Direct Observation of Rotational Brownian Motion of Spheres by NMR

Abstract

We present a new NMR method for the investigation of rotational Brownian motion based on the appearance of dipolar relaxation as the particles rotate. Our experimental results obtained on submicrometer polystyrene spheres in water confirm quantitatively the basic laws of rotational Brownian motion. The effects of hydrodynamic interactions and long-time tails are discussed. PACS numbers: 05.40. +j, 76.60.EsBrownian motion, discovered by R. Brown in 1828, refers to the random movement of small objects like pollen particles dispersed in a fluid. The phenomenon affords an ideal way of studying the evolution of a system with a small number of degrees of freedom coupled to a heat bath, a central question in statistical mechanics. For example, it gave rise to the well-known Einstein relation which was a first form of the general fluctuationdissipation theorem. More recently, Brownian motion has provided the first experimental evidence 1 for the so-called long-time tails, a very general phenomenon in dynamical systems. Despite its importance, Brownian motion has not been fully investigated however. In particular, not much experimental work has been done on rotational Brownian motion, 2 mainly by reason of the great difficulty in observing the rotation of small objects, especially spherical ones.The aim of this Letter is to present an experimental study of the rotational Brownian motion of well defined objects-submicrometer polystyrene spheres in water-using a new NMR method which gives direct access to the diffusion coefficient D.Analysis of the method.-This method is based on the nonadiabatic thermodynamic evolution of a nuclear-spin system ( ! H belonging to the polystyrene spheres here) as its Hamiltonian changes with time. This change in our case results from the dependence of the secular dipolar Hamiltonian H d on the orientation of the internuclear vectors with respect to the applied field. The lack of adiabaticity is detected by following the evolution of the system after a Jeener-Broekaert 3 sequence which creates a density matrix p(0) = 1-/3(0)H d (0) corresponding to an equilibrium state with respect to H d (0). As the Hamiltonian evolves, p remains close to an adiabatic form p=l-fi(t)H d (t), but the inverse dipolar temperature /3 (t) decreases with a relaxation rate which includes a contribution Tf l due to the motion.Using the following formal solution for p (t), P (t) =Jeener 4 has derived the following general expression for T d x which is valid as long as the line shape retains its rigid form (i.e., for a rotational Brownian motion with DT 2 « 1, where T 2 , 10~5 s in polystyrene, is the transverse relaxation time):where the angular brackets denote a statistical average. We anticipate that the correlations between H(0) and Hit) die out fast enough that the departure in (2) of the spin evolution operator U(t, 0) from the identity operator can be fully neglected. Within this limit, it can be shown that one obtains for a rotating object Tf x = 2$~dt(S{Q)-£i(t)) 91180

Spin 3/2 Zeeman perturbed NQR in the presence of slow sample rotation

“…The time dependence of the Hamiltonian during a simple echo measurement gives rise to an echo decay envelope which depends on the quadrupole broadening. More rapid rotational motion has been observed by Esteve et al [7] who investigated the effects of rotational Brownian motion of polystyrene spheres and showed that the resulting nuclear dipole relaxation provides a means to study that motion. Tycko [8] investigated very fast rotation (1-4 kHz) for a pure NQR measurement and observed a spectral splitting associated with Berry's phase.…”

Brownian Motion in Biological Sensing

“…An alternative approach for the detection of Brownian relaxation is nuclear magnetic resonance, since the relaxation time for nuclear spins is modified by the rotational motion of the particles [12]. In addition to using intrinsic anisotropies, it is also possible to add anisotropy artificially.…”

Brownian motion studies of viscoelastic colloidal gels by rotational single particle tracking

“…Ensemble methods such as depolarized dynamic light scattering or fluorescence correlation spectroscopy also require optical or physical anisotropy (Berne & Pecora, 1975;Pieper & Enderlein, 2011;Rogers et al, 2012). NMR can measure the rotation of isotropic ensembles but only at longer timescales (Esteve et al, 1984). The resolution of most optical techniques is of the order of 1 , still far larger than the rotational motion of particles in a colloidal gel.…”