Peptide mediated formation of hierarchically organized solution and solid state polymer nanostructures

Biologically-inspired peptide sequences have been explored as auxiliaries to mediate self-assembly of synthetic macromolecules into hierarchically organized solution and solid state nanostructures. Peptide sequences inspired by the coiled coil motif and "switch" peptides, which can adopt both amphiphilic alpha-helical and beta-strand conformations, were conjugated to poly(ethylene glycol) (PEG). The solution and solid state self-assembly of these materials was investigated using a variety of spectroscopic, scattering and microscopic techniques. These experiments revealed that the folding and organization properties of the peptide sequences are retained upon conjugation of PEG and that they provide the driving force for the formation of the different nanoscale structures which were observed. The possibility of using defined peptide sequences to direct structure formation of synthetic polymers together with the potential of peptide sequences to induce a specific biological response offers interesting prospects for the development of novel self-assembled and biologically active materials.

Electrospinning atactic polystyrene: a neutron scattering study

Electrospinning is a method used to produce nanoscale to microscale sized polymer fibres. In this study we electrospin 1:1 blends of deuterated and hydrogenated atactic- Polystyrene from N,N-Dimethylformamide for small angle neutron scattering experiments in order to analyse the chain conformation in the electrospun fibres. Small angle neutron scattering was carried out on randomly orientated fibre mats obtained using applied voltages of 10kV-15kV and needle tip to collector distances of 20cm and 30cm. Fibre diameters varied from 3μm – 20μm. Neutron scattering data from fibre samples were compared with bulk samples of the same polymer blend. The scattering data indicates that there are pores and nanovoiding present in the fibres; this was confirmed by scanning electron microscopy. A model that combines the scattering from the pores and the labelled polymer chains was used to extract values for the radius of gyration. The radius of gyration in the fibres is found to vary little with the applied voltage, but varies with the initial solution concentration and fibre diameter. The values for the radius of gyration in the fibres are broadly equivalent to that of the bulk state.

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

Multiple sol-gel transitions of PEG-PCL-PEG triblock copolymer aqueous solution

An aqueous solution of a poly(ethylene glycol)-polycaprolactone-poly(ethylene glycol) (PEG-PCL-PEG) with a composition of EG13CL23EG13 undergoes multiple transitions, from sol-to-gel (hard gel)-to-sol-to-gel (soft gel)-to-sol, in the concentration range 20.0∼35.0 wt.-%. Through dynamic mechanical analysis, UV-vis spectrophotometry, small angle X-ray scattering, differential scanning calorimetry, microcalorimetry and 13C NMR spectroscopy, the mechanism of these transitions was investigated. The hard gel and soft gel are distinguished by the crystalline and amorphous state of the PCL. The extent of PEG dehydration and the molecular motion of each block also played a critical role in the multiple transitions. This paper suggests a new mechanism for these multiple transitions driven by temperature changes.

Modelling small angle neutron scattering data from electrospun fibres

Electrospinning is a technique employed to produce nanoscale to microscale sized fibres by the application of a high voltage to a spinneret containing a polymer solution. Here we examine how small angle neutron scattering data can be modelled to analyse the polymer chain conformation. We prepared 1:1 blends of deuterated and hydrogenated atactic-polystyrene fibres from solutions in N, N-Dimethylformamide and Methyl Ethyl Ketone. The fibres themselves often contain pores or voiding within the internal structure on the length scales that can interfere with scattering experiments. A model to fit the scattering data in order to obtain values for the radius of gyration of the polymer molecules within the fibres has been developed, that includes in the scattering from the voids. Using this model we find that the radius of gyration is 20% larger than in the bulk state and the chains are slightly extended parallel to the fibre axis.

A new frequency-uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star-combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L2(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high-frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains.

Puroindoline-a, a lipid binding protein from common wheat, spontaneously forms prolate protein micelles in solution

The self-assembly in solution of puroindoline-a (Pin-a), an amphiphilic lipid binding protein from common wheat, was investigated by small angle neutron scattering, dynamic light scattering and size exclusion chromatography. Pin-a was found to form monodisperse prolate ellipsoidal micelles with a major axial radius of 112 +/- 4.5 A ˚ and minor axial radius of 40.4 +/- 0.18 A ˚ . These protein micelles were formed by the spontaneous self-assembly of 38 Pin-a molecules in solution and were stable over a wide pH range (3.5–11) and at elevated temperatures (20–65 degC). Pin-a micelles could be disrupted upon addition of the non-ionic surfactant dodecyl-b-maltoside, suggesting that the protein self-assembly is driven by hydrophobic forces, consisting of intermolecular interactions between Trp residues located within a well-defined Trp-rich domain of Pin-a.

Wave scattering by an axisymmetric ice floe of varying thickness

The problem of water wave scattering by a circular ice floe, floating in fluid of finite depth, is formulated and solved numerically. Unlike previous investigations of such situations, here we allow the thickness of the floe (and the fluid depth) to vary axisymmetrically and also incorporate a realistic non-zero draught. A numerical approximation to the solution of this problem is obtained to an arbitrary degree of accuracy by combining a Rayleigh–Ritz approximation of the vertical motion with an appropriate variational principle. This numerical solution procedure builds upon the work of Bennets et al. (2007, J. Fluid Mech., 579, 413–443). As part of the numerical formulation, we utilize a Fourier cosine expansion of the azimuthal motion, resulting in a system of ordinary differential equations to solve in the radial coordinate for each azimuthal mode. The displayed results concentrate on the response of the floe rather than the scattered wave field and show that the effects of introducing the new features of varying floe thickness and a realistic draught are significant.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory

We investigate Fréchet differentiability of the scattered field with respect to variation in the boundary in the case of time–harmonic acoustic scattering by an unbounded, sound–soft, one–dimensional rough surface. We rigorously prove the differentiability of the scattered field and derive a characterization of the Fréchet derivative as the solution to a Dirichlet boundary value problem. As an application of these results we give rigorous error estimates for first–order perturbation theory, justifying small perturbation methods that have a long history in the engineering literature. As an application of our rigorous estimates we show that a plane acoustic wave incident on a sound–soft rough surface can produce an unbounded scattered field.

Scattering by infinite one-dimensional rough surfaces

We consider the Dirichlet boundary-value problem for the Helmholtz equation in a non-locally perturbed half-plane. This problem models time-harmonic electromagnetic scattering by a one-dimensional, infinite, rough, perfectly conducting surface; the same problem arises in acoustic scattering by a sound-soft surface. ChandlerWilde & Zhang have suggested a radiation condition for this problem, a generalization of the Rayleigh expansion condition for diffraction gratings, and uniqueness of solution has been established. Recently, an integral equation formulation of the problem has also been proposed and, in the special case when the whole boundary is both Lyapunov and a small perturbation of a flat boundary, the unique solvability of this integral equation has been shown by Chandler-Wilde & Ross by operator perturbation arguments. In this paper we study the general case, with no limit on surface amplitudes or slopes, and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including the incident plane wave, the Dirichlet boundary-value problem for the scattered field has a unique solution.

Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers

We consider a two-dimensional problem of scattering of a time-harmonic electromagnetic plane wave by an infinite inhomogeneous conducting or dielectric layer at the interface between semi-infinite homogeneous dielectric half-spaces. The magnetic permeability is assumed to be a fixed positive constant. The material properties of the media are characterized completely by an index of refraction, which is a bounded measurable function in the layer and takes positive constant values above and below the layer, corresponding to the homogeneous dielectric media. In this paper, we examine only the transverse magnetic (TM) polarization case. A radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as an equivalent mixed system of boundary and domain integral equations, consisting of second-kind integral equations over the layer and interfaces within the layer. Assumptions on the variation of the index of refraction in the layer are then imposed which prove to be sufficient, together with the radiation condition, to prove uniqueness of solution and nonexistence of guided wave modes. Recent, general results on the solvability of systems of second kind integral equations on unbounded domains establish existence of solution and continuous dependence in a weighted norm of the solution on the given data. The results obtained apply to the case of scattering by a rough interface between two dielectric media and to many other practical configurations.

A uniqueness result for scattering by infinite rough surfaces

Consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane with an unbounded, piecewise Lyapunov boundary. This problem models time-harmonic electromagnetic scattering in transverse magnetic polarization by one-dimensional rough, perfectly conducting surfaces. A radiation condition is introduced for the problem, which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi-periodic, and allows a variety of incident fields including an incident plane wave to be included in the results obtained. We show in this paper that the boundary value problem for the scattered field has at most one solution. For the case when the whole boundary is Lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle.

Acoustic scattering by an inhomogeneous layer on a rigid plate

The problem of scattering of time-harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in R2 is considered. The density is assumed to be unity in the media: within the layer the sound speed is assumed to be an arbitrary bounded measurable function. The problem is modelled by the reduced wave equation with variable wavenumber in the layer and a Neumann condition on the plate. To formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate. Under additional assumptions on the wavenumber in the layer, uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established. General results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data.

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.