High Resolution I3C Nuclear Magnetic Resonance in Cured Epoxy Polymers Rotating Frame Relaxation BY A. N. GARROWAY, W. B. MONIZAND H. A. RESING Chemistry Division Naval Research Laboratory Washington D.C. 20375 U.S.A. Received 27th July 1978 The combination of cross-polarization dipolar decoupling and magic angle spinning produces high resolution l3C n.m.r. spectra in organic solids. Carbon rotating frame relaxation is complicated by the presence of the strongly interacting proton spin system spin-spin and spin-lattice processes will both promote relaxation. Dipolar order is created when the proton system is spin locked and is rapidly destroyed by magic angle spinning. Some implications of these spin-spin mechanisms on determination of I3C TIPare examined and illustrated by an experimental study on a model cured epoxy polymer diglycidyl ether of bisphenol A (DGEBA) cured with piperidine.For this specimen at room temperature we find that for r.f. fields above z 40 kHz the decay of the carbon spin lock magnetization after the first 100ps or so is not dominated by spin-spin relaxation and can reflect molecular motion. Over the temperature range 242-324 K the TIPfor the methyl carbon is described by an activation energy of 11 kJ mol-'. From the truncation of the dipolar hamiltonian by Van Vleck' to magic angle spinning2 to the rather complex multiple pulse sequence^,^,^ it is well established that coherently driving nuclear spin-spin interactions can selectively average them away. It has almost become an article of faith that a suitable combination of rather robust averaging schemes can convert a solid into a liquid insofar as coherent averaging in spin or real space can substitute for rapid and random isotropic motion.High reso-lution I3C n.m.r. in organic solids is a case in point; here the solid state methods of cross-p~larization,~~~ dipolar decoupling 'and magic angle spinning have generated reasonably narrow spectra&" (in amorphous polymers linewidths of a few p.p.m. and substantially less in crystalline solids). Such encouraging results should not blind us to the presence of the strongly interacting proton spin system. It is appealing' to examine I3C rotating frame relaxation rates. In the absence of strong proton-proton coupling these reflect molecular motional fluctuations in a rather interesting frequency regime (say 25-100 kHz) and further as the carbon- carbon coupling can generally be ignored the carbon nuclei will monitor local motions rather than performing the sort of averaging implicit in a proton relaxation rate.Here we shall examine some aspects of the complications of tight proton coup- ling in the determination of I3C rotating frame relaxation and specifically how spin- spin relaxation is to be distinguished from spin-lattice relaxation. While these are rather general concerns we shall concentrate on organic solids under conditions of magic angle spinning and apply some of these notions to the results of a preliminary study on a model cured epoxy. Different considerations apply when the sample is not spun and will be examined elsewhere.12 l3cN.M.R.IN POLYMERS EXPERIMENTAL A Bruker SXP spectrometer modified for cross-polarization and magic angle spinning at a carbon frequency of 15 MHz was employed. Both proton and carbon resonances can be independently observed and variable r.f. fields up to 70 kHz can be applied to each species. Unless otherwise noted a magic angle spinning speed of 2 kHz was used. The model epoxy was prepared from commercial diglycidyl ether of bisphenol A (DOW DER 332) and 5 % by weight of piperidine and cured for 16 h at 393 K. The specimen was cast in a mould into the form of a Lowe-Kes~emeier-Norberg~~-~~ type spinner. THEORETICAL In this section we look for manifestations of spin-spin coupling on 13C rotating frame relaxation.The plan is to decide how TIPshould be measured and also to estimate the spin-spin contribution to the relaxation rate. (Here Tip is defined as the observed rotating frame relaxation time constant.) We shall consider spin-spin processes under rigid lattice conditions and use the results as an upper bound on spin-spin rates at higher temperature; we adopt the naive view that higher tempera- ture introduces spin-lattice processes and reduces purely spin-spin couplings. In the following we assemble a number of results for spin-spin coupling and specialize them to organic solids. The hamiltonian governing the spin system is comprised of proton-proton (11) and proton-carbon (IS) dipolar coupling as well as interactions with the static and r.f.irradiation. Carbon-carbon coupling is ignored. If each species is irradiated on resonance with r.f. fields wll,coIS(=yIBII,ysB,,) then in the doubly rotating frame5*15 where HII(O'is the usual (high field) homonuclear dipolar hamiltonian' and HII(ns) is a nonsecular term which when viewed in this frame will involve frequencies at ~2~0,~. Similarly the nonsecular heteronuclear term HIS("')involves & wlI & qS; in the event that wlI = wls (Hartmann-Hahn ~ondition),~ part of the IS interaction is restored to secularity. All nonsecular terms corresponding to oscillations at the static field Larmor frequency cool coosare dropped. We wish to examine the entire time evolution of the S spin lock magnetization (S,) under the influence of the spin-spin hamiltonian of eqn (1).It is most con- venient to divide the behaviour into short and long time domains although one can pass smoothly16 through both regimes but at the expense of computational com-plexity which is unwarranted here. In the short time period the subject of transient oscillations has been treated in detail for the case of sudden application of static and r.f. fields to dipolar ordered spin locking by r.f. pulses in which the magnetization is obtained by a 71/2 preparatory pulse 21 p22 and for heteronuclear cros~-polarization.~~~~ These effects can be estimated by a simple argument. A large static or r.f. field splits the hamil- tonian into secular and nonsecular terms. A sudden change in the hamiltonian introduces coherences in the nonsecular terms but as a taste of this non-equilibrium state is transferred to more and more spins via strong dipolar coupling the oscilla- tions die out in prelude to the establishment of a spin temperature determined solely by the secular part of the hamiltonian.If the (appropriate) initial inverse spin temperature is pi,then the energy associated with such a coherence is" A. N. GARROWAY W. B. MONIZ AND H. A. RESING By conservation of (total) energy the influence of this decay can be calculated. This energy is associated with the “ step ” in the “ step plus oscillation ” description” of transient oscillations. First consider spin locking the proton (I) system. A 42 preparatory pulse is followed at t = 0 by the spin lock pulse.At t = 0 the initial magnetization is (Iz)i and the inverse temperature of the spin lock state is Pli= ~ooor/co,r where Podescribes the lattice temperature. The energy associated with the oscillation of the nonsecular hamiltonian is then for on resonance irradiation E = PIi(tr [H(yi)]Iz + tr [H(yl)l2}/tr I (34 where M(;J and M(f2 are the second moments of the I1 and IS interactions and N, Ns are the number of spins. This energy is supplied by the proton spin lock reservoir and so both the spin lock signal and spin temperature are diminished according to in agreement with explicit calculations.22 The IS contribution may be ignored in this expression and in the following. As these oscillations dephase the energy ~(~owoI/wl,)A4(~~ is transferred into the (secular) dipolar reservoir.As the local field under irradiation is +coL = 3[3M($]+ this implies an effective dipolar inverse spin temperature during irradiation of P; = 3Pooor/coo,l = 3p1i. When the r.f. field is removed the full (high field) dipolar hamil- tonian becomes secular and the effective dipolar spin temperature is then Thus the dipolar system is described by a temperature not very different from the initial spin lock temperature (DIi)or the spin lock temperature after the transient oscil- lation eqn (4). These circumstances are indicated schematically in fig. 1. We have implicitly assumed wII is sufficiently large to preclude any further cross-relaxation between the I dipolar and I spin lock systems during this interval.We also inquire into the transient arising when the S spins are spin locked following a 7t/2 pulse. Here only the IS term becomes nonsecular and after these oscillations die out in something like the T2of the I1 coupling the S spin lock magnetization is fractionally reduced as 22 In the experiments to be presented the S magnetization was prepared not by a n/2 pulse but rather by a single contact cross-polarization between carbon and proton spin lock states matching the Hartmann-Hahn condition; we shall refer to this as a matched SL cross-polarization. The pulse sequence is also shown in fig. 1. The cross-polarization contact times (1 ms) are sufficiently long to allow any transient oscillation during cross-polarization to die out. As the heat capacity of the S spin lock system is negligible in comparison to that of I the final inverse spin temperature of all the reservoirs is approximately PI given by eqn (4).Removing the I r.f. field at the start of the S TIPmeasurement makes the IS interaction nonsecular. Two types of oscillations ensue. The first produces a decrease in the S spin lock magnetization identical to eqn (6). The second arises from the IS dipolar order established during l3C N.M.R. IN POLYMERS cross-polarization and is analogous to the transient expected when S spins are cross- polarized from a dipolar ordered ~tate.~.'~ We find that the second mechanism fractionally increases the spin lock magnetization by x+[Mfi)/co&]and so partly com- pensates the decrease of eqn (6).We therefore expect a transient when the carbon magnetization is prepared either by a n/2pulse or by cross-polarization. lHRF~ 13C RF -------c 1 'Hi pc -W Wl" 1H FIG. 1.-R.f. pulse sequence and spin temperatures appropriate to a TIPmeasurement. The carbon magnetization is prepared by cross-polarization against a proton spin lock state. The inverse spin temperatures of the dipolar proton and carbon spin lock systems are indicated as &-, BH and BC,respectively; PHiis the initial inverse temperature. If the sample were mechanically rotated the dipolar order would die out rapidly (see text and fig. 2). The implications of these step reductions are (i) the initial rapid drop in the carbon magnetization during a spin lock pulse is related only to the strength of the proton- carbon dipolar coupling and (ii) in the case the carbon magnetization is prepared by SL cross-polarization the protons will be in a state of dipolar order at the start of the carbon spin lock pulse.This last observation will be later modified if the sample is spun. In the long time regime a simple thermodynamic can be applied. The rate (Ts-D)-l at which the S spin lock state decays through its coupling to the I dipolar state (D) has been discussed at length3*16*23*24 and under the assumption that the correlation function for the spin fluctuations responsible for the coupling is lorentzian then where the fluctuation time is24 A. N. GARROWAY W. B. MONIZ AND H. A. RESING and where are defined conventionally.(This relaxation time is also called9*16 TAyFF,though the dipolar state need not be prepared by adiabatic demagnetization in the rotatingframe.) The cross-relaxation rate (TI-s)-l when both I and S are irradiated has also been examined :3-5~16there a gaussian correlation function seems more appropriate. In this case =+n*MMfg)zI-s {+exp [-$(co, -~o,~)~.r~-,] + 3 exp [-$(% +~ls)2~21-s1L(8) where the correlation time zI-s is given el~ewhere~*~*'~ and is of the same order of magnitude as zS-,. Interestingly the time constant Ti-s with wls =0 is appropriate for the T2,of the free induction decay of the S spins under I deco~pling.~~ The stronger r.f. field dependence in eqn (8) ensures that for fields beyond some size the T2effects governing the carbon linewidth will diminish more rapidly than those cross-relaxing the S spin lock state :narrow lines do not provide a bound on the spin-spin contribu- tion to TIP.(However if spin-lattice effects do indeed predominate then T2s= TIPs when the relevant r.f. fields are The actual effect of the S spin lock to I dipolar state cross-relaxation depends on the relative spin temperatures of the two reservoirs. What is the appropriate dipolar temperature? If the sample is not spun then as we have just seen the carbon and dipolar temperatures are not too different during the time that only the carbons are spin locked. However spinning the sample about any axis not parallel to the static magnetic field will contribute to the dipolar spin lattice relaxation rate (T',,) -1.27-29 For perpendicular spinning the contribution is,27-29for a powder of spin 4 where SZ -Lf,,,/27ris the spinning speed and the factor [3n]arises29 by the assumption of a lorentzian correlation function.The appropriate dipolar fluctuation time is based on ref. (30) I1 G2=3lW2'* (9b) The theory is not yet presented for spinning at the magic angle but for slow spinning one expects the same qualitative features. We remark that magic angle spinning an organic solid at 2 kHz will reduce the effective T, down to 100 pus or less. So magic angle spinning ensures that any proton dipolar order (created by the transient oscillation during the proton spin lock) is dissipated over the order of 100 ,us and for times much longer than this the carbon spin lock state sees a completely dis- ordered proton system which is tied strongly to the lattice.Hence this pathway a carbon-proton spin flip costing the carbons wlSin energy followed by rapid dissipa- tion of the proton dipolar order will compete on an equal footing with the (motional) spin-lattice contribution to TIP. (If the sample were not spun then T, might become rate limiting.)12 We have ignored sample spinning except for its influence in T,,. Provided the spinning is slow compared to the various fluctuation frequencies of the order of the proton-proton local field coL,the effect of sample spinning on other spin-spin processes can be regarded as introducing side-bands via the amplitude modulation of HIS:the narrowing of the proton-proton fluctuation spectrum is not ~ignificant.~' Hence the l3C N.M.R.IN POLYMERS frequencies cols should be replaced by cols & 52 this does not qualitatively alter any conclusions for the regime at hand in which cols col > coL 9 52. With this qualitative theoretical framework we turn now to some experimental results on a model epoxy diglycidyl ether of bisphenol A (DGEBA) cured with piperi- dine. RESULTS AND DISCUSSION We wish to establish experimentally the fate of dipolar order under mechanical spinning. Fig. 2 shows the contribution to the dipolar spin-lattice relaxation rate due , 0.l0.1 0.2 0.5 1.0 2.0 5.0 to mechanical rotation perpendicular to the static field for the cured epoxy at room temperature. The dipolar state was prepared by the two pulse Jeener-Broekaert sequence32 and the remaining magnetization monitored by a 71/4 proton inspection pulse.For reference the " static " TI,is 1.1 ms while the extrapolated spinning con- tribution at 2 kHz is ~665ps; though at that speed one should really examine how dipolar order is created under spinning conditions. In the figure the straight line depicts a square law dependence on rotation speed and reasonably represents the data. The correlation time extracted from the data by means of eqn (9a)gives a value for the dipolar fluctuation time of Z = 19 ps. From the initial slope of the proton free induction decay also at room temperature we find a proton second moment of 8.4 & 0.4 GL. From this value and the relation between the moment and dipolar fluctuation time eqn (9b),we estimated zD = 22 ps in qualitative agreement with the above sample spinning result.(For this epoxy we find that at room temperature the A. N. GARROWAY W. B. MONIZ AND H. A. RESING proton second moment is not except for the methyl contribution significantly motionally averaged; such averaging is seen at 325 K and above.) The foregoing establishes a dilemma. We would like to compare the observed Tlpwith the spin-spin relaxation time TS-, measured by preparing a state of dipolar order and using it to cross-polarize the S spin lock state. However the sample spin- ning required for high resolution spectra also rapidly destroys dipolar order. Now high speed spinning is not mandated; we have performed some experiments at spin- ning speeds of 400 Hz for which the effective T, is only 700 ps.As we shall find the observed carbon Tlpvalues range from about 2.5 to 60 ms hence T's-D B TIDand under these conditions it is quite difficult to measure Ts-Ddirectly. Instead we shall appeal to the r.f. field dependence of the observed Tlp,noting the exponential depend- ence embodied in Ts-D,eqn (7a) compared to the weaker dependence expected for motional effects.33 To compare the field dependence of the various I3C Tlpvalues in the epoxy we should like to normalize by dividing out the interaction strength Mi;) which appears in Ts-Dor indeed in the equivalent expression for the relaxation induced by isotropic motion. Accordingly we normalize by TI-s(0) the matched SL cross-polarization time given in eqn ($a)with wI1-wls = 0.Hence -~ The correlation times z,-~ z ~ are comparable and are determined largely by I1 rather than IS couplings. [We do not take eqn (8) too seriously especially when M\i) N -MI:) e.g. for protonated carbons for then the initial transient' will play a large role in the cross-polarization.] There is however an effect of sample spinning on T,-,(O) which should be examined. For reference the 13Cspectrum of the piperidine cured DGEBA epoxy is shown in fig. 3. For each of the seven resolved peaks the dependence on spinning of the matched SL cross-polarization times Tl-s(0)[TCH= T,-,(O)] is shown in fig. 4 and has been discussed elsewherez6 in a different light. Here the r.f. fields were 38 kHz DGEBA + PIP I 2 00 100 0 p.p.rn.FIG.3.-I3C spectrum of the piperidine (PIP) cured DGEBA epoxy polymer at room temperature. The assignments have been discussed" and the structure indicates a possible polymerization me~hanism.~~ l3cN.M.R. IN POLYMERS and the proton T,,found to be 2.6ms. The relaxation behaviour of the non-protonated carbons shows some averaging from the magic angle spinning. The protonated carbons relax so rapidly due to their larger interaction M,',2),that the cross-polariza- tion process is largely completed in the time the specimen rotates say one-half revo- lution. We shall not attempt to fit these data to any model of narrowing but note rather that the reported time constants reflect an averaging for the non-protonated but not for the protonated carbons.To normalize in a more consistent fashion we shall estimate the non-spinning relaxation times by using the values for the 1 kHz rotation. I I I 500 DGEBA + PIP X AA x 4 oc 0 0 non -protonated v) 300 I \ 4 I h-200 T + + + I 4 100 P Q protonated U 0 0 1 2 3 froti k ti z FIG.4.-Magic angle spinning alters the cross-polarization rates. Here TCH[!T,-40)] is the time constant for spin lock cross-polarization under the Hartmann-Hahn condition at r.f. fields of 38 kHz. We then display in fig. 5 the r.f. field dependence of the 'jC TIPtimes (at 2 kHz spinning) normalized by the TCHvalue (at 1 kHz) of the previous figure. The TIP relaxation times were measured mindful of the concern over the initial transient no data within the first 500 ps were used to determine TIP.The decay of the magnetiza- tion of the protonated aromatic carbon is certainly non-exponential and the reported time constants are a fair description of the behaviour over the first 80 yo of the decay. In the absence of a concrete theory for the decay a single parameter must suffice. From fig. 5 we see that the values T,,/T,- [or equivalently the ratios of the spec- tral densities J(0)/J(culs)],cluster within about a factor of three at each field and depend rather weakly on r.f. field. We can make a crude estimate of the expected A. N. GARROWAY W. B. MONIZ AND H. A. RESING r.f. field variation of the spin-spin contribution Ts-D to the observed spin-lattice relaxation.The correlation time governing the dipolar fluctuations which relax the carbons is given in eqn (7b). To overestimate the spin-spin effects we produce an underestimate of the correlation time zest,by assuming that in this epoxy the correla- tion time is independent of the details of the local IS coupling; that is We have both the experimental proton second moment and the fluctuation time z, = 19 ps from the TI sample spinning experiment; we use the latter value to infer from eqn (96) z~~~= 3+2D = 33 ps. This fluctuation rime will determine the r.f. field dependence of the pure spin-spin processes. In fig. 5 the broken line represents the ~ spin-spin contribution predicted by eqn (lo) with T~ =-rest= 33 ps and assuming that T1-s = 7S-D.This curve presumes that TCHis correctly given by eqn (8) which is certainly not the case for protonated carbons. As a second and more restrictive 1000 500 200 x 100 I LU . 0. L- P g 50 I DGEBA+ PIP I 20 10 I 1 I 30 50 70 f (=YB,/2n) /kHz FIG.5.-R.f. field dependence of the 13C TIPtimes. The Tlp values have been normalized by TCH (at 1 kHz spinning). The broken line estimates the field variation expected if the observed rotating frame relaxation were exclusively determined by spin-spin coupling TS-,; see eqn (10) in the text. The dashed line represents the same field dependence and has been drawn through the 32 kHz data as an even more restrictive estimate; there is no evidence whether or not the low field data are deter- mined exclusively by spin-spin effects.As the relaxation times at 43 and 66 kHz are shorter than those predicted for purely spin-spin effects the high field results (and perhaps even at 32 kHz) indicate molecular motion. l3cN.M.R. IN POLYMERS estimate suppose that the relaxation at low field were due solely to spin-spin events the dashed curve in fig. 5 represents that hypothesis with the same exponential field ~ dependence as before (T =~ 33~ ps). Clearly the T, values at higher field (43 and 66 kHz) cannot be due to spin-spin effects; they are too short. We have no experi- mental indication either way about the 32 kHz data. Hence the conclusion of this rather lengthy argument is that above around 40 kHz for this epoxy at room tempera- ture the I3C TIPvalues are not determined exclusively by spin-spin events but reflect spin-lattice effects.This is not an idle exercise; in oriented polyethylene at room temperature the apparent 13C TIPis dominated by spin-spin effects even at an r.f. field of 80 kHz." We now examine the temperature dependence of the I3C TIPmeasured with an r.f. field of 55 kHz for which the relaxation times should indicate molecular motions. In fig. 6 the T, data are presented directly without normalization by TCH. Over 242-324 K the only significant temperature dependence arises from the methyl group. The activation energy inferred from the data (dashed line) is 11 kJ mol-' (2.6 kcal mol-') in quite fortuitous agreement with the value (2.6 kcal mol-') ob-tained from rather complete proton relaxation studies3' in long alkanes.The other spectral line (0) showing a rather weak temperature variation is unfortunately an un- resolved peak comprising methylene and methine resonances and it is unclear how to 100 0 0 0 0 50 X x x 800 0 0 8 X < X x x L4 \ 20 ul \ \ E 3. \+ Q 0 B k- 10 P 4 4 a DGEBA + PIP 5 ~~ 2 3.0 3.5 4.0 4.5 103/ T FIG.6.-Variation in I3C TIPvalues with temperature. Only the methyl resonance shows a well- represent defined dependence with an apparent activation energy of 11 kJ mol-'. The squares (0) an unresolved methylene-methine resonance and little can be inferred from its behaviour. A.N. GARROWAY W. B. MONIZ AND H. A. RESING 73 interpret the resulting composite relaxation rate. It may be that this dependence reflects the rather labile nature of the epoxy molecule at the polymerization reaction site. In conclusion the determination of 13CTIPis complicated in solids by the presence of strongly coupled proton system. Transient oscillations will fractionally diminish the carbon spin lock magnetization over the time scale of approximately the proton- proton T2and create dipolar order in the case where carbon magnetization was created by cross-polarization. Spin-spin fluctuations can transfer order from the carbon spin lock to the dipolar reservoir ; sample spinning rapidly destroys dipolar order preventing the dipolar system from becoming a bottleneck.Hence under sample spinning spin-spin fluctuations which reduce the carbon spin lock magnetization will compete on an equal footing with the carbon (motional) spin-lattice coupling. A direct comparison of this spin-spin relaxation time Ts-Dis preferable but quite diffi- cult when Ts-D TID. Instead for a cured DGEBA epoxy we observe the r.f. field dependence of the I3CTIPvalues to be much weaker than that crudely estimated for a purely spin-spin relaxation mechanism. Hence the observed 13CTIPvalues are not dominated by spin-spin effects at least for spin lock fields above z 40 kHz. Over the range of temperature 242-324 K the methyl carbon TIPshows an activation energy of 11 kJ mol”. Conversations with D. L. VanderHart have helped delineate the role of dipolar order in these experiments.This work is sponsored in part by the Naval Air Systems Command. J. H. Van Vleck Phys. Rev. 1948 70 1 168. E. R. Andrew Progr. N.M. R. Spectroscopy 1971 8 1. M. Mehring High Resolution N.M.R. Spectroscopy in Solids N.M.R. Basic Principles and Progress 1976 11 1. U. Haeberlen High Resolution NMR in Solids Selective Averaging Adv. Mag. Resonance 1976 Supplement 1. S. R. 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