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Journal of Magnetic Resonance 181 (2006) 29–34

Thermal diffusivity and nuclear spin relaxation: A continuouswave free precession NMR study

Tiago Venancio a,b, Mario Engelsberg c, Rodrigo B.V. Azeredo d, Luiz A. Colnago b,*

a Universidade de Sao Paulo, Instituto de Quımica de Sao Carlos, Avenida Trabalhador Sao-Carlense 400, Sao Carlos, SP 13560-590, Brazilb Embrapa Instrumentacao Agropecuaria, Rua 15 de Novembro 1452, Sao Carlos, SP 13560-970, Brazil

c Universidade Federal de Pernambuco, Departamento de Fısica, Avenida Professor Luiz Freire s/n� Cidade Universitaria, Recife, PE 50670-901, Brazild Universidade Federal Fluminense, Instituto de Quımica, Outeiro de Sao Joao Batista s/n�, Campus do Valonguinho, Niteroi, RJ 24020-150, Brazil

Received 12 December 2005; revised 8 March 2006Available online 17 April 2006

Abstract

Continuous wave free precession (CWFP) nuclear magnetic resonance is capable of yielding quantitative and easily obtainable infor-mation concerning the kinetics of processes that change the relaxation rates of the nuclear spins through the action of some externalagent. In the present application, heat flow from a natural rubber sample to a liquid nitrogen thermal bath caused a large temperaturegradient leading to a non-equilibrium temperature distribution. The ensuing local changes in the relaxation rates could be monitored bythe decay of the CWFP signals and, from the decays, it was possible to ascertain the prevalence of a diffusive process and to obtain anaverage value for the thermal diffusivity.� 2006 Elsevier Inc. All rights reserved.

Keywords: Thermal diffusivity; Natural polymer; CWFP; SSFP; Freezing

1. Introduction

Steady-state free precession (SSFP) [1–4] has been veryeffective in fast magnetic resonance imaging and in thestudy of flow and diffusion [5–7]. Recently, we have demon-strated that continuous wave free precession (CWFP), aspecial case of SSFP, can be used in several applications.For example: for signal to noise ratio improvement inquantitative determinations using low-resolution NMRequipment [8,9], to monitor flow [10], and for fast simulta-neous measurements of T1 and T2 [11].

The CWFP regime is attained when a train of short p/2pulses, with period Tp small compared to T �2 ¼ 1=D-0, isapplied to a liquid or motionally narrowed solid obeyingBloch’s equations. Here D-0 denotes the dispersion of pre-cession frequencies about a central frequency -0 caused byan inhomogeneous magnetic field. Furthermore, the

1090-7807/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmr.2006.03.011

* Corresponding author. Fax: +55 1633725958.E-mail address: colnago@cnpdia.embrapa.br (L.A. Colnago).

applied radiofrequency is shifted from the central frequen-cy -0 by a frequency offset chosen such thatx0Tp = (2n + 1)p, where n is an integer. This causes the sig-nal amplitude just preceding a p/2 pulse to be equal to theamplitude following the pulse. Since the condition T p < T �2implies that dephasing of isochromats in each interval Tp isrelatively small a continuous wave periodic signal withpractically constant amplitude, displaying n nodes withineach Tp interval, is obtained [8]. If the condition Tp� T2,T1 is satisfied the amplitude of the transverse CWFP mag-netization, under steady-state conditions, can be shown tobe given by [8] |Ms+| = M0/(1 + T1/T2). Here T2 is thespin–spin relaxation time and M0 denotes the thermal equi-librium magnetization.

When a CWFP steady state is established under equilib-rium conditions its amplitude is determined by the time-in-dependent ratio C = T1/T2. Moreover, in a non-equilibrium situation, the steady-state signal can evolve intime as changes in the relaxation times, which could becaused by a variety of processes, take place. Such processesmight include: temperature changes due to heat transfer,

30 T. Venancio et al. / Journal of Magnetic Resonance 181 (2006) 29–34

fast polymerization reactions, phase changes, etc. Providedthe response of the CWFP signal is fast enough, valuableinformation about the kinetics of these processes couldbe obtained.

CWFP can be used effectively to monitor such processesprovided some specific conditions are met. Denoting by sthe correlation time of the relevant molecular motionCWFP can be sensitive to the kinetics of the process pro-vided the motion is in the slow correlation time regime[12] characterized by -0s > 1. In this regime, the spin–lat-tice relaxation time can become appreciably longer thanthe spin–spin relaxation time. Conversely, in extreme nar-rowing regime characterized by -0s� 1, the conditionT1 � T2 prevails and the CWFP signal becomes indepen-dent of the relaxation times.

The response time of the CWFP signal to changes in therelaxation times is an important factor for the proposedapplication. The establishment of a steady-state CWFP sig-nal starting from thermal equilibrium has been studied inconsiderable detail. It has been shown to consist of an ini-tial oscillatory transient [11], with a short time constantdetermined by T �2 superimposed on an exponentialapproach to the steady state with time constant [13]Ts = 2T1T2/(T1 + T2). One could expect that, if the rateof change of the relaxation ratio C = T1/T2 for the ongoingprocess is smaller than 1/Ts the CWFP signal should beable to follow the kinetics of the process. In a regime char-acterized by -0s > 1 the spin–lattice relaxation time T1 maybecome quite longer than T2 and the response timeapproaches the value Ts � 2T2.

Although several applications can be conceived we hereconsider, as an example, the heat transport process inspheres of a polyisoprene elastomer (natural unvulcanizedlatex rubber without any filler). In spite of being able, inprinciple, to monitor faster processes, there exists someinterest in measuring thermal transport properties byNMR [14–16].

Starting from a spatially non-uniform temperature dis-tribution the system under study, with a well-defined geo-metrical shape, is allowed to evolve towards equilibriumvia thermal diffusion. As the local temperature varies thechanges in 1H relaxation times that take place can be mon-itored through the decay of the CWFP signal. By varyingthe size of the system it is possible to make sure that thescaling of the decays obeys a diffusion equation and toascertain that the response time comfortably permits tomonitor the kinetics of the process.

Using low-resolution bench-top NMR equipment, andcombining CWFP measurements under non-equilibriumconditions with equilibrium CWFP measurements at vari-ous temperatures, we are able to obtain thermal diffusivitydata.

2. Theory

In several elastomers, such as natural rubber, at Larmorfrequencies in the range -0/2p � 5 MHz, the condition

-0s > 1 is satisfied for T1 somewhat below room tempera-ture. Here s represents the correlation time of the motion,which in the present case, corresponds to the reorientationof elementary polymer segments.

If a uniform polymer sample of spherical shape, initiallyin equilibrium at a temperature h0, is placed in contact witha thermal bath at a lower temperature hb, the time evolu-tion of the local temperature H (r, t) resulting from heatflow to the bath can be described by a diffusion equation.For spherical symmetry the local temperature H (r, t) obeysthe equation

oHot¼ 1

r2

o

orkðHÞr2 oH

or

� �; ð1Þ

where the thermal diffusivity k (H) is, in general, a functionof temperature and is related to the heat conductivity j, thespecific heat at constant pressure cp, and the mass density qthrough [17]

k ¼ jcpq

. ð2Þ

In particular, if the thermal diffusivity can be assumed tobe temperature independent in the interval of interest, thediffusion equation takes the simpler form

oHot¼ k

ro2

or2ðrHÞ. ð3Þ

Denoting by a the radius of the sphere, Eq. (3) permits ananalytical solution [17], satisfying the boundary conditionsof our problem, which is given in closed form by

Hðr; tÞ � h0

hb � h0

¼ 1þ 2X1n¼1

ð�1Þn expð�n2kp2t=a2Þ sinðpnr=aÞpnr=a

;

ð4Þin the range t > 0 and r < a.

On the other hand, when the thermal diffusivity changesappreciably in the temperature interval of interest, Eq. (2)can be solved numerically using, for example, the finite dif-ferences method of Crank–Nicholson [17,18] with a tem-perature dependent k (H).

As the local temperature within the sphere decreasesfrom its initial uniform value h0 towards its final equilibri-um value hb the relaxation ratio C = T1/T2 rapidly increas-es and the CWFP signal amplitude decays to zero. Strictlyspeaking not only the relaxation ratio increases but also thethermal equilibrium magnetization increases at a rate lim-ited by 1/T1. However, given that the changes in M0 aresmall in the present problem, the effect is practicallynegligible.

If the rate of change of the ratio T1/T2 is small com-pared with 1/Ts the CWFP signal amplitude as a functionof time can be assumed to be determined by the time evo-lution of a succession of steady-state regimes. Denoting byC (H) = T1 (H)/T2 (H) the temperature-dependent relaxa-tion ratio, the normalized CWFP signal amplitude S (t)would be given simply by an integral over all sphericalshells of form

T. Venancio et al. / Journal of Magnetic Resonance 181 (2006) 29–34 31

SðtÞ ¼ ð1þ Cðh0ÞÞð4=3Þpa3

Z a

0

4pr2 dr1þ CðHðr; tÞÞ . ð5Þ

The correction arising from the variation of M0 withtemperature, as predicted by Curie’s law, can be taken intoaccount by multiplying the integrand in Eq. (5) by h0/H (r, t). However, given the steep variation with tempera-ture of the relaxation ratio C (H), the resulting correctionis quite small.

Eq. (5) can be used to determine the average thermal dif-fusivity of the polymeric material. First the functional formof C (H) must be determined experimentally by measuringthe relaxation ratio C at various temperatures. A simpleway to do that is to measure the ratio |Ms+|/M0 for thepolymer in equilibrium at various temperatures directlyfrom the CWFP signal amplitude. Moreover, since thefunctional dependence H (r, t) is furnished by the solutionof the diffusion equation it is possible, using Eq. (5), to cal-culate the time dependence of the CWFP signal decayunder non-equilibrium conditions.

Fig. 1. CWFP signal amplitude decays for rubber spheres of variousmasses, initially at h0 = 295 K, after contact with a bath at hb = 77.8 K.

Fig. 2. Normalized CWFP signal decays for rubber spheres with fourdifferent masses.

3. Experimental results and discussion

1H CWFP measurements were performed at a frequencyof 7.24 MHz using a home-made [8] system with a bench-top 0.17 T Alnico V permanent magnet. The dephasingcaused by the inhomogeneity of the magnetic field couldbe characterized by a characteristic time T �2 ¼ 1:0 ms andthe period of the sequence of p/2 pulses was Tp = 0.3 ms.The frequency offset, D = - � x0, was chosen such thatDTp = 3p and the width of a p/2 pulse was 9 ls. Pulseimperfections can have some effect on the CWFP signalamplitude, as demonstrated in [11]. Good homogeneity ofthe RF field over the volume sample volume is generallynecessary but the required conditions can be met withoutgreat difficulty.

The probe consisted of a single receive/transmit coil25 mm in diameter and 25 mm in height and the electronichardware consisted of a Apollo transceiver (Tecmag) usedin conjunction with a Miteq AU1448 preamplifier and aAMT2053 power amplifier.

The rubber spheres were introduced in a glass tube 23 mmin diameter and 200 mm in height which was not in directcontact with the RF coil. A steady-state CWFP regime wasfirst established at h0 = 295 K by applying a train of p/2 puls-es for approximately 3 s. At this point a non-equilibrium ini-tial condition was created by establishing contact betweenthe rubber spheres and a liquid nitrogen bath at temperaturehb = 77.8 K. Liquid nitrogen was poured on the samples,keeping the spheres immerse, while signal acquisition wasstarted simultaneously with the addition of the liquid nitro-gen. Initial boil-off instabilities appeared to have little effecton the CWFP signal. Furthermore, detuning and mismatch-ing caused by the addition of liquid nitrogen was also foundto be negligible. The CWFP pulse sequence applied in thisexperiment consisted of a train of 9 · 104 pulses separatedby a time interval Tp of 300 ls.

Bath temperatures closer to h0 could yield more accuratethermal diffusivity data but, since longer decays would pre-vail, the time response of the CWFP method would not beadequately tested.

Fig. 1 shows CWFP signal decays for spherical samplesof various masses. Each of the 10K data points in everydecay was acquired at 9Tp intervals (2.7 ms) and corre-sponds to an average of 64 points sampled with a dwelltime of 4 ls.

Fig. 1 shows various decays in spheres of different mass-es, indicating that the decay rates increase as the mass ofthe spheres decreases. This can be seen more clearly inFig. 2 where normalized decays have been plotted. Thedecay of Fig. 1 corresponding to a mass of 0.47 g, with

32 T. Venancio et al. / Journal of Magnetic Resonance 181 (2006) 29–34

relatively poor signal to noise ratio, has been removed fromFig. 2 to improve clarity.

According to Eq. (4), and also from the more generalEq. (1), the time dependence of the decays of Fig. 2 shouldbe governed by the variable t/a2. Hence, if one neglectsvariations in the mass density q, and the data of Fig. 2are plotted as a function of the variable t/a2, one shouldobtain a universal decay curve. Equivalently, all decaysshould also collapse into a universal decay if one plots asa function of t/m2/3, where m denotes the mass of a sphereand m2/3 = a2 (4pq)2/3.

Fig. 3 shows a universal decay obtained by plotting eachof the four decays of Fig. 2 as a function of 1/m2/3 whichconfirms the prediction of the diffusion equation. For thedecay corresponding to m = 0.47 g, not shown in Fig. 3,the poorer signal to noise ratio only permits to determinethe value of the mass exponent as 0.66 ± 0.05.

Given that the decays of Fig. 2 scale with time accordingto the prediction of the diffusion equation one could nowattempt to employ Eq. (5) to fit the universal curve ofFig. 3 by first assuming a constant diffusivity. To thatend one needs to first determine the temperature depen-dence C (H) of the relaxation ratio. Since the motions rele-vant for T1 and T2 are thermally activated one can expect,in the region -0s > 1, a functional dependence of formC (H) = A + Bexp (EC/kBH), where the exponent EC con-tains contributions from the motions relevant for T1 atthe given frequency as well as of T2. Hence the temperaturedependence of C (H) is expected to be, in general, steeperthan that of T1 or T2.

The parameters A, B, and EC could be determined frommeasurements of the ratio |Ms+|/M0 of the CWFP signal inthe steady state to the thermal equilibrium signal ampli-tude. At a temperature of h0 = 295 K values of

Fig. 3. Universal CWFP signal decay for rubber spheres with fourdifferent masses obtained by plotting as a function the scaled time variablet/m2/3.

T1 = 13 ms and T2 = 5.5 ms were independently measuredby inversion recovery [19] and Carr–Purcell–Meiboom–Gill [20] sequences respectively, yielding a ratioC (h0) = 2.36 for -0/2p = 7.24 MHz.

Fig. 4 shows measurements of the relaxation ratio C as afunction of 1/H obtained from CWFP measurements inrubber at various temperatures under thermal equilibriumconditions. For temperatures below the T1 minimum atH � 298 K the ratio C (H) rises quite steeply and theCWFP signal amplitude |Ms+|/M0 = 1/(1 + C (H)) reachesnoise level at H � 268 K. Hence, only spins in a relativelynarrow temperature range below h0 are effectively involvedin the decays of Fig. 1. From a best fit to the data of Fig. 4one obtains A = 1.846, EC/kB = 5600 K, andB = 0.607 · 10�8. It is worth pointing out that for -0/2p = 5 MHz the spin–lattice relaxation time in polyiso-prene [21] exhibits, in the region -0s > 1, an exponentialgrowth parameter E1/kB � 3775 K, considerably smallerthan EC/kB.

It is now possible to employ Eqs. (4) and (5) to calculatethe universal decay curve and obtain an average value �k ofthe thermal diffusivity in the interval h0 to hb from anadjustment to the data of Fig. 3. Fig. 5 shows the theoret-ical decay corresponding to �k ¼ 1:684� 10�3 cm2=sobtained from Eqs. (4) and (5) including the small Curielaw correction.

Given the simple theory employed and the approxima-tions involved, the overall agreement is quite rewarding.Deviations from spherical shape in our samples as well asdensity variations with temperature may explain somesmall but systematic departures in Fig. 5. It is instructive

Fig. 4. Relaxation ratio C (H) as a function of inverse temperature 1/Hobtained from CWFP steady-state signal amplitudes in thermal equilib-rium. The solid line is an empirical fit of the data in the region of interestH < h0.

Fig. 5. Universal CWFP decay for rubber spheres. The circles arecalculated values assuming an average thermal diffusivity�k ¼ 1:684� 10�3 cm2=s and a constant mass density q = 1.1 g/cm3.

T. Venancio et al. / Journal of Magnetic Resonance 181 (2006) 29–34 33

to compare the value of �k obtained from Fig. 5 with valuesfound in the literature for related materials. For vulcanizedrubber, for example, the heat conductivity [22] j and thespecific heat at constant pressure[23] cp have been separate-ly measured in the temperature range 100 K < H < 290 Kyielding j (290 K) = 1.36 · 10�3 W/cm K and cp

(290 K) � 1.8 J/g K. Assuming q � 1.1 g/cm3 one obtainsfrom Eq. (2) k (290 K) = 0.69 · 10�3 cm2/s. This valueappears to remain approximately constant from 290 K toapproximately 200 K and drop rather abruptly below200 K.

Although the decay rate of the theoretical curve of Fig. 5is quite sensitive to the value of the average thermal diffu-sivity �k, its shape alone is not very useful to determine thetemperature dependence of the diffusivity. To prove thispoint, we solved Eq. (1) numerically by the implicit finitedifference method of Crank–Nicholson [17,18] assuming,for example, a linear variation with temperature of k (H).The calculations indicated that, after the integration overthe sphere implied by Eq. (5), it is possible, for realisticparameters, to find an average constant value �k that yieldsessentially the same curve as the linearly temperaturedependent k (H). Other monotonic temperature dependenc-es yielded similar conclusions. Thus, if one wishes toextract some information concerning the temperaturedependence of k (H) from Fig. 5 it would be necessary todetermine thermal diffusivity values at some narrow tem-perature intervals.

4. Conclusions

We have shown that the CWFP technique is capable ofyielding quantitative and easily obtainable informationconcerning the kinetics of processes that change the relax-

ation rate of the nuclear spins through the action of someexternal agent. In this particular application heat flowcaused by a large temperature gradient led to local changesin the relaxation rates that could be monitored by thedecay of the CWFP signal. From the decays it was possibleto ascertain the prevalence of a diffusive process and toobtain an average value for the thermal diffusivity.

Acknowledgments

We thank FAPESP (02/05409-0 and 04/11904-9),CNPq, and FINEP/RBT for support.

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