Kell Mortensen

Analytical treatment of the resolution function for small-angle scattering

Analytical expression for the resolution function for small-angle scattering in pinhole geometry are derived. The contributions to the resolution function due to wavelength spread, finite collimation and detector resolution are determined separately using Gaussian functions to approximate the contributions. A general resolution function is derived which is the result of the combined effect of the three contributions. An azimuthal-integrated resolution function, which can be applied to scattering from a material with a circular symmetric scattering cross section, is calculated. This resolution function contains in addition a contribution from the averaging procedure itself. The analytical results are compared with the results of computer simulations. The comparison shows that Gaussian functions give a good description of the resolution function and that the widths agree with those calculated by the analytical expressions. The resolution function is applied in the analysis of two experimental examples: neutron scattering from latex particles [Wignall, Christen & Ramakrishnan (1988). J. Appl. Cryst. 21, 438–451] and neutron scattering from lamellar structures of bilayer lipid membranes (Mortensen, Pfeiffer, Sackmann & Knoll, unpublished). The analytical expressions for the resolution function allow a least-squares analysis to be performed and excellent agreement between experimental and theoretical scattering patterns are obtained.

Fluctuations, conformational asymmetry and block copolymer phase behaviour

Phase behaviour near the order–disorder transition (ODT) of 58 model hydrocarbon diblock copolymers, representing four different systems, is summarized. Six distinct ordered-state microstructures are reported, including hexagonally modulated lamellae (HML), hexagonally perforated layers (HPL) and a bicontinuous cubic morphology with Iatext-decoration:overline3d space group symmetry. Two non-classical parameters, Iµ and text-decoration:overlineN, control the occurrence and distribution of these phases, in addition to the classical variables ƒ and χN, where ƒ, χ and N are the composition, segment–segment interaction parameter and degree of polymerization, respectively. Iµ accounts for differences in the conformational and volume-filling characteristics of each block. Conformational asymmetry, Iµ≠ 1, produces an asymmetric phase diagram around ƒ= 1/2. The importance of fluctuation effects are inversely related to the magnitude of text-decoration:overlineN, a type of Ginzburg parameter that is proportional to N. As text-decoration:overlineN decreases, the bicontinuous Iatext-decoration:overline3d phase appears adjacent to the ODT. Development of this cubic phase can be rationalized based on chain-packing frustration near the lamellar hexagonal state. Apparently the Iatext-decoration:overline3d cubic state is stabilized by fluctuations since it disappears when text-decoration:overlineN becomes large. These findings provide new insights into the origins of phase complexity in condensed soft matter.

Structural studies of aqueous solutions of PEO - PPO - PEO triblock copolymers, their micellar aggregates and mesophases; a small-angle neutron scattering study

The structural characteristics of aqueous solutions of the Pluronic triblock copolymers of poly(ethylene oxide) - poly(propylene oxide) - poly(ethylene oxide), PEO - PPO - PEO, and their self-associated assemblies are reviewed. It is shown by small-angle neutron scattering that at low temperatures and/or concentration the individual copolymers exist in solution as individual unimers. Thermodynamically stable micelles are formed with increasing copolymer concentration and/or temperature. The unimer-to-micelle transition is not sharp, however. Micelles of well defined spherical shape and size coexist with unimers over a relatively wide temperature/concentration range. The micellar volume fraction increases accordingly with increasing temperature, increasing copolymer concentration and decreasing hydrostatic pressure. The copolymer suspension undergoes as a result a transition from a Newtonian liquid to a soft solid material when the micellar volume fraction crosses the critical value for hard-sphere crystallization. Crystallographic investigations on shear-aligned monodomain samples prove that the micelles in the solid phase are organized on a body-centred cubic lattice. As a result of an increasing micellar size upon increasing the temperature, the micelles themselves undergo a sphere-to-rod transition at elevated temperatures. In a shear field these rod-like micelles form a macroscopic nematic phase for low copolymer concentration, and a hexagonal solid phase for higher concentrations. For even higher concentrations, lamellar phases are observed: one lamellar type which is still governed by the hydrophobic interactions, and one type which appears as a result of crystallization of the PEO blocks.

Can a single function for χ account for block copolymer and homopolymer blend phase behavior

Most theoretical treatments of polymer–polymer phase behavior assume that homopolymer mixtures and block copolymer melts are controlled by a common segment–segment interaction parameter knows as χ. This publication describes the results of small-angle neutron scattering (SANS) experiments conducted as a function of temperature and composition from homogeneous mixtures of poly(ethylene) (PE) and poly(ethylenepropylene) (PEP) of equal molecular weight. Analysis of these SANS measurements based on the random phase approximation indicates that χPE/PEP is independent of composition and linear in T−1. The associated symmetric phase diagram calculated with Flory–Huggins theory contains a stability curve that is consistent with the divergence in single phase susceptibility obtained by SANS. This function χPE/PEP(T) is compared with functions for χPE–PEP(T) associated with the homologous PE–PEP diblock copolymers, extracted both from the temperature dependence of the disordered state scattering structure factor, a...

Compound refractive optics for the imaging and focusing of low-energy neutrons

Low-energy neutrons are essential for the analysis and characterization of materials and magnetic structures. However, both continuous (reactor-based) and pulsed (spallation-based) sources of such neutrons suffer from low fluence. Steering and lensing devices could improve this situation dramatically, so increasing spatial resolution, detectable sample volume limits and even perhaps opening the way for the construction of a neutron microscope. Neutron optics have to date exploited either Bragg diffraction,, such as bent crystals, or reflection, as in mirror guides or a Kumakhov lens,. Refractive optics remain an attractive alternative as they would permit full use of the beam cross-section, allow a compact and linear installation and, because of similarity to conventional optics, enable the use of commercial design and simulation tools. These advantages notwithstanding, single-element refractive optics have previously been considered impractical as they are too weakly focusing, too absorptive and too dispersive. Inspired by the recent demonstration of a compound refractive lens (CRL) for high-energy X-rays, we have designed, built and tested a prototype CRL for 9–20-Å neutrons by using readily available optical components: our CRL has gains greater than 15 and focal lengths of 1–6 m, well matched to small-angle neutron scattering.

Epitaxial growth and shearing of the body centered cubic phase in diblock copolymer melts

Two poly(ethylenepropylene)–poly(ethylethylene) (PEP‐PEE) diblock copolymer melts, containing 25% and 83% by volume PEP, were investigated using small‐angle neutron scattering (SANS) and rheological measurements. The SANS measurements were performed with the aid of an in situ shearing device operated directly in the neutron beam. Each sample was observed to possess three equilibrium phases: two ordered phases at low temperature and a disordered phase at elevated temperatures. The low and high temperature ordered phases have been evaluated to be hexagonally packed (hex) cylinders and body centered cubic (bcc) spheres, respectively. Application of a large amplitude dynamic shear deformation to the hex phase leads to well‐aligned cylinders, with a specific crystallographic orientation relative to the shear plane. Upon heating through the cylinder‐to‐sphere transition, the bcc phase grows epitaxially, with the [111] direction coincident with the original cylinder axis, leading to a well‐defined twinned micros...

Complex Phase Behavior in Solvent-Free Nonionic Surfactants

Unsolvated block copolymers and surfactant solutions are “soft materials” that share a common set of ordered microstructures. A set of polyethyleneoxide-polyethylethylene (PEO-PEE) block copolymers that are chemically similar to the well-known alkane-oxyethylene (CnEOm) nonionic surfactants was synthesized here. The general phase behavior in these materials resembles that of both higher molecular weight block copolymers and lower molecular weight nonionic surfactant solutions. Two of the block copolymers exhibited thermally induced order-order transitions and were studied in detail by small-angle scattering. The fundamental microstructural spacing was determined to be a crucial parameter in these transitions. Transitions from one ordered state to another occur only when the lattice spacing is nearly matched. These materials highlight the importance of epitaxy and molecular conformation in the phase transformations of soft material.

Elliptical Structure of Phospholipid Bilayer Nanodiscs Encapsulated by Scaffold Proteins: Casting the Roles of the Lipids and the Protein

Phospholipid bilayers host and support the function of membrane proteins and may be stabilized in disc-like nanostructures, allowing for unprecedented solution studies of the assembly, structure, and function of membrane proteins (Bayburt et al. Nano Lett. 2002, 2, 853-856). Based on small-angle neutron scattering in combination with variable-temperature studies of synchrotron small-angle X-ray scattering on nanodiscs in solution, we show that the fundamental nanodisc unit, consisting of a lipid bilayer surrounded by amphiphilic scaffold proteins, possesses intrinsically an elliptical shape. The temperature dependence of the curvature of the nanodiscs prepared with two different phospholipid types (DLPC and POPC) shows that it is the scaffold protein that determines the overall elliptical shape and that the nanodiscs become more circular with increasing temperature. Our data also show that the hydrophobic bilayer thickness is, to a large extent, dictated by the scaffolding protein and adjusted to minimize the hydrophobic mismatch between protein and phospholipid. Our conclusions result from a new comprehensive and molecular-based model of the nanodisc structure and the use of this to analyze the experimental scattering profile from nanodiscs. The model paves the way for future detailed structural studies of functional membrane proteins encapsulated in nanodiscs.

Structural properties of self-assembled polymeric aggregates in aqueous solutions

Scattering experiments by X-ray and neutrons have proven to be important techniques to unveil the thermodynamics and the structural characteristics of self-assembled block copolymer systems in the melt as well as in selective solutions. The present Review describes briefly experimental details of the small-angle scattering technique, including the use of in situ auxiliary devised and procedures for data analysis. The use of scattering techniques is exemplified through a number of experimental examples on aqueous block copolymer systems like the PEO/PPO-based Pluronics, which associate into micellar aggregates of various form and size, depending on thermodynamic parameters. The intermicellar interactions are also discussed, and shown to be the basis for ordered colloidal-like structures on the nanometer length-scale, including classical cubic, hexagonal and lamellar ordered phases and network structures. Copyright

PEO-related block copolymer surfactants

Abstract Non-ionic block copolymer systems based on hydrophilic poly(ethylene oxide) and more hydrophobic co-polymer blocks are used intensively in a variety of industrial and personal applications. A brief description on the applications is presented. The physical properties of more simple model systems of such PEO-based block copolymers in aqueous suspensions are reviewed. Based on scattering experiments using either X-ray or neutrons, the phase behavior is characterized, showing that the thermo-reversible gelation is a result of micellar ordering into mesoscopic crystalline phases of cubic, hexagonal or lamellar nature. Shear has major effect on the crystalline texture, but seems not to change the thermodynamic stable phases significantly.