the Equations 3, 4, and 5 in the form: c


600


PROCESSES

Using the temperature dependent values for the second moment from Figure 3 and 4 it is possible to calculate the correlation frequencies v from Equation 7. Assuming that the correlation frequencies v obtained this way were identical to the tunneling frequencies ντ, the calculated values for v were inserted in Figure 5. The correlation frequencies from Equation 7 are close to the theoretical curves at low temperatures. At high temperatures the v values differ more and more from the quantum mechanical model. This is understandable since the neighboring groups which constitute the real hindrance for methyl-group rotation become more mobile with increasing temperature. Therefore, methyl-group rota tion is similar to a classical describable rotation diffusion. This result shows clearly that the motion of methyl groups as they change from a frozen state to one of free motion can be described not only by the model of a quantum-mechanical rotator. Still the tunneling effect seems to give a possible explanation for the low rotation fre quencies of the methyl groups and the corresponding hindering potential at low temperatures. Previously, it has been noted that the observed low activation energies compared with the classical model for a hindered methyl-group rotation point to a correlation spectrum. It is very difficult to determine correlation functions. Therefore, we will study in a way in which the behavior of the motion of the methyl groups can be described. c

c

c

c

Separation of the Absorption Curves The shape of the absorption curves shows that the lateral extremes differ from the inner extremes. Extrapolation of the inner side of the lateral extremes to the value H gives curves which incline concavely to the abscissa. In performing a convex extrapolation corresponding to the inner extremes an intersection with the abscissa alongside H is obtained. This indicates that the absorption curves arise from the superposition of several curves, whose centers of symmetry are shifted with respect to H , at least for the lateral components. As shown before, the splitting of the absorption curves to several components is caused by the freezing of methyl-group rotation. Andrew and Bersohn (2) have calculated the corresponding line shapes for an isolated rigid three-spin system in a powderlike substance. From this one expects for rigid methyl groups having no interactions between each other a spectrum of three line groups which are symmetrical to a center. If the spin groups interact with their neighboring groups, the lines broaden. The resulting line shape can be described approximately by three gaussian curves, two of which have identical shapes lying sym metrically to the third one. Gutowsky and Pake (5) and Kakiuchi et al. (6) confirmed these results of Andrew and Bensohn (2). 0

0

0


39.

KOSFELD E T A L .

601


Corresponding to these results it is assumed that the line shape of the absorption curves of PC and PMST at low temperatures can be described by the expression: 7 = C exp [- a (H - H ) 2 ] + C {exp [- a (H - H + H ) ] + exp (H — H — H i ) ] } 0

x

e

0

±

0

x

2

(8)

2

0

I is the intensity of the absorption curves depending on the field strength H; C describes the intensity of the inner gaussian curve and C i the intensity of the lateral gaussian curves; a and «i are the corresponding shape parameters of the curves. H determines the position of the inner curve, and H i determines the shift of the lateral curves corresponding to H . In this expression it is important that the inner gaussian curve is a superposition of several curves. It is composed of a contribution of the frozen methyl groups and a contribution of the still rotating methyl groups and benzol-ring protons. In PMST there is an additional contribution of the C H groups. The contribution of the rigid C H groups must be separated from this. 0

0

0

0

2

3

In the experiment the differentiated absorption curves are obtained. Therefore, we begin with the differentiated form of Equation 8. The differentiation of the field strength H is marked here by a comma. Ac cordingly, the intensity of the lateral extremes for the differentiated absorption curves is marked (J/) 1/2 and the intensity of the inner ex tremes (Ι ')ι/2· The linewidths resulting from the distance of the ex tremes are marked ( Δ Η ι ) for the lateral gaussion curves and (ΔΗ )ι/2 for the inner curve. In this way the areas under the different gaussian curves defined by Equation 8 are determinable. 0

1 / 2

0

F designates the area under the central gaussian curve and F i that under one of the lateral curves. The total number of protons Ν in the sample is proportional to the area F under the complete absorption curve. F follows from the summation of F and 2F . From this, the total number of protons Ν is: 0

0

N~(F

+

0

a

(9)

2F ). 1

Since the protons in PC are bound at 3/7 in methyl groups and in PMST at 3/10, the number of protons in the methyl groups N are: M

N

u

= jN

and


(10)

602


PROCESSES

The number of protons in the frozen methyl groups IVME can be defined as being proportional to an area consisting of 2Fi and a contribution F contained in F : 2

0

(2 F + F )

NME~

For the relation N /N 13: MB

M

x

(12)

2

for PC from Refs. 9, 10, and 12 and Equation N N

M

M

E

_

2F + F t

2

3/7(F + 2F ) 0

^

1

and for PMST from Refs. 9, 11, and 12 and Equation 14: N 2F + F N ~ 3/10 (F + 2F ) M E

X

M

(14)

2

0

;

1

is obtained. Andrew and Bersohn (2) have shown that for rigid methyl groups the intensity of the spectrum in the central part of the absorption curve is twice as high as that of the lateral parts—i.e., (15)

F = 2F 2

1

If we determine the different areas by values of (1/) i , (Ιό) i/ , ( Δ Η ) i , and ( Δ Η ) ι , we can transform Equation 13 by Equation 15, and we obtain the relation for PC: /2

0

2

1

/2

/2

From Equations 14 and 15 we obtain the corresponding relation for PMST:

With Equations 16 and 17 it is possible to determine the transition of methyl groups from the rotating state to the frozen state as functions of temperature from the experimental data. ( Δ Η ) ι corresponds to the linewidths for the inner component of the absorption curves as shown in Figures 1 and 2. These values are false, owing to the superposition of the lateral components. A correction is reported later. The values of the lateral components of the absorption curves marked in Figures 1 and 2—in these figures these parts of the curves were marked as the broad components—have lost the meaning of a linewidth. These values give the position of the lateral gaussian curves only. That means it is possible to calculate the shift of the lateral gaussian curves with respect to H from these values. A correction is also necessary to determine the 0

/2

0


39.


KOSFELD E T A L .

( δ Η

603

1 1/2 )

5x δ x

no

120

HO

130

150

Τ °K Figure 6. Temperature dependence of the corrected (AH ) 'Values for PC 1

1/Z

"G~

I00

150

200

Figure 7. Temperature dependence of the corrected (AHJ^g-values for PMST relation between the ordinates (Ιχ') /'{Ι ')i/2 of the lateral and inner extremes. At the position of the extreme value a contribution, which belongs to the dropping side of the component of the curve must be 1/2

0


ADDITION A N D CONDENSATION POLYMERIZATION PROCESSES

604

100

2

0

0

JL

Figure 8. Intensity relation (1/) / /βό)ι/2 °f the lateral and central extremes for the differentiated absorption curves as a function of ternperature for PMST 1 2

taken into account. Therefore, this correction must start with the dif ferentiated form of Equation 8, where a , «i, C , and C must be replaced by (J/)i/2, Uo')i/2, and ( Δ ί / ι ) and (AH ) . Thus, the values of (h')i/2 and (I ')i/2 taken from the experimental curves are used to perform the correction mentioned above, according to differentiated Equation 8. Hence, corrected (I/)i/2 and (10) 1/2 values are obtained, which are used for a second correction. In this way the true intensity 0

1 / 2

0

0

±

1/2

0


39.

KOSFELD E T A L .

605


values are iteratively approximated. This iteration must of course con sider the correction of the linewidths. First there is a question as to how ( Δ / / ) ι , the linewidth of the lateral gaussian curves, can be determined. A determination is not pos sible by the inflection point distance because the sides of the lateral curves near H are covered by the middle part of the spectrum. There fore, a point of the absorption curve must be found which is shifted by 0.5 ( Δ ί / ! ) ι with respect to the position of the lateral extremes. In determining this point the contribution of the central gaussian curve similar to the determination of the intensities must be considered. 1

/2

0

/2

110

120

130

HO

150

T

Figure 9. Intensity relation (Ι/Λ/*/(Ιο'λ/* °f lateral and central ex tremes for the differentiated absorption curves as function of temperature for PC t n e

As mentioned above an increase of the linewidth occurs by the superposition of the lateral and central gaussian curves or by a shift of the extremes of the differentiated absorption curves, which is essentially the same. Again the real linewidths can be obtained iteratively only. For this purpose we begin with the twice-differentiated form of Equation 8 because the question after the shift of the extremes for the differentiated curves is identical to the question after the shift of the zeros for the twice-differentiated absorption curves. There the twice-differentiated Equation 8 is developed in a Taylor series at the positions of the extremes breaking off after the second term. At these positions the function is


ADDITION A N D CONDENSATION POLYMERIZATION PROCESSES

606

approximated by straight lines, but it is easy to determine the shift of the zero by superimposing the straight lines. This indicates the direction and size of shift of the zeros for the twice-differentiated equation (Equa tion 8) and also gives a correction for the linewidths. Repetition of this performance gives finally the true linewidth. The corrected (AH ) values are nearly identical with the dashed extrapolation to low tempera tures for the small linewidths in Figures 1 and 2. Figures 6 and 7 give the corrected ( Δ / / ι ) values for PC and PMST. Figures 8 and 9 show the corrected intensity relation ( Z i ) i / (io')i/2 for PC and PMST. The relation N /N can be calculated from Equations 16 and 17 by the data obtained from Figures 1, 6, and 8 for PC and from Figures 2, 7, and 9 for PMST. Figure 10 displays these results for PC and for PMST. 0

1/2

1/2

/

ME

100

150

/2

M

200

°K Figure 10. Number (N ) of frozen methyl groups with respect to the total number (N ) of methyl groups as a function of temperature for PC and PMST ME

M

For low temperatures the curves must approach asymptotically the value 1. The physical meaning of these curves is as follows. It is possible to determine for each temperature in the transition range how many methyl groups in relation to the total number of methyl groups change from a frozen state to one of free rotation during a temperature increase


39.

KOSFELD E T A L .


607

ST. Yet it must be remembered that the decision as to when a methyl group is in a frozen or a state of free rotation is made by the N M R experiment in such a way that the methyl group under consideration does or does not cross a specific rotational frequency. This deciding rotational frequency has been introduced in Equation 7 as the correlation frequency v . According to Equation 7 v is temperature dependent. Figure 5 shows the v values calculated from Equation 7 as a function of the reciprocal temperature. c

c

c

Figure 11. Number of methyl groups changing from a frozen state to rotation during a temperature variation as function of temperature for PC and PMST


608


PROCESSES

The NMR experiment not only answers the question of how many methyl groups are frozen and how many are moving freely but also how many cross the threshold defined by v during a small temperature varia tion δΤ. Therefore, differentiating Ν /Ν = f(T) as a function of temperature as shown in Figure 10 relates the methyl group distribution to the different frequencies v . c

ΜΈ

Μ

c

The result of this differentiation is shown in Figure 11 for PC and for PMST. The functional relation given in thesefigurescan be described as follows: For a rise in the temperature ST at a defined temperature Τ the NMR experiment forces the number of methyl groups to change from a rotational movement below v to a rotational movement above v accord ing to δΤ. This number is related to the number of methyl groups occur ring in a frequency interval δν above the frequency v given by the temperature Γ. c

c

c

It is possible to demonstrate the relation between ν and Τ giving an arbitrary assumed frequency distribution function f(T,v). This is done in Figure 12 for two temperatures Γ and Τ + δΓ according to the function: ^=/(T,v)8v

(18)

Ν is the total number of methyl groups, AN the number of methyl groups occurring in the frequency interval δν. The number of methyl groups crossing the frequency v is represented by (AN/N)ST. Figure 12 dem onstrates this number by the area difference F — F . From this it follows: c

x

2

(19)

F -F 1

2

' C

= J Ο = - f

c

f(T,v)

dv- f

/(Τ + δΓ,ν)^ν

Ό [ / ( Γ + δΤ,ν) - / ( T , v ) ] d v

For small δΤ f(T + δΓ, ν) can be approximated by a Taylor series breaking off after the first term: / (Τ + βΤ, ν) = / (Τ, ν) +

8T


(20)

39.

KOSFELD E T A L .


609

J_ AN. όν Ν

Figure 12. Frequency distribution function f(T, v) vs. temperature at two different temperatures Equation 20 with Equation 19 leads to: */(T,v)

civ

(21)

When AN

_/AN\

(22)

~Ν~~~\Ν~)δτ

it follows from Equations 18 and 21:

f{T,v) 8v = T

I

— —

dv

(23)

Assuming for a further interpretation (this is arbitrary) the unknown frequency distribution corresponding to a Maxwell distribution, from Equation 23 a relationship between δν and δΤ of the form: δΤ _ δν 2Γ ~~ is obtained.


(24)

ADDITION AND CONDENSATION POLYMERIZATION PROCESSES

610

Therefore, if it is possible to find a frequency distribution function corresponding to experimental results, it would be possible to find a relation using Equation 24 which would correspond to a Maxwell distri bution. With this equation the variable 1/δΓ X N M E / N of Figure 11 should be transformed into the variable 1/δν Χ N E / N . Κ P °f values using this performance corresponding to the intersection points of the correlation frequencies with the theoretically determined distribution (cf. Figure 5) are gained, the theoretically determined distribution func tion would be confirmed by experimental results. m

M

m

a i r s

The distinct maximum in Figure 11 at 140°K. for PC and at 186°K. for PMST demonstrates that the change of the number of methyl groups crossing the limiting frequency which is critical for line splitting occurs most frequently at this temperature. The temperature of 140 °K. marked as T has been used already to extrapolate the second moment reported in Figure 3. For PMST this extrapolation was not necessary because the measurements could be made to sufficiently low temperatures. E

It is not possible to derive the frequency distribution curves from the curves reported in the Figure 11. The only value of these demonstrations is that they uncover experimental criteria which are useful for examining a model describing the methyl group motion. Literature Cited (1) Andrew, E. R., Phys. Rev. 91, 425 (1953). (2) Andrew, E. R., Bersohn, R., J. Chem. Phys. 18, 159 (1950). (3) Glasstone, S., Laidler, F. K. J., Eyring, Ε. H., "The Theory of Rate Proc esses," McCraw-Hill, New York, 1941. (4) Gutowsky, H. S., Meyer, L. H., J. Chem. Phys. 21, 2133 (1953). (5) Gutowsky, H. S., Pake, G. E., J. Chem. Phys. 18, 162 (1950). (6) Kakiuchi, Y., Shono, H., Kigoshi, K., Komatsu, H., J. Chem. Phys. 19, 1069 (1951). (7) Kovarskaja, E. M., Zigunova, I. E., Slonin, J. J., Urman, J. G., Nejman, M. B., "Sammelband Chimiceskie Svojstva: modificacija, polimerov," SSSR, 1964. (8) Landolt, H., Börnstein, R., "Zahlenwerte und Funktionen," Teil 3, II, Springer-Verlag, Berlin, 1951. (9) Murakami, I., Kawai, Α., Yamamura, H., J. Sci. Hiroshima Univ., Ser. A-II, 27, 141 (1964). (10) Odajima, Α., Woodward, A. E., Sauer, J. Α., J. Polymer Sci. 55, 181 (1961). (11) Perlman, M. M., Bloom, M., Phys. Rev. 88, 1290 (1952). (12) Powles, J. G., Polymer 1, 219 (1960). (13) Powles, J. G., Gutowsky, H. S., J. Chem. Phys. 18, 162 (1950). (14) Ibid., 23, 1962 (1955). (15) Prietzschk, Α., Kolloid-Z. 156, 8 (1958). (16) Sinnot, Κ. M., J. Polymer Sci. 42, 3 (1960). (17) Slichter, W. P., Makromol. Chem. 34, 67 (1959). (18) Slichter, W. P. Mandell, E. R., J. Appl. Phys. 30, 1473 (1959).


39. (19) (20) (21) (22) (23)

KOSFELD E T AL.

Mobility

of Methyl

Groups

611

Slonin, I., Urman, J. G., Konovalov, A. G., Sammelband Chimiceskie Svostva: modificacija polimerov lzd. AN SSSR (1964). Steijskal, E. O., Gutowsky, H. S., J. Chem. Phys. 28, 388 (1958). Stuart, H. A., "Die Physik der Hochpolymeren," Vol. III, Springer-Verlag, Berlin, 1955. Van Vleck, H. J., Phys. Rev. 74, 1168 (1948). Visweswaramurthy, S., Indian J. Pure Appl. Phys. 3, 261 (1965).

RECEIVED March 25,

1968.


Mobility of Methyl Groups in Polycarbonate and in Poly--methylstyrene

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