Inelastic coherent neutron scattering by small particles

Dynamic structure factor of undulating vesicles: finite-size and spherical geometry effects with application to neutron spin echo experiments

We consider the dynamic structure factor (DSF) of quasi-spherical vesicles and present a generalization of an expression that was originally formulated by Zilman and Granek (ZG) for scattering from isotropically oriented quasi-flat membrane plaquettes. The expression is obtained in the form of a multi-dimensional integral over the undulating membrane surface. The new expression reduces to the original stretched exponential form in the limit of sufficiently large vesicles, i.e., in the micron range or larger. For much smaller unilamellar vesicles, deviations from the asymptotic, stretched exponential equation are noticeable even if one assumes that the Seifert-Langer leaflet density mode is completely relaxed and membrane viscosity is neglected. To avoid the need for an exhaustive numerical integration while fitting to neutron spin echo (NSE) data, we provide a useful approximation for polydisperse systems that tests well against the numerical integration of the complete expression. To validate the new expression, we performed NSE experiments on variable-size vesicles made of a POPC/POPS lipid mixture and demonstrate an advantage over the original stretched exponential form or other manipulations of the original ZG expression that have been deployed over the years to fit the NSE data. In particular, values of the membrane bending rigidity extracted from the NSE data using the new approximations were insensitive to the vesicle radii and scattering wavenumber and compared very well with expected values of the effective bending modulus ([Formula: see text]) calculated from results in the literature. Moreover, the generalized scattering theory presented here for an undulating quasi-spherical shell can be easily extended to other models for the membrane undulation dynamics beyond the Helfrich Hamiltonian and thereby provides the foundation for the study of the nanoscale dynamics in more complex and biologically relevant model membrane systems.

Fluctuation dynamics of bilayer vesicles with intermonolayer sliding: experiment and theory

The presence of coupled modes of membrane motion in closed shells is extensively predicted by theory. The bilayer structure inherent to lipid vesicles is suitable to support hybrid modes of curvature motion coupling membrane bending with the local reorganization of the bilayer material through relaxation of the dilatational stresses. Previous experiments evidenced the existence of such hybrid modes facilitating membrane bending at high curvatures in lipid vesicles [Rodríguez-García, R., Arriaga, L.R., Mell, M., Moleiro, L.H., López-Montero, I., Monroy, F., 2009. Phys. Rev. Lett. 102, 128201.]. For lipid bilayers that are able to undergo intermonolayer sliding, the experimental fluctuation spectra are found compatible with a bimodal schema. The usual tension/bending fluctuations couple with the hybrid modes in a mechanical interplay, which becomes progressively efficient with increasing vesicle radius, to saturate at infinity radius into the behavior expected for a flat membrane. Grounded on the theory of closed shells, we propose an approximated expression of the bimodal spectrum, which predicts the observed dependencies on the vesicle radius. The dynamical features obtained from the autocorrelation functions of the vesicle fluctuations are found in quantitative agreement with the proposed theory.

Bending stiffness of biological membranes: what can be measured by neutron spin echo?

Large vesicles obtained by the extrusion method represent adequate membrane models to probe membrane dynamics with neutron radiation. Particularly, the shape fluctuations around the spherical average topology can be recorded by neutron spin echo (NSE). In this paper we report on the applicable theories describing the scattering contributions from bending-dominated shape fluctuations in diluted vesicle dispersions, with a focus on the relative relevance of the master translational mode with respect to the internal fluctuations. Different vesicle systems, including bilayer and non-bilayer membranes, have been scrutinized. We describe the practical ranges where the exact theory of bending fluctuations is applicable to obtain the values of the bending modulus from experiments, and we discuss about the possible internal modes that could be alternatively contributing to shape fluctuations.

Fluctuation dynamics of spherical vesicles: frustration of regular bulk dissipation into subdiffusive relaxation

Spherical lipid vesicles obtained by the extrusion method are nonequilibrium membrane structures more curved than the zero spontaneous curvature equilibrium state of the bilayer. Furthermore, these structures are quite rigid as compared to spontaneous vesicles or microemulsion droplets made of soluble surfactants. The dynamical description of the shape fluctuations derived by Milner and Safran (MS) [Phys. Rev. A 36, 4371 (1987)], which is based on the elastic Helfrich energy referred to the equilibrium state, could be misleading in these cases. In the present contribution, shape fluctuations of unilamellar palmitoyl-oleyl-phosphocholine (POPC) vesicles (radius Rh<or=100 nm ) prepared by extrusion are studied by means of neutron spin echo (NSE) in combination with dynamic light scattering (DLS). The relaxation of the fluctuation modes is inferred from the DLS autocorrelation functions and from the intermediate NSE scattering functions measured for several different values of the wave vector. The observed relaxations are compatible with a stretched-exponential decay rather than the single-exponential behavior. Dynamical frustration of the bulk dissipation mechanism in the way described by Zilman and Granek (ZG) for weak amplitude fluctuations [Phys. Rev. Lett. 77, 4788 (1996)] is invoked as a plausible scenario for explaining the subdiffusive nonexponential relaxations experimentally observed. The combined analysis of NSE and DLS data allows a calculation of the bending elastic constant of POPC vesicles kappa=19+/-2 kBT , in excellent agreement with literature data. The ZG approach is revealed as the adequate extension of the MS theory to describe fluctuation dynamics of rigid membranes.

Concentration dependence of shape and structure fluctuations of droplet microemulsions investigated by neutron spin echo spectroscopy

We describe dynamic modes that originate from shape and structure fluctuations in a droplet microemulsion system. The modes are decoupled by a contrast variation neutron scattering technique using the relative intermediate form factor method. The strategy of the method is analogous to the relative form factor method, which decouples the form and structure factors from the small-angle neutron scattering intensity [M. Nagao, Phys. Rev. E 75, 061401 (2007)]. First, we will briefly explain theoretical and experimental approaches to understanding neutron spin echo (NSE) data from droplet microemulsion systems. Then we will introduce the relative intermediate form factor method, which decouples shape and structure fluctuations. The concentration dependence of the droplet dynamics in a microemulsion system is used to elucidate the strengths of this method. The intermediate form and structure factors are successfully decoupled from an observed intermediate scattering function by NSE. The decay rate of the shape fluctuation modes linearly decreases, while the fluctuation amplitude increases as the droplet concentration increases. The first cumulant of the obtained intermediate structure factor shows a clear de Gennes narrowing behavior at a length scale corresponding to the interdroplet distance. However, in the high-momentum-transfer and longer-time regions, the first cumulant deviates from the intermediate structure factor. This result suggests the existence of other dynamic modes of structure fluctuations rather than the center-of-mass diffusion mode.

Interpretation of static and dynamic neutron and light scattering from microemulsion droplets: effects of shape fluctuations

The theory of static and dynamic scattering of neutrons and light on microemulsion droplets is developed. The droplets are modeled by double-layered fluid spheres immersed in another fluid. The surface layer of arbitrary thickness thermally fluctuates in the shape. The scattering functions are consistently calculated up to the second order of the fluctuations. The bulk fluids and the layer are characterized by different scattering length densities (or dielectric constants). Involving the Helfrich's concept of interfacial elasticity, the theory is applied for the description of small-angle neutron scattering (SANS), neutron spin echo (NSE), and dynamic light scattering (DLS) experiments on dilute microemulsions. From the fits to the experimental data the bending elasticity and the Gaussian modulus are extracted. Due to the corrected account for the fluctuations, their values differ markedly from those obtained in the original works. The theory well describes the SANS experiments. In the case of DLS, we had to assume the shell of the solvent molecules to be built of several layers. Previous theories were in a sharp disagreement with the NSE experiments. A better agreement with these experiments is obtained if the dissipation in the surface layer is included into the consideration. From the experiments, the viscosity of the layer is estimated for a concrete microemulsion system.