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The measurement of nuclear magnetic relaxation time T1 in polymers by means of spin echo technique
Trappeniers, Gerritsma, Oosting,
N. J.
Physica
C. J.
30
997-1017
P. H.
1964
THE MEASUREMENT OF NUCLEAR MAGNETIC RELAXATION TIME T, IN POLYMERS BY MEANS OF SPIN ECHO TECHNIQUE by N. J. TRAPPENIERS,
C. J. GERRITSMA
180th publication Van der Waals-laboratorium,
and P. H. OOSTING
of the Van der Waals
Universiteit
Fund
van .I\msterdam,
.4msterdam,
Nederland.
Synopsis A brief description solid
materials,
about
is given
especially
The
spectrometer recovery
6 ps after a 90” pulse of 1 ,US.The spin-lattice
polytetrafluoroethylene to 280°C. The observed results
and polyoxymethylene
minima
and one for P.O.M.
from mechanical
Spin diffusion
in the log Ti vs i/T
are discussed
of spin diffusion
curves,
of
the
to the study
of
spectrometer
is
time T1 of polyethylene, in the range
three for P.E., theory
: - 196°C
three for P.T.F.E. and related
to the
loss measurements.
to be the dominating lattice,
relaxation
of relaxation
in the three polymers.
in a cubic
adapted
time
was measured
in the light
and dielectric
is thought
Tr at low temperatures first
of a spin echo
of polymers.
sprinkled
factor,
governing
In the appendix, with
relaxation
the behaviour
therefore, centra,
of
the model
is treated
to a
approximation.
3 1. Introduction. A considerable amount of work has been done on the study of polymers by nuclear magnetic resonance. Most authors have been concerned solely with the measurement of line width as function of temperature, using steady-state methods, and a number of review articles have been devoted to the results thus obtained [Slichter r), Powles a), Sauer and Woodward a)]. It was pointed out, however, that the direct measurement of nuclear magnetic relaxation times, especially of the spin-lattice relaxation time T1, would provide more insight into the motional behaviour of polymers [Powles 4) 5), Smith 6)]. Furthermore, there would be much interest in correlating nuclear magnetic relaxation times with the results obtained from dielectric and mechanical relaxation measurements. The comparative lack of data on T1 in polymers must be attributed to the experimental difficulties which, in general, impair the accurate determination of Tr in solids. Until recently the only practical method of measuring T1 in polymers was the progressive saturation method as first described by Bloembergen, Purcell and Pound 7). The drawbacks of -
997 -
.
998
N. J. TRAPPENIERS,
c. 5. GERRITSMA
AND
r. H. OOSTING
this technique are well known. For example, it does not yield absolute values for T1; the determination of T1 necessitates a knowledge of I’2 (spinspin interaction time); the stability of the marginal oscillator becomes a problem when T1 is long, while in the case of broad resonance lines the n.m.r. signal is weak and the accuracy poor [\I:ard S)]. In spite of these difficulties some results have been obtained
by the satu-
ration method for polyethylene
9) investiga-
and teflon. Wilson
and Pake
ted teflon and found between 170°K and 290°K two distinct spin-lattice relaxation times. Ward 8) measured T1 for the methyl proton and also for the aromatic proton in poly(ethylene trideuteromonohydrogen terephthalate). Recently Slichter 10) found two values for T1 in linear polyethylene crystallised from a solution, in contradiction with earlier results of Wilson and Norberg 9), using a free induction and Pake 9). However, Bloom method, stated that they found two T1 values at room temperature in thrl sample used by Wilson and Pake. The spin echo technique, invented by Hahn 11) and further developed by Carr and Purcell 12), constitutes a much more elegant and powerful method for measuring relaxation times. Thusfar, however, its application has been limited mainly to liquids. The reason for this lies in the fact that in solids the dipole-dipole coupling between magnetic nuclei is strong, resulting in a short spin-spin interaction time Tp. Consequently, the free induction decay tail, following after a 90” radio-frequency pulse, will also be very short and in order to observe a free induction signal a number of stringent conditions have to be met. 1. R.f. pulses have to be short compared to Tz of the sample but at the same time sufficiently intense to obtain the desired 90” or 180” rotation of the magnetisation vector of the sample. 2. The recovery time of amplifier and detector, after severe overloading by the r.f. pulse, has to be short compared to T;?. 3. The bandwidth of the amplifier has to be sufficient so as to avoid signal deformation. Up to the present spin echo measurements in solid polymers have been made only in a number of favourable cases where 1’2 was comparatively long. Thus Lowe e.u.13) measured T1 in polyethylene, Powles e.u.5) determined T1 and T2 in polyisobutylene and later 14) also in silicon polymers. Nolle and Billings 15) measured T1 in polyisobutylene under high Kawai e.a.16) 17) have applied the spin echo method pressure. Recently to the measurement of T1 in amorphous polymers such as polymethyl methacrylate etc. In this paper results will be given of measurements of T1 in three polymers which may be partly crystalline: namely polyethylene, polytetrafluoroethylene and polyoxymethylene. This was achieved with the use of a new spin echo spectrometer which is equipped with a powerful trans-
N~CLE!AR s~iN-~4TTIcti ~~I~LAXATIO~~ ~ti POLYMERS
mitter,
so that a 90” pulse is performed
499
in 1 ,LLS.By the use of a pulsed dam-
ping circuit, which precedes the broadband receiver, the spectrometer has a recovery time of 6 ,us. This means that signals can be observed in solids displaying
an n.m.r.
3 2. The
line width of approximately
spin echo spectrometer.
a. General
40 gauss. principles.
All the mea-
surements were made with a phase coherent spin echo spectrometer, which in its general design is similar to the apparatus described by Bucht a, Gutowsky and Woessner 18). PULSE
TRIGGER GENERATOR 4 100 KC/S OSCILLATOR
GENERATOR 4,
+_____ lb.L
MC/S
_
,,
FREQUENCY DOUBLER
FREOCHANGER
+ ZdGATE
t
+ POWER \I
AMPLIFIER $ C.R.O.
DETECTOR
, -
BROAOBANO AMPLIFIER
RF
HEAD
t
II
I
Fig. 1. Block diagram of spin echo spectrometer
In fig. 1 a block diagram of the instrument is shown. By means of the 100 kHz crystal oscillator, the trigger generator, the trigger selector, the three pulse generators and the pulse mixer a pulse train is generated consisting of 3 pulses which are variable in length, height and mutual spacing. Alternatively a pulse train may be produced consisting of a large number (10, 100 or 1000) of identical pulses, to be used for example in measurements of Tz according to the method of Carr and Purcell. Briefly, the working of these units is as follows. The trigger generator, which is monitored with a frequency of 100 kHz, generates sharp pulses with repetition interval of either 1,2,4,6,8 or 10 ps. These pulses are counted by the trigger selector, which is essentially a decimal counter fitted with 8 beam switching tubes (Burroughs). By means of 3 coincidence circuits the trigger selector is made to transmit a trigger pulse at preset intervals to each of the three pulse generators and to repeat this pattern at a well defined repetition rate.
1000
_~
Ii.
J. TRAPPENIERS,
The chief advantage
C. J. GERRITSMA
in using a counter
AND
P. H.
OOSTING
lies in the fact that
the intervals
between successive pulses are accurately preset by the coincidence circuits (the accuracy being 1, 2, 4, 6, 8 or 10 /is according to the setting of the trigger generator), so that they do not have to be determined from readings on the oscilloscope. A fourth coi’ncidencc circuit is used to trigger the oscilloscope at any chosen time. The pulses, delivered
by the pulse generators,
pass through
the p~~lsc
mixer and then operate the first gating circuit. This gate is fed with radio frequency at 14.4 MHz, i.e. half the Larmor frequency used in the measurements, which is obtained through frequency multiplication from the basic 100 kHz oscillator. The radio-frequency pulses at 14.4 MHz from the first gate pass through a frequency changer which doubles the frequency to 28.8 MHz, but which unavoidably alters the shape of the pulses to some extent. A second gate, which is also operated by the pulse mixer, is used therefore to reshape the r.f. pulses. By working at half the Larmor frequcncy the problem of leaking through radiation is avoided. The power amplifier increases the voltage of the r.f. pulses to the desired level, which is such that when applied to the r.f. sample coil of the tank circuit, the pulses produce an r.f. field 2H1 a 120 gauss. The single coil at the same time serves the purpose of receiver coil for the induced n.m.r. signals. These signals pass through a cathode follower followed by a broadband amplifier and after detection they are displayed on a 545 Tektronix oscilloscope. (A more detailed description of the apparatus will be published shortly). b. Circuit diagram of r.f. head. The tank circuit, the cathode follower together with two accessory circuits: the “pulsed damping circuit” and a \%ual tuning indicator are housed in a separate, well shielded. r.f. head. The wiring diagram is shown in fig. 2. Condensor Cl is used to tune the sample coil to resonance while the two Lrariable condensers
Cz and C3 couple
the
coil to the
power
amplifier.
The choice of the capacity of condensers Cl, Ca and C3 is dictated by a compromise between two factors: on the one hand, the aim of optimum transmission of r.f. energy to the coil will fix the ratio between the coupling and tuning capacity; on the other hand, the best condition for detection of n.m.r. signals is obtained with total coupling capacity small in comparison with the tuning capacity. The choice adopted is such that with a tank circuit Q factor of 100 the transmitted r.f. energy is sufficient to obtain a 90” flip of the sample magnetization vector with a pulse of 1 ,a, As stated in the introduction a pulse of 1 ,US is sufficiently short to allow relaxation measurements in most solids. There remains, however, one big difficulty. The intense r.f. pulse causes “ringing” of the tank circuit, meaning that the applied voltage will decay comparatively slowly in a characteristic time which is proportional to the
NUCLEAR
SPIN-LATTICE
RELAXATION
IN POLYMERS
1001
Q factor of the circuit. As a result, the broadband amplifier is severely overloaded for some time and no signal can be observed immediately after the pulse, a fact which largely
obliterates
Fig. 2. Radio-frequency
the advantage
of working
head.
with short pulses. The disastrous effect of tank circuit “ringing” could be avoided in part by decreasing the Q factor of the circuit, but as this would reduce at the same time the signal to noise ratio the sensitivity of the method would be impaired. A better solution of this problem was found by equiping the tank circuit with a “pulsed damping circuit” (p.d.c.) which works on the following principle. Pulses taken from the pulse mixer are amplified to -200 V with the use of an E180F pentode (V3) and an EC50 thyratron (V2). An RC filter partly differentiates this negative pulse so as to produce at the end of the pulse a sharp positive peak of sufficient amplitude to fire an EC50 thyratron (Vl) in parallel over the tank circuit. When conducting, this thyratron short circuits the sample coil, thus strongly damping the “ringing” of the tank circuit, while the negative part of the pulse prevents the thyratron from conducting during the application of the r.f. pulse. Fig. 3 clearly demonstrates the beneficial effect of the p.d.c. in limiting the r.f. pulse over the sample coil. Fig. 4 shows the Bloch decay following a 90” pulse without p.d.c. in a solution of Fe(NOa)a containing 5 x 101s ions/cma: it is evident that “ringing” of the tank circuit blocks the ampli-
1002
N. I.
TRAPPENIERS,
C. J. GERRITSMA
AND
P. H. OOSTING
fier for about 20 jrs. In fig. 5 the same decay tail is taken with p.d.c. showing that blocking of the amplifier is limited to about 6 jts. The great advantage of the 1j.d.c. for solid state work may be judged from fig. 6 in which is shown Bloch decay following a 90” pulse in polyoxymethylene \Vithout 1l.d.c. the decay tail would not have been visible.
at -.- 165°C.
When the relaxation time 7’1 of the specimen is long, tuning of the tank circuit to exact rtlsonance becomes difficult and time consuming. The procedure may be greatly facilitated by incorporating in the r.f. head a “magic eye” EM34 (V4) as tuning indicator. The r.f. pulses, after rectification, charge condenser C+ The magic eye is sensitive to the voltage over this condenser which will attain its maximum value when the tank circuit is tuned to resonance. For a satisfactory operation the repetition rate of the pulses should not be smaller than 10 Hz. Lastly, the E180F (7’5)) serves the purpose of a cathode follower, which couples the r.f. head to the broadband amplifier. c. Temperature control. A conventional gas-flow thermostat is used to control the temperature of the sample between room-temperature and 280°C. Below roomtemperature, down to - 196”C, a liquid nitrogen cryostat is used. Temperature is measured with copper-constantan thermocouples. Although a gas-flow thermostat is easy in use it has the drawback of producing a temperature gradient over the specimen. At the highest temperature, for example, a gradient as large as 2°C can exist over a sample of 5 mm in
bfU6Ll?A#
length.
SPIN-LATTICE
In the liquid nitrogen
RELAXATION
cryostat
1603
Iti Pc~L~WERS
the gradient
over the sample is less
than 0.05”C. 93. Measurement of TI in polymers. a. Procedure In the present investigation the spin-lattice relaxation
of measurement. time T1 is determined
from the free induction signals produced by a 180”-90” pulse sequence. It is well known la) that the magnitude of the induction signal M(t), which follows after the 90” pulse, depends on the time interval the 180” and 90” pulses in the following manner :
M(t) = Mo[ 1 -
t between
2 exp (-t/Tl)]
(1)
where MO is the equilibrium value of M(t), i.e. MO = M(t) for t > T1. If M(t) is measured for a number of values of t, T1 can be deternimed graphically from the plot: log [MO - M(t)]/2Mo versus t. processes with different In some polymers, however, two relaxation relaxation times Ti and T; may operate simultaneously and the procedure for their determination is somewhat more complicated. Equation (1) now reads :
M(t) = Mb [I -2 The equilibrium clei participating
exp (-t/Ti)]
+ M”,[l-2
exp (-t/T;)].
values MI, and M”, are proportional to the number in the two respective processes, while
(2) of nu-
Mb + Mj = M(t) when t > Ti, T;. From
(2) 4 [Mb + Mb -
M(t)] = Mb exp (-t/T;)
+ M”, exp (-t/T;).
(3)
If T; is small with respect to Ti the second term in (3) will become negligeable for sufficiently large t. A plot of log [MI, + M, - M(t)] versus t will yield a curve which tails off into a straight line. From this Ti and Mb are found and after substitution in (3) T; and M”, can also be calculated. Provided Ti and T; differ at least by a factor 4, the accuracy in the determination of the longer relaxation time Ti is about 5% and in the shorter relaxation time T; about 10%. If the difference between Ti and T; becomes smaller the accuracy becomes much poorer up to a point when the logarithmic plot scarcely shows the existence of two different relaxation times at all. Thus, when the difference between Ti and T; is smaller than a factor 2, the experimental points, within their margin of error, lie on one straight line with an apparent relaxation time which is an average of Ti and T;. It is interesting to note that from the decay signal, after the 90” pulse, it can be seen immediately whether the sample shows one or more simultaneous relaxation times. In the first case only the initial height of the decay signal varies with t ,while in the case of two or more values of T1 the shape as well as the initial height of the Bloch decay signal change with t.
1004
N. J. TRAPPENIERS,
b. Results
and
C. J.
GERRITSMA
qualitative
.4ND P. H. OOSTING
interpretation.
Three
polymers
are
used for the T1 measurements: 1. polyethylene (-CHB--)~ : a sample of PE 75 was obtained from du Pont *) with following characteristics: optical m,p. 104”C, density 0.912 g/cm3 at 24.5”C and 3.3 methyls/lOOC. 2. polytetrafluoroethylene (--CFZ-)~~ : a sample of the standard commercial “Teflon”, made by Montecatini was used. Specification: melting begins at 327X, density 2.14 g/cm3 at 23.5”C. 3. polyoxymethylene (-O-CHz-) II: a sample of “Delrin” 500 .X was obtained from du Pont *) ; melting begins at 174”C, density 1.405 g/cm3 at 24.5”C. Readings of T1 were made at atmospheric pressure, as function of temperature in the range from ~ 196°C to about 280°C. Later, a second series of T1 values was measured between roomtemperature and ~ 196”C, by using three degassed samples. For this purpose the samples were vacuum pumped for 48 hours and sealed off under vacuum in glass tubes. In every case the working frequency of the spin echo spectrometer was 28.8 MHz. The results obtained for the three polymers are shown respectively in fig. 7, 8 and 9 as a plot of log 7’1 against reciprocal absolute temperature 200
100
30
'Ol s
(3 -30 I'
-60 1
-90 '
-120
-150 I
-180
-19OOC -f
TI
/' 4 5:T
/ Y
32-
d' /
A'
I-
/
0.5 0.4 0.3 1
t
0.2
l
d' /
LJ-=++
t 0.1
I
I
1
2
I,
m.p. 3
Fig. l
_
T
T1 in normal
o 271in degassed
lOOO/
Tabs
II
.&
I
I
I
1
4
5
6
7
7. Relaxation sample sample
time
8
for protons
I
I
I
I
I
9
IO
II
12
13
in polyethg’lene
(solid curve). (dashed
curve).
*) The authors arc 111uch iMlcl,ted to lb. N. G. McCru,,, of E.I. du Pout de Ncrnours and Comp., Polychemicals Dept., Research and I)?\ r’lopmvnt I)ivisioll, for providing thr samples.
Received
26-l l-63
NUCLEAR
200 I
I 101
100 I
30 0 I,,,
SPIN-LATTICE
-30
-60
-90 I
RELAXATION
-120 I
1005
IN POLYMERS
-160 I
-190 I
-190% I
5
51 41 31 2[
1c
1
‘
1
0.5 0.4 0.3 0.2
0.1
Fig. .
8. Relaxation
T1 in normal
o T1 in degassed
sample sample
time for fluorine
in polytetrafluoroethylene.
(solid curve). (dashed
curve).
l/T. The most striking feature of these curves is the appearance of one or several minima, shaped like a V. This aspect suggests that it should be possible to give a qualitative interpretation of the behaviour of T1 in polymers in the light of the now generally accepted theory of nuclear magnetic relaxation. A given proton or fluorine nucleus experiences a local magnetic field created by neighbouring nuclei. This field is a rapidly fluctuating function of time due to the movement of reorientation of polymer chain segments, and as such it is responsible for spin-lattice relaxation. In a first approximation only the local field produced by one nearest neighbour need be considered, as the -CHs--, -CFsor -0-CHssegments provide fairly isolated two-spin systems of hydrogen or fluorine nuclei. This means that the influence of nuclei belonging to another segment, either in the same or in a different polymer molecule, is neglected. It also implies that
1006
K;. J, TRAPPENIERS,
0 200 1
100,
s
100 I
-30
-60 I
30 III
C. J.
GERRITSMA
-120 I
-90 I
AND P. H. OOSTING
-19OOC ,
-180 1
-150 !
-----+f
T
50 LO 30 -
5
__9-----_cr-
20 -
r
/-
I 0
/ d’ d’
10 -
5L32-
I
/’
I
I-
0.5 0.1 0.3 0.2 -
---+ lOOO/
!v
0.1 I
1
1
I
I
I
1
I
1
I
1
I
2 m.p.
3
L
5
6
7
6
9
10
II
Relaxation
time
Fig.
.
7
’
I
9.
T1 in normal
c, T1 in degassed
sample
sample
for protons
es ,
12
,3
in l~~~l~ox~methylene.
(solid curve) (dashed
curve).
no account is taken of translational movement of one molecule with respect to another. The simplest way of characterising the fluctuating local field of the twospin system is to introduce a single correlation time TV, which sets a time scale to the reorientation rate of the chain segments. To this model we can apply the general relation 20) which expresses T1 as function of 7C for intramolecular relaxation in the case of a two-spin system in spatial isotropic motion : T ; ‘=
2/5,I(Z +
1) /%zy4b-‘3
where I is the nuclear spin, y the gyromagnetic ratio, b the distance tween the two spins and o the resonance frequency. A plot of T1 versus according to (4), results in a V-shaped graph with a minimum value of when : cwrc = 0.6158.
(4) beTV, T1
(5)
NUCLEAR
SPIN-LATTICE
RELAXATION
This is the value of 7c for which the relaxation
1007
IN POLYMERS
mechanism
is most effective.
The movement of reorientation of polymer chain segments is likely to be a thermally activated process so that it seems reasonable to assume that: TV = TOexp (E/RT), where E is the activation
(6)
energy of the process and TO a constant.
ciently high temperature, 7c is small and w%“, < 1, so that with (6), reduces to log TI = -log (5 Go) - E/RT, where C = 2/5 1(1 + 1)Rsy4b-6. At low temperature, when &%-z >
At suffi-
(4), together (7)
1, (4) reduces to
log TI = log (0%0/2C) + E/RT.
(8)
Hence, a plot of log 1‘1 vs l/T, according to this theoretical model, also results in a V-shaped graph; the slopes are directly proportional to the activation energy and the minimum in log T1 corresponds to relation (5). If the simplified behaviour of the two-spin system is applied to the measured T1 curves for the three polymers, the following conclusions may be drawn. 1. On heating polyethylene from - 196°C (fig. 7), TI versus 1/T goes for both samples (normal and degassed) through a first V-shaped minimum III. This means that a relaxation mechanism becomes active, which reaches its maximum efficiency at about - 15O”C, where T1 goes through its minimum value. According to (5) the correlation time in this minimum is given by: TV (min) = 0.6158/m. In correspondence
to TV a correlation
frequency
may be defined as:
Yc = 1/(27UTc) which, in the minimum, takes the value Ye (min) = 46.77 MHz. As the temperature increases this relaxation mechanism becomes less efficient for its oc, according to (4), increases rapidly. A second relaxation process now takes over, causing T1 again to decrease and to pass through a second minimum II at - 11 “C where in turn the correlation frequency of the second process has the optimal value Ye (min) = 46.77 MHz. At still higher temperatures a third process becomes active, giving rise to minimum I at 43”C, again with Ye (min) = 46.77 MHz. From fig. 7 it is seen that at low temperature T1 is larger for the degassed than for the normal samples. The difference increases with TI, while at room temperature the relaxation times become equal. 2. In teflon (fig. 8) two relaxation times occur simultaneously for the spin-lattice relaxation of F 19. Following previous authors 9) we attribute
1008
N.
J. TRAPPENIERS,
C. J. GERRITSMA
AND
P. H. OOSTISG
the upper curves with the longer relaxation times to the crystalline region of the polymer, the lower curves to the amorphous region. In total three minima were found: in the crystalline region at 168°C (I) and at 25°C (II); in the amorphous region at 21°C (III). At the highest temperatures available in our thermostat the lower Tr curve (amorphous phase) is still descending. It seems likely that it will eventually pass through another minimum (IV?), which, however, we were unable to observe. In each minimum, the correlation frequency of the associated mechanism is given by (5) i.e. vc (min) = 46.77 MHz. In th e 1ow temperature range the solid curve (normal sample) as well as the dashed curve (degassed sample) become flat. The latter sample shows longer Tl’s, which, however, become equal below about - 160°C. At roomtemperature T1 has the same value for both samples. 3. The graph of T1 in polyoxymethylene (fig. 9) only shows one minimum (I) at 12°C. Apparently a single relaxation mechanism is operative with vc (min) = 46.77 MHz. The two TI curves flatten out at lower temperatures. As before, the relaxation times are the same for the normal and degassed sample at roomtemperature. In table I the characteristics of the T1 curves are given.
i Minimum
Polymer
Polyethylene
--em
TWllp. C i-.z_:-imm
r-1
1 Ii--’ i 103 I I ;_;z:---pm~
I‘[
’ ~~ 0.13; - 0.03
I --I,,, ;&;.Hiph i rrgirm *) [(Yl11.(8)]
III Polytetrafluoro~ ethylene
8.1
.t0.3
0.42LO.02
1
16815
2.27 2: 0.03
0.6210.05
II
2512
3.36_: 0.02
0.42 AL0.02
III IV? Polyoxymethylene
-150LS
I
! 2115 ‘>220
3.40 !;0.05 .’ 2.03
12,5
3.51+_0.06
0.14,O.Ol
Icqll.(7): 5.0
1.S
I ~
0.08 10.01
I 1
i
I
0.14_!_0.03
I 1
trmp. rrgion
1.2
I.0 3.6
/
8.8
J.il I I
4.4
6.4
*) In this region slopes are given of the dashed curws.
The simple theory of a two-spin system with single correlation time, though of value for the qualitative interpretation of TI curves, cannot be expected to give a quantitative agreement with the measurements. Moreover, the molecular movements of chain segments is not likely to be isotropic as implied in the foregoing equations. The discrepancy between the simple theory and the experimental results is apparent from the logarithmic plot of log TI vs l/T in fig. 7, 8 and 9, which does not result in straight lines as demanded by (7) and (8). Thus, the simple equations (4), (7) and
NUCLEAR
SPIN LATTICE
RELAXATION
IN POLYMERS
1009
(8) are not reliable for the computation of correlation times and activation energies from measured Ti values. The most obvious way of improving the theory would be to introduce a spectrum of correlation times as suggested by 0 da j im a 21). The fitting of a spectrum to a T1 curve with several minima is a tedious affair as one needs to adjust a large number of parameters, and it was not attempted in this investigation. However, the work of Odaj ima has shown the important fact that the occurrence of a correlation time distribution does not effect the position of the minima in the T1 curve, even though the shape of the curve is altered considerably. Equation (5), therefore, is of general validity and the correlation frequency of 46.77 MHz deduced from it for each individual minimum as well as the corresponding temperature given in table I, are reliable data for the further discussion of relaxation processes. S 4. Discussion. a. Polyethylene. From fig. 7 it can be seen that only a single TI curve was found in the temperature range from - 196°C to 130°C. This, apparently, is in contradiction with results of Bloom and Norberga) and Slichter la), who found two Tl’s in samples of linear polyethylene and concluded to the existence of a crystalline and an amorphous phase. The discrepancy may arise from the fact that our sample of PE 75 is strongly branched, which would preclude the formation of a crystalline phase. In order to identify the relaxation processes, associated with the three minima in the T1 curve of polyethylene, it is necessary to establish a correlation with the studies of mechanical and dielectric relaxation. The measurement of mechanical and dielectric losses in polyethylene yields three distinct absorption maxima at three critical frequencies. The associated relaxation processes, taken in order of increasing frequency, are termed cc, b and y. A rise in temperature produces a shift of each maximum to a higher value of the frequency. K abin ss), by varying the temperature from -80°C to 115°C was able to determine the critical frequency of the ,I!?and y process from mechanical losses between 0.5 and 5 MHz. His results are in agreement with those of Michailov, Kabin and Sazhin 33) who measured the CL, p and y process from dielectric losses between 50 Hz and 10 GHz in the temperature range from - 140°C to 110°C. When all their results are plotted on a logarithmic scale against 1/T the points fall on three reasonably straight lines, shown in fig. 10, from which the following activation energies are estimated: TVprocess 28 kcal/mole, /I process 16 kcal/mole and y process 11 kcal/mole. Since Kabin e.a. gave no specifications of their polyethylene samples, a number of additional measurements on PE 75 are plotted in fig. 10. These points include mechanical loss measurements [Kline, S a u e r and Woodward s4)] and n.m.r. linewidth data [Sauer and Woodward 3); Fuschillo and Sauer 25)].
1010
S. J. TRAPPENIERS,
2
C. J. GERRITSMA
,
/
3
L
.4ND
I’. H. OOSTING
I
6
-+1000/
6 ‘abs
Fig, 10. Z, fl ant1 ;I relaxation
in polyrthylene
If we assume that the observed spin-lattice relaxation in polymers is due to the same fundamental molecular processes, which are responsible for mechanical and dielectric relaxation, we may compare the correlation freof maxiquency vc (min) at temperatures of minimal T 1, i.e. temperatures mal spin-lattice relaxation, with the critical frequencies at temperatures of maximum mechanical and dielectric losses. The points corresponding to the minima I and II have also been plotted in fig. 10. It is fairly certain that minimum II at - 11°C is due to the y relaxation process and minimum I at 43°C to the p relaxation process. The expected minimum, which would correspond to the CCprocess at vc (min) = 46.77 MHz, has not been observed in our measurements, as it would fall at a temperature above the melting point of polyethylene. In this paper we shall not attempt to identify the specific molecular mechanism responsible for minima I and II in T1 as this would require a lengthy theoretical analysis. We may quote, however, the opinion of Sauer and Woodward 3) that the y process is due to diffusional motion of short chain segments while the /3process arises from movements involving branch .points as well. The last minimum III, at about - 15O”C, does not fit in with any of the drawn loss-curves and it is probably due to a new relaxation mechanism which has not been detected thusfar. A plausible explanation, we think, is to attribute it to the relaxation of terminal CHa groups. However, as the proportion of CHa groups in branched P.E. is not larger than about 3.3 per 100 C atoms, the relaxation through rotation of CHa groups can only
NUCLEAR
be effective
SPIN-LATTICE
if it is associated
RELAXATION
IN POLYMERS
with a spin diffusion
through the mutual flip of neighbouring
process.
1011 This operates
spins and is thus capable
of propa-
gating over large distances the influence of CHs relaxation. A rough estimate of the efficiency of the spin diffusion process can be made as follows. As a crude model, assume the CHs and CHs segments to be distributed over the lattice points of a simple cubic lattice. If each lattice point were occupied by a single spin the transition probability W of a spin flip between neighbouring spins could be related n.m.r. absorption line 26)
directly
to the second moment
of the
In order to obtain an average probability W,, for a spin exchange between neighbouring CHs, in our model, the experimental second moment of polyePZ 25 y?-2 at low temperatures has to be diminished with thylene (dws)_ the contribution of the spin-spin interaction within the CHz segment. This correction can be easily evaluated once the interproton distance is fixed by assuming tetrahedral angles and a C-H distance of 1.09 A:
The time interval 7-l =
w,, =
between successive
[(Aw2)oxp -
(?iZ&J~/30
flips T is then given by = (14y2)$/30 s-1 m 3300 s-i.
Spin diffusion is now treated as a discrete 3-dimensional random walk problem. The diffusing nuclear magnetization in a given lattice point (CHs) has a choice of 6 equally probable directions, since on an average in each lattice point there is always one of the two CHs protons having the right spin orientation for mutual flip with a neighbouring spin. Whenever magnetization reaches a site occupied by a CHs group it disappears through the rotational mechanism of spin-lattice relaxation. The average lifetime of the diffusing magnetization can then be calculated to a good approximation (see appendix A-5) as f = 6 715 C in which C is the concentration of CHa groups. Taking C = l/30 the average lifetime is found to be 0.011 sec. This means that nuclear magnetization travels to the site of a CHs group in a time which is small compared to the measured value of Tr = 0.42 s in the minimum III. Spin diffusion could thus be a very efficient mechanism in propagating the influence of the proposed CHa rotational relaxation to all the protons in the sample. Degassing of the sample results in longer Ti values (fig. 7). From this it may be concluded that, besides the CHa groups, the paramagnetic oxygen molecules also constitute effective relaxation centra. The contribution of the latter to spin-lattice relaxation depends only on their concentration, since relaxation at these centra is very fast. It may be concluded, however,
1012 that
S. 1. TRAPPENIERS, in the case of branched
C. J. GERRITSMrl
polyethylene
AND I’. H. OOSTIKG
the CHa-mechanism
minating factor at low temperature. b. Polytetrafluoroethylene. Contrary it found that P.T.F.E. two the temperature covered in degassed teflon - 160°C. mentioned in of fig.
with the
I and
to the
is the do-
the case polyethylene values of are present investigation, except 3, we the upper of a
phase and lower curves minimum III IV?) to amorphous region the polymer. mechanical and measurements, as as from line width two relaxation termed glass or /I glass II y relaxation known to operative in amorphous phase, only one relaxation process has thusfar detected. Fig. shows the of a by E and Sinno t 27) data for critical frequencies maximum mechanical dielectric losses for the frequency from line width for the and y plotted on log scale function of The points approximately on straight lines, activation cncrgies 34
Fig.
Crystalline
(a)
I (/?)
glass 11
relaxation
in polytetrafluoroethylene.
NUCLEAR
SPIN-LATTICE
RELAXATION
IN POLYMERS
1013
; Illers and Jenckelsa); McCrumse)] and at 12 MHz [Eby and Sinnott 27)]. The results of T1 measurements by Wilson and Pake 9) are also plotted. The correlation frequencies, corresponding to the minima I and III in our T1 curves, have been indicated in fig. 11. It may be concluded that minimum III at 21”C, is due to glass II relaxation in the amorphous phase and that minimum I, at 168°C is caused by the crystalline relaxation. There is no T1 minimum corresponding to the glass I relaxation, probably because the temperature at which it should occur lies above our available range. Tentatively we might assume that the hypothetical minimum IV, mentioned in 9 3, is related to the glass I relaxation. The behaviour of the relaxation curves in teflon below - 160°C differs markedly from the case of polyethylene, for the Tr curves become entirely flat. Now, teflon in contradistinction to PE 75 has a linear structure and a high molecular weight (380 000 to 9 000 000), which means that the proportion of CFs to CFs is small. In a sample with average molecular weight of 4 000 000 it is of the order of 1 in 40 000. A random walk computation similar to that for polyethylene shows that the average lifetime of the magnetization is about 20 s. For this calculation use was made of the fact that the total second moment of teflon at low temperatures is 11.7 y2 s-2, while for the determination of the correction (d~s),:~, a C-F distance of 1.35 A and tetrahedral angles were assumed. A lifetime of 20 s means that conveyance of nuclear magnetism to the CFs groups by spin diffusion is far too slow to account for the observed values of Ti in the normal teflon sample (resp. 0.73 and 3.95 s for the two phases). Moreover, the fact that T1 is temperature independent suggests that the relaxation is determined by a process of spin diffusion towards oxygen impurities. This is strikingly confirmed by the measurements on degassed samples, where T1 is much longer and becomes equal for both phases (about 25 s). The fact that T1 is now of the order of the lifetime calculated above: i = 20 s and again temperature independent might indicate that in the degassed sample relaxation is due to rotation of CFs groups coupled with a spin diffusion process. Minimum II, which occurs in the crystalline region at about 25°C is new. It is probably closely associated with the mechanism which is responsible for the first order transition in teflon at 19”C, discovered by Rigby and Bun n 31). From X-ray evidence C 1ark and Mu u s 32) concluded that below
1014
N. J. TRAPPENIERS,
19°C the
crystal
shows
C. J. GERRITSMA
3-dimensional
AND P. H. OOSTING
order
while
above
the
transition
temperature the molecules execute torsional vibrations about their longitudinal axis. It is also interesting to note that Eby and Sinnott 27) observed an n.m.r. line width narrowing in the region around 19”C, which they ascribe to the transition c. Polyoxymethylene.
point. Our measurements
of Ti
(fig. 9) were made
from - 196°C to 180°C. Only one reading could be taken in the liquid phase, just above the melting point (175”C), as the polymer is decomposed at higher temperature. In the entire temperature range only single values of Tr were found: the log Ti versus 1/Tcurves show one minimum. From mechanical loss measurements at about 1 Hz McCrum 33) concludes to the existence of three relaxation processes in P.O.YI., which he calls a, /? and y relaxation in analogy with the case of polyethylene. The loss peak of the p process is weak in comparison with the other two. The same author also mentions more extensive mechanical measurements of thr y process between 1 Hz and 10 MHz by He?) boer and Eb y, Their results can be represented by the relation : log (1~~= 29.26 - 5.822 (103/T,), where frequency of maximum mechanical loss and T, is the Wrn is the angular corresponding temperature. The activation energy is 27 kcal/mole. This functional relation, together with the three points of McCr urn are drawn in fig. 12. Also shown are a set of dielectric loss measurements between 20 Hz and 1 JIHz by Ishida 34) with activation energy of about 20 kcal/ mole
Fig. 12. CL,/3and 7 relaxation
in polyoxymethylene.
NUCLEAR
SPIN-LATTICE FQ?LAXATION IN POLYMERS
1015
It can be seen from fig. 12 that our own correlation frequency for P.O.M. lies close to the line of H eij b oe r and E by and there can be little doubt that it belongs to the process of y relaxation. According to McCrum the CLprocess has an activation energy of 24 kcal/mole; this process would certainly be inobservable at our correlation frequency of 46.77 MHz, as the corresponding temperature would fall far into the range of decomposition. No further
particulars
are known about the ,!? process and there is no trace
of it in our measurements. It may also be pointed out that Ishida’s dielectric measurements for the y process do not agree very well with the mechanical data, nor is there any mention of the cc or /3process in his paper. The behaviour of P.O.M. at low temperatures is similar to that of P.T.F.E. which is not surprising since P.O.M. is also a linear polymer. Again it is likely that T1 is determined by spin diffusion to oxygen impurities, for degassing leads to an increase in TI from 9.1 to 27.5 s. A calculation of the lifetime f of the diffusing magnetization was not possible, as the required experimental data for this polymer are not known. Acknowledgement. This investigation is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)“, supported by the “Organisatie voor Giver Wetenschappelijk Onderzoek (Z.W.O.)“. The authors are much indebted to Mr. P. K. Korver, Mr. P. J. Rens and Mr. C. P. Valkering for assistence with the measurements, to Dr. C. A. ten Seldam for the computation of the average spin lifetime and to the Mathematisch Centrum, Amsterdam, for permission to use the computer X- 1.
APPENDIX
Three-dimensional random walk with sinks. The diffusing nuclear magnetization behaves like a particle, which travels by steps through a simple cubic lattice. Each step consists of a jump from a lattice point to one of its six neighbours, the direction being chosen at random. The time interval between steps is called T. Let C be the fraction of lattice points, at which relaxation occurs; they are treated as randomly distributed “sinks” in which the wandering particle dies. Define n,, the number of particles surviving after having made s steps, and u, the mortality rate i.e. the fraction of particles which die after the sth step. Then n, = n,_i (1 - u). (A-l) A first approximation for u is obtained by observing that in each lattice point the particle has a chance C of hitting a sink in five out of the six equal-
N. 5. TRAPPENIERS,
1016
C. J. GERRITSM.4
AND P. H. OOSTING
ly probable directions (the sixth direction is the particle, which is certainly not a sink). Thus B: is a constant
to this
contribution its tracks.
order”
of the
(A-2)
This
paths
Integrating (A-l) with constant and s can be treated as continuous
station
SC‘!6
approximation.
of “higher
previous
elementary
in which
approach
the particle
C, and using variables,
neglects
doubles
the approximation
lls = 9~0exp (-US). It follows that its destruction,
the average is
number
of steps
back
in
that
n,?
(A-3)
performed
by a particle,
before
(A-4) The mean
lifetime
of a particle
i is given
by I = .+r or with
(A-2) and
t = 67/X‘.
(A-4)) (A-5)
The simple approximation (A-2) was put to the test by a Monte Carlo calculation on a computer. For a simple cubic lattice, containing 163 lattice points, but infinitised by using Uorn-Von IGrmLn boundary conditions, a 1000 random walks were computed. Sinks were distributed at random over the sample in the proportion c‘ = 13 1/I 63 =-= l/31.3. The decay relation (A-3) for n, was found to be closely obeyed; the calculated value of u-1 was 41.7, which, for our purpose, is in satisfactory agreement with the value from (A-2): n-1 = 37.6. Received
1)
Slich
2)
Powlcs,
3)
Sauer,
4)
Powles,
5)
Powles,
6)
Smyth,
26-1
l-63
ter,
IV. J.
J.
Bloembergen, Ward,
I.
Wilson,
A.
11)
Hahn, Carr,
13)
Lowe,
14)
Powles,
15)
Nolle,
and S.,
A.
Gen~ve
1;. L., Y.
Trans.
1. J., J.
J.
Phys. and
I<.,
Faraday
SW.
Isowen,
1%‘. and
SK.
Pake,
G.
appl.
Phys.
Rev.
80
Billings,
J.
32
(1950)
II. II.,
J. J.,
Ph!_<.
(SC Coil. 25
(1955)
3s.
them. (1961)
:I”
(1960)
~8.
Amp&r).
(1959)
45.5.
207.
Pound, (1960)
1:.
\‘.,
Phyi.
Rev.
i:l
(1948)
(1955)
1243.
648. Phys.
27
(1957)
115.
2339.
580. Phys.
Rev.
Norberg, A.,
nwd. 182
Physica 19
3(i
E.,
L. J. and Hartland,
Rev.
B. >I. and
1:araday
Purcell,
G. and
Ii.,
9 (1956)
Luszczynaki, Disc.
\V. P.,
A.
Sci.
I (1958.601
219.
Woodward,
Arch.
C. LV. and
II.
Hochpol\lllerr,ll-I;orscII. 1 (1960)
.U., Purcell, RI.,
Slichter,
12)
and
J. G. J.
7)
9)
A.
l;ortvhr.
Polywr
J. (;.,
8)
IO)
P.,
(;.,
9r.
1~. E.,
Nature
(London)
J. them.
Phys.
(1954)
630.
Phys.
Rev. 180
30
100
(1960)
(1959)
84.
26.
b79.
NUCLEAR
16)
Kawai,
T., Sasaki,
SPIN-LATTICE
M., Hirai,
RELAXATION
A., Hashi,
T. and Odajima,
1017
IN POLYMERS
A., J. phys. Sot. Japan 15 (1960)
1700. 17)
Kawai,
18)
Buchta,
T., J. phys. Sot. Japan 19 (1961) 1220. J. C., Gutowsky,
19) See f.i. Abragam, 20) See f.i. Abragam,
H. S. and Woessner,
D. E., Rev. sci. Instr. 29 (1958) 55.
A., Nuclear
Magnetism,
Oxford
Univ. Press (1961) Chapter
III.
A., Nuclear
Magnetism,
Oxford
Univ. Press (1961) Chapter
VIII.
21) 22)
Odajima,
23)
Mikhailov,
24)
Kline, D. E., Sauer, J. A. and Woodward, A. E., J. Polymer Sci. 22 (1956) 455. Fuschillo, N. and Sauer, J. A., J. appl. Phys. 28 (1957) 1073.
25)
Kabin,
A., Suppl.
Progr.
S. P., Zh. tekhn.
theor.
Phys. 10 (1959)
142.
Fiz. 26 (1956) 2628.
G. P., Kabin,
S. P. and Sazhin,
A., Nuclear
26)
See f.i. Abragam,
27)
Eby,
28) 29)
Schultz, A. K., J. Chim. phys. 53 (1956) 933. Illers, K. H. and Jenckel, E., Kolloid Z. 160 (1958) 97.
30)
McCrum,
31)
Rigby,
32)
Clark,
R. K. and Sinnott,
E. S. and Muus,
Hochpolymeren-Forsch. 33)
JlcCrum,
34)
Ishida,
35)
Ming
Chen
Wang
Univ.
Press (1961) Chapter
V.
Sci. 34 (1959) 355.
C. W., Nature (London) 1~. T., see f.i. Sperati, 2 (1960-61)
N. G., J. Polymer Y., Kolloid
Oxford
Fiz. 25 (1955) 590.
K. M., J. appl. Phys. 32 (1961) 1765.
N. G., J. Polymer H. A. and Bunn,
magnetism,
B. I., Zh. tekhn.
Z. 171
164 (1949) 583.
C. A. and Starkweather,
H. W., Jr. Fortschr.
465.
Sci. 54 (1961) 561. (1960)
and Uhlenbeck,
149. G. E., Rev. mod. Phys. 17 (1945) 323.
N. J.
Physica
C. J.
30
997-1017
P. H.
1964
THE MEASUREMENT OF NUCLEAR MAGNETIC RELAXATION TIME T, IN POLYMERS BY MEANS OF SPIN ECHO TECHNIQUE by N. J. TRAPPENIERS,
C. J. GERRITSMA
180th publication Van der Waals-laboratorium,
and P. H. OOSTING
of the Van der Waals
Universiteit
Fund
van .I\msterdam,
.4msterdam,
Nederland.
Synopsis A brief description solid
materials,
about
is given
especially
The
spectrometer recovery
6 ps after a 90” pulse of 1 ,US.The spin-lattice
polytetrafluoroethylene to 280°C. The observed results
and polyoxymethylene
minima
and one for P.O.M.
from mechanical
Spin diffusion
in the log Ti vs i/T
are discussed
of spin diffusion
curves,
of
the
to the study
of
spectrometer
is
time T1 of polyethylene, in the range
three for P.E., theory
: - 196°C
three for P.T.F.E. and related
to the
loss measurements.
to be the dominating lattice,
relaxation
of relaxation
in the three polymers.
in a cubic
adapted
time
was measured
in the light
and dielectric
is thought
Tr at low temperatures first
of a spin echo
of polymers.
sprinkled
factor,
governing
In the appendix, with
relaxation
the behaviour
therefore, centra,
of
the model
is treated
to a
approximation.
3 1. Introduction. A considerable amount of work has been done on the study of polymers by nuclear magnetic resonance. Most authors have been concerned solely with the measurement of line width as function of temperature, using steady-state methods, and a number of review articles have been devoted to the results thus obtained [Slichter r), Powles a), Sauer and Woodward a)]. It was pointed out, however, that the direct measurement of nuclear magnetic relaxation times, especially of the spin-lattice relaxation time T1, would provide more insight into the motional behaviour of polymers [Powles 4) 5), Smith 6)]. Furthermore, there would be much interest in correlating nuclear magnetic relaxation times with the results obtained from dielectric and mechanical relaxation measurements. The comparative lack of data on T1 in polymers must be attributed to the experimental difficulties which, in general, impair the accurate determination of Tr in solids. Until recently the only practical method of measuring T1 in polymers was the progressive saturation method as first described by Bloembergen, Purcell and Pound 7). The drawbacks of -
997 -
.
998
N. J. TRAPPENIERS,
c. 5. GERRITSMA
AND
r. H. OOSTING
this technique are well known. For example, it does not yield absolute values for T1; the determination of T1 necessitates a knowledge of I’2 (spinspin interaction time); the stability of the marginal oscillator becomes a problem when T1 is long, while in the case of broad resonance lines the n.m.r. signal is weak and the accuracy poor [\I:ard S)]. In spite of these difficulties some results have been obtained
by the satu-
ration method for polyethylene
9) investiga-
and teflon. Wilson
and Pake
ted teflon and found between 170°K and 290°K two distinct spin-lattice relaxation times. Ward 8) measured T1 for the methyl proton and also for the aromatic proton in poly(ethylene trideuteromonohydrogen terephthalate). Recently Slichter 10) found two values for T1 in linear polyethylene crystallised from a solution, in contradiction with earlier results of Wilson and Norberg 9), using a free induction and Pake 9). However, Bloom method, stated that they found two T1 values at room temperature in thrl sample used by Wilson and Pake. The spin echo technique, invented by Hahn 11) and further developed by Carr and Purcell 12), constitutes a much more elegant and powerful method for measuring relaxation times. Thusfar, however, its application has been limited mainly to liquids. The reason for this lies in the fact that in solids the dipole-dipole coupling between magnetic nuclei is strong, resulting in a short spin-spin interaction time Tp. Consequently, the free induction decay tail, following after a 90” radio-frequency pulse, will also be very short and in order to observe a free induction signal a number of stringent conditions have to be met. 1. R.f. pulses have to be short compared to Tz of the sample but at the same time sufficiently intense to obtain the desired 90” or 180” rotation of the magnetisation vector of the sample. 2. The recovery time of amplifier and detector, after severe overloading by the r.f. pulse, has to be short compared to T;?. 3. The bandwidth of the amplifier has to be sufficient so as to avoid signal deformation. Up to the present spin echo measurements in solid polymers have been made only in a number of favourable cases where 1’2 was comparatively long. Thus Lowe e.u.13) measured T1 in polyethylene, Powles e.u.5) determined T1 and T2 in polyisobutylene and later 14) also in silicon polymers. Nolle and Billings 15) measured T1 in polyisobutylene under high Kawai e.a.16) 17) have applied the spin echo method pressure. Recently to the measurement of T1 in amorphous polymers such as polymethyl methacrylate etc. In this paper results will be given of measurements of T1 in three polymers which may be partly crystalline: namely polyethylene, polytetrafluoroethylene and polyoxymethylene. This was achieved with the use of a new spin echo spectrometer which is equipped with a powerful trans-
N~CLE!AR s~iN-~4TTIcti ~~I~LAXATIO~~ ~ti POLYMERS
mitter,
so that a 90” pulse is performed
499
in 1 ,LLS.By the use of a pulsed dam-
ping circuit, which precedes the broadband receiver, the spectrometer has a recovery time of 6 ,us. This means that signals can be observed in solids displaying
an n.m.r.
3 2. The
line width of approximately
spin echo spectrometer.
a. General
40 gauss. principles.
All the mea-
surements were made with a phase coherent spin echo spectrometer, which in its general design is similar to the apparatus described by Bucht a, Gutowsky and Woessner 18). PULSE
TRIGGER GENERATOR 4 100 KC/S OSCILLATOR
GENERATOR 4,
+_____ lb.L
MC/S
_
,,
FREQUENCY DOUBLER
FREOCHANGER
+ ZdGATE
t
+ POWER \I
AMPLIFIER $ C.R.O.
DETECTOR
, -
BROAOBANO AMPLIFIER
RF
HEAD
t
II
I
Fig. 1. Block diagram of spin echo spectrometer
In fig. 1 a block diagram of the instrument is shown. By means of the 100 kHz crystal oscillator, the trigger generator, the trigger selector, the three pulse generators and the pulse mixer a pulse train is generated consisting of 3 pulses which are variable in length, height and mutual spacing. Alternatively a pulse train may be produced consisting of a large number (10, 100 or 1000) of identical pulses, to be used for example in measurements of Tz according to the method of Carr and Purcell. Briefly, the working of these units is as follows. The trigger generator, which is monitored with a frequency of 100 kHz, generates sharp pulses with repetition interval of either 1,2,4,6,8 or 10 ps. These pulses are counted by the trigger selector, which is essentially a decimal counter fitted with 8 beam switching tubes (Burroughs). By means of 3 coincidence circuits the trigger selector is made to transmit a trigger pulse at preset intervals to each of the three pulse generators and to repeat this pattern at a well defined repetition rate.
1000
_~
Ii.
J. TRAPPENIERS,
The chief advantage
C. J. GERRITSMA
in using a counter
AND
P. H.
OOSTING
lies in the fact that
the intervals
between successive pulses are accurately preset by the coincidence circuits (the accuracy being 1, 2, 4, 6, 8 or 10 /is according to the setting of the trigger generator), so that they do not have to be determined from readings on the oscilloscope. A fourth coi’ncidencc circuit is used to trigger the oscilloscope at any chosen time. The pulses, delivered
by the pulse generators,
pass through
the p~~lsc
mixer and then operate the first gating circuit. This gate is fed with radio frequency at 14.4 MHz, i.e. half the Larmor frequency used in the measurements, which is obtained through frequency multiplication from the basic 100 kHz oscillator. The radio-frequency pulses at 14.4 MHz from the first gate pass through a frequency changer which doubles the frequency to 28.8 MHz, but which unavoidably alters the shape of the pulses to some extent. A second gate, which is also operated by the pulse mixer, is used therefore to reshape the r.f. pulses. By working at half the Larmor frequcncy the problem of leaking through radiation is avoided. The power amplifier increases the voltage of the r.f. pulses to the desired level, which is such that when applied to the r.f. sample coil of the tank circuit, the pulses produce an r.f. field 2H1 a 120 gauss. The single coil at the same time serves the purpose of receiver coil for the induced n.m.r. signals. These signals pass through a cathode follower followed by a broadband amplifier and after detection they are displayed on a 545 Tektronix oscilloscope. (A more detailed description of the apparatus will be published shortly). b. Circuit diagram of r.f. head. The tank circuit, the cathode follower together with two accessory circuits: the “pulsed damping circuit” and a \%ual tuning indicator are housed in a separate, well shielded. r.f. head. The wiring diagram is shown in fig. 2. Condensor Cl is used to tune the sample coil to resonance while the two Lrariable condensers
Cz and C3 couple
the
coil to the
power
amplifier.
The choice of the capacity of condensers Cl, Ca and C3 is dictated by a compromise between two factors: on the one hand, the aim of optimum transmission of r.f. energy to the coil will fix the ratio between the coupling and tuning capacity; on the other hand, the best condition for detection of n.m.r. signals is obtained with total coupling capacity small in comparison with the tuning capacity. The choice adopted is such that with a tank circuit Q factor of 100 the transmitted r.f. energy is sufficient to obtain a 90” flip of the sample magnetization vector with a pulse of 1 ,a, As stated in the introduction a pulse of 1 ,US is sufficiently short to allow relaxation measurements in most solids. There remains, however, one big difficulty. The intense r.f. pulse causes “ringing” of the tank circuit, meaning that the applied voltage will decay comparatively slowly in a characteristic time which is proportional to the
NUCLEAR
SPIN-LATTICE
RELAXATION
IN POLYMERS
1001
Q factor of the circuit. As a result, the broadband amplifier is severely overloaded for some time and no signal can be observed immediately after the pulse, a fact which largely
obliterates
Fig. 2. Radio-frequency
the advantage
of working
head.
with short pulses. The disastrous effect of tank circuit “ringing” could be avoided in part by decreasing the Q factor of the circuit, but as this would reduce at the same time the signal to noise ratio the sensitivity of the method would be impaired. A better solution of this problem was found by equiping the tank circuit with a “pulsed damping circuit” (p.d.c.) which works on the following principle. Pulses taken from the pulse mixer are amplified to -200 V with the use of an E180F pentode (V3) and an EC50 thyratron (V2). An RC filter partly differentiates this negative pulse so as to produce at the end of the pulse a sharp positive peak of sufficient amplitude to fire an EC50 thyratron (Vl) in parallel over the tank circuit. When conducting, this thyratron short circuits the sample coil, thus strongly damping the “ringing” of the tank circuit, while the negative part of the pulse prevents the thyratron from conducting during the application of the r.f. pulse. Fig. 3 clearly demonstrates the beneficial effect of the p.d.c. in limiting the r.f. pulse over the sample coil. Fig. 4 shows the Bloch decay following a 90” pulse without p.d.c. in a solution of Fe(NOa)a containing 5 x 101s ions/cma: it is evident that “ringing” of the tank circuit blocks the ampli-
1002
N. I.
TRAPPENIERS,
C. J. GERRITSMA
AND
P. H. OOSTING
fier for about 20 jrs. In fig. 5 the same decay tail is taken with p.d.c. showing that blocking of the amplifier is limited to about 6 jts. The great advantage of the 1j.d.c. for solid state work may be judged from fig. 6 in which is shown Bloch decay following a 90” pulse in polyoxymethylene \Vithout 1l.d.c. the decay tail would not have been visible.
at -.- 165°C.
When the relaxation time 7’1 of the specimen is long, tuning of the tank circuit to exact rtlsonance becomes difficult and time consuming. The procedure may be greatly facilitated by incorporating in the r.f. head a “magic eye” EM34 (V4) as tuning indicator. The r.f. pulses, after rectification, charge condenser C+ The magic eye is sensitive to the voltage over this condenser which will attain its maximum value when the tank circuit is tuned to resonance. For a satisfactory operation the repetition rate of the pulses should not be smaller than 10 Hz. Lastly, the E180F (7’5)) serves the purpose of a cathode follower, which couples the r.f. head to the broadband amplifier. c. Temperature control. A conventional gas-flow thermostat is used to control the temperature of the sample between room-temperature and 280°C. Below roomtemperature, down to - 196”C, a liquid nitrogen cryostat is used. Temperature is measured with copper-constantan thermocouples. Although a gas-flow thermostat is easy in use it has the drawback of producing a temperature gradient over the specimen. At the highest temperature, for example, a gradient as large as 2°C can exist over a sample of 5 mm in
bfU6Ll?A#
length.
SPIN-LATTICE
In the liquid nitrogen
RELAXATION
cryostat
1603
Iti Pc~L~WERS
the gradient
over the sample is less
than 0.05”C. 93. Measurement of TI in polymers. a. Procedure In the present investigation the spin-lattice relaxation
of measurement. time T1 is determined
from the free induction signals produced by a 180”-90” pulse sequence. It is well known la) that the magnitude of the induction signal M(t), which follows after the 90” pulse, depends on the time interval the 180” and 90” pulses in the following manner :
M(t) = Mo[ 1 -
t between
2 exp (-t/Tl)]
(1)
where MO is the equilibrium value of M(t), i.e. MO = M(t) for t > T1. If M(t) is measured for a number of values of t, T1 can be deternimed graphically from the plot: log [MO - M(t)]/2Mo versus t. processes with different In some polymers, however, two relaxation relaxation times Ti and T; may operate simultaneously and the procedure for their determination is somewhat more complicated. Equation (1) now reads :
M(t) = Mb [I -2 The equilibrium clei participating
exp (-t/Ti)]
+ M”,[l-2
exp (-t/T;)].
values MI, and M”, are proportional to the number in the two respective processes, while
(2) of nu-
Mb + Mj = M(t) when t > Ti, T;. From
(2) 4 [Mb + Mb -
M(t)] = Mb exp (-t/T;)
+ M”, exp (-t/T;).
(3)
If T; is small with respect to Ti the second term in (3) will become negligeable for sufficiently large t. A plot of log [MI, + M, - M(t)] versus t will yield a curve which tails off into a straight line. From this Ti and Mb are found and after substitution in (3) T; and M”, can also be calculated. Provided Ti and T; differ at least by a factor 4, the accuracy in the determination of the longer relaxation time Ti is about 5% and in the shorter relaxation time T; about 10%. If the difference between Ti and T; becomes smaller the accuracy becomes much poorer up to a point when the logarithmic plot scarcely shows the existence of two different relaxation times at all. Thus, when the difference between Ti and T; is smaller than a factor 2, the experimental points, within their margin of error, lie on one straight line with an apparent relaxation time which is an average of Ti and T;. It is interesting to note that from the decay signal, after the 90” pulse, it can be seen immediately whether the sample shows one or more simultaneous relaxation times. In the first case only the initial height of the decay signal varies with t ,while in the case of two or more values of T1 the shape as well as the initial height of the Bloch decay signal change with t.
1004
N. J. TRAPPENIERS,
b. Results
and
C. J.
GERRITSMA
qualitative
.4ND P. H. OOSTING
interpretation.
Three
polymers
are
used for the T1 measurements: 1. polyethylene (-CHB--)~ : a sample of PE 75 was obtained from du Pont *) with following characteristics: optical m,p. 104”C, density 0.912 g/cm3 at 24.5”C and 3.3 methyls/lOOC. 2. polytetrafluoroethylene (--CFZ-)~~ : a sample of the standard commercial “Teflon”, made by Montecatini was used. Specification: melting begins at 327X, density 2.14 g/cm3 at 23.5”C. 3. polyoxymethylene (-O-CHz-) II: a sample of “Delrin” 500 .X was obtained from du Pont *) ; melting begins at 174”C, density 1.405 g/cm3 at 24.5”C. Readings of T1 were made at atmospheric pressure, as function of temperature in the range from ~ 196°C to about 280°C. Later, a second series of T1 values was measured between roomtemperature and ~ 196”C, by using three degassed samples. For this purpose the samples were vacuum pumped for 48 hours and sealed off under vacuum in glass tubes. In every case the working frequency of the spin echo spectrometer was 28.8 MHz. The results obtained for the three polymers are shown respectively in fig. 7, 8 and 9 as a plot of log 7’1 against reciprocal absolute temperature 200
100
30
'Ol s
(3 -30 I'
-60 1
-90 '
-120
-150 I
-180
-19OOC -f
TI
/' 4 5:T
/ Y
32-
d' /
A'
I-
/
0.5 0.4 0.3 1
t
0.2
l
d' /
LJ-=++
t 0.1
I
I
1
2
I,
m.p. 3
Fig. l
_
T
T1 in normal
o 271in degassed
lOOO/
Tabs
II
.&
I
I
I
1
4
5
6
7
7. Relaxation sample sample
time
8
for protons
I
I
I
I
I
9
IO
II
12
13
in polyethg’lene
(solid curve). (dashed
curve).
*) The authors arc 111uch iMlcl,ted to lb. N. G. McCru,,, of E.I. du Pout de Ncrnours and Comp., Polychemicals Dept., Research and I)?\ r’lopmvnt I)ivisioll, for providing thr samples.
Received
26-l l-63
NUCLEAR
200 I
I 101
100 I
30 0 I,,,
SPIN-LATTICE
-30
-60
-90 I
RELAXATION
-120 I
1005
IN POLYMERS
-160 I
-190 I
-190% I
5
51 41 31 2[
1c
1
‘
1
0.5 0.4 0.3 0.2
0.1
Fig. .
8. Relaxation
T1 in normal
o T1 in degassed
sample sample
time for fluorine
in polytetrafluoroethylene.
(solid curve). (dashed
curve).
l/T. The most striking feature of these curves is the appearance of one or several minima, shaped like a V. This aspect suggests that it should be possible to give a qualitative interpretation of the behaviour of T1 in polymers in the light of the now generally accepted theory of nuclear magnetic relaxation. A given proton or fluorine nucleus experiences a local magnetic field created by neighbouring nuclei. This field is a rapidly fluctuating function of time due to the movement of reorientation of polymer chain segments, and as such it is responsible for spin-lattice relaxation. In a first approximation only the local field produced by one nearest neighbour need be considered, as the -CHs--, -CFsor -0-CHssegments provide fairly isolated two-spin systems of hydrogen or fluorine nuclei. This means that the influence of nuclei belonging to another segment, either in the same or in a different polymer molecule, is neglected. It also implies that
1006
K;. J, TRAPPENIERS,
0 200 1
100,
s
100 I
-30
-60 I
30 III
C. J.
GERRITSMA
-120 I
-90 I
AND P. H. OOSTING
-19OOC ,
-180 1
-150 !
-----+f
T
50 LO 30 -
5
__9-----_cr-
20 -
r
/-
I 0
/ d’ d’
10 -
5L32-
I
/’
I
I-
0.5 0.1 0.3 0.2 -
---+ lOOO/
!v
0.1 I
1
1
I
I
I
1
I
1
I
1
I
2 m.p.
3
L
5
6
7
6
9
10
II
Relaxation
time
Fig.
.
7
’
I
9.
T1 in normal
c, T1 in degassed
sample
sample
for protons
es ,
12
,3
in l~~~l~ox~methylene.
(solid curve) (dashed
curve).
no account is taken of translational movement of one molecule with respect to another. The simplest way of characterising the fluctuating local field of the twospin system is to introduce a single correlation time TV, which sets a time scale to the reorientation rate of the chain segments. To this model we can apply the general relation 20) which expresses T1 as function of 7C for intramolecular relaxation in the case of a two-spin system in spatial isotropic motion : T ; ‘=
2/5,I(Z +
1) /%zy4b-‘3
where I is the nuclear spin, y the gyromagnetic ratio, b the distance tween the two spins and o the resonance frequency. A plot of T1 versus according to (4), results in a V-shaped graph with a minimum value of when : cwrc = 0.6158.
(4) beTV, T1
(5)
NUCLEAR
SPIN-LATTICE
RELAXATION
This is the value of 7c for which the relaxation
1007
IN POLYMERS
mechanism
is most effective.
The movement of reorientation of polymer chain segments is likely to be a thermally activated process so that it seems reasonable to assume that: TV = TOexp (E/RT), where E is the activation
(6)
energy of the process and TO a constant.
ciently high temperature, 7c is small and w%“, < 1, so that with (6), reduces to log TI = -log (5 Go) - E/RT, where C = 2/5 1(1 + 1)Rsy4b-6. At low temperature, when &%-z >
At suffi-
(4), together (7)
1, (4) reduces to
log TI = log (0%0/2C) + E/RT.
(8)
Hence, a plot of log 1‘1 vs l/T, according to this theoretical model, also results in a V-shaped graph; the slopes are directly proportional to the activation energy and the minimum in log T1 corresponds to relation (5). If the simplified behaviour of the two-spin system is applied to the measured T1 curves for the three polymers, the following conclusions may be drawn. 1. On heating polyethylene from - 196°C (fig. 7), TI versus 1/T goes for both samples (normal and degassed) through a first V-shaped minimum III. This means that a relaxation mechanism becomes active, which reaches its maximum efficiency at about - 15O”C, where T1 goes through its minimum value. According to (5) the correlation time in this minimum is given by: TV (min) = 0.6158/m. In correspondence
to TV a correlation
frequency
may be defined as:
Yc = 1/(27UTc) which, in the minimum, takes the value Ye (min) = 46.77 MHz. As the temperature increases this relaxation mechanism becomes less efficient for its oc, according to (4), increases rapidly. A second relaxation process now takes over, causing T1 again to decrease and to pass through a second minimum II at - 11 “C where in turn the correlation frequency of the second process has the optimal value Ye (min) = 46.77 MHz. At still higher temperatures a third process becomes active, giving rise to minimum I at 43”C, again with Ye (min) = 46.77 MHz. From fig. 7 it is seen that at low temperature T1 is larger for the degassed than for the normal samples. The difference increases with TI, while at room temperature the relaxation times become equal. 2. In teflon (fig. 8) two relaxation times occur simultaneously for the spin-lattice relaxation of F 19. Following previous authors 9) we attribute
1008
N.
J. TRAPPENIERS,
C. J. GERRITSMA
AND
P. H. OOSTISG
the upper curves with the longer relaxation times to the crystalline region of the polymer, the lower curves to the amorphous region. In total three minima were found: in the crystalline region at 168°C (I) and at 25°C (II); in the amorphous region at 21°C (III). At the highest temperatures available in our thermostat the lower Tr curve (amorphous phase) is still descending. It seems likely that it will eventually pass through another minimum (IV?), which, however, we were unable to observe. In each minimum, the correlation frequency of the associated mechanism is given by (5) i.e. vc (min) = 46.77 MHz. In th e 1ow temperature range the solid curve (normal sample) as well as the dashed curve (degassed sample) become flat. The latter sample shows longer Tl’s, which, however, become equal below about - 160°C. At roomtemperature T1 has the same value for both samples. 3. The graph of T1 in polyoxymethylene (fig. 9) only shows one minimum (I) at 12°C. Apparently a single relaxation mechanism is operative with vc (min) = 46.77 MHz. The two TI curves flatten out at lower temperatures. As before, the relaxation times are the same for the normal and degassed sample at roomtemperature. In table I the characteristics of the T1 curves are given.
i Minimum
Polymer
Polyethylene
--em
TWllp. C i-.z_:-imm
r-1
1 Ii--’ i 103 I I ;_;z:---pm~
I‘[
’ ~~ 0.13; - 0.03
I --I,,, ;&;.Hiph i rrgirm *) [(Yl11.(8)]
III Polytetrafluoro~ ethylene
8.1
.t0.3
0.42LO.02
1
16815
2.27 2: 0.03
0.6210.05
II
2512
3.36_: 0.02
0.42 AL0.02
III IV? Polyoxymethylene
-150LS
I
! 2115 ‘>220
3.40 !;0.05 .’ 2.03
12,5
3.51+_0.06
0.14,O.Ol
Icqll.(7): 5.0
1.S
I ~
0.08 10.01
I 1
i
I
0.14_!_0.03
I 1
trmp. rrgion
1.2
I.0 3.6
/
8.8
J.il I I
4.4
6.4
*) In this region slopes are given of the dashed curws.
The simple theory of a two-spin system with single correlation time, though of value for the qualitative interpretation of TI curves, cannot be expected to give a quantitative agreement with the measurements. Moreover, the molecular movements of chain segments is not likely to be isotropic as implied in the foregoing equations. The discrepancy between the simple theory and the experimental results is apparent from the logarithmic plot of log TI vs l/T in fig. 7, 8 and 9, which does not result in straight lines as demanded by (7) and (8). Thus, the simple equations (4), (7) and
NUCLEAR
SPIN LATTICE
RELAXATION
IN POLYMERS
1009
(8) are not reliable for the computation of correlation times and activation energies from measured Ti values. The most obvious way of improving the theory would be to introduce a spectrum of correlation times as suggested by 0 da j im a 21). The fitting of a spectrum to a T1 curve with several minima is a tedious affair as one needs to adjust a large number of parameters, and it was not attempted in this investigation. However, the work of Odaj ima has shown the important fact that the occurrence of a correlation time distribution does not effect the position of the minima in the T1 curve, even though the shape of the curve is altered considerably. Equation (5), therefore, is of general validity and the correlation frequency of 46.77 MHz deduced from it for each individual minimum as well as the corresponding temperature given in table I, are reliable data for the further discussion of relaxation processes. S 4. Discussion. a. Polyethylene. From fig. 7 it can be seen that only a single TI curve was found in the temperature range from - 196°C to 130°C. This, apparently, is in contradiction with results of Bloom and Norberga) and Slichter la), who found two Tl’s in samples of linear polyethylene and concluded to the existence of a crystalline and an amorphous phase. The discrepancy may arise from the fact that our sample of PE 75 is strongly branched, which would preclude the formation of a crystalline phase. In order to identify the relaxation processes, associated with the three minima in the T1 curve of polyethylene, it is necessary to establish a correlation with the studies of mechanical and dielectric relaxation. The measurement of mechanical and dielectric losses in polyethylene yields three distinct absorption maxima at three critical frequencies. The associated relaxation processes, taken in order of increasing frequency, are termed cc, b and y. A rise in temperature produces a shift of each maximum to a higher value of the frequency. K abin ss), by varying the temperature from -80°C to 115°C was able to determine the critical frequency of the ,I!?and y process from mechanical losses between 0.5 and 5 MHz. His results are in agreement with those of Michailov, Kabin and Sazhin 33) who measured the CL, p and y process from dielectric losses between 50 Hz and 10 GHz in the temperature range from - 140°C to 110°C. When all their results are plotted on a logarithmic scale against 1/T the points fall on three reasonably straight lines, shown in fig. 10, from which the following activation energies are estimated: TVprocess 28 kcal/mole, /I process 16 kcal/mole and y process 11 kcal/mole. Since Kabin e.a. gave no specifications of their polyethylene samples, a number of additional measurements on PE 75 are plotted in fig. 10. These points include mechanical loss measurements [Kline, S a u e r and Woodward s4)] and n.m.r. linewidth data [Sauer and Woodward 3); Fuschillo and Sauer 25)].
1010
S. J. TRAPPENIERS,
2
C. J. GERRITSMA
,
/
3
L
.4ND
I’. H. OOSTING
I
6
-+1000/
6 ‘abs
Fig, 10. Z, fl ant1 ;I relaxation
in polyrthylene
If we assume that the observed spin-lattice relaxation in polymers is due to the same fundamental molecular processes, which are responsible for mechanical and dielectric relaxation, we may compare the correlation freof maxiquency vc (min) at temperatures of minimal T 1, i.e. temperatures mal spin-lattice relaxation, with the critical frequencies at temperatures of maximum mechanical and dielectric losses. The points corresponding to the minima I and II have also been plotted in fig. 10. It is fairly certain that minimum II at - 11°C is due to the y relaxation process and minimum I at 43°C to the p relaxation process. The expected minimum, which would correspond to the CCprocess at vc (min) = 46.77 MHz, has not been observed in our measurements, as it would fall at a temperature above the melting point of polyethylene. In this paper we shall not attempt to identify the specific molecular mechanism responsible for minima I and II in T1 as this would require a lengthy theoretical analysis. We may quote, however, the opinion of Sauer and Woodward 3) that the y process is due to diffusional motion of short chain segments while the /3process arises from movements involving branch .points as well. The last minimum III, at about - 15O”C, does not fit in with any of the drawn loss-curves and it is probably due to a new relaxation mechanism which has not been detected thusfar. A plausible explanation, we think, is to attribute it to the relaxation of terminal CHa groups. However, as the proportion of CHa groups in branched P.E. is not larger than about 3.3 per 100 C atoms, the relaxation through rotation of CHa groups can only
NUCLEAR
be effective
SPIN-LATTICE
if it is associated
RELAXATION
IN POLYMERS
with a spin diffusion
through the mutual flip of neighbouring
process.
1011 This operates
spins and is thus capable
of propa-
gating over large distances the influence of CHs relaxation. A rough estimate of the efficiency of the spin diffusion process can be made as follows. As a crude model, assume the CHs and CHs segments to be distributed over the lattice points of a simple cubic lattice. If each lattice point were occupied by a single spin the transition probability W of a spin flip between neighbouring spins could be related n.m.r. absorption line 26)
directly
to the second moment
of the
In order to obtain an average probability W,, for a spin exchange between neighbouring CHs, in our model, the experimental second moment of polyePZ 25 y?-2 at low temperatures has to be diminished with thylene (dws)_ the contribution of the spin-spin interaction within the CHz segment. This correction can be easily evaluated once the interproton distance is fixed by assuming tetrahedral angles and a C-H distance of 1.09 A:
The time interval 7-l =
w,, =
between successive
[(Aw2)oxp -
(?iZ&J~/30
flips T is then given by = (14y2)$/30 s-1 m 3300 s-i.
Spin diffusion is now treated as a discrete 3-dimensional random walk problem. The diffusing nuclear magnetization in a given lattice point (CHs) has a choice of 6 equally probable directions, since on an average in each lattice point there is always one of the two CHs protons having the right spin orientation for mutual flip with a neighbouring spin. Whenever magnetization reaches a site occupied by a CHs group it disappears through the rotational mechanism of spin-lattice relaxation. The average lifetime of the diffusing magnetization can then be calculated to a good approximation (see appendix A-5) as f = 6 715 C in which C is the concentration of CHa groups. Taking C = l/30 the average lifetime is found to be 0.011 sec. This means that nuclear magnetization travels to the site of a CHs group in a time which is small compared to the measured value of Tr = 0.42 s in the minimum III. Spin diffusion could thus be a very efficient mechanism in propagating the influence of the proposed CHa rotational relaxation to all the protons in the sample. Degassing of the sample results in longer Ti values (fig. 7). From this it may be concluded that, besides the CHa groups, the paramagnetic oxygen molecules also constitute effective relaxation centra. The contribution of the latter to spin-lattice relaxation depends only on their concentration, since relaxation at these centra is very fast. It may be concluded, however,
1012 that
S. 1. TRAPPENIERS, in the case of branched
C. J. GERRITSMrl
polyethylene
AND I’. H. OOSTIKG
the CHa-mechanism
minating factor at low temperature. b. Polytetrafluoroethylene. Contrary it found that P.T.F.E. two the temperature covered in degassed teflon - 160°C. mentioned in of fig.
with the
I and
to the
is the do-
the case polyethylene values of are present investigation, except 3, we the upper of a
phase and lower curves minimum III IV?) to amorphous region the polymer. mechanical and measurements, as as from line width two relaxation termed glass or /I glass II y relaxation known to operative in amorphous phase, only one relaxation process has thusfar detected. Fig. shows the of a by E and Sinno t 27) data for critical frequencies maximum mechanical dielectric losses for the frequency from line width for the and y plotted on log scale function of The points approximately on straight lines, activation cncrgies 34
Fig.
Crystalline
(a)
I (/?)
glass 11
relaxation
in polytetrafluoroethylene.
NUCLEAR
SPIN-LATTICE
RELAXATION
IN POLYMERS
1013
; Illers and Jenckelsa); McCrumse)] and at 12 MHz [Eby and Sinnott 27)]. The results of T1 measurements by Wilson and Pake 9) are also plotted. The correlation frequencies, corresponding to the minima I and III in our T1 curves, have been indicated in fig. 11. It may be concluded that minimum III at 21”C, is due to glass II relaxation in the amorphous phase and that minimum I, at 168°C is caused by the crystalline relaxation. There is no T1 minimum corresponding to the glass I relaxation, probably because the temperature at which it should occur lies above our available range. Tentatively we might assume that the hypothetical minimum IV, mentioned in 9 3, is related to the glass I relaxation. The behaviour of the relaxation curves in teflon below - 160°C differs markedly from the case of polyethylene, for the Tr curves become entirely flat. Now, teflon in contradistinction to PE 75 has a linear structure and a high molecular weight (380 000 to 9 000 000), which means that the proportion of CFs to CFs is small. In a sample with average molecular weight of 4 000 000 it is of the order of 1 in 40 000. A random walk computation similar to that for polyethylene shows that the average lifetime of the magnetization is about 20 s. For this calculation use was made of the fact that the total second moment of teflon at low temperatures is 11.7 y2 s-2, while for the determination of the correction (d~s),:~, a C-F distance of 1.35 A and tetrahedral angles were assumed. A lifetime of 20 s means that conveyance of nuclear magnetism to the CFs groups by spin diffusion is far too slow to account for the observed values of Ti in the normal teflon sample (resp. 0.73 and 3.95 s for the two phases). Moreover, the fact that T1 is temperature independent suggests that the relaxation is determined by a process of spin diffusion towards oxygen impurities. This is strikingly confirmed by the measurements on degassed samples, where T1 is much longer and becomes equal for both phases (about 25 s). The fact that T1 is now of the order of the lifetime calculated above: i = 20 s and again temperature independent might indicate that in the degassed sample relaxation is due to rotation of CFs groups coupled with a spin diffusion process. Minimum II, which occurs in the crystalline region at about 25°C is new. It is probably closely associated with the mechanism which is responsible for the first order transition in teflon at 19”C, discovered by Rigby and Bun n 31). From X-ray evidence C 1ark and Mu u s 32) concluded that below
1014
N. J. TRAPPENIERS,
19°C the
crystal
shows
C. J. GERRITSMA
3-dimensional
AND P. H. OOSTING
order
while
above
the
transition
temperature the molecules execute torsional vibrations about their longitudinal axis. It is also interesting to note that Eby and Sinnott 27) observed an n.m.r. line width narrowing in the region around 19”C, which they ascribe to the transition c. Polyoxymethylene.
point. Our measurements
of Ti
(fig. 9) were made
from - 196°C to 180°C. Only one reading could be taken in the liquid phase, just above the melting point (175”C), as the polymer is decomposed at higher temperature. In the entire temperature range only single values of Tr were found: the log Ti versus 1/Tcurves show one minimum. From mechanical loss measurements at about 1 Hz McCrum 33) concludes to the existence of three relaxation processes in P.O.YI., which he calls a, /? and y relaxation in analogy with the case of polyethylene. The loss peak of the p process is weak in comparison with the other two. The same author also mentions more extensive mechanical measurements of thr y process between 1 Hz and 10 MHz by He?) boer and Eb y, Their results can be represented by the relation : log (1~~= 29.26 - 5.822 (103/T,), where frequency of maximum mechanical loss and T, is the Wrn is the angular corresponding temperature. The activation energy is 27 kcal/mole. This functional relation, together with the three points of McCr urn are drawn in fig. 12. Also shown are a set of dielectric loss measurements between 20 Hz and 1 JIHz by Ishida 34) with activation energy of about 20 kcal/ mole
Fig. 12. CL,/3and 7 relaxation
in polyoxymethylene.
NUCLEAR
SPIN-LATTICE FQ?LAXATION IN POLYMERS
1015
It can be seen from fig. 12 that our own correlation frequency for P.O.M. lies close to the line of H eij b oe r and E by and there can be little doubt that it belongs to the process of y relaxation. According to McCrum the CLprocess has an activation energy of 24 kcal/mole; this process would certainly be inobservable at our correlation frequency of 46.77 MHz, as the corresponding temperature would fall far into the range of decomposition. No further
particulars
are known about the ,!? process and there is no trace
of it in our measurements. It may also be pointed out that Ishida’s dielectric measurements for the y process do not agree very well with the mechanical data, nor is there any mention of the cc or /3process in his paper. The behaviour of P.O.M. at low temperatures is similar to that of P.T.F.E. which is not surprising since P.O.M. is also a linear polymer. Again it is likely that T1 is determined by spin diffusion to oxygen impurities, for degassing leads to an increase in TI from 9.1 to 27.5 s. A calculation of the lifetime f of the diffusing magnetization was not possible, as the required experimental data for this polymer are not known. Acknowledgement. This investigation is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)“, supported by the “Organisatie voor Giver Wetenschappelijk Onderzoek (Z.W.O.)“. The authors are much indebted to Mr. P. K. Korver, Mr. P. J. Rens and Mr. C. P. Valkering for assistence with the measurements, to Dr. C. A. ten Seldam for the computation of the average spin lifetime and to the Mathematisch Centrum, Amsterdam, for permission to use the computer X- 1.
APPENDIX
Three-dimensional random walk with sinks. The diffusing nuclear magnetization behaves like a particle, which travels by steps through a simple cubic lattice. Each step consists of a jump from a lattice point to one of its six neighbours, the direction being chosen at random. The time interval between steps is called T. Let C be the fraction of lattice points, at which relaxation occurs; they are treated as randomly distributed “sinks” in which the wandering particle dies. Define n,, the number of particles surviving after having made s steps, and u, the mortality rate i.e. the fraction of particles which die after the sth step. Then n, = n,_i (1 - u). (A-l) A first approximation for u is obtained by observing that in each lattice point the particle has a chance C of hitting a sink in five out of the six equal-
N. 5. TRAPPENIERS,
1016
C. J. GERRITSM.4
AND P. H. OOSTING
ly probable directions (the sixth direction is the particle, which is certainly not a sink). Thus B: is a constant
to this
contribution its tracks.
order”
of the
(A-2)
This
paths
Integrating (A-l) with constant and s can be treated as continuous
station
SC‘!6
approximation.
of “higher
previous
elementary
in which
approach
the particle
C, and using variables,
neglects
doubles
the approximation
lls = 9~0exp (-US). It follows that its destruction,
the average is
number
of steps
back
in
that
n,?
(A-3)
performed
by a particle,
before
(A-4) The mean
lifetime
of a particle
i is given
by I = .+r or with
(A-2) and
t = 67/X‘.
(A-4)) (A-5)
The simple approximation (A-2) was put to the test by a Monte Carlo calculation on a computer. For a simple cubic lattice, containing 163 lattice points, but infinitised by using Uorn-Von IGrmLn boundary conditions, a 1000 random walks were computed. Sinks were distributed at random over the sample in the proportion c‘ = 13 1/I 63 =-= l/31.3. The decay relation (A-3) for n, was found to be closely obeyed; the calculated value of u-1 was 41.7, which, for our purpose, is in satisfactory agreement with the value from (A-2): n-1 = 37.6. Received
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