Nuclear Spin Relaxation in Solid n-Alkanes

NUCLEAR SPINRELAXATION IN SOLID%-ALKANES

3099

Nuclear Spin Relaxation in Solid n-Alkanes

by J. E. Anderson and W. P. Slichter

J. Phys. Chem. 1965.69:3099-3104. Downloaded from pubs.acs.org by RUTGERS UNIV on 09/04/15. For personal use only.

Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey

(Received April 6,1966)

A study has been made of the role of specific molecular motions in effecting nuclear spin relaxation in normal alkanes in the solid state. Steady-state measurements of n.m.r. absorption exhibited the onset of premelting motion in certain members of a series of n-alkanes ranging between CeH14and C40Hs2,notably in the odd-numbered members studied in the series and in even-numbered members of suffciently high molecular weight. The contribution of the rotation of methyl groups to the n.m.r. spin-lattice relaxation time TI has also been determined in these compounds and in various solid solutions of C ~ H and I ~ CID18 to assess the intermolecular contribution to the spin-lattice relaxation. All of these studies were carried out between 100'K. and the respective melting points. In the studies of spinlattice relaxation, TI minima were seen in all the compounds at 150 f 5°K.) when measured at a radiofrequency of 50 Mc./sec., and the temperature dependence of TI corresponded to an activation energy of 2.6 f 0.2 kcal./mole for all the compounds. These Ti minima are ascribed to the threefold reorientation of the methyl groups. The magnitude of 2'1 at the minimum depends upon the relative abundance of methyl protons. It is concluded from steady-date measurements that the methylene protons are essentially immobile at these temperatures. The spin-lattice relaxation of the methylene protons is ascribed to spin diffusion to the methyl protons. A kinetic model for this process has been developed and is found to account satisfactorily for the observed dependence of the minimum TI value on the CHJCH2 ratio.

r,O exp(AE/RT), where AE is the activation energy o the relaxation process and roois a constant of the sys The n.m.r. spin-lattice relaxation time, Ti, has been tem having the dimensions of time. widely used to characterize molecular motions in both The derivation of eq. 1 is based on a model of nonsolids and liquids. TI measurements are commonly interacting relaxing units. This model is unsatisfactory analyzed in terms of the simple expression de~elopedl-~ for solids wherein spin exchange between the relaxing for relaxation among pairs of magnetic dipoles units is rapid. The use of eq: 1 to analyze relaxation data for solids therefore often predicts TI values that are much smaller than those experimentally observed. In this paper we shall explore the consequences of spin worO 4w0r0 (1) diffusion in a system in which a fraction af the nuclei (1 (wo70)2 1 (20070)2 couple strongly to the lattice, while the remainder experience weak coupling. We shall demonstrate by I n eq. 1, y is the magnetogyric ratio of the nuclear spin I, wo is the angular frequency of the applied radiofrequency field, r f j represents the distance between mag(1) N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev., netic nuclei, K is a constant that depends on the process 73, 679 (1948). (2) R. Kubo and K. Tomita, J . Phys. SOC.Japan, 9 , 888 (1954). of molecular reorientation, and r0 is the correlation (3) I. Solomon, Phys. Rev., 99, 559 (1955). time for this motion. Differentiation of (1) with re(4) E. 0.Stejskal and H. S. Gutowsky, J . Chem. Phys., 2 8 , 388 spect to worO shows the existence of a maximum when (1958). war, = 0.62. It is commonly assumed that r0 de(5) E.0.Stejskal, D. E. Woessner, T. C. Farrar, and H. S. Gutowsky, pends on temperature according to the relation ro = ibid., 31, 55 (1959).

Introduction

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Volume 60, Number 0 September 1066

J. E. ANDERSONAND W. P. SLICRTER

3 100


means of a kinetic argument how the magnitude of TI is affected by the extent of this coupling to the lattice. Experimental All of the protonated n-alkanes studied were reagent chemicals obtained from commercial sources. Perdeuterio-n-octane with an isotopic purity of 90% H2 and a 98% chemical purity was obtained from Merck Sharpe and Dohme of Canada, Ltd. The samples were used without subsequent purification and were sealed under vacuum. The n.m.r. spin-lattice relaxation times were determined by the 180-90" pulse technique of Carr and PurcelllB using the null method. Pulse lengths were of the order of 1-4 psec. The recovery time of the system was about 10 psec. Further details of the pulse equipment may be found elsewhere.? The steady-state measurements were carried out on a Varian dud-purpose spectrometer operating under wide-line conditions. Values of the spin-spin relaxation time Tz were obtained from the line width, 6H, using the expression T2 = l/y(6H).' Species of Motion in the Solid The n.m.r. second moment, M2,provides a quantitative description of certain kinds of molecular motion in solids. Values of M2for immobile nuclei can be calculated: as well as M z values for nuclei undergoing reorientations about specific molecular axes.g Experimental second moments are then compared with the calculated quantities. Using the X-ray data of Miiller,'O Andrew" has shown that the second moment of a completely rigid n-alkane CnH2%+2may be expressed as Mz = 26.3 19.l/(n 1) gauss2. Below their rotational phase transitions, experimental MZ values for C28H58and C3zHasagreed closely with the calculated values, and it was concluded by Andrew that these molecules were essentially motionless. However, Andrew found the second moment for Cl8H88 at low temperatures t o be much smaller than the value calculated for immobile molecules. Similarly, the second moments for C1&Is and C16H, at low temperatures were found12 to be smaller than those calculated from Andrew's expression for rigid n-alkanes. Moreover, as is seen from Table I, which includes both the foregoing results and additional results from the present study, the M2 values are found to increase with chain length, contrary to Andrew's equation for rigid n-alkanes. The disparity in the case of ClsHa was ascribed by Andrew to the presence of some effective motion even at the lowest temperatures used. In the present investigations by steady-state n.m.r., two peaks were seen in the derivative curves for the lower alkanes at 125"K., the lowest temperature

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The Journal of PhysicaL Chemislry

Table I: Wide-Line Data at 148OK. T Z(wide),

T z(narrow)

Mn, gauss2

pee.

pseo.

17.1

3.2 3.2 3.0 3.1 3.1 3.1 3.1 3.1 3.0 2.9 2.8 2.8 2.7

24 22 18 30 29 31 26

19.7 20.7 19.5 21.5" 21.6 23.5" 23. gb 25.7 26.9 24.9 See ref. 11.

studied. The derivative curve for CVH16, taken at 150°K., is illustrated in Figure 1. The narrow component decreases in intensity with increasing chain length and cannot be clearly identified in alkanes higher than CIBao. With the exception of CllHu and C l & I ~ , to be discussed subsequently, these derivative curves were essentially independent of temperature over the temperature range studied, up to the temperature of the rotational transition (see below). Experimental 4

1.63 GAUSS

+

If--i2

GAUSS

--A

Figure 1. The derivative curve for n-C,Hls measured at 150°K.

(6) H.Y.Carr and E. M. Purcell, Phys. Rev., 94, 630 (1954). (7) W.P. Slichter and D. D. Davis, J . A p p l . Phys., 35, 10 (1964). (8) J. H.Van Neck, Phys. Rev., 74, 1168 (1948). (9) H.5. Gutowsky and G. E. Pake, J. C h m . Phys., 18, 162 (1950). (10) A. Muller, Proc. Roy. SOC.(London), A120, 437 (1928). (11) E.R. Andrew, J . Chem. Phys., 18, 607 (1950). (12) D. F.R. Gilson and C. A. McDowell, Mol. Phys., 4,125 (1961).

NUCLEAR SPINRELAXATION IN SOLID~-A.LI(ANES

3101

M 2 values, obtained at 150 f 5"K., increased with chain length and appeared to approach an asymptotic value of 25-27 gauss2. These data suggest the existence of a low temperature crystal structure wherein the methylene protons are held rigidly on the molecular backbone, while the methyl protons appear to be freely reorienting on the time scale of the steady-state experiment. If this is the case, the intramolecular contribution to M2 is calculated to be Mz = 18.5 - 46.3/(n 1) gauss2. Assuming that the intermolecular contribution has the same dependence on methyl concentration, 1) gauss2. the total theoretical M Zis 26.3 - 65.7/(n The measured M z values agree with this expression within experimental uncertainty. It is well known13-15 that the odd-numbered nalkanes and their even-numbered homologs above CBH38 undergo a phase transition below their melting points, which marks the onset of considerable motion in the solid. In the present work, this transition was by manifest in C1Jh, C13HZ,C%H58,C&€&B,and C40H82 narrow absorption lines that were first observable somewhat below the Curie temperature. These lines grew more intense as the temperature was raised. There was a sharp decrease in TI in the same temperature interval. The T1 data for ClaH, are shown in Figure 2. Curves for the other four compounds are similar. All of the samples melted before T1 minima were reached. Similar behavior has also been observed in CsJLao by McCall and Doug1ass.l6 In the course of such a phase transition, a wide distribution of frequencies of molecular motion will occur, causing simultaneous changes in TI and Tz. Each of the thirteen alkanes studied gave a plot of log T1 vs. reciprocal temperature similar to those illustrated in Figure 3. Every member of the series exhibited a T1 minimum at 148 f 5°K. The activation energy for the correlation time rc, mentioned above, was determined to be 2.6 f 0.2 kcal./mole in all samples. The M z studies have shown the methyl protons to be the only mobile nuclei at this temperature. We therefore assign these minima to the threefold reorienta-

TO K

125 1

200 I

I50 1

250 1

+


+

Table II: Values of the 148°K. 2'1 M i n i " in n-Alkanes

t 0VTOn

Figure 2. Temperature dependence of the spin-lattice relaxation time in n-ClsHm. T OK

0

:.::I 0.04 IO

I

I

9

8

I

I

,

7

6

5

1O3/T0K

Alkane

T I(rain.), msec.

Alkane

TI(min)., meo.

C6H14 C7Hl6 CsHia CioHzz CllH24 Cl&6 CUE28

58 f 2 61 =!= 2 66 f 2 77 f 2 87 f 2 92 i 2 99 i 2

Cl4H80 Cl&4 CisHaa CzaHm C8ZH66 C40He.a

106 i 2 130 f 5 140 i 5 214 f 7 238 f 15 300 f 15

Figure 3. Temperature dependence of the spin-lattice and O n-CtsHss. , relaxation time in n-CTHlo, ~ - C I ~ % (13) A. R. Ubbelonde, Trans. Faraday Soc., 34, 282 (1938). (14) W.F. Seyer, R. B. Bennett, and F. C. Williams,J. Am. Cham. SOC.,66, 179 (1944). (15) H.L. Finke, et al., ibid., 76, 333 (1964). (16) D.W.McCall and D. C. Douglsss, Polymer, 4,433 (1963).

Volume 69,Number 9 September 1966

J. E. ANDERSONAND W. P. SLICHTER

3102


tion of the terminal CHI groups. The magnitude of TI at the minimum increases with chain length as indicated in Table 11. Terms corresponding to those within the sum shown in eq. 1 are generated by both intermolecular and intramolecular local fields. For reasons that will be discussed subsequently, it is of interest to discriminate between intramolecular and intermolecular contributions to TI. One method of singling out these contributions is to reduce the intermolecular effects by isotopic substitution. If attention is confined to linear terms in composition, TIcan be represented by 1

-=A+Bx

Ti

12

0

In eq. 2, A and B refer to the total intra- and intermolecular proton-proton interactions, and x represents the mole fraction of the protonated molecules. The intermolecular relaxation caused by the deuterium nuclei is smaller than the corresponding proton relaxa) ~ and can be ignored over the tion by ( y ~ / y=~0.0236 concentration range studied. A plot of (l/Tl)maxvs. mole fraction of CsHls in CsDle is shown in Figure 4. The extrapolated value of A is 12.6 set.-', and B is found to be 2.55 set.-'. From the ratio B / ( A B ) , we find that intermolecular local fields are responsible for 17% of the spin-lattice relaxation in pure CsHls.

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Discussion of TI Results Perhaps the most interesting feature of the spinlattice relaxation data is the variation of the methyl TI minimum with chain length. Equation 1, which was derived for a system of noninteracting pairs of nuclei, implicitly presumes that aII the nuclei are equivalent in their capability for spin-lattice relaxation. This situation does not occur in the n-alkanes. The steadystate experiments cited above show that the second moments of all members of the series are large at -150”K., the temperature of the TIminimum. The conclusion is that the methylene groups are effectively motionless and therefore should be inefficient in direct transfer of spin energy to the lattice. The methyl groups, however, are comparatively efficient in spinlattice relaxation, owing to their rotation at appropriate frequencies in this temperature range. It appears that the mechanism for the dissipation of the spin energy of the methylene protons, following irradiation by the radiofrequency magnetic field, is that described by McCall and Douglass16 in n.m.r. studies of polyethylenes. If Tz > k ~ kn. ,

NUCLEAR SPINRELAXATION IN SOLID TZ-ALKANES

3103

kd3 may be evaluated in terms of the equilibrium populations: at equilibrium kl(Ao+) = Ll(Ao-), etc. Since (Ao+/Ao-) = (Bo+/Bo-), we find k3 = L 3 . By straightforward but lengthy algebra, the rate expressions for the two-component system can be expressed as

d dt

-(a

- CYO)

=

+ P ) - + b0)l

-2k1(a - ao) -

[(a

~ N[(a B-


d -(p dt

- Po)

=

ao)

- (P - Po) I

A + - A - ‘ p = P+ - P-. etc, A + A-’ P+ b-’ The quantities a and 6 are proportional to the mag-

+

+

netizations and therefore to the observed signals. Equations 6 and 7 have a general solution (a - ao) =

(p -- Po)

=

+ P> + Po)linitiale

= [(a

-t’T1

(10)

where

where a =

(a0

((YO

(6)

- Po) -

-21cz(P

son in the present case, and considering the range of values of n that occur here, we can estimate that k8 is comparable to 1/T2 within two orders of magnitude. On the basis of the TZdata in Table I, it then follows that k3 >> kl, k2 in all the alkanes at 150°K. In this case, the system effectively relaxes by the expression”

C+e-*+t

+ C-e-’-’

(Sa)

C’+e-*+‘t + C’-e-*-t

(8b)

Equation 10 can be seen to revert to (1) whenever kl = h. If kl >> kz, BS is the case for the n-alkanes, this treatment predicts that the minimum values of TI ought to vary linearly with the relative abundance of methyl sites. On the other hand, it shows that the position of the TIminimum along the temperature axis and the activation energy of the relaxation process are independent of the number of methyl sites. We have already mentioned the close agreement among the activation energies of the various alkanes and among

where @* are the roots of the equation

- [2(kl +

k2)

+

k3NPl@*

+

2k,(k&~

BNA)

+ 4kikz = 0

and

The absolute values of C+ and C- are established by the boundary conditions. The experimental sign8cance of lc1 and kz may now be seen by considering systems composed entirely of methyl sites or isolated methylene protons. The r e laxation process is then described by exp(-2klt) or exp(-2k2t), and we can associate 1/(21cr) and 1/(2k2) with the TI values of the isolated methyl and methylene protons, respectively. The term represents spinlattice relaxation caused by spin Epping between the methylene and methyl sites. The relaxation process envisaged here is formally similar to the relaxation of the nuclear spin system by paramagnetic impurities, which waa treated by B1oembergen.l’ In his kinetic model, le3 can be approximated 1

k8

G

I

(9)

-1- 7(n - 2)]T2 where n is the number of carbon atoms in the alkane. Taking his model to be an adequate basis for compari-

TOTAL NUMBER OF PROTONS NUMBER OF METHYL PROTONS

Figure 5. The methyl TIminimum value in the n-alkanes aa a function of the relative number of methyl protons. (20) Equations 10 and 11 may be obtained from eq. 6 and 7 directly by imposing the condition that a aommon spin temperature is maintained throughout the system at all times.

Voluma 69,Number 9 Septtmber l06b


3104

J. E. ANDERSON AND W. P. SLICHTER

msec. at a radiofrequency of 50 Mc. However, the the temperatures at which Tlminima occur. Figure 5 seven protons in the camphor molecule that are imshows the linear relationship between the experimental mobile at low temperatures were not adequately acTlminimum values and the N p / N ratio. ~ The agreecounted for in calculating the magnitude of the TI ment between theory and experiment is excellent. The TIminimum for a system composed entirely of methyl minimum: properly, their presence would raise the magnitude of the TIminimum to be expected in camsites can be determined from the slope of the line in phor. Figure 5. A value of 22 msec. is obtained by this The ratio of 22 msec., the experimental value for the method. We extended the work of Gutowsky, et T1 minimum, to 28 msec., the calculated value arising to obtain a 24 f 1 msec. TIminimum in C(CH& at from interactions within a CH3group, shows that about 114 f 2”K.,which can be attributed to CH3 rotation. 7901, of the methyl spin-lattice relaxation in the nWe now examine the separation of the intra- and intermolecular contributions to Tl.Using eq. 2, the TI alkanes is generated within the CHa group. The rel& tive magnitudes of these intra- and extra-group contristudy of the CsHlgCsDls system indicated that a probutions should not differ greatly for any hydrocarbon. tonated n-octane molecule completely surrounded by its This suggests that the methyl T1minimum, when obdeuterated analogs would give a T1 minimum of 79 tainable, could provide a qualitative measure of the msec. at 50 Mc. Application of eq. 11 eliminates the number of reorienting CH3 groups present in a solid effect of methylene relaxation through the methyl sample. In principle, this figure can be measured “sink” and yields a value of 26 msec. for the intraquantitatively by the n.m.r. second moment. As molecular contribution to methyl relaxation. Hubillustrated in the M2 study of the n-alkanes, this method bard21 has calculated the time variation of the nuclear becomes increasingly insensitive as the relative conmagnetization of three equidistant, identical nuclei, centration of methyl protons decreases. In contrast, using a semiclassical density-matrix theory of relax+ the Tl minimum is probably more sensitive to the tion, and has found that the relaxation consists of the number of reorienting CHI groups as their concentrasum of two terms that decay exponentially with diftion diminishes; e.g., a sample with a 10% concentraferent time constants. Practically speaking, however, tion of mobile CHS protons will have a much larger the decay of the magnetization is adequately repreminimum than a sample with a 15% concentration. sented22by a single exponential of the form appropriate Using the slope of Figure 5, we have obtained satisto relaxation in the absence of correlation between relafactory determinations of mobile CH3 content in a tive motions of pairs of As calculated for a number of polyolefim7 and ethylene-propylene coradiofrequency of 50 Mc./sec., the magnitude of the T1minimum for relaxation via the rotation of methyl p o l y m e r ~ . ~Of~ course, the treatment set forth here for mobile CHa groups is applicable to any mobile groups about the threefold axis, in a polycrystalline unit, once its intrinsic relaxation time is known. sample, is 28 msec. This calculation involves only pairwise intera~tions~l-~ with no cross terms, and is in Acknowledgments. We are grateful to Professor R. close agreement with our experimental results. Using Bersohn, Columbia University, for critical comments an essentially classical calculation, Stejskal and Gutowon the interpretation of relaxation in three-spin syssky4 have developed an expression that predicts a TI tems. We are also grateful to our colleagues, Dr. minimum for methyl rotation of 57 msec., at a resonant D. C. Douglass and Dr. D. W. McCall, for helpful disfrequency of 50 Mc./sec. It seems that the latter calcussions of this work. culation of the cooperative influence of the two neighbors that relax the third methyl proton is fundamentally (21) P. S. Hubbard, Phys. Rev., 109, 1153 (1958). an overestimate. Thus, it appears that the agreement (22) A. Abragam, ref. 18, p. 293 ff. between the 57-msec. value and some of our earlier ex(23) J. E. Anderson and W. P. Slichter, J . C h m . Phys., 41, 1922 perimental results on relaxation in camphors3 was (1964). fortuitous. The methyl TI minimum in camphor is 50 (24) E. G. Kontos and W. P. Slichter, J . Polymer Sci., 61,631 ( I 62)

The Jou~nulof Physical C h k t r y

Nuclear Spin Relaxation in Solid n-Alkanes

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