High-Resolution I3CNuclear Magnetic Resonance in Solids BY EDWARD0. STEJSKAL AND THEODORE JACOB SCHAEFER R. STEGER Monsanto Company 800 N. Lindbergh Blvd. St. Louis Missouri 63166 U.S.A. Received 1st August 1978 A procedure is demonstrated for determining the relative contributions of spin-lattice and spin- spin interactions to TIp(C), the relaxation time characterizing the loss of spin-locked carbon polariza- tion. This involves the characterization of relaxation processes associated with several cross-polariza- tion experiments. Three glassy polymer Tlp(C)values are shown to be dominated by spin-lattice effects while a crystalline polymer relaxes through a spin-spin mechanism. The combination of cross-polarization,' high-powered heteronuclear decoupling2 and magic-angle spinning3p4 has made it possible to obtain high-resolution natural- abundance 13Cn.m.r.spectra in solid^.^*^ These spectra are sufficiently detailed to permit the relaxation behaviour of individual carbon lines to be studied. In particular there are several relaxation processes which can occur in cross-polarization n.m.r. experiments with the potential to yield information about molecular motions and interactions. In order to make use of these relaxation phenomena it is necessary to establish first whether they are the result of spin-spin or spin-lattice interaction^.^ To answer this question we have made 'H-l3C (1-S) cross-polarization (CP) measurements on both glassy and partially crystalline polymers and have charac- terized the following relaxation times T,,(SL) the time for polarization transfer in a matched CP experiment; T,,(ADRF) the time for polarization transfer when the protons have been adiabatically aligned in the dipolar field; Tlp(H) the time for loss of spin-locked proton polarization; Tlp(C),the time for loss of spin-locked carbon polarization; and TI, the time for loss of proton polarization aligned in the dipolar field.(Although it is conventional to refer to these relaxation processes by the relaxation times indicated we seldom find that they are all simple first-order processes.) We have devised a straightforward phenomenological analysis of these strongly related phenomena which has the ability to separate the fundamental pro- cesses which are occurring and identify the contributions of molecular motion.The results of this analysis agree well with an assessment of the relaxation phenomena based on a variety of qualitative physical arguments.' EXPERIMENTAL The "C n.m.r. experiments were performed at 22.6 MHz on a home-built 13C-'H double resonance spectrometer with an external 19F field-frequency lock. We employ quadrature detection * and spin temperature alternation for increased sensitivity and to obtain distortion- free spectra. The sample probe has a single r.f. coil which is double-tuned'O~l' for both I3C and 'H. This provides greater uniformity and conformity of the two r.f. fields. Spin-ning experiments are based on the Lowe ge~metry.~ E. 0. STEJSKAL J. SCHAEFER AND T. R. STEGER The relaxation phenomenon that interests us most is Tlp(C),both with and without sample ~pinning.~ However since spinning complicates the T,,(ADRF) process by interact- ing strongly with the proton dipolar order we have chosen to study the five relaxation pro- cesses on stationary samples.The further effects due to spinning may then be assessed with little trouble.12 Fig. 1 depicts the three CP pulse sequences that,were used. The SL se-.ao? I-S decoupling 'H 9Q ; SL 13C 0, .-1 .e G i 1 H l3 c IH I hold I l3C P time (schematic 1 FIG.1.-Schematic representations of the three cross-polarization (CP) pulse sequences used to study CP relaxation processes. In the first sequence (SL) polarization is transferred between carbons and protons each spin-locked in its own r.f.field. The r.f. fields have been adjusted to satisfy the Hartmann-Hahn condition. In the second sequence (ADRF) polarization is transferred after an adiabatic demagnetization in the rotating frame has aligned the protons in the dipolar field. The final sequence (TIP)depicts the sequence used to measure Tlp(C)after carbon polarization has been generated through a spin-locked contact. quence was used to obtain T,,(SL) and TIP(H). The TIPsequence was used to obtain Tlp(C). The ADRF sequence was used both to obtain TID(by varying the time beween the ADRF step and the start of the contact) and to obtain data from which T,,(ADRF) was ultimately derived. Table 1 summarizes the experimental data and the several parameters derived from them.It also identifies the four polymer systems that were studied. As may be seen in fig. 2 the methyl-carbon resonance in poly(2,6-dimethylphenyleneoxide) is clearly distinguishable in the spectrum. Similarly the a-methyl carbon in poly(methy1 methacrylate) and the aliphatic carbons in polystyrene can be isolated even without spinning. The crystalline component of polyoxymethylene is considerably broader than the rubbery component and may be identified on that basis. We chose one crystalline polymer and three glassy polymers to provide a variety of degrees of molecular motion for contrast. HIGH-RZSOLUTION l3cN.M.R. IN SOLIDS TABLE 1 .-lH AND I3CRELAXATION PARAMETERS AT 30°CFOR FOUR SOLID STATIONARY MOULDED POLYMER PLUGS OF DIMENSIONS COMPARABLE WITH THOSE OF R.F.COIL polyoxymethylene polystyrene poly(methy1 POlY(2,6-(crystalline (aliphatic methacrylate) dimethylphenyl-component) carbons) (a-methyl ene oxide) carbon) (methyl carbons) 36 36 36 36 10 7.2 5.9 3 .O 1000 1000 1000 1000 167 52 14 4.8 50 2.8 7.7 30 50 1.2 8.O 20 20(85%) 30(85%) 1 20(85%) 500(85%) 2( 15%) 3( 15%) W5%) 50(15%) 8.5 12.8 51 213 12 15 56 210 5.5(90%) 12(90%) 1OO(UO%) 0.4( 10%) 2.4(10%) 5(10%) -2.4 8.6 34 9.0 3.0 10 32 0 81 83 85 . 0.01 10 20 30 tlms FIG.2.-Typical Tlp(C)relaxation data. The insert spectrum of poly(2,6-dimethylphenyleneoxide) is typical for the solid in the absence of magic-angle spinning; the broad aromatic region is dearly distinguishable from the narrower methyl region.The data points represent the peak amplitude S of the methyl region in arbitrary units as a function of hold time. The average scatter is <l%. For these measurements the contact time was I ms and H1 was 28 kHz. Each point represents 800 replications accumulated in 1600 s. The sample was in the form of a moulded plug with a volume of 0.9 cm3. E. 0. STEJSKAL J. SCHAEFER AND T. R. STEGER DATA ANALYSIS The ADRF experiment contains all of the elements of relaxation to be c~rnpared.'~ The equation governing the evolution of the carbon signal S during the contact time is dS -S0e-'/'1D -S S dt-TP TPL' (1) where we have used T:s for TIs(ADRF) TpLto represent the (spin-lattice) relaxa-tion process that dissipates S through contact with the lattice and Soto represent the total carbon polarization available in the absence of dissipative relaxation processes.If S =Si,when t =0 then Sirepresents carbon polarization developed during the rise of the carbon r.f. field presumably during the short time when that field and the proton local field nearly match. As we shall see below we may make the substitution l/TlP(C)=VTPL +1/T& (3) with the result that eqn (2) becomes We note that The Tlpexperiment is analysed in similar terms.' If no net proton polarization winds up in the dipolar field we can set So to zero. Si now represents the carbon polarization developed during the initial CP process. Thus as anticipated by eqn (3) Note that Tis still governs the spin-spin coupling between the spin-locked carbons and the protons in the dipolar field even though the latter have no net polarization.In the event that all proton dipolar order is not suppressed that is if Sodoes not vanish exactly the magnitude of Socan be estimated in a T, experiment from which the spin- locked carbon-proton contact has been omitted. Eqn (4) then justifies removing the contribution of a non-vanishing So from normal Tlp data by subtraction. This correction was made on the Tlpdata reported in this paper although it was scarcely necessary except for polyoxymethylene. One further caution should be mentioned. Following the removal of the proton r.f. field for a period of several proton T2values the carbon polarization is perturbed; after that it settles down into a smooth Tlpdecay.This can be seen in the first point in fig. 2 taken after 10 ,us hold. We do not normally begin collecting data in this time domain but wait 20-50 ,us. The SL experiment is similar to the ADRF experiment except that several of the parameters are of different 01-igin.I~ The appropriate differential equation is 7' where we have used HIGH-RESOLUTION l3C N.M.R. IN SOLIDS for T,,(SL) and replaced So and T, by S and T,,(H) respectively. TIPLreplaces TpLsince the protons are spin-locked by the r.f. field. Since S = 0 when t = 0 The relationship between S and Sois given by5 sM(YS Hl/yl HL)E (9) where ys and y1 represent the carbon and proton gyromagnetic ratios Hl the r.f. field applied to the carbons and HLthe local field of the protons coupled to the carbon of interest.The local field HL can be estimated from either the proton second moment (Hi z 3M2)or T2(by assuming a reasonable line ~hape).'~ The factor E represents the efficiency of the ADRF transfer. In the development just given it has been assumed that T,,(SL) T,,(ADRF) Tl,(C) and TpL(C)are simple first-order time constants. For polymers this is generally not the case a fact we ascribe to dynamic structural and orientational heterogeneity in the solid. [TIDand Tl,(H) have spin diffusion to help eliminate this heterogeneity; carbon relaxation processes do not.] The easiest way to deal with this complication is by a multiple relaxation-time model. (Since we are dealing with polycrystalline or amorphous systems we do not see transient oscillation^^^ in the various CP processes.) For example TlPmight be represented by a two-phase model TlPSa(fraction fa) and Tlp,b (fraction fb) where fa +f = 1 and (Tip) = dfa/Tlp,a +fb/Tlp,b)-l* The average value (Tl,) also corresponds to the relaxation time derived from the initial slope of the relaxation curve.We prefer this quantity when a single number is to be used to describe the relaxation process since it is the most completely representa- tive of all the processes going on and can be defined by a relatively few data points. (It does however require high-quality data to measure accurately.) Our relaxation data clearly required that both TpL(C)and T,,(ADRF) be treated as non-exponential processes. We chose to use a two-phase model for each giving rise to a system of four different phases as the most comprehensible way to introduce heterogeneity.In this case eqn (5) must be modified to Note that although after a few milliseconds the T,,(C) decay seems to become mono- disperse this does not mean that the effects of the T,,(C) and T,,(ADRF) dispersions disappear early in the ADRF experiment. As the polarization builds up through the influence of T,,(ADRF) it is simultaneously pulled down by the T,,(C) process. This competition goes on throughout the entire ADRF experiment. RESULTS Fig. 3 portrays the results of our T,,(ADRF) measurements. For each system we also measured the following quantities H1 SM,Si,Tl,(H) TID,(Tlp(C))and the rest of the Tl,(C) decay.We estimated HL from proton n.m.r. data and chose E = 0.75 which gave an estimate of So. These quantities (except for SMand with So normalized to 1000) are given in table 1. Also given is (T,s(ADRF))t = 0 derived from eqn (10) and the slope of the data near t = 0. Note that we do not take the slopes directly from fig. 3 which is a logarithmic plot but from a linear plot. To obtain the curves plotted in fig. 3 we adjusted the parameters of the T,,(ADRF) and E. 0. STEJSKAL J. SCHAEFER AND T. R. STEGER TpLrelaxation processes until a reasonable fit was obtained. It became apparent early that TpLwould not contribute significantly to the polyoxymethylene relaxation data. This simplified the analysis of that particular curve. From the polyoxy- methylene fit we obtained the particular two-phase TI,(ADRF) pattern (1 5% re-laxing ten times as fast as the rest) and the value of E both of which were used in -----0 5 10 15 tlrns FIG.3.-Evolution of carbon polarization S in the TIs(ADRF) experiment for four solid stationary polymers polyoxymethylene poly(methy1 methacrylate) poly(2,6-dimethylphenylene oxide) and polystyrene.Further details are given in table 1 including the parameters determining the curves fit to the data. all four fits. The breakdown of the TpLtwo-phase pattern into 10-90% is charac- teristic of most of our T,,(C) data-as the fitting procedure progressed it also seemed appropriate for TpL. The final parameters arrived at are given in table 1. We also calculated (TIs(ADRF)) and (TpL) for comparison with (T,,(ADRF) jt= 0 and (Tlp(C)).Finally a comparison between (TI,(ADRF)/ and <TPL)gave rise to the estimate for the spin-lattice part of (T1,(C)} the last entry in table 1. For all three glassy polymers the spin-lattice contribution to (T1,(C)) is far more important than the spin-spin contribution. DISCUSSION We have presented a procedure for analysing the origins of the Tl,(C) relaxation process. The weakest part of this analysis from a quantitative standpoint is the estimation of So relative to Si. However changing So by 20% does not change the fact easily seen in fig. 3 that in those systems in which spin-lattice processes dominate S never rises to anywhere near So during the ADRF experiment even if the T, decay is allowed for.Clearly the dissipative effects of spin-lattice interactions dominate the constructive effects of spin-spin interactions in these systems. In general this behaviour is diagnostic of significant amounts of spin-lattice processes contributing to Tl,(C). No doubt a more elaborate fitting procedure could have produced better fits to the data. That was not the point of this exercise. The point was to show that a HIGH-RESOLUTION l3cN.M.R. IN SOLIDS phenomenological examination of the relaxation data could show that in some systems Tl,(C) must be dominated by a spin-lattice process even though in others a spin-spin process may be dominant. No appeal to detailed physical estimates of the relative importance of these two modes of relaxation is needed.A finer adjustment of the fitting parameters would not change these conclusions. There are other methods for reaching similar conclusions that make use of specific physical models (such as H1 dependence of relaxation) but they are less reIiable because of the oversimplifications implicit in the models. Naturally we do not expect to apply our full procedure often only often enough to identify classes of systems and ranges of experimental conditions where the origin of T,,(C) is clear enough to make it a useful parameter for characterizing molecular motion or structure. For ~ instance so long as HI > HL,a system with motion for which T~ -kHz will prob- ably be dominated by spin-lattice processes. This will include most glassy polymers and biopolymers at room temperatureI5 and probably plastic crystals near the melting point.On the other hand crystalline polymers and small organic molecules well below the melting point will show little effect of spin-lattice processes on T,,(C). It is possible that heavily cross-linked polymers will form an intermediate case to be decided individually. Although we have applied this analysis to non-spinning samples the general conclusion as to the importance of spin-lattice interactions does not change if the samples are spun at high speed (-kHz). Spinning perturbs the T,,(C) experiment by collapsing the dispersion of relaxation rates due to orientation relative to H, by plac- ing the sample under dilational stress which can alter molecular motions by modulat- ing the spin-spin interaction and by adding additional (usually negligible) mechanical motion to the spin-lattice interaction.l29 l5 We conclude by recommending that T,,(C) be used in much the same way that other n.m.r. relaxation times are. Seldom is it true that TI,for instance is determined solely by a single motion of a single internuclear interaction; nevertheless if the dominant source of relaxation is known Tl can provide useful information. Similarly Tlp(C)can be used to understand those classes of systems in which its dominant source is clear even if a full theoretical analysis does not yet exist. 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