Pulsed gradient spin-echo nuclear magnetic resonance of confined Brownian particles

Transcription

1 Pulsed gradient spin-echo nuclear magnetic resonance of confined Brownian particles E. M. Terentjev, P. T. Callaghan, a) and M. Warner Cavendish Laboratory, University of Cambridge, Cambridge CB3 HE, United Kingdom Received 2 October 1994; accepted 9 December 1994 We present here a model calculation of the spin-echo gradient nuclear magnetic resonance signal of a Brownian particle diffusing in a confined environment, described by a harmonic potential well. The general expression reduces to characteristic scaling laws in limiting cases: long and short pulses, comparative to the spin-echo duration, and long, intermediate, and short durations of the spin phase reversal cycle, comparative to the internal system characteristic times scales. Spin-echo measurements in these limits enable one to detect independently the particle s mass, viscous drag, and the confining potential strength American Institute of Physics. I. INTRODUCTION a Department of Physics, Massey University, Palmerston North, New Zealand. The determination of molecular or Brownian particle self-migration with the help of nuclear magnetic resonance NMR has several advantages over other traditional experimental techniques. In particular, the method is noninvasive, it does not require materials being studied to be optically transparent, it requires no tracer label other than the magnetism of the atomic nucleus, and it involves a well-defined ensemble average. In particular, the pulsed gradient spinecho PGSE NMR method provides a direct measurement of nuclear spin translational motion over a predetermined time interval. 1,2 This method relies on the use of magnetic field gradient pulses to impart a spatial signature to the Larmor precession frequencies of the nuclear spins which thereby label their host molecules. In its sensitivity to selfmotion rather than to relative displacements the method is similar to incoherent inelastic neutron scattering. Indeed there exists a formal equivalence to neutron scattering in the mathematical description of PGSE NMR using wave vector terminology. This formalism enables the motion to be described in very simple terms using a well-defined propagator, provided that the magnetic field gradient pulses are sufficiently narrow. The propagator description is particularly helpful in cases where diffusion is restricted, and yields analytic solutions to the NMR signal amplitude which directly relate to the dynamical parameters in a number of simple geometries. By contrast, when the gradient pulse is sufficiently long that significant Brownian motion occurs during its application, the mathematical formalism is severely complicated. This is illustrated by the fact that no exact analytic solutions for the NMR signal amplitude are known which apply at all gradient pulse widths and all time scales except for the somewhat trivial case of unrestricted self-diffusion. This article deals with a special case of confined motion for which such an analytic solution is possible, namely the case of Brownian motion in an harmonic well. This problem is of interest because of its relevance to the diffusion of polymer segments in a crosslinked network. More generally the harmonic well provides a good model for hindered diffusion in the presence of a soft wall. An example of the latter would be the rapid and restricted diffusion transverse to the Doi Edwards tube in an entangled polymer system. 3 In both applications the distance scale is sufficiently short that in order to measure translational motions using currently available magnetic field gradients, it is necessary to employ pulses of extended width, comparable in duration with the spin-echo time. A narrow gradient pulse solution for the PGSE NMR echo amplitude due to spins undergoing Brownian motion in an harmonic well has been obtained by Stejskal 4 whose expression relates to times sufficiently long that particle inertia may be neglected. Another related problem, recently solved by Stepisnik, 5 concerns the PGSE NMR signal which results from particles subject to a linear Langevin equation incorporating a memory function. His solution, which is obtained using a spectral density analysis, applies for finite width gradient pulses. While the memory function mimics diffusion of a free particle in the limit of short memory time and the undamped harmonic oscillator in the long-time limit, it is essentially phenomenological. In the present article we will solve the damped harmonic oscillator problem exactly and at all time scales, retaining the natural dynamical parameters in an explicit manner. II. THE ECHO ATTENUATION FUNCTION Our description of the PGSE NMR experiment begins with the narrow pulse limit. Following radio-frequency excitation of the spin system, the application of a field gradient, B, in the form of a pulse of duration, results in a helical phase twist for spins distributed along the gradient direction. The spin-echo method relies on a subsequent 18 rf pulse to reverse all these phases before the application of a second gradient pulse, delayed by time T as shown in Fig. 1. In the absence of translational motion of spins along the gradient direction, this second pulse causes all isochromats of the spin system to have their phases returned to zero at time T. Generally, the presence of weak background gradients means that the echo will be formed perfectly only when the evolution times before and after the 18 rf pulse are equal. For the purpose of this article we shall ignore such effects, taking J. Downloaded 6 Sep 26 to Redistribution subject to AIP license or copyright, see Chem. Phys. 12 (11), 15 March /95/12(11)/4619/6/$ American Institute of Physics 4619

2 462 Terentjev, Callaghan, and Warner: Spin-echo NMR of confined Brownian particles G xd T T xd, 3 while the PGSE NMR signal amplitude is again given by the ensemble average shown in Eq. 1. III. SOLUTION FOR HARMONIC POTENTIALS FIG. 1. Timing sequence for the standard pulsed gradient spin-echo NMR experiment, showing the gradient pulses of duration and amplitude G. The effect of any transverse relaxation additional to that arising from the pulsed gradient may be treated separately and is therefore neglected. Hence the echo amplitude is determined by the spin phase distribution at the time T following the start of the first gradient pulse. the origin of phase evolution time at the start of the first gradient pulse and the time of the spin-echo formation at the end of the second gradient pulse. The effect of motion is to introduce residual phase shifts to the spin ensemble. For a single spin, motion by an amount X will result in a local phase shift G X, where G is the amplitude of the gradient pulse and is the nuclear gyromagnetic ratio. The effective scattering wave vector is therefore given by qg. The signal measured in the spin echo is the ensemble average of the transverse magnetization at the echo time TT. The ratio of this magnetization, (q,t), to that which obtains under zero gradient conditions therefore results in a normalized echo amplitude which directly reflects the ensemble-averaged phase distribution as Eq,T q,t,t ei. The narrow gradient pulse condition leads to a simple scattering expression for the echo attenuation in terms of the Fourier transform of the averaged propagator P s (X,T), namely Eq,T P s X,TexpiqXdX, where P s (X,t) describes the probability of a particle moving a distance Xx(t)x over a time t, x being the starting displacement of the spin. By contrast, when the gradient pulse is finite, the concept of a scattering wave vector is ill-defined and the phase history of individual spins must be integrated over the echo duration. Thus the phase of a spin at time t will be given by (t) t B()x()d, where x is the instantaneous position of the labeled particle and B is the effective field gradient. The phase reversal associated with the 18 rf pulse can therefore be represented by a change in sign of the effective gradient. In the case of arbitrary the phase of the labeled spin precession in the pulsed gradient spin-echo experiment shown in Fig. 1 is given by 1 2 For Brownian motion in a potential well, the instantaneous position of the labeled particle is given by the stochastic variable x(t), which satisfies the Langevin equation mẍẋ U x t, 4 where m is the mass, the viscous drag coefficient, (t) the delta-correlated Brownian noise, and U(x) represents the confining potential for the particle. In the general case, when U(x) reflects a more realistic environment, Eq. 4 is nonlinear and it is impossible to perform the statistical averaging in Eq. 1 analytically. Numerical analysis can be helpful, but there are important advantages of having the NMR signal expressed through compact analytic formulas, dependent on parameters of the system. We, therefore, model U(x) bya harmonic well potential, U 2a(xx 1 ) 2, in which the strength of confinement is given by a single parameter a. In this case the statistical problem remains Gaussian and is amenable to easy solution, although somewhat tedious analytically. As for any Gaussian process the statistical average in Eq. 1 has only one independent moment: expigytexp G 2 y 2 t, where in our case the stochastic variable y(t) is defined by Eqs. 1 and 3, y(t) x()d T T x()d. The position of the particle is given by the linear equation ẍẋ 2 x(1/m)(t) with /m, 2 a/m, and with initial conditions x and ẋ governed by Boltzmann and Maxwell distributions, respectively: Wx exp ax 2 Wẋ 2k B T; exp mẋ 2 6 2k B T and where, by the fluctuation-dissipation theorem, the -correlated noise has t2k B Tt. 7 The method of solution of such a problem is straightforward and is due to Chandrasekhar. 6 The general solution for x(t) of the linear Langevin equation 4 is xt 1 2 x e 1 t e 2 t 1 2N x 2ẋ e 1 t e 2 t 1 t mn e 1 t e 2 t d, 8 where , , and N see Ref. 6. Asy(t) is a linear function of x(t), it can be represented through the same three 5 J. Chem. Phys., Vol. 12, No. 11, 15 March 1995 Downloaded 6 Sep 26 to Redistribution subject to AIP license or copyright, see

3 Terentjev, Callaghan, and Warner: Spin-echo NMR of confined Brownian particles 4621 stochastic variables, x, ẋ, and, which are governed by their Gaussian distributions 6. Taking Eq. 8 and repeatedly integrating by parts in the last term, we obtain with y tt x KT, N LT, ẋ 1 N mn T T T T d, K 2 1 1e 1 1e 1 T 1 2 1e 2 1e 2 T, L 1 1 1e 1 1e 1 T 1 2 1e 2 1e 2 T, q 1 1 1e 1 q 1 2 1e 2 q. From Eq. 5 it is clear that we require y 2 (t), which is easily taken by squaring Eq. 9 and using the average 7 of the squared random force, all cross terms vanishing. The initial conditions x 2 and ẋ 2 are averaged over their distributions 6, again cross terms x ẋ vanishing. Explicitly one obtains N 2 y 2 T, k BT a K 2 k BT m 9 L 2 2 k BT m 2 BT,, 1 where B TT 2 d TTT 2 d T T 2 Td. 11 After some algebra we then obtain the NMR signal Eexp G 2 y 2 : Eexp 1 m G 2 k B T 2 4am 1 a K 2 1 m L 2 2 m 2 B. 12 This is a complete analytic expression for E(t), however it is somewhat long and less illuminating than the limiting cases. These limiting cases fall into two different groups given by internal parameters and controlled by experimental conditions. First, there are two characteristic times in our system, v m/ and x /a. v determines the relaxation of the particle velocity distribution to the equilibrium Maxwell form, while x controls the relaxation to the equilibrium Boltzmann distribution of positions. Typically for molecular objects there is a wide interval between the times at which these two equilibrium distributions are established, and hence v x. Taking molecular parameters of the particlesize R1 Å, mass m1 22 g, viscosity of the fluid 1 2 P so that the friction constant 31 8 ergscm 2, and the characteristic interaction constant ak B T/R 2 4 erg cm 2, we obtain a very crude estimate: v 1 13 s and x 1 8 s. For a suspension of micron-size colloid particles in water, R1 4 cm and density 1 gcm 3, one obtains v 1 5 s. Taking the confinement energy of the order of flocculation minimum, a1 6 erg cm 2, the position relaxation time is x 1 1 s, or even greater. However, in a rigidly confined environment, with high a, and in a lowviscosity solvent, with low, one may expect the reversal of the inequality v x. When the duration of spin phase reversal cycle T is much smaller, in between, or much larger than v and x the resulting signal characterizes the particle motion, controlled by inertial, viscous, or potential forces, respectively. The second group of limiting cases of the general equation 12 is determined by the experimentally controlled parameters of magnetic field pulses. These are the cycle duration T and the pulse length, which can be chosen much smaller, of the order of, or greater than v or x. IV. VERY SHORT PULSES, T Expanding our functions K, L, and B in powers of /T we find that the leading term is proportional to 2. The complete expression for the exponent of the general NMR signal 12 in the limiting case v, x, i.e., for a very short gradient pulses, can be written in a relatively compact form. Leaving aside the common factor G 2 (k B T/a)m 2 /[ 2 4am] and using, as in Eq. 8, the parameters 1 and 2, we obtain the exponent in Eq. 12: 2 1 1e 2 T 2 1e 1 T 2 a m e 1 T e 2 T 2 a m e 2 2 T e 2 1 T e 1 2 T. 13 This expression applies for an arbitrary time T and any relation between the characteristic times x and v.ineq.13 one can clearly distinguish the three terms, with K 2, L 2, and B, which are present in the general equation 12. If, as happens in most typical systems, the velocity relaxation time v is much shorter than the time x, required for the particle positional distribution to relax to its equilibrium Boltzmann form, Eq. 13 dramatically simplifies. In J. Chem. Phys., Vol. 12, No. 11, 15 March 1995 Downloaded 6 Sep 26 to Redistribution subject to AIP license or copyright, see

4 4622 Terentjev, Callaghan, and Warner: Spin-echo NMR of confined Brownian particles this case 2 am and, therefore, 2 1 v 1 1 x. We have then, keeping only the leading terms for the three entries: K e 1 T 2 ; L 2 2 e 2 1 T, 14 B 1 1e2 1 T It is important to emphasize that the neglect of terms of the order v / x 1 effectively corresponds to the assumption that the Maxwell distribution of the particle velocities is already achieved. This restricts us from taking the time T to be too short, T v. In order to examine this limit one has to expand the initial equation 13 in powers of T/ v 1, the leading term coming from L 2 2 T Substituting asymptotics 14 to the exponent 12 and returning to parameters, we obtain the simple result for the case v T: Eexp 2 G 2 k BT 2 a 1eT/ x. 15 When, in some very special system with heavy particles in a low-viscosity matrix, the limit T v is achieved, the resulting signal is determined solely by inertial effects the term L 2 in Eq. 12 and takes the form Eexp G k BT 2 m 2 T In the short gradient pulse limit, the PGSE NMR amplitude is more naturally expressed in terms of the scattering wave vector amplitude, qg. We may rewrite the echo attenuation in the three characteristic limiting cases in terms of the dynamical parameters intrinsic to each time scale, namely, Eexpq 2 ẋ 2 T 2 /2 expq 2 DT for T v for v T x expq 2 x 2 for x T. 17 The first expression is that of ballistic motion of the particle before the effect of the viscous drag, or thermal noise are felt. The second is simple diffusion, with Dk B T/, before the effect of confining potential starts affecting the particle relaxation. For T x, the labeled particle achieves the equilibrium distributions of its position in the potential well. At this long time scale the averaged propagator P s (X,T) is simply the autocorrelation function of the position distribution. The scattering theory, applicable when x, T, therefore predicts that EF P s (X,T) 2 expq 2 x 2, where F is the Fourier transform operator. This is exactly the result found above. As required, the latter two expressions along with Eq. 15 are consistent with the narrow gradient pulse expression of Stejskal. 4 V. LONG PERIODS, T V, X Another important limiting case is achieved when the delay time of the PGSE experiment T is sufficiently long. In particular the regime x T may be very interesting since at such long times the position of the particle essentially follows its equilibrium Boltzmann distribution, potential forces dominate, and the effects of restricted diffusion are obvious. When T v, x the general expression 12 simplifies so that it can also be presented in a reasonably compact form. Three parameters in the exponent of the NMR signal, expressed through parameters 1 and 2, now take the form B e e e e e e e , K e e 1 2, L e 2 2 1e 1 2, 19 2 which are valid for an arbitrary gradient pulse duration and for any relation between x and v. As in the previous section with Eq. 13, a great simplification can be achieved by assuming that v x. Neglecting the corrections of the order 1 / 2 1, one obtains B e e 2 1, K e 1 2, L e Changing to parameters 1 x 1, 2 v 1, the resulting expression for the NMR signal takes a very simple form: Eexp 2 G 2 k BT a 2 x x 2 x 2 e / x, 22 quite independent of the relation between and both characteristic times. J. Chem. Phys., Vol. 12, No. 11, 15 March 1995 Downloaded 6 Sep 26 to Redistribution subject to AIP license or copyright, see

5 Terentjev, Callaghan, and Warner: Spin-echo NMR of confined Brownian particles 4623 Naturally, the short pulse limit x will return the corresponding portion of Eq. 15 at T x. As before, by neglecting terms of the order 1 / 2 we effectively disregard the ballistic aspect of the particle motion. The fact that the x limit of Eq. 22 coincides with the long-time limit of the short pulse expression 15 means that at long times the pulse duration need not be so short it was assumed that v in Eq. 15 in order to have the NMR signal in the form of the Fourier transform E(q) with the exponent proportional to q 2. VI. FINITE GRADIENT PULSES For completeness we examine the case when the magnetic gradient pulse duration is finite, not infinitesimally short as it was assumed in Sec. IV. For long periods T v, x the complete result is given in Sec. V. Here we present similar results for two other characteristic regions: very short times, T v, and intermediate times, v T x. In order to avoid unnecessary complicated equations, the limit v x is assumed throughout this section. At very short times the result is Eexp G 2 k BT m 2 T 2 1 T v x 2 6T v, 23 where we kept only the leading terms in v / x 1 in each respective correction. When the period T is much greater than v the NMR signal for arbitrary pulse length T takes the form Eexp 2 G 2 k BT a 2 x x 2 x 2 e / x Eexp 1 2q 2 ẋ 2 T 2 1T/ v / x 2 /6T v for T v expq 2 DT1/3TT/2 x for v T x expq 2 x 2 2 x 2 / 2 / x 1e / x for x T. 26 The echo attenuation expressions reduce to the corresponding results for T. Furthermore, the intermediate time scale (T x ) expression agrees precisely with the wellknown Stejskal Tanner equation. 1 In each case the effect of the finite gradient pulse width is to reduce the mean squared value of the displacement conjugate to q in the echo attenuation exponent. This displacement attenuation effect of finite pulse widths has been pointed out in the recent work of Mitra and Halperin. 7 These authors have independently obtained an expression equivalent to our result for x T. In this case the effect of the finite pulse width is to reduce the apparent well dimensions through the term 2 (/ x 1 e / x) 2 x / 2 via the corrections of order / x. Note that the dimensionless parameter / x may also be written D/x 2 and therefore depends on the relative distance diffused during the application of the gradient pulse. VII. THE STEADY GRADIENT CASE Finally, we write down the limiting expressions which apply in the case of the steady gradient, namely T. Although these represent a trivial extension of the previous section they have particular relevance to other results in the NMR literature. Clearly x 2 e T/ x1cosh/ x. 24 Eexp 2 G 2 ẋ 2 T 4 /2 for T v As in the two previous sections, by implementing the limiting case v x we effectively lose information about the ballistic stage of the particle motion and the result depends only on x. At long times T x we return, as expected, to Eq. 22. Equation 24, however, is valid for an arbitrary relation between T and x. Therefore, when the period T lies inside the large interval between v and x, we obtain, by expanding Eq. 24: Eexp 2 G 2 k BT a 2 T x 1 3T T 2 x, 25 with the limiting case 17. While the scattering wave vector is not well defined under conditions of finite width gradient pulses, we shall find it convenient to continue using the symbol q to represent the gradient pulse area G. Using Eqs. 23, 25, and 22 for the long times, we may rewrite the echo attenuation for each characteristic time scale as exp2 2 G 2 DT 3 /3 for v T x exp2 2 G 2 x 2 x T for x T. The physical meaning of the three characteristic time regimes of the steady gradient case is quite transparent. When the spin echo takes place within an interval T v the signal is collected from labeled particles that did not achieve statistical equilibrium and whose registered motion is inertial. In this case, qualitatively, xẋt with the velocity distribution corresponding to the starting Maxwellian expression for which ẋ 2 k B T/m. Since the NMR signal in the Gaussian approximation is EexpG 2 y 2 /2, for which the exponent involves two integrations over time of x, this exponent naturally scales as t 4. This fourth power dependence on time is consistent with the expression obtained by Nalciolgu and Cho 8 for stationary random flow. The intermediate limit reflects a particularly simple physical situation, corresponding to the long-time limit in the case of free diffusion. For such a regime the second moment x 2 Dt and, after two time integrations in the Gaussian exponent y 2, we obtain a t 3 scaling. Both the finite and J. Chem. Phys., Vol. 12, No. 11, 15 March 1995 Downloaded 6 Sep 26 to Redistribution subject to AIP license or copyright, see

6 4624 Terentjev, Callaghan, and Warner: Spin-echo NMR of confined Brownian particles steady gradient results in this time scale are identical with the classical NMR expressions for free diffusion. Finally, for x T, the labeled particle achieves its positional equilibrium. In this regime the linear time dependence of the exponent for the diffusive motion of the particle constrained in the parabolic potential well describes simple T 2 relaxation. The apparent relaxation rate T G 2 x 2 x corresponds to the well-known 9 fast motion rate, 2 c, where 2 is the mean squared frequency spread and c is the motional correlation time. VIII. CONCLUSIONS We have obtained here, for a well-defined dynamical problem, an expression for the PGSE NMR echo attenuation which is exact for all values of dynamical parameters and for any gradient pulse spacing and gradient pulse duration. The theory reproduces all the known limiting cases as required and provides an exact analytic description of all transition regimes. To our knowledge this is the first such PGSE NMR problem solved completely. The ability to represent the dependence of the PGSE NMR signal on echo time T, over the transition T x to T x, and especially for finite gradient pulse widths, should prove of particular value in studies of polymer dynamics. In particular it will enable the harmonic well model to be tested and the characteristic relaxation times to be measured. While the motivation for development of this theory concerns the problem of measuring restricted polymer motion, we note that the transitions described here may relate to other problems of interest. For example, the notion of a Gaussian distribution of velocities is also of importance in describing perfusion in porous media and biological tissue. For such systems v represents the velocity correlation time and may be easily accessible to PGSE NMR studies. Hence the transition between T v and T v which, for molecular collision processes, is unlikely to be observed by PGSE NMR, becomes of real significance in the case of perfusion. ACKNOWLEDGMENTS We are grateful to P. P. Mitra and B. I. Halperin for sending us a copy of their preprint during the preparation of this article. E. M. T. and M. W. acknowledge support from Unilever PLC while P. T. C. acknowledges support from the New Zealand Foundation for Research, Science, and Technology. 1 E. O. Stejskal and J. E. Tanner, J. Chem. Phys. 42, P. T. Callaghan, Principles of NMR Microscopy Clarendon, Oxford, M. Doi and S. F. Edwards, The Theory of Polymer Dynamics Clarendon, Oxford, E. O. Stejskal, J. Chem. Phys. 43, J. Stepisnik, Physica 198, S. Chandrasekhar, Rev. Mod. Phys. 15, P. P. Mitra and B. I. Halperin, J. Magn. Reson. in press. 8 O. Nalcioglu and Z. H. Cho, IEEE Trans. Med. Imag. MI-6, A. Abragam, Principles of Nuclear Magnetism Clarendon, Oxford, J. Chem. Phys., Vol. 12, No. 11, 15 March 1995 Downloaded 6 Sep 26 to Redistribution subject to AIP license or copyright, see