J. Phys. Chem. 1981, 85,3031-3040
3031
FEATURE ARTICLE Study of Dynamical Processes in Liquids by Electron Spin Echo Spectroscopy Arthur
E. Sllllman
John Stuart Research Laboratories, Quaker Oats Company, Barrington, Illinois 600 10
and Robert N. Schwartz” Department of Chemistry, Universi& of California, Los Angeles, California 90024 (Received: May 4, 1961; In Final Form: June 16, 198 1)
The theoretical and experimental development of electron spin echo spectroscopy for the study of dynamical processes in liquids is described. This technique makes it possible to measure directly the transverse and respectively, of free radicals which are used as probes of rotational longitudinal relaxation times, T2and TI, and translational motion in liquids. The applicationsof the spin echo method to dilute and concentrated isotropic solutions of free radicals and to paramagnetic spin probes incorporated in model membrane systems are reviewed.
Introduction Electron spin relaxation measurements of paramagnetic molecules in normal and ordered liquids such as model and biological membranes remain a powerful tool for probing a wide variety of dynamic pro~esses.~,~ Almost all of the measurements to date have been carried out in the frequency domain by using continuous wave (CW) electron paramagnetic resonance (EPR) techniques. From a careful line shape analysis of the CW EPR spectrum, dynamical information about the paramagnetic species may be obtained. In motionally narrowed systems, the spectral widths are related to the dynamical information, whereas, in slow motional systems, the entire spectral line shape must be analyzed in order to study the molecular dynamics. The analysis of EPR line shapes, however, is often complicated because frequently they consist of a superposition of spin packets (Le., inhomogeneous broadening) due to static local magnetic field difference^.^-' This complicates the problem in that spin relaxation theories are valid only for the individual spin packets and not for the whole inhomogeneously broadened spectral line.8t9 Therefore, one must in some way deconvolute the inhomogeneously broadened line in order to extract the transverse electron spin relaxation time T,,which in turn is related to the various dynamical processes affecting the spins.“’ This requires knowledge of all magnetic interactions responsible for the inhomogeneous broadening; however, this information is available for only a relatively small number of system^."^ One method for obtaining T2from an inhomogeneously broadened spectral line is to utilize spin echo techniques (1) On leave from the Department of Chemistry, University of Illinois at Chicago Circle, Chicago, IL 60680. (2) Muus, L. T.; Atkins, P. W., Ed. “Electron Spin Relaxation in Liquids”; Plenum Press: New York, 1972. (3) Berliner, L. J., Ed. “Spin Labeling Theory and Applications”; Academic Press: New York, 1976. (4) Mims, W. B. In “Electron Paramagnetic Resonance”;Geschwind, S., Ed.; Plenum Press: New York, 1972. (5) Poggi, G.; Johnson, C. S., Jr. J.Mag. Reson. 1970, 3, 436. (6) Stillman, A. E.; Schwartz, R. N. J. Mag. Reson. 1976, 22, 269. (7) Bales, B. L. J. Mag. Reson. 1980, 38,193. (8) Kivelson, D. J. Chem. Phys. 1960, 33, 1094. (9) Freed, J. H.; Fraenkel, G . K. J. Chem. Phys. 1963, 39, 326. 0022-3654/81/2085-3031$01.25/0
similar to the ones which have proven invaluable in nuclear magnetic resonance (NMR) spectro~copy.4J*’~Spin echo and other time-domain techniques are also well suited for the measurement of additional dynamic properties of the electron spin system such as the spin lattice relaxation time Tla4 The major difficulty encountered in time-domain EPR experiments has been the technology for producing short high-power microwave pulses and the availability of sensitive low-noise microwave amplifiers and fast data acquisition equipment. However, recent advances in commercial microwave and digital instrumentation have made it possible to perform time-domain EPR experiments with relative ease. Considerable progress over the last 10 years has been made in the use of time-domain EPR techniques such as electron spin echoes (ESE) to obtain structural and dynamical information about organic free radicals and paramagnetic transition metal ions in ~olids.~J’-’~ However, very little attention has been paid to the use of time-domain techniques to study free radicals in liquid solution. The lack of impetus to develop pulsed EPR, in general, probably stems from the fact that EPR is not as versatile and wide ranging in application as NMR spectroscopy. Nevertheless, it is possible to obtain structural and dynamical information through the use of EPR spectroscopy which is difficult or impossible to obtain by other methods. The first reported observation of electron spin echoes was from solvated electrons formed in solutions of sodium metal in liquid ammonia.14 Since this first observation of ESEs in 1958 the authors are aware only of 11papers which report experimental studies of liquid-phase electron T1and T2 relaxation times by the ESE technique.’&24 (10) Farrar, T. C.; Becker, E. D. ‘‘Pulse and Fourier Transform NMR”, Academic Press: New York, 1971. (11) Kevan, L. J. Phys. Chem. 1981,85, 1628. (12) Salikov, K. M.; Semenov, A. G.; Tsvetkov, Yu. D. “Electron Spin Echoes and Their Applications”; Nauka: Novosibirsk, 1976. (13) Kevan, L.; Schwartz, R. N., Ed. “Time Domain Electron Spin Resonance”; Wiley-Interscience: New York, 1979. (14) Blume, R. J. Phys. Reu. 1958, 109, 1867. (15) Briindle, R.; Kruger, G. J.; Miiller-Warmuth, W. Z. Naturforsch. A 1970, 24, 1.
@ 1981 Amerlcan Chemical Society
3032 The Journal of Physical Chemistty, Vol. 85, No. 21, 1981
Although some of the earlier ESE spectrometers had response times of approximately 30 ns, these instruments were used to study radical solutions approaching the exchange narrowed limit and, consequently, were somewhat limited in their applicability. However, with the newer generation of pulsed EPR spectrometers,26the sensitivity has been greatly improved without sacrificing the time response of the instrument so that ESE studies on dilute solutions of free radicals can now be made quite routinely. In this review we will only be concerned with the use of the ESE technique to measure directly the electron Tl and T2of free radicals which are used as probes of the rotational and translational dynamics of liquids. A thorough and important review of other applications of the ESE method for time-resolved spectroscopy has recently been presented by Norris, Thurnauer, and Bowman.26 Other literature sources which also provide a comprehensive description of the ESE technique are the review by Mims4 and Kevan” and the books by Salikov et del2 and Kevan and Schwartz.13
Physical Description of ESE Experiments An ESE is a spontaneous emission from a superradiant state which is created by the phase coherence of the microwave source.26 An applied microwave field produces coherence of the spin levels and, therefore, the local magnetic moments become phase correlated. Without the driving field of the microwave pulse, the coherence decays and the local magnetic moments gradually again become uncorrelated.n Since the bulk magnetization is given by the vectorial s u m of the local magnetic moments, the bulk magnetization in the X-Y plane decreases with the loss of phase coherence. This physical picture has been formulated mathematically by Kivelson who demonstrated through linear response theory that the time dependence of the bulk magnetization M(t) is that of the spin autocorrelation function (S(t)S(O)),where S = c,ySj, Sj is the spin angular momentum of the jth molecule in a system containing N molecules, and the pointed brackets indicate an average over an equilibrium ensemble.2s The basic idea behind the ESE experiment is to prepare the spin system in such a way that the only loss of phase coherence is due to dynamic rather than static phenomena. This would correspond in a CW motionally narrowed EPR experiment to measuring the homogeneous or spin-packet line width without the inhomogeneous contribution.m It is the spin-packet line width which contains the dynamical information about the spin ~ y s t e m . ~ ? ~ An ESE experiment can be described quantum me~hanically,2~~*~~ by absorbed thermodynamical(16) Milov, A. D.; Salikov, K. M.; Tsvetkov, Yu. D. Chem. Phys. Lett. 1971.8. 523. --(17) Brown, I. M. Chem. Phys. Lett. 1972,17, 404. (18) Brown, I. M. J. Chem. Phys. 1974,60,4930. (19) Milov, A. D.; Mel’nik, A. D.; Tsvetkov, Yu. D. Theor. Elcpt. Chem. 1975,11, 790. (20) Kaksal, F.; Kriiger, G. J. Z . Naturforsch. A 1975,30, 883. (21) Schwartz, R. N.; Jones, L. L.; Bowman, M. K. J. Phys. Chem. 1979,83, 3429. (22) Madden, K.; Kevan, L.; Morse, P. D.; Schwartz, R. N. J. Chem. Phys. 1980,84, 2691. (23) Stillman, A. E.; Schwartz, L. F.; Freed, J. H. J. Chem. Phys. 1980, 73, 3502. (24) Madden, K.; Kevan, L.; Morse, P. D.; Schwartz, R. N. J . Am. Chem. SOC.In press. (25) Norris, J. F.; Thurnauer, M. D.; Bowman, M. K. Adu. Biol. Med. Phys. 1980,17, 365. (26) Dicke, R. H. Phys. Reu. 1954,93, 99. (27) Hahn,E. L. Phys. Reu. 1950,80, 580. (28) Kivelson, D. In “Electron Spin Relaration in Liquids”; Muus, L. T.; Atkins, P. R., Ed.; Plenum Press: New York, 1972; Chapter 10. (29) Stoneham, A. M. Rev. Mod. Phys. 1969,41,82. (30) Stillman, A. E.; Schwartz, R. N. Mol. Phys. 1976,32, 1045. I
- I
Stillman and Schwartz
l ~or through , ~ a semiclassical vector dipole m ~ d e l . ~ J ~ f l $ ~ For simplicity we will consider the latter description for a 7r/2-7r ESE sequence. Initially, the equilibrium bulk magnetization is directed along the 2 axis defined by the dc magnetic field. A 7r/2 pulse directed along the X axis in the rotating frame applies a torque on the magnetic moment, tilting the magnetization so that it lies along the rotating axis. After the microwave pulse is turned off, there is no coherent radiation field to maintain the phase coherence among the local magnetic moments. Each local magnetic moment then precesses at a different rate about the 2 axis because of varying local magnetic fields. As a result, the net projection of the bulk magnetization along the Y axis decreases. The spin resonance signal intensity is proportional to the Y magnetization, so there is a corresponding decrease of signal intensity. This phenomenon is called the free induction decay (FID). The local m a g netic fields vary for two reasons: (1)the magnetic field is inhomogeneous (often largely due to unresolved hyperfine interactions) and (2) there is stochastic modulation of the local magnetic fields due to dynamical processes. For example, in liquids the stochastic modulation is associated with the rotational and translational motion of the molecule bearing the unpaired spin. The inhomogeneous magnetic field gives rise to a static distribution of precessional rates for the local magnetic moments, whereas the stochastic local magnetic field modulation produces a time-dependent distribution of precessional rates. The phase coherence can be reestablished by the application of a 7r microwave pulse. This rotates the magnetization by 180”about the X axis. Those local magnetic moments which precessed for a static rate must then rephase at the same rate at which they dephased. That is, if the spins were allowed to precess at a time 7 before the application of the 7r pulse, then 7 seconds following this pulse they must rephase, but only if the local magnetic moments had precessed at a constant rate during the time 27. The rephasing at the time 27 gives rise to an increase of magnetization along the Y axis and, hence, also signal intensity. This phenomenon is called a spin echo. There are several points to be made here. The stochastic changes of the precession rates are not the only reason that a local magnetic moment will not contribute to the ESE. Dynamical local magnetic fields also induce spin transitions which will redirect the local magnetic moments to the 2 axis. This is called longitudinal or spin-lattice relaxation and is characterized by a rate Ti’. TI processes, as well as imperfect R pulses, give rise to the FID following the second pulse. It should also be noted that, by virtue of the stochastic time-dependent nature of the field fluctuations, the ESE intensity is a decreasing function of the time 27. The phase memory time, TM,may be defined as the time required for the ESE intensity to be reduced to e-l of its initial value.4 If the ESE intensity is an exponential function of the time 27, then TM = T2,the transverse relaxation time.4 It should be clear then that the phase memory time contains only dynamicd information and is not affected by the sources of inhomogeneous broadening which complicate the analysis of CW EPR line shapes. There are two commonly used pulse sequences for determining T1.4124In the stimulated ESE (7r/2-1/2-~/2) (31) Stillman, A. E.; Schwartz, R. N. J . Chem. Phys. 1978,69,3632. (32) Stillman, A. E.; Schwartz, R. N. In “Time Domain Electron Spin
Resonance”; Kevan, L.; Schwartz, R. N., Ed.; Wiley-Interscience: New York, 1979; Chapter 5. (33) Gordon, R. A. Am. J. Phys. 1977,45,563. (34) Mayer, J. E.; Mayer, M. G. “Statistical Mechanics”; Wiley-Interscience:. New York, 1977; p 133.
Feature Article
pulse sequence the local magnetic moments are allowed to precess for a time 7 following the initial a12 microwave pulse. Another a12 pulse is then applied which nutates the magnetization to the -2 direction. Longitudinal relaxation processes will affect the magnetization, but T2 processes cannot since there is no transverse magnetization. A time T i s allowed to pass before the application of the final 7r/2 pulse. Since no dephasing could occur during the time T, an ESE is observed 7 seconds following the 2 Tl is determined by observing the ESE final ~ / pulse. intensity as a function of T. In the inversion-recovery experiment a 7r microwave pulse is applied which nutates the magnetization to the -2 direction. Longitudinal relaxation acts on the magnetization for a time 7. This magnetization is then detected by utilizing a 7r12-a pulse sequence; observation of the ESE intensity as a function of 7 provides Tl. One frequently finds that Tlas determined by a stimulated ESE sequence is shorter than that measured by the method of inversion recovery.% This observation may be readily explained by noting that longitudinal relaxation pathways involve electron spin transitions, nuclear spin transitions, and mixed electron-nuclear transition^.^^ However, nuclear (and mixed) spin transitions will not be manifest in the inversion-recovery experiment. The reason is as follows: A nuclear transition during the time T of a stimulated ESE sequence alters the Larmor frequency of the corresponding spin packet and, hence, the affected local magnetic moment will not have the correct phase relation to be refocused by the final microwave pulse. These nuclear transitions will therefore contribute to TI. They cannot affect the inversion-recovery experiment, however, since the ESE does not depend upon the relative phase relations of the local magnetic moments prior to the application of the detecting 7r/2-7r pulse sequence. Thus, Tl as determined by the inversion-recoverytechnique must therefore be (2 WJ1, where We is the pure electron spin flip rate. The T1determined by the stimulated ESE sequence has additional contributions. Clearly, care must therefore be taken to specify the method by which T;s are obtained.
Instrumentation Presently no commercially available ESE spectrometer exists, although both Bruker and Varian have expressed an interest in marketing one in the near future. An ESE spectrometer may be constructed from available microwave components and relatively simple logic driving circ u i t ~ . ~ We ~ *are~ most ~ ~ familar ~ ~ * with ~ ~ the ESE spectrometer at Cornel1 which one of us (A.E.S.) assembled. It is largely based on the design developed at Argonne National Laboratory. A detailed description of this spectrometer has been presented by Norris et al.= We will therefore briefly outline the salient and novel features of the Cornell ESE spectrometer. A block diagram of the microwave circuit is displayed in Figure la. Low microwave power (-200 mW) is supplied by either a Gunn diode or a klystron source. A small fraction of this power is tapped for a phase reference to a double-balanced mixer, but most of the microwave power passes through a pair of single-pole double-throw waveguide switches which permit bypass of the pulsed portion of the microwave circuit for conventional CW use. The (35) Bowman, M. K., private communication. (36) Freed, J. H.In “Electron Spin Relaxation in Liquids”; Muss, L. T.; Atkins, P. W., Ed.; Plenum Press: New York, 1972; Chapter 18. (37) Brown, I. M.; Sloop, D. J. Rev. Sci. Instrum. 1970, 41, 1774. (38) (a) Blumberg, W. E.; Mims, W. B.; Zuckerman, D. Rev. Sci. Instrum. 1973,44,546. (b) Davis, J. L.; Mims, W. B. Ibid. 1981,52,131.
The Journal of Physlcal Chemistry, Vol. 85, No. 2 1, 198 1 3033
high-power microwave pulses are obtained by modulating the input of a pulsed traveling wave tube (TWT) amplifier while the TWT has been switched on with a logic trigger pulse (cf. the pulses displayed on the lower right hand corner of Figure la). The amplified microwave pulses pass through a four-port circulator to a standard TEllo reflection microwave cavity with an enlarged iris opening. This modification of the iris permits overcoupling in order to reduce the cavity Q and, hence, cavity ringing from the high-power pulses. A small fraction of the reflected microwave power is detected by a crystal for tuneup purposes, whereas most of the reflected power passes through a diode limiter and GaAs FET amplifier. This serves as a low-noise microwave preamplifier for the ESE signal. The G d s FET amplifier is protected by the microwave limiter from high-power transients. The high-power pulses also can saturate the video amplifiers further in the circuit; however, this is circumvented by the use of a PIN diode receiver protect switch. The ESE signal is phase detected at the double-balanced mixer and then further amplified by a series of dc fast amplifiers. A block diagram of the pulse triggering and data acquisition system is shown in Figure lb. The heart of the system is a pair of digital delay generators. These operate in series by each generating a pair of TTL pulses of selectable relative delay whenever they are triggered. The TTL pulses determine the relative times that the microwave pulses are generated. The pulse programmer selects the delay times which can automatically be incremented through communication with a microprocessor. The pulse sequencer permits one to select a one-, two-, or three-pulse sequence with adjustable pulse widths and provides trigger pulses for the PIN diode switches and the TWT amplifier. A fast oscilloscope is necessary for displaying ESE signals (when strong enough) and for purposes of tuneup of the spectrometer. The ESE signals are time averaged in a dual-channel boxcar integrator; one channel averages the ESE while the other integrates a time window of the baseline. These are subtracted from each other in order to correct for baseline drift. The microprocessor also serves as a multichannel analyzer for the boxcar integration and permits automation of the ESE experiment. It is interfaced to a larger time-shared computer (PRIME) for data storage and analysis. The echo envelope decay curve is displayed on a graphics terminal, although a hard copy may also be obtained from the PRIME computer. Experiments which require a laser or electron beam may conveniently be performed with this system by incorporating an additional digital delay generator. This permits varying of the time delay between the laser or electron pulse and the microwave pulses for kinetic studies, An ESE spectrometer has a dead time during which the echo signals either cannot be measured or else are interfered with. The dead time is especially troublesome for ESE signals with short phase memory times as much of the signal may have decayed before observation. One form of interference commonly found in ESE signals in liquids results from the FID’s following the microwave pulses. This problem may be eliminated by decreasing the magnetic field homogeneity so that the FID decays more rapidly. The ESE signal may then be reduced, however, since fewer spins are on resonance. This difficulty is better dealt with by phase modulating the microwave pulses. A digital 0-180° microwave phase shifter can be placed before the PIN diode switch which modulates the input power of the TWT amplifier (cf. Figure la). By performing, for example, a two-pulse ESE experiment with no phase modulation and subtracting the results from another
3034
The Journal of phvsical Chemistry, Vol. 85, No. 2 1, 198 1
Stlliman and Schwartz
a FIELDIFREQUENCY
ISOLATOR
I TERY(NATK)N
CHASE
'
OR POWER METER
SPDT LIMITER
SWITCH
+j
ATTENUATOR
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I
I BALANCED
T W T INPUT
P I N DIODE
SWITCH
1-L
TWT T R l e Q E R RECEIVER PROTECT
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SIGNAL F R O M PRE-AMP
qmm&KyJ--7
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PROCESSOR
Flgure 1. The Cornell University ESE spectrometer (see text for details): (a) block diagram of the microwave bridge; (b) block diagram of the pulse triggering and data acquisition system.
ESE experiment where each a12 microwave pulse is phase shifted by 180°,the FID will be removed, leaving only the ESE signal. This method will also remove any baseline irregularities from switching transients. Another source of dead time results from the finite time required for the pulsed TWT amplifier to shut off following a trigger pulse. During this time white noise power is emitted from the TWT amplifier which may be many orders of magnitude greater than the ESE signal. This problem is particularly evident with the Litton Model 624 pulsed TWT microwave amplifiers used at Argonne and Cornell, although some obsolete TWT amplifiers do not have this The dead time resulting from TWT noise is typically -200 ns but may be reduced by using a transmission- rather than a reflection-type microwave cavity since the noise will then be filtered. A better design would make use of a bimodal induction-type cavity since the pump and observation modes are isolated from one another. We expect to significantly improve upon the minimum measurable phase memory time of about 50 ns
for the Cornell spectrometer by use of an induction cavity. The ultimate limitation of dead time should result from cavity ringing following the microwave pulses. The ring down time decreases with decreasing cavity Q.4J1 This is why one typically overcouples the cavity to reduce the Q to about 100. A lower Q reduces the magnitude of themicrowave B1field with a concommitant loss of ESE signal intensity so that some compromise must be made. Calculations of the optimal cavity Q for an ESE spectrometer have been presented by Salikov et a1.12 We expect that a bimodal cavity design should be preferred in this regard, since one could improve upon the sensitivity by increasing the relative Q of the observation mode. Moreover, a bimodal cavity may provide convenient CW use of acceptable sensitivity, not presently possible because of the low coupling Q. Further reduction of the cavity ringing and dead time is possible through the use of a microwave delay line. In fact Davis and Mims have recently reported a 113 reduction of their spectrometer dead time by using this technique.38b
The Journal of Physical Chemistty, Vol. 85, No. 21, 1981 3035
Feature Article
With the growing interest in ESE spectroscopy we expect that improvements in spectrometer design will permit the measurement of phase memory times in magnetically dilute solutions as short as 10 ns. This would permit, for example, studies of nitroxide free radicals over the entire motional range from the motional narrowing to the rigid limit and, hence, provide a powerful experimental tool for molecular dynamics studies.
Theory of Electron Spin Echoes in Liquids In this section we present a brief outline of the theory of ESE phenomena in liquids based on the recent work of Stillman and S c h ~ a r t z These . ~ ~ ~authors ~ have used both density matrix-relaxation operatorm and generalized cumulant expansion31formalisms to describe liquid phase ESE’s. The two approaches are intimately related and offer their own particular advantages; the former approach proves to be more useful for computational purposes, whereas the cumulant theory provides considerable insight into the nature of time-domain relaxation behavior and therefore will be discussed below. Consider a two-pulse experiment in which the equilibrium spin system is perturbed by a 7r/2 pulse about the Y axis at t = 0 and subsequently, at t = 7,by a 7r pulse of microwave radiation about the X axis. The effect of the 7r pulse is to negate the phase accumulation that each of the spins has acquired during the time interval 7. Therefore, for times t > 7 the phase 4(t)acquired is given by30
essary to augment eq 5 as follows:
S , ( t ) = iS(t)%(t)”S,
(7)
To facilitate the solution of this equation it is convenient to transform to the interaction picture:31 = exp[-i7fo
J t S(t? dt’]v(t) exp[i7fo itS(t? dt’]
(8)
where it is noted that at t = 27, d(27) = 427). In the interaction picture eq 7 transforms to 4,t(t) =
i7fl*(t)”Sx* This equation can be integrated formally to give
S>(t) = expo[i
f dt’S(t?7fl*(t?”]#>(O)
(9)
(10)
so that for the ensemble average:
S , t ( t ) = (expo[;
xt
dt’S(t?%*(t?’])S>(O)
(11)
In eq 10 and 11 the subscript 0 denotes a specific time ~ r d e r i n g ~for l.~ any ~ expansion of the exponential. This is necessary since in general ?f?(t)”will not commute with itself for arbitrary time. By using a cumulant expansion method based on partial time ordering the quantity in angular brackets in eq 11can be evaluated
where where S ( t ) = -1
0 T ~ therefore ; the shape of the decay signal cannot be expected to be very sensitive to the nature of the molecular dynamics. Yet, the phase memory times (TM, related to the inverse of the elements of R(2n)’)may have a model dependence in the slow motional region. The rate of convergence of the cumulant expansion depends on the relative magnitude of 17fl(t)7,( so that higher order terms (n > 2) are important for 17f1(t)7J22 These higher order cumulants may have different magnitudes as well as different signs depending on the choice of the stochastic model used to describe motion, thus leading to the model dependence of T M . ~ ~ ~ ~ ~ l.41142
Rotational Motion Rotational dynamics of spin S = 1/2 free radicals may be studied indirectly through its contributions to electron spin r e l a x a t i ~ n .This ~ ~ ~arises ~ ~ ~from ~ ~ motional modulation of the magnetic tensors which appear in the spin Hamiltonian N
h%1
= P,Bo.g‘.S
+ h J*CS (20) + h k = l S*A”k’*I(k)
where h is Planck’s constant divided by 2a, P, is the Bohr magneton, Bois the applied magnetic field strength,‘g is the traceless part of the g tensor, S is the electron spin angular momentum operator, A ’@) is the electron-nuclear (46) Fraenkel, G. K. J. Chem. Phys. 1965,42,4275. (47) Freed, J. H. In “Spin Labeling Theory and Applications”; Berliner, L. J., Ed.; Academic Press: New York, 1976. (48) Atkins, P. W. Adu. Mol. Relaxation Processes 1972, 2, 121.
dipolar hyperfine tensor for the kth nucleus, Ifk)is the nuclear spin angular momentum operator for the kth nucleus, J is the rotational angular momentum operator, and C is the spin-rotational interaction tensor. The first attempt to use ESE spectroscopy for the study of rotational diffusion was made by Brown.17J8 His analysis made use of only the second term in eq 20. This is inadequate as we have shown in our more general t h e ~ r y which ~ ~ J ~has recently been experimentally The neglect of the electronic Zeeman (first) term results in the incorrect dependence of the electron spin relaxation times TI and T2on the nuclear spin quantum state. The last term in eq 20 is due to the spin-rotation interactionBswand gives rise to contributions to the electron spin relaxation times that are independent of the nuclear spin quantum state. This type of contribution can be eliminated from consideration by studying appropriate linear combinations of the relaxation rates? As discussed below, however, the spin-rotational contribution to the relaxation times does contain important dynamical information not found in the contributions from the other terms of the spin Hamiltonian. The rotational diffusion of small molecules in liquids is a complex phenomenon. Simple Brownian diffusion is expected to be obeyed for relatively large and heavy particles so that the surrounding bath appears as a continuous media.s1 This clearly need not be true on a molecular scalee2and indeed a number of experiments have confirmed non-Brownian behavior for small molecules in liq u i d ~ .Other ~ ~ diffusion models of various levels of sophistication have been proposed to deal with different sources of deviation from Brownian b e h a v i ~ r . ~ ~ - ~ While EPR spectroscopy is routinely used to determine single particle rotational correlation times, it remains one of the few experiments by which molecular dynamics may be studied. The motionally narrowed region is defined by )%1TRl2 1, i.e., WHE >> We,eq 29 simplifies to
which indicates that the spin-lattice relaxation rate is constant and independent of the radical concentration. It is important to note that in this limit the absolute value of TlaI(HE)is determined by T1*JO), which, in turn, is determined by other relaxation mechanisms such as spin-rotationapm@ and rotational modulation of the radical’s g‘ and A’ magnetic interaction tensor^.^^^ In addition to the exchange interaction described above it is important also to include the effects of the intermolecular electron spin-electron spin magnetic dipolar interaction (dipole-dipole) in the analysis of the concentration-dependent contribution to the relaxation times Tl (70) Bales, B. L.; Swenson, J. A.; Schwartz, R. N. Mol. Cryst. Liq. Cryst. 1974,28, 143. (71) Kriiger, G . J. Adu. Mol. Relaxation fiocesses 1972,3, 235. (72) Johnson, C. S., Jr. Mol. Phys. 1967;12, 25.
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The Journal of Physical Chemlstry, Vol. 85, No. 21, 1981 3039
and TPu-'l In moderately concentrated radical solutions, and especially at high viscosities, dipole-dipole interactions contribute significantlyto the relaxation times and, hence, make it difficult to separate these two concentration-dependent mechanisms. This leads to a considerablesource of error in the determination of the translational diffusion constants of the radical from analysis of the CW EPR line widths.64@ In the region of high microwave frequencies o, the spectral densities and, hence, T1 and T2, depend to a considerable degree on the character of molecular displacements over small times of 10-lo-lO-ll For a nitroxide free radical (5' = 1/2, I = 1) in the motional narrowed region the intermolecular dipole-dipole contribution to the homogeneous line width is given by12165b@9'7193'
-
s.65bJ1373
where y is the electron spin gyromagnetic ratio, a is the hydrodynamic radius of the free radical, d is the distance of closest approach of the two interacting electron spins, and N, is the number of radicals per cm3. This expression is based on the following assumptions: (1)the dipoles are treated as arising from single points, (2) 0 7 >~ 1,where ?D is a time characteristic of translational relaxation, (3) the mean square jump distance ( r 2 )is less than the distance of closest approach, (4) the Stokes-Einstein relation, D = k ~ T / 6 n q ais, valid, and ( 5 ) that the interacting radicals may have either the same (like spin) or different (unlike spin) nuclear spin configuration. These approximations lead to a dipole-dipole contribution to the spinlattice relaxation time given by56b971J3
(i) D-D
(y 4 y 9 ( -$)(;) -
(33)
An imDortant feature to note here is that (T,-')Ln 0: nlT. . ., whereas (T1-l)WD a (T/q). The ESE method has the advantage over a CW line width measurement for studying modeiately concentrated solutions of free radicals that the random modulation of the dipole-dipole and Heisenberg exchange interaction by the diffusion of the particles is manifested differently in the phase memory decay of the two-pulse and three-pulse (stimulated) E S E ' S . ~ ~ JIn ~ Jthe ~ slow-exchange region, where the exchange frequency w m is smaller than the maximum splitting in the CW EPR spectrum of the radical, an exponential decay is observed for two pulse and stimulated ESE's with the exchange mechanism making an identical contribution to Tl and T2 For a single inhomogeneously broadened line the two-pulse ESE signal intensity at 27 is given by12J9v74 E(27) = Eo exp(-2~(u)
, - -
,
(34)
where = TC'(0)
+ WHE + (T2-l)D-D
(35) and T2(0) is the phase memory time in the absence of exchange and dipole-dipole interactions. For the stimulated ESE the signal intensity at 27 + Tis described by12J9 E(27 + T ) = E(27) exp(-/3T) (36) where (73) Freed, J. H. J. Chem. Phys. 1978, 68,4034. (74)The more general case of multiple inhomogeneouslybroadened lines has been treated by Salikov et al. in ref 12, Chapter 4.
(37) In eq 37 T ~ l ( 0is) the spin-lattice relaxation rate in the absence of Heisenberg exchange and dipolar interactions. An important feature to notice here is that the contribution of the dipole-dipole interaction to the decay of the stimulated ESE may be neglected in comparison to the exchange contribution. For a given N, and T/q, (T
