Pulsed NMR in dynamically heterogeneous systems

JOURNAL OF MAGNETIC RESONANCE

4,40-46(1971)

Pulsed NMR in Dynamically Heterogeneous Systems J. CHARVOLIN~ AND

P. RICNY

Dipartement de Physico-Chimie Centre d’Etudes Nuclkaires de Saclay 91-Gif s/Yvette, France

Received June 8, 1970; accepted August 27, 1970 Some particular properties of dynamically heterogeneous nuclear spin systems, as studied by pulsed NMR, are discussed and illustrated by experiments on the lamellar and cubic phases of the potassium laurate-D,O system. Relaxation data obtained on this system show evidence of two types of motions, molecular diffusion occurring in addition to rapid deformations of the paraffinic chains.

INTRODUCTION

NMR properties of materials are largely determined by the motions of the nuclei; this gives rise to complex NMR signals whenever a system is heterogeneouswith respect to its dynamical structure. Partially ordered materials, as for instance, polymers, liquid crystals, biological membranes, or molecules adsorbed on surfaces, are examples of such systems. Motions manifest themselvesin particular through the modulation of the spin-spin dipolar interactions (la): In a rigid spin system these interactions, which are static, give the absorption resonanceline an approximatively Gaussian shape, while in the presenceof a motion, which averagesthem out, the resonanceline is narrowed and has a Lorentzian shape. To affect the resonanceline a motion must be faster than a critical frequency 60, which is of the order of the line width of the Gaussian line, expressedin set- ‘. Liquids provide, of course, typical examples of motional narrowing. In heterogeneoussystems, the characteristic frequencies 6w, can be distributed over a range of values, as can be the rates of the motions: Only part of the resonance line will then eventually be motionally narrowed. The resonanceline shapescan be used to analysethesesystems,at least very qualitatively, but as will be amplified below, pulsed NMR methods are much more powerful (lb). To interpret these experiments, one must keep in mind that the resonanceline width is due to the transversespinlattice relaxation in “liquid-like” (motionally narrowed) materials only; in “solidlike” samplesit is due to the spin-spin dipolar interactions. As a result, the formation of spin echoesfollowing a sequenceof two radiofrequency pulses is very different for the two types of samples.Thus, if the external field is much more homogeneousthan t Permanent address: Laboratoire de Physique des Solides, Faculte des Sciences, 91.Orsay, France. 40

PULSED NhfR SPECTRA

41

the line width, there is no echo with a “liquid-like” sample but one can be obtained with a “solid-like” one. This property will be used below to analyse heterogeneous systems. To illustrate these points we present measurementsobtained on lamellar and cubic mesophasesof the D,O-potassium laurate [CH,(CH2)10C02K, hereafter abbreviated C12K] system, prepared as indicated in Ref. (2). Here, as in systems with similar structures (3), the paraffinic chains of the surfactant molecules are highly disordered and previous NMR studies have shown this disorder to be dynamical (4). The degree of motion of a CH2 group is supposed to depend strongly on its distance to the polar head, the methyl end of the chain being expected to move rather freely while the polar one should be more rigid. The study of this compound with pulsed NMR has been undertaken to gain knowledge on the types and rates of the motions. The preliminary measurements, reported here, indicate that the surfactant molecules undergo simultaneously rapid deformations along the carbon-carbon bonds and slow translations over macroscopical distances. The measuredvalues of the relaxation times allow the characteristic times of these motions to be estimated. FREE

PRECESSION

SIGNALS

The free precession signal, the response of a nuclear spin system to a pulse of resonant radiofrequency, contains in principle the same information as the absorption line which is its Fourier transform (lb). However, in heterogeneous systems, the absorption line can contain narrow and broad components at the same time. The latter, frequently lost in the noise, are difficult to analyse but appear as narrow components in the free precession signal; their observation is then limited only by the recovery time of the pulsed apparatus. Figure 1 shows, in semilogarithmic plot, a typical free precession signal from the paraffinic protons of a C12K-D,O sample in the lamellar phase. This decay can be analysed into three components; its complex shape indicates the coexistence of motionally nonequivalent protons, some of which exhibit residual dipolar interactions. Becauseof the lamellar structure, the motions of the paraffinic chains must be anisotropic, which prevents complete averaging of the dipolar interactions betweenprotons. It is easy to see that under these conditions the free precession given by protons of a definite CH, group will have approximately the form (la)

Scrj = Scoj w

-!p-;.

( 2) The second moment m2 is due to the residual dipolar interaction experiencedby the two protons (not averaged out by the motion), while the transverse relaxation time T2 is due, on the contrary, to the modulated part of this interaction’; m2 and T, are functions of the distance of the CH, group to the polar head, their expression depending on the details of the motions. Of course, in our approximation, the free precession 1 Tz describes here the exponential decay of the transverse relaxation due to spin-lattice coupling; it should be distinguished from the inverse line width expressed in set-I, which is also often called Tz. The two definitions are equivalent for liquids where the line width is indeed due to the spin-lattice interactions. They are of course very different for solids or heterogeneous systems where the line width is at least partially due to static spin-spin interactions.

42

CHARVOLIN AND RIGNY

FIG. 1. Logarithmic plot of the proton free precession decay, .S,,,,obtained at 30 MHz after a n/2 pulse, from a (&K-28 ‘A DzO sample in the lamellar phase at 90°C. to is the sum of pulse width plus apparatus recovery time. qtJ, expanded in the inset as a function of t2, is the difference between the total decay SC0and its exponential part. The shaded area defines the superior limit of experimental accuracy.

-

of “solid-like” protons (T,J m2 % 1) is of Gaussian shape while that of “liquid-like” protons is Lorentzian. Proper distributions of m2 and T2 could naturally fit the decay shown in Fig. 1. However, a priori calculations of m2 and T,, as functions of the distance of the CH2 group to the polar head, are very complex and furthermore depend on the details of the motions. On the other hand, as shown in Fig. 1, the experimental decay can be analysed as the superposition of two “solid-like” and one “liquid-like” components. This does not necessarily imply discontinuities in the behaviour of the CH, groups along the chain but, considering three regions in the chain, must be a good approximation. The motion of the chain end is almost isotropic and the residual second moment is so small as not to be apparent on the corresponding free decay. The reverse situation prevails for protons near the polar head for which the very anisotropic motions result in a large second moment which makes the exponential part of the decay unobserved. Only an indicative value of 6 G2, determined at 90°C for a 14% D20 sample, can be attributed to this second moment, since the corresponding free decay is partially lost in the apparatus recovery time. In addition, an intermediate

43

PULSED NMR SPECTRA

region is observed which corresponds to protons having a D,O content and temperature-independentsecond moment of 0.5710.09 G2 over the domain of existence of the lamellar phase. The percentagesof CH2 groups in each part of the chain arp indicated in Fig. 2 for several temperatures and D,O concentrations. “The liquid-like” component cannot be accounted for either by the methyl protons, which represent only 13 % of the total number of protons, or by the residual protons in the heavy water used. The generaltrend in Fig. 2 is, as expected,an increaseof the mobile part of the molecule

70

0 ILL’C) 0 (55’C)

60

2

50

c II ..!? 2

LO

0 (73-c)

0

+ +

c b z

+ a

30

.o_ 5

L;:

0 r90%, + + (90-c)

0

0

+ (73’C)

+

+ (55°C) + (UT)

+ 20 0

10

l . I

0 10

I

I

I

I

I

15

D20

1,

I

I,

20

concentration

,

1

,

I

I

I

lU.9O”C) I

25

I

LI

I

/

I

30

020

weight

total

weight

>

FIG. 2. Percentages of protons of the paraffinic chains in “solid-like” te, mz=6G2, 0, ma = 0.57 GZ) and liquid-like (+T, = 270 p(s) regions of the paratlinic chains. The C&K-D20 samples are all in the lamellar phase. The effect of DzO concentration has been studied at WC, and temperature dependence has been studied at one concentration (28 % D,O).

with temperature and water content. The absenceof second moment variations with temperature and concentration indicates that the anisotropic motion responsible for the partial averaging of the dipolar interactions is fast compared to the corresponding second moment reduction. The shapesof the free precession signals have been interpreted here as coming from partially averaged dipolar interactions. Similar shapes can result from molecular diffusion in inhomogeneous magnetic fields. The following pulse experiment gives a test of whether the free precession decay of a heterogeneoussystem is due to dipolar interactions or to other effects. We take advantage of the very different properties of “liquid-like” or “solid-like” samples relative to the formation of a spin echo after a sequence[0,(0,),] of two radiofrequency pulses of angles 8, and 0, out of phase by an angle cp. If the free precession decay is due to field inhomogeneities an echo can be

44

CHARVOLIN

AND RIGNY

observed either after

71 nn [-01

or

2 z sequences.On the contrary, if it is due only to dipoIar interactions an echo can be created only after a

sequence(5). Figure 3 shows that a dipolar echo is indeed present in our case and that no effect of field inhomogeneities is apparent.

Time L 100 ps I grad. I

FIG. 3. Dipolar echo in lamellar C&K-D20

are compared. (a)

n2% R 2 [ 01

(90°C; 28 % DzO). The effects of two pulse sequences

[ 01 g

n I i

sequence. The second pulse does not alter the free precession decay. (b)

sequence. The second pulse is followed by a dipolar echo.

Other workers have reported, with similar systems, frequency-dependent line widths (6) or free induction decays (7), while we have not been able to see differences between 7 and 30 MHz. We believe these discrepancies to be due to differences in sample preparations, the compactness of the sample being perhaps an important factor. Another influence of sample preparation will be encountered below. The test proposed above indicates very clearly the dipolar origin of the decay, but unfortunately it does not give straightforward quantitative information. It also stresses the complexity of the response of a heterogeneous system to radiofrequency pulsed excitations, thus the conventional spin echo methods (e.g., effects of Carr-Purcell trains) should be considered with extreme care, for the dipolar echo can create artificial nonexponential decay.

45

PULSED NMR SPECTRA

RELAXATION

TIMES

AND

MOLECULAR

MOTIONS

The transverse spin-lattice relaxation time T, of the “liquid-like” protons is obtained by direct analysis of the free precession decay and is 270 & 20 psec at 90°C. It increases slightly with temperature, the activation energy being roughly 0.5 kcal/mole. The T, of “solid-like” protons should be obtained by other methods (e.g., TIP measurements). The spin-lattice longitudinal relaxation times are measured by usual

sequences.The single exponential recovery curve of the longitudinal magnetization, as well as the constant shape of the free precession decay following the second pulse, shows that a single Tl describes the relaxation of both “solid-like” and “liquid-like” protons. Tl is temperature dependent and its value at 90°C is 0.6810.04 sec. The measurements have been made at frequencies ranging from 7 to 30 MHz, but no frequency dependenceof the relaxation times was observed. A T2 very much smaller than T,, together with a frequency independent T,, imply two different relaxation mechanisms for Tl and T,, respectively. A fast (compared to the inverse Larmor frequency) motion must be responsible for the longitudinal relaxation, while a slow motion must cause transverse relaxation. Under these conditions, accurate values of the characteristic times of the motions and of the strength of the interactions they modulate cannot be given without further experiments, and will be reported elsewherewith detailed justification of the model. The rapid motion probably consists of internal rotations about the C-C bonds of the paraffinic chains. The effects of these rotations are cumulative all along the chain so that the resulting motion has a smaller amplitude close to the polar head than near the chain end; this can explain the decreaseof the second moment from 6 G2 [value appropriate to alkanes rotating about their longitudinal axis (S)] to zero at the end of the chain. Also, an activation energy of 5.5 kcal *mole- ‘, deduced from the variation of T, between 90°C and 44°C is consistent with the heights of the barriers hindering internal rotations in alkanes (9). Assuming these rotations to be described by a harmonic potential, one finds a rough value of 10m9set for their characteristic time. A single Tl behaviour, as observed, is somewhat inconsistent with such a model and might indicate some spin-diffusion limited relaxation (IO). The slow motion responsible for the transverse relaxation is probably a twodimensional diffusion of the laurate molecule in the plane of its lamella; taking 2 G2 as an average intermolecular interaction, one deduces from the experimental T, a characteristic time z of roughly 10m6sec. A diffusion coefficient

where 1 is the mean distance between neighbouring polar heads, can be ascribed to this motion. I- 5Ayields DN 1.5~10-~crn~~sec-‘. A similar diffusion is seen very clearly in the cubic mesophaseof the same system C,,K--D20. In this phase, which is isotropic, the free precession decay is purely exponential and the residual second moment is zero for all the protons. The decay of the amplitude of the spin echo obtained in an inhomogeneous external magnetic

46

CHARVOLIN AND RIGNY

field is shown in Fig. 4. Its variation, proportional to exp (- kt 3), reveals molecular diffusion over distances of microns, with a diffusion coefficient of about 2 x 10e6 cm2 asec-’ at 90°C. Some of our cubic samples did not show similar echo decays, probably because of the very small size of the crystallites in these samples. This would prevent the diffusion from acting over distances large enough to cause irreversible decay of the transverse magnetization.

Time

(5 ms / grad. )

FICX 4. Echo envelope, E,s at 30 MHz, obtained in an external field gradient for a C&-D20

sample in the cubic phase (38 % DzO, 90°C).

It is interesting to stress that the relatively very rapid diffusion takes place in spite of a very high macroscopic viscosity. Such a diffusion is clearly in favour of the structure recently proposed for the cubic phase of lipid-water systems (11) where the lipidic regions are connected over the whole crystal. It is very likely that diffusion also occurs over macroscopic distances in the lamellar phase, as has been proposed previously to interpret T2. However, in this case evidence similar to that for the cubic phase cannot be given, owing to the very fast decay of the transverse magnetization. ACKNOWLEDGMENTS We thank Miss A. Tardieu, Prof. Luzzati (C.N.R.S., 91-Gif s/Yvette), and Dr. M. G&on Polytechnique, Paris) for helpful discussions of this work.

(Ecole

REFERENCES Principles of Nuclear Magnetism,” Oxford University Press, 1961, (a) Chapter X, (b) Chapter III. J. CHARVOLIN AND P. RIGNY, J. de Physique 3OC4 76 (1969) (publication of the Societe Francaise de Physique). V. LUZZATI, in “Biological Membranes” (D. Chapman, Ed.), Academic Press, London, 1968. K. D. LAWSON AND T. J. FLAUTT, J. Phys. Chem. 72,2066 (1968). J. G. POWLES AND J. H. STRANGE, PUOC. Phys. Sac. 82, 6 (1963). S. A. PENKE~, A. G. FLOOK, AND D. CHAPMAN, Chem. Phys. Lipids 2,273 (1968). J. R. HANSEN AND K. D. LAWSON, Nature 225, 542 (1970). E. R. ANDREW, J. Chem. Phys. 18,607 (1950). N. P. BORI~OVA AND V. M. VOLKENSTEIN, Zh. Strukt. Khim. 2,469 (1961). J. E. ANDERSON AND W. P. SLIGHTER, J. Phys. Chem. 69,3099 (1965). V. LIJZZATI, A. TARDIEU, T. GULIK-KRYWICKI, E. RIVAS, AND F. REINS-HUSSON, Nature 220, 485 (1968).

1. A. ABRAGAM, “The 2. 3. 4. 5.

6. 7. 8. 9.

10. 11.