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Atactic polypropylene at high pressure. II. Methyl group and segmental motion from deuteron NMR spectra and relaxation times
Journal of Non-Crystalline Solids 286 (2001) 12±24
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Atactic polypropylene at high pressure. II. Methyl group and segmental motion from deuteron NMR spectra and relaxation times A.G.S. Hollander, K.O. Prins * Van der Waals±Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018XE Amsterdam, The Netherlands Received 13 April 2000
Abstract Deuteron NMR spectra and relaxation times have been used in an investigation of the eect of high pressure (up to 5000 bar) on the molecular motion in atactic deuterated polypropylene, below and above its glass-transition temperature Tg . Fast axial reorientation of the methyl groups is found, which is neither aected by the glass transition, nor by the application of high pressure. Small-angle motion of the main-chain segments is observed. This motion is identi®ed with the mechanical b-relaxation process. In the glass the character of this process is maintained up to the glasstransition temperature found at each pressure value, while its pressure dependence is very weak. Above Tg this process is strongly pressure-dependent while its character is mainly determined by the distance to Tg . Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 6140; 6250; 6470P; 7660
1. Introduction In polypropylene the available information on the molecular dynamics via mechanical-relaxation studies [1] has been obtained at ambient pressure in materials, composed of chains of dierent tacticities, and in specimens of dierent crystallinities. In a nearly completely atactic material, the largest peak in the mechanical-loss curve (measured at about 1 c/s) appears to be centered at about 270 K, somewhat above the glass transition Tg 253 K. A broad, weak peak centered at
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about 200 K is observed in the glass. Presumably, these two peaks are due to the two main (a and b)-relaxation processes [2,3] generally occurring in amorphous hydrocarbon polymers of simple structure. The a-process, corresponding to cooperative motion of the chains, slows down very rapidly as Tg is approached and is completely arrested at Tg . The temperature dependence of the mean frequency of the a-process is usually well described by the Williams±Landel±Ferry (WLF) function [4]. The b-process continues in the glass and corresponds to limited motions of the chain segments allowed in the quasi-static environment of the neighboring chains. An additional mechanical relaxation, observed at higher temperature (350 K), is a process occurring only in crystalline regions, formed by stereoregular chains
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 0 ) 0 0 3 0 3 - 3
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
[5,6]. Not observed in mechanical-relaxation studies, but evident from proton magnetic resonance investigations [7,8], is the presence of rapid methyl-group reorientation down to very low temperature. Whereas in the macroscopic dynamic properties the underlying microscopic relaxation processes cannot clearly be identi®ed, the techniques of nuclear magnetic resonance oer the possibility of studying the nature of a relaxation process on a molecular level. In particular, deuteron magnetic resonance 2 H-NMR represents a powerful tool. Through the nuclear spin relaxation rates and the spectra in one or more dimensions, one is able to probe molecular motion on a time scale ranging from 10 10 to 102 s. By using two- and three-dimensional exchange 2 H-NMR [9,10], Spiess and co-workers have shown that in atactic polystyrene [11] and also in atactic polypropylene [6,12±14] the mechanism of the a-process is the orientation relaxation of the main chain, which can essentially be described as small-step rotational diusion. It appears in their investigations that, on approaching the glass transition from a higher temperature, the reorientation of the chain slows down dramatically and the mean correlation time characterizing the orientation relaxation increases according to the WLF function. The application of high pressure has distinct eects on the properties of polymers in the glass and in the supercooled liquid. First, the glass transition shifts to a higher temperature. In the preceding paper [15] (paper I) we have shown how the rate of decay of a 2 H-NMR quadrupole echo can be used in monitoring the glass transition. While in the glass the echo decay rate is determined by dipolar coupling, the onset of mainchain motion at the glass-transition temperature strongly accelerates the echo decay. This property was used to determine the pressure shift of Tg in aPP up to 5000 bar. Secondly, because of the large compressibility, particularly of the supercooled liquid, the application of pressure is expected to have a strong in¯uence on the extent and time scale of the molecular motion. In the present paper (paper II) we investigate the rate of reorientation of the methyl groups and
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the restricted motion of the main-chain segments in deuterated aPP on passing from the glassy state to the melt, at a number of pressure values up to 5000 bar. The information on these relatively fast types of motion is obtained from the (one-dimensional) 2 H-NMR spectrum and the spin-lattice relaxation rates of the deuterons of the methyl groups and of the main chain. In the next paper [17] (paper III) we present an investigation of the eect of high pressure on the slow dynamics of the main chain above the glass transition (corresponding to the a-process), using two-dimensional exchange 2 H-NMR. The NMR principles and methods used are well established [10] and are entered in this paper only brie¯y, when necessary. 2. Experimental We used the same high-pressure NMR equipment and sample material as in paper I [15]. Deuteron NMR spectra of aPP were obtained using the quadrupole echo technique [17,18], by applying a p=2x s1 p=2y s1 -acquisition sequence, using an inter-pulse spacing s1 of 30 ls. The time domain signals, with a length of 1024 or 2048 points, were obtained using quadrature detection with a 2000 kHz sampling rate. To minimize spectral distortions a phase cycling scheme [19] was used in combination with the quadrupole echo pulse sequence. Prior to Fourier transformation the time domain signal was shifted to start from the echo maximum using spline functions. Deuteron spin-lattice relaxation was investigated using a saturation pulse sequence consisting of ten p=2 pulses at 1 ms intervals which, after a variable time delay t ranging from 2 ms to 20 s, was followed by a quadrupole echo sequence with s1 30 ls. 3. 2 HNMR quadrupole echo spectra The deuteron quadrupole echo spectrum obtained in aPP in the glassy state at ambient pressure at 246 K has been discussed in the preceding
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A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
paper [15]. As was shown schematically in Fig. 1, the spectrum consists of a superposition of two Pake doublets. The broad doublet with a distance between the singularities Dchain of about 123 kHz originates from the main-chain deuterons. The narrow doublet with a peak distance Dmethyl of about 37 kHz is the contribution of the methylgroup deuterons. Spectra obtained at ambient pressure as a function of temperature are presented in Fig. 1.
The repetition time of the quadrupole echo sequence used was always at least 3hT1 ichain (for de®nition: see below). The Pake line shapes corresponding to a quasi-rigid solid are maintained up to about 280 K Tg 30 K. Closer inspection shows that both Dmethyl and Dchain decrease with increasing temperature, while their ratio Dmethyl =Dchain slowly decreases from 0.30 at 174 K to 0.28 at 284 K. These values are smaller than the ratio 1/3 expected for tetrahedral bond angles,
Fig. 1. 2 H-NMR spectra of aPP at 1 bar as a function of temperature.
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
indicating the presence of additional motion of the methyl C±2 H bonds superimposed on the methylgroup reorientation. At about 280 K we observe the eect of additional molecular motion on the line shape, indicating that the rate of this motion becomes the order of the total line width of about 100 kHz. Presumably, the motion observed is the reorientation of the chain segments over larger angles. The reorientation is visible both from the reorientation of the axes of the EFGs at the chain deuterons and of the axes of the averaged EFGs at the methyl deuterons. Not directly visible due to the scaling of the spectra to the same peak intensity, but noticeable in the increase of the noise in the spectrum, is the decrease in intensity as a result of loss of coherence by molecular reorientation [15,20,21]. Above 290 K Tg 40 K we observe the spectra corresponding to averaging by close to isotropic motion. Above 300 K Tg 50 K the motion becomes isotropic on the time scale of the quadrupole echo sequence. The EFGs average to zero and we did not detect any quadrupole echoes. Instead, the spectra were obtained from free induction decays using single p=2 pulses.
15
4. Restricted chain motion in the glass Fig. 2 contains a part of the spectra of the chain deuterons, obtained at ambient pressure and at temperature values below and slightly above Tg . Shown in detail is the region of the singularity corresponding to orientations of the static magnetic ®eld B with h p=2. The slight deviation from a pure Pake pattern indicates the presence of a small non-axiality in the EFG seen by the chain deuterons. This must be due to a limited non-axial motion of the chain C±2 H bonds. Presumably, in the glass such motion is restricted to local small-angle torsional motion of the chain segments, in which the chain C±2 H bonds participate; their motion occurs approximately in the plane, perpendicular to the plane de®ned by the carbon atom, involved in the bond, and its two main-chain neighbors. Averaging an axial EFG tensor (with principle values eq=2; eq=2; eq) over two C±2 H bond orientations, related by rotations about the Yaxis over angles and #, indeed results [22] in a non-axial tensor with principle values eq 3 sin2 # 1=2, eq=2, eq 3 cos2 # 1=2 and
Fig. 2. Chain deuteron line shape in the region of the singularities, at ambient pressure and 174, 222, 242 and 262 K.
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asymmetry parameter g 3 sin2 #= 3 cos2 # 1. In the spectrum the non-axiality causes the Pake singularity occurring for H p=2 to split up into two [23]. The frequencies mX and mY , indicated in the ®gure, correspond to orientations of the static magnetic ®eld B parallel to the principal axes X and Y of the average EFG tensor. Clearly visible is how the singularity at mX shifts to lower (negative) frequencies at increasing temperature. The edges of the spectra (denoted by mY ) do not shift. Rather than the typical line shape of a powder spectrum corresponding to a single, non-zero value of the asymmetry parameter g, the shape of the spectra between mX and mY indicates a distribution of g values. By reading the ratio mX =mY 1 3 sin2 # from Fig. 2 we can estimate the rms value of #. We ®nd that this value increases from 4:7° at 174 K to 7:8° at 242 K. In this analysis it is assumed that the reorientation rate is in the fast motion limit, i.e., much larger than dQ 3=8 2p 169 103 s 1 . As will be seen below, this assumption can be justi®ed from the values of the chain deuteron spin-lattice relaxation time. Similar behavior is found in the spectra obtained at high pressure. The reduction of Dchain noted above can now be identi®ed as the gradual shift in the line-shape singularity, caused by the non-axiality of the average EFG tensor. Due to the limited spectral resolution, we cannot determine whether similar distortions of the methyl Pake doublet occur. Since Dmethyl =Dchain < 1=3, it is clear that the methyl groups, apart from their axial reorientation, take part in additional, close to axial, motion (probably) of the C±C2 H3 bonds. We note that a similar narrowing of a Pake pattern due to fast small-angle motion has been observed before, for instance in chain-deuterated polystyrene, slightly above the glass transition [24]. Also in the 2 H-NMR spectrum of deuterated polyethylene [25,26] in the crystalline (orthorhombic) state, which at room temperature consists of a Pake doublet, the singularities split up into two on increasing the temperature, due to a single non-zero value of the asymmetry parameter, also caused by small-angle reorientation of the C±2 H bonds in a plane.
5. Deuteron spin-lattice relaxation 5.1. Experimental We investigated the deuteron spin-lattice relaxation as a function of temperature at about ambient pressure and pressure values of 1, 2 and 5 kbar, using the saturation recovery method. In these experiments the temperature was increased through the glass transition, except at 5 kbar where the temperature was lowered through the glass transition. The temperature was changed at a rate of 2 K h 1 , and stabilized for about 30 min prior to each new measurement. We separately monitored the recovery of the methyl deuteron and the chain deuteron longitudinal magnetization with the following procedure. After Fourier transformation of the quadrupole echoes, obtained at each delay time, the noise in the spectra obtained was reduced by averaging the intensities at x and x. The chain deuteron magnetization was determined as a function of the delay time by integrating the area around the Pake singularity in the spectrum of the chain deuterons in the frequency range +55 to +65 kHz, outside the range of the spectrum of the methyl deuterons (see Fig. 2 of this paper and paper I [15], Fig. 1). We assume that the integrated intensity is a measure for the total chain deuteron magnetization. We obtained the methyl deuteron magnetization by integrating each spectrum from 22 to 22 kHz and by subtracting the contribution of the chain deuterons in that interval. From a fully relaxed experimental spectrum obtained in the glass we determined that this contribution is 19.3% of the total intensity integrated in the frequency interval from 22 to 22 kHz. The result, again as a function of the delay time, gives the recovery of the methyl-deuteron magnetization. We note that in this procedure we neglect the small eects of the anisotropy of the quadrupole spin-lattice relaxation rate to be discussed below. The recovery of the deuteron magnetization M t in aPP was found to behave strongly nonexponentially, both for the chain and for the methyl deuterons. Such behavior is usually observed in glass-forming systems [27]. The experimental relaxation curves can be represented very well by a
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
stretched exponential, the Kohlrausch±Williams± Watts (KWW) function [28,29] M 1 M t e M 1
t=Nb
;
1
where N is a measure of the time constants involved in the recovery and 1 b is a measure of the non-exponentiality. In the ®t of the KWW function to the experimental relaxation curves, we followed the recovery of the magnetization up to 1 e 2 , i.e., up to 86% of its equilibrium value. When the non-exponential recovery function is considered as a superposition of exponential functions exp t=T1 , distributed according to q T1 Z 1 M 1 M t q T1 e t=T1 dT1 ; 2 M 1 0 the time constant T1av , de®ned as an average over this distribution, can be expressed [30] in the KWW parameters after ®tting the data according to Eq. (1): Z 1 N T1av q T1 T1 dT1 C b 1 ; 3 b 0 where C is the gamma function. 5.2. The spin-lattice relaxation of the methyl-group deuterons We ®rst discuss the experimental results concerning the motion of the methyl groups. The average relaxation time T1av methyl of the methyl deuterons (as de®ned in Eq. (3)) in the temperature range from 180 to 340 K and at pressure values up to 5 kbar is shown in Fig. 3(a). The glass-transition temperature at each pressure value, indicated in this ®gure, is obtained from paper I. It appears that T1av methyl is hardly dependent on pressure and that no eect of the glass transition on T1av methyl is visible at any pressure value. Fig. 3(b) shows bmethyl as a function of T Tg and pressure. The non-exponentiality decreases with increasing temperature while, at constant T Tg , the relaxation becomes more exponential at higher pressure. We note, however, that the estimated errors in the b values obtained are about 0:05, due to inaccuracy in the determination M 1 M t at the longest delay times t.
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The coupling of the deuteron electric quadrupole moment with the EFG present at its site is two orders of magnitude stronger than the magnetic dipole±dipole coupling between the nuclei. Therefore, the mechanism for spin-lattice relaxation of the methyl deuterons is the modulation of the deuteron electric quadrupole coupling caused by the rapid axial reorientation of the methyl groups. The general formalism for this mechanism has been presented by Abragam [31]. It is important to note that for a single deuteron, because of its spin quantum number I 1, the relaxation is exponential. The deuteron spins relax independently from each other, as the quadrupole coupling Hamiltonian depends on the variables of single spins. Spin diusion [32] between neighboring deuterons is very much slower than the methyl deuteron spin-lattice relaxation. The orientation dependence of the relaxation rate has been analyzed by Torchia and Szabo [33] for a number of types of anisotropic motion. They give an analysis of a model, in which a C±2 H bond reorients over three equivalent directions making an angle H with the axis of reorientation, with a mean jump rate k. For the spin-lattice relaxation rate of a methyl deuteron we obtain from their results in equation (39) and Table 2, with H arccos 1=3: 2 1 1 e2 qQ s 2 sin2 h h 1 x20 s2 T1Q h; / 36 p 0:5 sin2 2h 4 2 sin3 h cos h cos 3/ s 6 sin2 2h 8 cos2 h 1 4x20 s2 p 3 4 2 cos h 4 2 sin h cos h cos 3/ ; 4 where s is de®ned as 3k 1 . In this expression h and / are the spherical polar angles of the static magnetic ®eld B in a methyl-group ®xed axis system, h being the angle between B and the axis of the methyl group. As a consequence of the anisotropic motion, the relaxation rate is dependent on both h and /. For a system of methyl deuterons,
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Fig. 3. (a) T1av methyl of methyl deuterons as derived from KWW ®ts. (b) bmethyl as derived from KWW ®ts as a function of T
Tg .
due to the / dependence the relaxation is slightly non-exponential for each frequency in the spectrum, even for a single value of s, as is noted in [33]. In the `extreme narrowing' limit x0 s 1 Eq. (4) reduces to 2 1 s e2 qQ 1 cos2 h: 5 h T1Q h 4
individual deuterons will result, in general, in a non-exponential recovery of the magnetization. The average recovery rate can be approximated by averaging Eq. (4) over an isotropic distribution of orientations, resulting in 2 1 e2 qQ s 4s h1=T1Q i : 6 15 h 1 x20 s2 1 4x20 s2
For a macroscopic sample the orientation dependence of the rate of exponential relaxation of the
In the extreme narrowing regime the recovery of the total magnetization for an isotropic distribu-
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
tion of orientations, which corresponds to the integrated intensity of the methyl-deuteron spectrum, has the form Z p sin h exp t 1 cos2 h dh: 0
It appears that a ®t of a KWW function to this recovery function results in b 0:98, so its shape is very close to exponential. We obtained h1=T1Q i from Eq. (6) by substituting e2 qQ= h 2p 169 103 rad s 1 and x0 2p 41:4 106 rad s 1 . In Fig. 4, h1=T1Q i as a function of s is compared with the experimental values of 1= T1av methyl as a function of 1=T . The solid line represents a ®t of h1=T1Q i to 1= T1av methyl obtained by adjusting the s-scale. The comparison shows that even at the lowest temperature the relaxation rate does not fully reach the maximum value predicted by Eq. (6): the factor in square brackets has a maximum value of 1:4252=x0 at x0 s 0:6158 and, therefore, at s 3:84 10 9 s. Nearly all the experimental data occur in the extreme narrowing regime. It is important to note that, as a consequence, the experimentally observed non-exponentiality in the recovery of the magnetization of the methyl deuterons cannot be due to the orientation dependence of the relax-
19
ation rate. Instead, it must be the result of the presence of a distribution of methyl group reorientation rates. The non-exponential relaxation of the methyl deuterons is really due to a distribution of exponentials, as in Eq. (2). In the extreme narrowing regime the spin-lattice relaxation time is proportional to s 1 . Therefore, the distribution G s 1 of reorientation rates is directly related to the distribution of relaxation times q T1 , introduced in Eq. (3): the distribution g ln s sG s 1 is equivalent to the distribution q T1 =T1 . The latter can be obtained [30] from the KWW ®t (2) to the recovery function, via an inverse Laplace transform. The observed values of b between about 0.8 and 0.9, correspond to a halfwidth of the distribution g ln s of reorientation rates of about half a decade. It also follows that T1av methyl is the value of the spin-lattice relaxation av time corresponding to the average value s 1 , so we obtain the temperature dependence of the latter from the comparison in Fig. 4. The spin-lattice relaxation rate 1= T1av methyl appears to reach its maximum value at 147 K, where s 1 av is av 1=3:84 10 9 s 1 ; s 1 follows an Arrhenius av law s 1 s0 1 exp DE=RT , with s0 1 8:46 10 13 s 1 and an activation energy DE of 9:7 kJ mol 1 .
Fig. 4. The calculated values of h1=T1Q i for the three-site jump model, as a function of s, compared with the reciprocal of the measured values T1av methyl , as a function of reciprocal temperature.
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A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
Our result may be compared with the reorientation of the methyl groups in non-deuterated, partially crystalline, isotactic PP, which has been investigated before [7,8] with NMR, using the proton spin-lattice relaxation rates 1=T1H and H 1=T1q . At temperature values below about 150 K, these relaxation rates are also mainly determined by the methyl reorientation, modulating in this case the magnetic dipole±dipole coupling between the methyl protons. These studies also show that the reorientation is rapid, down to low temperature, leading to a maximum in 1=T1H at about 160 K (measured at a Zeeman frequency of 30 MHz). Our value DE 9:7 kJ mol 1 is in very good agreement with the value 2:3 kcal mol 1 found from proton spin-lattice relaxation [8]. This is the typical value of the potential energy barrier for methyl reorientation [34,35], determined by intrachain interactions. Therefore, it appears that the reorientation of the methyl groups can be considered as the process of activated jumps across this potential barrier. The average methyl reorientation rate is not dependent on pressure and is not aected by a volume decrease of several percent, which also indicates that the energy barrier for reorientation mainly consists of intra-chain interactions. The interactions dier locally as a consequence of the dierent chain conformations at the carbon atom to which the methyl groups are bound. In the glass these conformations are frozen, which leads to the distribution of reorientation rates. The chain conformations change only slowly [6,16] up to about 15 K above Tg , therefore, T1av methyl does not change at Tg while b remains smaller than 1. At higher temperature, the chain conformations start to change on the time scale of T1av methyl , the distribution becomes narrower and b approaches 1. We do not have an explanation for the distribution becoming narrower with increasing pressure at constant T Tg . 5.3. The spin-lattice relaxation of the chain deuterons Fig. 5(a) shows the relaxation time T1av chain of the chain deuterons. The glass-transition temperature at each pressure value, indicated in this ®g-
ure, is again obtained from paper I [15]. In the glass T1av chain is rather long, indicating that the motion of the chain deuterons is very restricted. The exponent bchain , as derived from ®tting the KWW function (1) to the recovery of the magnetization, is about 0.7 in the glass (see Fig. 5(b)). Below Tg ; T1av chain appears to be pressure independent. At increasing temperature T1av chain decreases slowly until, at a temperature slightly above Tg at each pressure value, hT1 ichain shows a sharp decrease, which is clearly the eect of the glass transition and indicates increased mobility of the chain segments. In the temperature range Tg 10 to Tg 30 K; bchain rapidly increases to unity, corresponding to fully exponential relaxation. Similar behavior has been observed in earlier experiments on atactic polystyrene [36]. In the glass, the dominant mechanism for spinlattice relaxation of the chain deuterons is the deuteron electric quadrupole coupling, modulated by the limited local reorientation motion of the chain C±2 H bond directions, leading to the eect of the non-axiality in the average EFG tensor observed in the spectra. This oers the possibility of determining the time scale of this motion from the measured spin-lattice relaxation time. As will be discussed in the following, this relaxation path is much more ecient than the other available paths, namely spin diusion to the methyl deuterons and cross-relaxation to both methyl and chain protons, the proton abundance in our sample being 12%. It is well known that spin diusion [32], the transport of magnetization between neighboring spins, is very slow in deuteron systems. In ®rst order it requires spin ¯ip±¯op transitions, caused by the homonuclear dipole±dipole coupling between the deuterons. Since in this process energy is conserved, it only occurs when the resonance lines of neighboring deuterons show overlap. For each deuteron the quadrupole shift depends on the orientation of the static magnetic ®eld B in the PAS of the EFG. Therefore, in an amorphous material the shifts of the resonances of neighboring deuteron spins are unlikely to be equal, and spin diusion is quenched to a large extent. However, spin diusion between inequivalent deuteron spins is not in®nitely slow, even in cases
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Fig. 5. (a) T1av chain of chain deuterons as derived from KWW ®ts. (b) bchain as derived from KWW ®ts as a function of T
where the dierence between the quadrupole splittings exceeds the resonance line widths [37]. As will be discussed in paper III [16], two-dimensional exchange 2 H-NMR experiments in aPP at temperatures around Tg 12 K indeed show that spin diusion occurs between methyl deuterons and chain deuterons, on a time scale of 700 ms. It occurs only for chain deuterons with frequency shifts within the range of the methyl-deuteron frequency
21
Tg .
shifts (from about 47 to 47 kHz). Spin diusion among the chain deuterons is at least one order of magnitude slower. We note that in the present experiment the values of hT1 ichain are obtained from the intensity of the spectrum outside the methyl frequency range and are, therefore, hardly aected by spin diusion to the methyl deuterons. This is also evident from the fact that T1av chain becomes largest at the temperature where
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T1av methyl approaches its lowest value and spin diusion would have the largest eect. The various mechanisms for spin-lattice relaxation of the chain deuterons, provided by the dipole±dipole coupling of the chain deuterons with chain protons and methyl protons present in the system, can all be shown to be much weaker than the direct quadrupole mechanism. The time scale on which all protons in the system are coupled among themselves by spin diusion can be estimated to be within 1 ms. The protons will have a common relaxation time, governed by the ecient relaxation mechanism of the methyl protons via the dipole±dipole coupling with proton and/or deuteron neighbors. The various contributions can be calculated individually [33,38], assuming the three-site jump model for the motion of the methyl group (the rate of which, as we have seen, is not aected by the glass transition). In the temperature region of interest the resulting common proton T1 can be estimated to decrease from about 5 s at 340 K to about 1 s at 180 K, where the methyl-group reorientation rate becomes close to the proton spin Zeeman frequency 2p 270 106 s 1 (we note, that the proton T1 determined [8] at 30 MHz in non-deuterated PP decreases from about 0.6 to about 0.05 s in this temperature range). The most eective cross-relaxation path for the chain deuterons is the dipole±dipole coupling with the available protons in the reorienting methyl groups. This coupling is largest for a deuteron and a methyl group at the same carbon atom. An estimate of the associated deuteron±proton cross-relaxation time can again be obtained using the results of Torchia and Szabo [33]. In the temperature region of interest its value is larger than 100 s. This time limits the process of cross-relaxation to the proton spin system since, as seen above, the latter has a faster spin-lattice relaxation mechanism. It may be concluded, therefore, that the contribution to T1av chain by relaxation through dipole±dipole interactions is minimal. We conclude that the spin-lattice relaxation rate of the chain deuterons is caused by the limited local reorientation motion of the chain C±2 H bonds. The relaxation rate of a single deuteron can be calculated in a simpli®ed model for this motion, by viewing it as a process of discrete reorientation
over an angle 2# with mean time sj between jumps. The results obtained by Torchia and Szabo [33] for this model lead to an orientation dependent relaxation rate. Therefore, for a macroscopic sample, the recovery of the total magnetization is non-exponential. However, since in the measurement of the relaxation of the chain deuterons we obtain the intensities from the area close to the Pake singularity in the spectrum, that is for orientations of B with azimuth angle h close to p=2, we may expect close to exponential recovery for single values of 2# and sj . We compare our data with the expression for the recovery rate, obtained by substituting the value h p=2 in Eq. (36) of Ref. [33]. In the slow-motion limit x0 sj > 1 we ®nd Q 1=T1chain
9 32
e2 qQ h
2
sin2 2# : x20 sj
7
It follows that the strong non-exponentiality observed is again the result of a superposition of close to exponential functions with time constants as in Eq. (7). Therefore, in principle the distribution of relaxation times q T1 re¯ects the distribution of sj = sin2 2#. However, since the longest relaxation times in the distribution are not observable because of spin diusion, the value of T1av chain is mainly determined by the contribution of the smaller values of sj = sin2 2#. Q From a comparison of T1av chain with T1chain in Eq. (7) we ®nd an estimate of the values of sj = sin2 2# giving the main contribution to the observable relaxation rate. At ambient pressure we derived the rms values of # from the spectra. With these we ®nd a value of sj of about 5:0 10 7 s in the temperature range 174 to 242 K; it decreases to about 1:3 10 7 s at 262 K (i.e., 10 K above Tg ). Presumably, these values merely represent the short time part of the sj distribution. These reorientation times indeed lead to motional-averaged spectra, as was assumed in the calculation of the shift of the line-shape singularity in the spectrum of the chain deuterons. At increasing temperature, the motion becomes gradually faster while, simultaneously, the average reorientation angle increases, resulting in a slowly decreasing T1av chain . The fast decrease of T1av chain at and above the glass transition indicates an increased mobility of
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
the main-chain segments, presumably, leading to reorientation over larger angles, together with a shift in the frequency distribution. It has to be noted that in the liquid the collective motion (the a-process) permits more isotropic segmental motion on which the small-angle reorientation is superimposed. However, from the experiments to be presented in the next paper, it follows that even at Tg 30 K the isotropic component in the reorientation is not fast enough to contribute to spinlattice relaxation. On application of pressure the distribution of sj = sin2 2#, characterizing the process of small-angle motion in the glass is maintained up to the glass-transition temperature at each pressure value, as appears from the behavior of T1av chain and bchain shown in Fig. 5. For instance at Tg at 5 kbar, T1av chain is only about a factor 1.8 shorter than at ambient pressure, indicating that also at this pressure value the time scale of this process at the glass transition is about 10 7 s. It also appears that in the glass the value of T1av chain does not dependent on pressure, but only on temperature. This is somewhat surprising, since on compressing one expects the increased inter-chain interactions, restricting the motion, to lead to a decrease and increase of the average values of # and of sj , respectively, which according to Eq. (7) would result in an increase of T1av chain . We conclude that the decrease in volume (about 10% at 5 kbar [15]) is not large enough to have a visible eect. In the liquid T1av chain is strongly dependent on pressure. However, at each pressure value, the decrease of T1av chain at and above the glass transition is very similar, showing that in the liquid the amplitudes and frequencies of the small-angle motion are mainly determined by the distance to the glass transition. Probably, the small-angle reorientation of the chain segments providing the spin-lattice relaxation path for the chain deuterons is the mechanism of the b-mechanical relaxation process. To our knowledge, in aPP only data at low frequency 1 s 1 are available [1]. A weak, broad peak (called the c-peak in these studies) is found with its maximum at about 200 K. Presumably, this peak corresponds to the low-frequency tail of a broad distribution of reorientation rates of which we
23
observe the component at about 106 s 1 in the deuteron spin-lattice relaxation rate. The observed behavior corresponds to the properties of the bprocess generally found in amorphous polymers, namely that it gives a very broad loss curve, while it is rather insensitive to pressure [3]. The observation that the small-angle reorientation resulting in spin-lattice relaxation is the mechanism of the b-process was also made in the study [24] on chain-deuterated polystyrene above the glass transition, mentioned above. There it was found, by employing a dynamic ®lter [39] in a 4D exchange experiment, that in spatial regions where slow components in the process occur, also the b-process proceeds with longer than average correlation times. 6. Conclusion In aPP the spin-lattice relaxation of the methyl group deuterons is caused by the high rate of methyl-group reorientation (between 109 and 1010 s 1 ). The energy barrier for reorientation mainly consists of intra-chain interactions. The reorientation rate is aected only very little by an increase in pressure of 5 kbar. In the glass the nonexponentiality in the recovery of the longitudinal magnetization indicates the presence of a distribution of reorientation rates, diering locally as a consequence of the dierent chain conformations. The chain conformations change only slowly up to about 15 K above Tg and, as a consequence, the methyl reorientation rate is not aected by the glass transition. In the glass, for temperatures up to Tg , smallangle reorientation of the chain segments occurs with an average amplitude up to 15°. This is concluded from the quadrupole echo spectra and spinlattice relaxation rates of the main-chain deuterons. The character of the motion is maintained up to the glass transition occurring at each pressure value. The eect of pressure on this process is very small. The motion is characterized by a broad rate distribution of which the component at about 107 s 1 is observed via the spin-lattice relaxation rate. This motion is identi®ed with the mechanism of the bprocess found in mechanical-relaxation studies.
24
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
In the liquid, above the glass transition, the amplitude of the reorientation increases, while it becomes somewhat faster. In the liquid this process is strongly dependent on pressure. It is mainly determined by the distance to the glass transition. This motion is superimposed on the much slower quasi-isotropic reorientation of the chain segments, the study of which is the subject of the next paper. References [1] N.G. McCrum, B.E. Read, G. Williams, Anelastic and Dielectric Eects in Polymeric Solids, Dover, New York, 1991. [2] G.P. Johari, J. Chem. Phys. 58 (1973) 1766. [3] G. Williams, Adv. Polym. Sci. 33 (1979) 59. [4] M.L. Williams, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc. 77 (1955) 3701. [5] N.G. McCrum, Polym. Lett. 2 (1964) 495. [6] D. Schaefer, H.W. Spiess, U.W. Suter, W.W. Fleming, Macromolecules 23 (1964) 3431. [7] M.P. McDonald, I.M. Ward, Proc. Phys. Soc. 80 (1962) 1249. [8] V.J. McBrierty, D.C. Douglas, D.R. Falcone, J. Chem. Soc. Faraday Trans. II 68 (1972) 1051. [9] C. Schmidt, S. We®ng, B. Bl umich, H.W. Spiess, Chem. Phys. Lett. 130 (1986) 84. [10] K. Schmidt-Rohr, H.W. Spiess, Multidimensional SolidState NMR and Polymers, Academic Press, London, 1994. [11] S. Kaufmann, S. We®ng, D. Schaefer, H.W. Spiess, J. Chem. Phys. 93 (1990) 197. [12] K. Zemke, B.F. Chmelka, K. Schmidt-Rohr, H.W. Spiess, Macromolecules 24 (1991) 6874. [13] D. Schaefer, H.W. Spiess, J. Chem. Phys. 97 (1992) 7944. [14] K. Zemke, K. Schmidt-Rohr, H.W. Spiess, Acta Polym. 45 (1994) 148. [15] A.G.S. Hollander, K.O. Prins, this issue, p. 1.
[16] A.G.S. Hollander, K.O. Prins, this issue, p. 25. [17] H.W. Spiess, H. Sillescu, J. Magn. Reson. 42 (1981) 381. [18] J.H. Davis, K.R. Jerey, M. Bloom, M.I. Valic, T.P. Higgs, Chem. Phys. Lett. 42 (1976) 390. [19] A.D. Ronemus, R.L. Vold, R.R. Vold, J. Magn. Reson. 70 (1986) 416. [20] H.W. Spiess, H. Sillescu, J. Magn. Reson. 42 (1981) 381. [21] M.S. Green®eld, A.D. Ronemus, R.L. Vold, R.R. Vold, P.D. Ellis, T.E. Raidy, J. Magn. Reson. 72 (1987) 89. [22] V.D. Fedotov, H. Schneider, NMR Basic Principles and Progress 21, Springer, Berlin, 1989. [23] U. Haeberlen, High Resolution NMR in Solids: Selective averaging. Advances in Magnetic Resonance, Supplement 1, Academic Press, New York, 1976. [24] J. Leisen, K. Schmidt-Rohr, H.W. Spiess, J. Non-Cryst. Solids 172±174 (1994) 737. [25] D. Hentschel, H. Sillescu, H.W. Spiess, Polymer 25 (1984) 1078. [26] M. De Langen, H. Luigjes, K.O. Prins, Polymer 41 (2000) 1183. [27] W. Schnauss, F. Fujara, K. Hartmann, H. Sillescu, Chem. Phys. Lett. 166 (1990) 381. [28] R. Kohlrausch, Ann. Phys. 12 (1847) 393. [29] G. Williams, D.C. Watts, Trans. Faraday Soc. 66 (1970) 80. [30] C.P. Lindsey, G.D. Patterson, J. Chem. Phys. 73 (1980) 3348. [31] A. Abragam, The Principles of Nuclear Magnetism, Clarendon, Oxford, 1961. [32] N. Bloembergen, Physica 15 (1949) 386. [33] D.A. Torchia, A. Szabo, J. Magn. Reson. 49 (1982) 107. [34] P.J. Flory, Statistical Mechanics of Chain Molecules, Hanser, New York, 1969. [35] R.T. Bailey, A.M. North, R.A. Pethrick, Molecular Motion in High Polymers, Clarendon, Oxford, 1981. [36] A.S. Kulik, K.O. Prins, Polymer 34 (1993) 4642. [37] D. Suter, R.R. Ernst, Phys. Rev. B 32 (1985) 5608. [38] R.L. Hilt, P.S. Hubbard, Phys. Rev. 134A (1964) 392. [39] U. Tracht, A. Heuer, H.W. Spiess, J. Non-Cryst. Solids 235±237 (1998) 27.
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Atactic polypropylene at high pressure. II. Methyl group and segmental motion from deuteron NMR spectra and relaxation times A.G.S. Hollander, K.O. Prins * Van der Waals±Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018XE Amsterdam, The Netherlands Received 13 April 2000
Abstract Deuteron NMR spectra and relaxation times have been used in an investigation of the eect of high pressure (up to 5000 bar) on the molecular motion in atactic deuterated polypropylene, below and above its glass-transition temperature Tg . Fast axial reorientation of the methyl groups is found, which is neither aected by the glass transition, nor by the application of high pressure. Small-angle motion of the main-chain segments is observed. This motion is identi®ed with the mechanical b-relaxation process. In the glass the character of this process is maintained up to the glasstransition temperature found at each pressure value, while its pressure dependence is very weak. Above Tg this process is strongly pressure-dependent while its character is mainly determined by the distance to Tg . Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 6140; 6250; 6470P; 7660
1. Introduction In polypropylene the available information on the molecular dynamics via mechanical-relaxation studies [1] has been obtained at ambient pressure in materials, composed of chains of dierent tacticities, and in specimens of dierent crystallinities. In a nearly completely atactic material, the largest peak in the mechanical-loss curve (measured at about 1 c/s) appears to be centered at about 270 K, somewhat above the glass transition Tg 253 K. A broad, weak peak centered at
* Corresponding author. Tel.: +31-20 525 5763; fax: +31-20 525 5788. E-mail address: [email protected] (K.O. Prins).
about 200 K is observed in the glass. Presumably, these two peaks are due to the two main (a and b)-relaxation processes [2,3] generally occurring in amorphous hydrocarbon polymers of simple structure. The a-process, corresponding to cooperative motion of the chains, slows down very rapidly as Tg is approached and is completely arrested at Tg . The temperature dependence of the mean frequency of the a-process is usually well described by the Williams±Landel±Ferry (WLF) function [4]. The b-process continues in the glass and corresponds to limited motions of the chain segments allowed in the quasi-static environment of the neighboring chains. An additional mechanical relaxation, observed at higher temperature (350 K), is a process occurring only in crystalline regions, formed by stereoregular chains
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 0 ) 0 0 3 0 3 - 3
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
[5,6]. Not observed in mechanical-relaxation studies, but evident from proton magnetic resonance investigations [7,8], is the presence of rapid methyl-group reorientation down to very low temperature. Whereas in the macroscopic dynamic properties the underlying microscopic relaxation processes cannot clearly be identi®ed, the techniques of nuclear magnetic resonance oer the possibility of studying the nature of a relaxation process on a molecular level. In particular, deuteron magnetic resonance 2 H-NMR represents a powerful tool. Through the nuclear spin relaxation rates and the spectra in one or more dimensions, one is able to probe molecular motion on a time scale ranging from 10 10 to 102 s. By using two- and three-dimensional exchange 2 H-NMR [9,10], Spiess and co-workers have shown that in atactic polystyrene [11] and also in atactic polypropylene [6,12±14] the mechanism of the a-process is the orientation relaxation of the main chain, which can essentially be described as small-step rotational diusion. It appears in their investigations that, on approaching the glass transition from a higher temperature, the reorientation of the chain slows down dramatically and the mean correlation time characterizing the orientation relaxation increases according to the WLF function. The application of high pressure has distinct eects on the properties of polymers in the glass and in the supercooled liquid. First, the glass transition shifts to a higher temperature. In the preceding paper [15] (paper I) we have shown how the rate of decay of a 2 H-NMR quadrupole echo can be used in monitoring the glass transition. While in the glass the echo decay rate is determined by dipolar coupling, the onset of mainchain motion at the glass-transition temperature strongly accelerates the echo decay. This property was used to determine the pressure shift of Tg in aPP up to 5000 bar. Secondly, because of the large compressibility, particularly of the supercooled liquid, the application of pressure is expected to have a strong in¯uence on the extent and time scale of the molecular motion. In the present paper (paper II) we investigate the rate of reorientation of the methyl groups and
13
the restricted motion of the main-chain segments in deuterated aPP on passing from the glassy state to the melt, at a number of pressure values up to 5000 bar. The information on these relatively fast types of motion is obtained from the (one-dimensional) 2 H-NMR spectrum and the spin-lattice relaxation rates of the deuterons of the methyl groups and of the main chain. In the next paper [17] (paper III) we present an investigation of the eect of high pressure on the slow dynamics of the main chain above the glass transition (corresponding to the a-process), using two-dimensional exchange 2 H-NMR. The NMR principles and methods used are well established [10] and are entered in this paper only brie¯y, when necessary. 2. Experimental We used the same high-pressure NMR equipment and sample material as in paper I [15]. Deuteron NMR spectra of aPP were obtained using the quadrupole echo technique [17,18], by applying a p=2x s1 p=2y s1 -acquisition sequence, using an inter-pulse spacing s1 of 30 ls. The time domain signals, with a length of 1024 or 2048 points, were obtained using quadrature detection with a 2000 kHz sampling rate. To minimize spectral distortions a phase cycling scheme [19] was used in combination with the quadrupole echo pulse sequence. Prior to Fourier transformation the time domain signal was shifted to start from the echo maximum using spline functions. Deuteron spin-lattice relaxation was investigated using a saturation pulse sequence consisting of ten p=2 pulses at 1 ms intervals which, after a variable time delay t ranging from 2 ms to 20 s, was followed by a quadrupole echo sequence with s1 30 ls. 3. 2 HNMR quadrupole echo spectra The deuteron quadrupole echo spectrum obtained in aPP in the glassy state at ambient pressure at 246 K has been discussed in the preceding
14
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
paper [15]. As was shown schematically in Fig. 1, the spectrum consists of a superposition of two Pake doublets. The broad doublet with a distance between the singularities Dchain of about 123 kHz originates from the main-chain deuterons. The narrow doublet with a peak distance Dmethyl of about 37 kHz is the contribution of the methylgroup deuterons. Spectra obtained at ambient pressure as a function of temperature are presented in Fig. 1.
The repetition time of the quadrupole echo sequence used was always at least 3hT1 ichain (for de®nition: see below). The Pake line shapes corresponding to a quasi-rigid solid are maintained up to about 280 K Tg 30 K. Closer inspection shows that both Dmethyl and Dchain decrease with increasing temperature, while their ratio Dmethyl =Dchain slowly decreases from 0.30 at 174 K to 0.28 at 284 K. These values are smaller than the ratio 1/3 expected for tetrahedral bond angles,
Fig. 1. 2 H-NMR spectra of aPP at 1 bar as a function of temperature.
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
indicating the presence of additional motion of the methyl C±2 H bonds superimposed on the methylgroup reorientation. At about 280 K we observe the eect of additional molecular motion on the line shape, indicating that the rate of this motion becomes the order of the total line width of about 100 kHz. Presumably, the motion observed is the reorientation of the chain segments over larger angles. The reorientation is visible both from the reorientation of the axes of the EFGs at the chain deuterons and of the axes of the averaged EFGs at the methyl deuterons. Not directly visible due to the scaling of the spectra to the same peak intensity, but noticeable in the increase of the noise in the spectrum, is the decrease in intensity as a result of loss of coherence by molecular reorientation [15,20,21]. Above 290 K Tg 40 K we observe the spectra corresponding to averaging by close to isotropic motion. Above 300 K Tg 50 K the motion becomes isotropic on the time scale of the quadrupole echo sequence. The EFGs average to zero and we did not detect any quadrupole echoes. Instead, the spectra were obtained from free induction decays using single p=2 pulses.
15
4. Restricted chain motion in the glass Fig. 2 contains a part of the spectra of the chain deuterons, obtained at ambient pressure and at temperature values below and slightly above Tg . Shown in detail is the region of the singularity corresponding to orientations of the static magnetic ®eld B with h p=2. The slight deviation from a pure Pake pattern indicates the presence of a small non-axiality in the EFG seen by the chain deuterons. This must be due to a limited non-axial motion of the chain C±2 H bonds. Presumably, in the glass such motion is restricted to local small-angle torsional motion of the chain segments, in which the chain C±2 H bonds participate; their motion occurs approximately in the plane, perpendicular to the plane de®ned by the carbon atom, involved in the bond, and its two main-chain neighbors. Averaging an axial EFG tensor (with principle values eq=2; eq=2; eq) over two C±2 H bond orientations, related by rotations about the Yaxis over angles and #, indeed results [22] in a non-axial tensor with principle values eq 3 sin2 # 1=2, eq=2, eq 3 cos2 # 1=2 and
Fig. 2. Chain deuteron line shape in the region of the singularities, at ambient pressure and 174, 222, 242 and 262 K.
16
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
asymmetry parameter g 3 sin2 #= 3 cos2 # 1. In the spectrum the non-axiality causes the Pake singularity occurring for H p=2 to split up into two [23]. The frequencies mX and mY , indicated in the ®gure, correspond to orientations of the static magnetic ®eld B parallel to the principal axes X and Y of the average EFG tensor. Clearly visible is how the singularity at mX shifts to lower (negative) frequencies at increasing temperature. The edges of the spectra (denoted by mY ) do not shift. Rather than the typical line shape of a powder spectrum corresponding to a single, non-zero value of the asymmetry parameter g, the shape of the spectra between mX and mY indicates a distribution of g values. By reading the ratio mX =mY 1 3 sin2 # from Fig. 2 we can estimate the rms value of #. We ®nd that this value increases from 4:7° at 174 K to 7:8° at 242 K. In this analysis it is assumed that the reorientation rate is in the fast motion limit, i.e., much larger than dQ 3=8 2p 169 103 s 1 . As will be seen below, this assumption can be justi®ed from the values of the chain deuteron spin-lattice relaxation time. Similar behavior is found in the spectra obtained at high pressure. The reduction of Dchain noted above can now be identi®ed as the gradual shift in the line-shape singularity, caused by the non-axiality of the average EFG tensor. Due to the limited spectral resolution, we cannot determine whether similar distortions of the methyl Pake doublet occur. Since Dmethyl =Dchain < 1=3, it is clear that the methyl groups, apart from their axial reorientation, take part in additional, close to axial, motion (probably) of the C±C2 H3 bonds. We note that a similar narrowing of a Pake pattern due to fast small-angle motion has been observed before, for instance in chain-deuterated polystyrene, slightly above the glass transition [24]. Also in the 2 H-NMR spectrum of deuterated polyethylene [25,26] in the crystalline (orthorhombic) state, which at room temperature consists of a Pake doublet, the singularities split up into two on increasing the temperature, due to a single non-zero value of the asymmetry parameter, also caused by small-angle reorientation of the C±2 H bonds in a plane.
5. Deuteron spin-lattice relaxation 5.1. Experimental We investigated the deuteron spin-lattice relaxation as a function of temperature at about ambient pressure and pressure values of 1, 2 and 5 kbar, using the saturation recovery method. In these experiments the temperature was increased through the glass transition, except at 5 kbar where the temperature was lowered through the glass transition. The temperature was changed at a rate of 2 K h 1 , and stabilized for about 30 min prior to each new measurement. We separately monitored the recovery of the methyl deuteron and the chain deuteron longitudinal magnetization with the following procedure. After Fourier transformation of the quadrupole echoes, obtained at each delay time, the noise in the spectra obtained was reduced by averaging the intensities at x and x. The chain deuteron magnetization was determined as a function of the delay time by integrating the area around the Pake singularity in the spectrum of the chain deuterons in the frequency range +55 to +65 kHz, outside the range of the spectrum of the methyl deuterons (see Fig. 2 of this paper and paper I [15], Fig. 1). We assume that the integrated intensity is a measure for the total chain deuteron magnetization. We obtained the methyl deuteron magnetization by integrating each spectrum from 22 to 22 kHz and by subtracting the contribution of the chain deuterons in that interval. From a fully relaxed experimental spectrum obtained in the glass we determined that this contribution is 19.3% of the total intensity integrated in the frequency interval from 22 to 22 kHz. The result, again as a function of the delay time, gives the recovery of the methyl-deuteron magnetization. We note that in this procedure we neglect the small eects of the anisotropy of the quadrupole spin-lattice relaxation rate to be discussed below. The recovery of the deuteron magnetization M t in aPP was found to behave strongly nonexponentially, both for the chain and for the methyl deuterons. Such behavior is usually observed in glass-forming systems [27]. The experimental relaxation curves can be represented very well by a
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
stretched exponential, the Kohlrausch±Williams± Watts (KWW) function [28,29] M 1 M t e M 1
t=Nb
;
1
where N is a measure of the time constants involved in the recovery and 1 b is a measure of the non-exponentiality. In the ®t of the KWW function to the experimental relaxation curves, we followed the recovery of the magnetization up to 1 e 2 , i.e., up to 86% of its equilibrium value. When the non-exponential recovery function is considered as a superposition of exponential functions exp t=T1 , distributed according to q T1 Z 1 M 1 M t q T1 e t=T1 dT1 ; 2 M 1 0 the time constant T1av , de®ned as an average over this distribution, can be expressed [30] in the KWW parameters after ®tting the data according to Eq. (1): Z 1 N T1av q T1 T1 dT1 C b 1 ; 3 b 0 where C is the gamma function. 5.2. The spin-lattice relaxation of the methyl-group deuterons We ®rst discuss the experimental results concerning the motion of the methyl groups. The average relaxation time T1av methyl of the methyl deuterons (as de®ned in Eq. (3)) in the temperature range from 180 to 340 K and at pressure values up to 5 kbar is shown in Fig. 3(a). The glass-transition temperature at each pressure value, indicated in this ®gure, is obtained from paper I. It appears that T1av methyl is hardly dependent on pressure and that no eect of the glass transition on T1av methyl is visible at any pressure value. Fig. 3(b) shows bmethyl as a function of T Tg and pressure. The non-exponentiality decreases with increasing temperature while, at constant T Tg , the relaxation becomes more exponential at higher pressure. We note, however, that the estimated errors in the b values obtained are about 0:05, due to inaccuracy in the determination M 1 M t at the longest delay times t.
17
The coupling of the deuteron electric quadrupole moment with the EFG present at its site is two orders of magnitude stronger than the magnetic dipole±dipole coupling between the nuclei. Therefore, the mechanism for spin-lattice relaxation of the methyl deuterons is the modulation of the deuteron electric quadrupole coupling caused by the rapid axial reorientation of the methyl groups. The general formalism for this mechanism has been presented by Abragam [31]. It is important to note that for a single deuteron, because of its spin quantum number I 1, the relaxation is exponential. The deuteron spins relax independently from each other, as the quadrupole coupling Hamiltonian depends on the variables of single spins. Spin diusion [32] between neighboring deuterons is very much slower than the methyl deuteron spin-lattice relaxation. The orientation dependence of the relaxation rate has been analyzed by Torchia and Szabo [33] for a number of types of anisotropic motion. They give an analysis of a model, in which a C±2 H bond reorients over three equivalent directions making an angle H with the axis of reorientation, with a mean jump rate k. For the spin-lattice relaxation rate of a methyl deuteron we obtain from their results in equation (39) and Table 2, with H arccos 1=3: 2 1 1 e2 qQ s 2 sin2 h h 1 x20 s2 T1Q h; / 36 p 0:5 sin2 2h 4 2 sin3 h cos h cos 3/ s 6 sin2 2h 8 cos2 h 1 4x20 s2 p 3 4 2 cos h 4 2 sin h cos h cos 3/ ; 4 where s is de®ned as 3k 1 . In this expression h and / are the spherical polar angles of the static magnetic ®eld B in a methyl-group ®xed axis system, h being the angle between B and the axis of the methyl group. As a consequence of the anisotropic motion, the relaxation rate is dependent on both h and /. For a system of methyl deuterons,
18
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
Fig. 3. (a) T1av methyl of methyl deuterons as derived from KWW ®ts. (b) bmethyl as derived from KWW ®ts as a function of T
Tg .
due to the / dependence the relaxation is slightly non-exponential for each frequency in the spectrum, even for a single value of s, as is noted in [33]. In the `extreme narrowing' limit x0 s 1 Eq. (4) reduces to 2 1 s e2 qQ 1 cos2 h: 5 h T1Q h 4
individual deuterons will result, in general, in a non-exponential recovery of the magnetization. The average recovery rate can be approximated by averaging Eq. (4) over an isotropic distribution of orientations, resulting in 2 1 e2 qQ s 4s h1=T1Q i : 6 15 h 1 x20 s2 1 4x20 s2
For a macroscopic sample the orientation dependence of the rate of exponential relaxation of the
In the extreme narrowing regime the recovery of the total magnetization for an isotropic distribu-
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
tion of orientations, which corresponds to the integrated intensity of the methyl-deuteron spectrum, has the form Z p sin h exp t 1 cos2 h dh: 0
It appears that a ®t of a KWW function to this recovery function results in b 0:98, so its shape is very close to exponential. We obtained h1=T1Q i from Eq. (6) by substituting e2 qQ= h 2p 169 103 rad s 1 and x0 2p 41:4 106 rad s 1 . In Fig. 4, h1=T1Q i as a function of s is compared with the experimental values of 1= T1av methyl as a function of 1=T . The solid line represents a ®t of h1=T1Q i to 1= T1av methyl obtained by adjusting the s-scale. The comparison shows that even at the lowest temperature the relaxation rate does not fully reach the maximum value predicted by Eq. (6): the factor in square brackets has a maximum value of 1:4252=x0 at x0 s 0:6158 and, therefore, at s 3:84 10 9 s. Nearly all the experimental data occur in the extreme narrowing regime. It is important to note that, as a consequence, the experimentally observed non-exponentiality in the recovery of the magnetization of the methyl deuterons cannot be due to the orientation dependence of the relax-
19
ation rate. Instead, it must be the result of the presence of a distribution of methyl group reorientation rates. The non-exponential relaxation of the methyl deuterons is really due to a distribution of exponentials, as in Eq. (2). In the extreme narrowing regime the spin-lattice relaxation time is proportional to s 1 . Therefore, the distribution G s 1 of reorientation rates is directly related to the distribution of relaxation times q T1 , introduced in Eq. (3): the distribution g ln s sG s 1 is equivalent to the distribution q T1 =T1 . The latter can be obtained [30] from the KWW ®t (2) to the recovery function, via an inverse Laplace transform. The observed values of b between about 0.8 and 0.9, correspond to a halfwidth of the distribution g ln s of reorientation rates of about half a decade. It also follows that T1av methyl is the value of the spin-lattice relaxation av time corresponding to the average value s 1 , so we obtain the temperature dependence of the latter from the comparison in Fig. 4. The spin-lattice relaxation rate 1= T1av methyl appears to reach its maximum value at 147 K, where s 1 av is av 1=3:84 10 9 s 1 ; s 1 follows an Arrhenius av law s 1 s0 1 exp DE=RT , with s0 1 8:46 10 13 s 1 and an activation energy DE of 9:7 kJ mol 1 .
Fig. 4. The calculated values of h1=T1Q i for the three-site jump model, as a function of s, compared with the reciprocal of the measured values T1av methyl , as a function of reciprocal temperature.
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A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
Our result may be compared with the reorientation of the methyl groups in non-deuterated, partially crystalline, isotactic PP, which has been investigated before [7,8] with NMR, using the proton spin-lattice relaxation rates 1=T1H and H 1=T1q . At temperature values below about 150 K, these relaxation rates are also mainly determined by the methyl reorientation, modulating in this case the magnetic dipole±dipole coupling between the methyl protons. These studies also show that the reorientation is rapid, down to low temperature, leading to a maximum in 1=T1H at about 160 K (measured at a Zeeman frequency of 30 MHz). Our value DE 9:7 kJ mol 1 is in very good agreement with the value 2:3 kcal mol 1 found from proton spin-lattice relaxation [8]. This is the typical value of the potential energy barrier for methyl reorientation [34,35], determined by intrachain interactions. Therefore, it appears that the reorientation of the methyl groups can be considered as the process of activated jumps across this potential barrier. The average methyl reorientation rate is not dependent on pressure and is not aected by a volume decrease of several percent, which also indicates that the energy barrier for reorientation mainly consists of intra-chain interactions. The interactions dier locally as a consequence of the dierent chain conformations at the carbon atom to which the methyl groups are bound. In the glass these conformations are frozen, which leads to the distribution of reorientation rates. The chain conformations change only slowly [6,16] up to about 15 K above Tg , therefore, T1av methyl does not change at Tg while b remains smaller than 1. At higher temperature, the chain conformations start to change on the time scale of T1av methyl , the distribution becomes narrower and b approaches 1. We do not have an explanation for the distribution becoming narrower with increasing pressure at constant T Tg . 5.3. The spin-lattice relaxation of the chain deuterons Fig. 5(a) shows the relaxation time T1av chain of the chain deuterons. The glass-transition temperature at each pressure value, indicated in this ®g-
ure, is again obtained from paper I [15]. In the glass T1av chain is rather long, indicating that the motion of the chain deuterons is very restricted. The exponent bchain , as derived from ®tting the KWW function (1) to the recovery of the magnetization, is about 0.7 in the glass (see Fig. 5(b)). Below Tg ; T1av chain appears to be pressure independent. At increasing temperature T1av chain decreases slowly until, at a temperature slightly above Tg at each pressure value, hT1 ichain shows a sharp decrease, which is clearly the eect of the glass transition and indicates increased mobility of the chain segments. In the temperature range Tg 10 to Tg 30 K; bchain rapidly increases to unity, corresponding to fully exponential relaxation. Similar behavior has been observed in earlier experiments on atactic polystyrene [36]. In the glass, the dominant mechanism for spinlattice relaxation of the chain deuterons is the deuteron electric quadrupole coupling, modulated by the limited local reorientation motion of the chain C±2 H bond directions, leading to the eect of the non-axiality in the average EFG tensor observed in the spectra. This oers the possibility of determining the time scale of this motion from the measured spin-lattice relaxation time. As will be discussed in the following, this relaxation path is much more ecient than the other available paths, namely spin diusion to the methyl deuterons and cross-relaxation to both methyl and chain protons, the proton abundance in our sample being 12%. It is well known that spin diusion [32], the transport of magnetization between neighboring spins, is very slow in deuteron systems. In ®rst order it requires spin ¯ip±¯op transitions, caused by the homonuclear dipole±dipole coupling between the deuterons. Since in this process energy is conserved, it only occurs when the resonance lines of neighboring deuterons show overlap. For each deuteron the quadrupole shift depends on the orientation of the static magnetic ®eld B in the PAS of the EFG. Therefore, in an amorphous material the shifts of the resonances of neighboring deuteron spins are unlikely to be equal, and spin diusion is quenched to a large extent. However, spin diusion between inequivalent deuteron spins is not in®nitely slow, even in cases
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
Fig. 5. (a) T1av chain of chain deuterons as derived from KWW ®ts. (b) bchain as derived from KWW ®ts as a function of T
where the dierence between the quadrupole splittings exceeds the resonance line widths [37]. As will be discussed in paper III [16], two-dimensional exchange 2 H-NMR experiments in aPP at temperatures around Tg 12 K indeed show that spin diusion occurs between methyl deuterons and chain deuterons, on a time scale of 700 ms. It occurs only for chain deuterons with frequency shifts within the range of the methyl-deuteron frequency
21
Tg .
shifts (from about 47 to 47 kHz). Spin diusion among the chain deuterons is at least one order of magnitude slower. We note that in the present experiment the values of hT1 ichain are obtained from the intensity of the spectrum outside the methyl frequency range and are, therefore, hardly aected by spin diusion to the methyl deuterons. This is also evident from the fact that T1av chain becomes largest at the temperature where
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A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
T1av methyl approaches its lowest value and spin diusion would have the largest eect. The various mechanisms for spin-lattice relaxation of the chain deuterons, provided by the dipole±dipole coupling of the chain deuterons with chain protons and methyl protons present in the system, can all be shown to be much weaker than the direct quadrupole mechanism. The time scale on which all protons in the system are coupled among themselves by spin diusion can be estimated to be within 1 ms. The protons will have a common relaxation time, governed by the ecient relaxation mechanism of the methyl protons via the dipole±dipole coupling with proton and/or deuteron neighbors. The various contributions can be calculated individually [33,38], assuming the three-site jump model for the motion of the methyl group (the rate of which, as we have seen, is not aected by the glass transition). In the temperature region of interest the resulting common proton T1 can be estimated to decrease from about 5 s at 340 K to about 1 s at 180 K, where the methyl-group reorientation rate becomes close to the proton spin Zeeman frequency 2p 270 106 s 1 (we note, that the proton T1 determined [8] at 30 MHz in non-deuterated PP decreases from about 0.6 to about 0.05 s in this temperature range). The most eective cross-relaxation path for the chain deuterons is the dipole±dipole coupling with the available protons in the reorienting methyl groups. This coupling is largest for a deuteron and a methyl group at the same carbon atom. An estimate of the associated deuteron±proton cross-relaxation time can again be obtained using the results of Torchia and Szabo [33]. In the temperature region of interest its value is larger than 100 s. This time limits the process of cross-relaxation to the proton spin system since, as seen above, the latter has a faster spin-lattice relaxation mechanism. It may be concluded, therefore, that the contribution to T1av chain by relaxation through dipole±dipole interactions is minimal. We conclude that the spin-lattice relaxation rate of the chain deuterons is caused by the limited local reorientation motion of the chain C±2 H bonds. The relaxation rate of a single deuteron can be calculated in a simpli®ed model for this motion, by viewing it as a process of discrete reorientation
over an angle 2# with mean time sj between jumps. The results obtained by Torchia and Szabo [33] for this model lead to an orientation dependent relaxation rate. Therefore, for a macroscopic sample, the recovery of the total magnetization is non-exponential. However, since in the measurement of the relaxation of the chain deuterons we obtain the intensities from the area close to the Pake singularity in the spectrum, that is for orientations of B with azimuth angle h close to p=2, we may expect close to exponential recovery for single values of 2# and sj . We compare our data with the expression for the recovery rate, obtained by substituting the value h p=2 in Eq. (36) of Ref. [33]. In the slow-motion limit x0 sj > 1 we ®nd Q 1=T1chain
9 32
e2 qQ h
2
sin2 2# : x20 sj
7
It follows that the strong non-exponentiality observed is again the result of a superposition of close to exponential functions with time constants as in Eq. (7). Therefore, in principle the distribution of relaxation times q T1 re¯ects the distribution of sj = sin2 2#. However, since the longest relaxation times in the distribution are not observable because of spin diusion, the value of T1av chain is mainly determined by the contribution of the smaller values of sj = sin2 2#. Q From a comparison of T1av chain with T1chain in Eq. (7) we ®nd an estimate of the values of sj = sin2 2# giving the main contribution to the observable relaxation rate. At ambient pressure we derived the rms values of # from the spectra. With these we ®nd a value of sj of about 5:0 10 7 s in the temperature range 174 to 242 K; it decreases to about 1:3 10 7 s at 262 K (i.e., 10 K above Tg ). Presumably, these values merely represent the short time part of the sj distribution. These reorientation times indeed lead to motional-averaged spectra, as was assumed in the calculation of the shift of the line-shape singularity in the spectrum of the chain deuterons. At increasing temperature, the motion becomes gradually faster while, simultaneously, the average reorientation angle increases, resulting in a slowly decreasing T1av chain . The fast decrease of T1av chain at and above the glass transition indicates an increased mobility of
A.G.S. Hollander, K.O. Prins / Journal of Non-Crystalline Solids 286 (2001) 12±24
the main-chain segments, presumably, leading to reorientation over larger angles, together with a shift in the frequency distribution. It has to be noted that in the liquid the collective motion (the a-process) permits more isotropic segmental motion on which the small-angle reorientation is superimposed. However, from the experiments to be presented in the next paper, it follows that even at Tg 30 K the isotropic component in the reorientation is not fast enough to contribute to spinlattice relaxation. On application of pressure the distribution of sj = sin2 2#, characterizing the process of small-angle motion in the glass is maintained up to the glass-transition temperature at each pressure value, as appears from the behavior of T1av chain and bchain shown in Fig. 5. For instance at Tg at 5 kbar, T1av chain is only about a factor 1.8 shorter than at ambient pressure, indicating that also at this pressure value the time scale of this process at the glass transition is about 10 7 s. It also appears that in the glass the value of T1av chain does not dependent on pressure, but only on temperature. This is somewhat surprising, since on compressing one expects the increased inter-chain interactions, restricting the motion, to lead to a decrease and increase of the average values of # and of sj , respectively, which according to Eq. (7) would result in an increase of T1av chain . We conclude that the decrease in volume (about 10% at 5 kbar [15]) is not large enough to have a visible eect. In the liquid T1av chain is strongly dependent on pressure. However, at each pressure value, the decrease of T1av chain at and above the glass transition is very similar, showing that in the liquid the amplitudes and frequencies of the small-angle motion are mainly determined by the distance to the glass transition. Probably, the small-angle reorientation of the chain segments providing the spin-lattice relaxation path for the chain deuterons is the mechanism of the b-mechanical relaxation process. To our knowledge, in aPP only data at low frequency 1 s 1 are available [1]. A weak, broad peak (called the c-peak in these studies) is found with its maximum at about 200 K. Presumably, this peak corresponds to the low-frequency tail of a broad distribution of reorientation rates of which we
23
observe the component at about 106 s 1 in the deuteron spin-lattice relaxation rate. The observed behavior corresponds to the properties of the bprocess generally found in amorphous polymers, namely that it gives a very broad loss curve, while it is rather insensitive to pressure [3]. The observation that the small-angle reorientation resulting in spin-lattice relaxation is the mechanism of the b-process was also made in the study [24] on chain-deuterated polystyrene above the glass transition, mentioned above. There it was found, by employing a dynamic ®lter [39] in a 4D exchange experiment, that in spatial regions where slow components in the process occur, also the b-process proceeds with longer than average correlation times. 6. Conclusion In aPP the spin-lattice relaxation of the methyl group deuterons is caused by the high rate of methyl-group reorientation (between 109 and 1010 s 1 ). The energy barrier for reorientation mainly consists of intra-chain interactions. The reorientation rate is aected only very little by an increase in pressure of 5 kbar. In the glass the nonexponentiality in the recovery of the longitudinal magnetization indicates the presence of a distribution of reorientation rates, diering locally as a consequence of the dierent chain conformations. The chain conformations change only slowly up to about 15 K above Tg and, as a consequence, the methyl reorientation rate is not aected by the glass transition. In the glass, for temperatures up to Tg , smallangle reorientation of the chain segments occurs with an average amplitude up to 15°. This is concluded from the quadrupole echo spectra and spinlattice relaxation rates of the main-chain deuterons. The character of the motion is maintained up to the glass transition occurring at each pressure value. The eect of pressure on this process is very small. The motion is characterized by a broad rate distribution of which the component at about 107 s 1 is observed via the spin-lattice relaxation rate. This motion is identi®ed with the mechanism of the bprocess found in mechanical-relaxation studies.
24
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In the liquid, above the glass transition, the amplitude of the reorientation increases, while it becomes somewhat faster. In the liquid this process is strongly dependent on pressure. It is mainly determined by the distance to the glass transition. This motion is superimposed on the much slower quasi-isotropic reorientation of the chain segments, the study of which is the subject of the next paper. References [1] N.G. McCrum, B.E. Read, G. Williams, Anelastic and Dielectric Eects in Polymeric Solids, Dover, New York, 1991. [2] G.P. Johari, J. Chem. Phys. 58 (1973) 1766. [3] G. Williams, Adv. Polym. Sci. 33 (1979) 59. [4] M.L. Williams, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc. 77 (1955) 3701. [5] N.G. McCrum, Polym. Lett. 2 (1964) 495. [6] D. Schaefer, H.W. Spiess, U.W. Suter, W.W. Fleming, Macromolecules 23 (1964) 3431. [7] M.P. McDonald, I.M. Ward, Proc. Phys. Soc. 80 (1962) 1249. [8] V.J. McBrierty, D.C. Douglas, D.R. Falcone, J. Chem. Soc. Faraday Trans. II 68 (1972) 1051. [9] C. Schmidt, S. We®ng, B. Bl umich, H.W. Spiess, Chem. Phys. Lett. 130 (1986) 84. [10] K. Schmidt-Rohr, H.W. Spiess, Multidimensional SolidState NMR and Polymers, Academic Press, London, 1994. [11] S. Kaufmann, S. We®ng, D. Schaefer, H.W. Spiess, J. Chem. Phys. 93 (1990) 197. [12] K. Zemke, B.F. Chmelka, K. Schmidt-Rohr, H.W. Spiess, Macromolecules 24 (1991) 6874. [13] D. Schaefer, H.W. Spiess, J. Chem. Phys. 97 (1992) 7944. [14] K. Zemke, K. Schmidt-Rohr, H.W. Spiess, Acta Polym. 45 (1994) 148. [15] A.G.S. Hollander, K.O. Prins, this issue, p. 1.
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