Higher Aliphatic Aldehyde Polymers. V. Cyclic Trimers of C9 to C12, Normal Aliphatic Aldehydes

Cyclic trimers of n-nonaldehyde (NA), n-decaldehyde (DA), n-undecaldehyde, (UA) and n-dodecaldehyde (DDA) were prepared by reacting the individual aldehydes with protic or Lewis acids. Higher aliphatic aldehydes whose long paraffinic chains dominate the general properties of these compounds do not trimerize readily the trimers, when formed, are purified with difficulty. The cyclic trimers, characterized by IR and NMR spectroscopy, are exclusively the cis isomers and commonly exist in all equatorial conformations. The melting behavior of the cyclic trimers was studied by DSC. The melting endotherms and the enthalpies of fusion increase with increasing chain length. Although the melting temperatures of the cyclic trimers are similar to the melting range of the side-chain crystallization of the corresponding crystalline isotactic polyaldehydes, the transition peaks of the trimers are single peaks and much sharper than the corresponding polymer peaks.     

A Comparison of Conventional Gel Stiffness Characterization Techniques with Cavitation Rheology

Experimental Mechanics - Interest in soft gels has arisen in recent years as they can be applied to many fields such as tissue engineering, food additives, and drug delivery. The importance of...     

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

Under suitable hypotheses, we prove that a form of a projective homogeneous variety G/P defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre’s Conjecture II in Galois cohomology for function fields over an algebraically closed field.     

About Thinning Invariant Partition Structures

Bernoulli-p thinning has been well-studied for point processes. Here we consider three other cases: (1) sequences (X ₁,X ₂,??); (2) gaps of such sequences (X _n+1?X ₁) _n???; (3) partition structures. For the first case we characterize the distributions which are simultaneously invariant under Bernoulli-p thinning for all p??(0,1]. Based on this, we make conjectures for the latter two cases, and provide a potential approach for proof. We explain the relation to spin glasses, which is complementary to important previous work of Aizenman and Ruzmaikina, Arguin, and Shkolnikov.     

Finite-Volume Excitations of the¶111 Interface in the Quantum XXZ Model

We show that the ground states of the three-dimensional XXZ Heisenberg ferromagnet with a 111 interface have excitations localized in a subvolume of linear size R with energies bounded by O(1/R²). As part of the proof we show the equivalence of ensembles for the 111 interface states in the following sense: In the thermodynamic limit the states with fixed magnetization yield the same expectation values for gauge invariant local observables as a suitable grand canonical state with fluctuating magnetization. Here, gauge invariant means commuting with the total third component of the spin, which is a conserved quantity of the Hamiltonian. As a corollary of equivalence of ensembles we also prove the convergence of the thermodynamic limit of sequences of canonical states (i.e., with fixed magnetization).     

Carbon films grown on Pt(1 1 1) as supports for model gold catalysts

Carbon films were grown on a Pt(1 1 1) single crystal by ethylene decomposition at elevated temperatures (1000-1300 K). Depending on the preparation conditions, different carbon structures formed on the metal surface such as flat and curved graphitic layers, carbon particles and carbon nanowires. Although these carbon films exhibited a high density of surface defects, gold interacted only weakly with the carbon surface. CO adsorption on the Au/carbon systems was very similar to that observed for various Au/oxide systems previously studied. This finding strongly indicates that CO adsorption on gold is essentially independent of the nature of support.     

Emptiness Formation Probability

We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix \({\mathcal{T}_0}\). Focusing on the eigenvalues of \({\mathcal{T}_0}\) of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required \({\mathcal{T}_0}\) to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some results by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular, we solve a problem posed in their work.     

The effect of nanoparticle shape on polymer‐nanocomposite rheology and tensile strength

Nanoparticles can influence the properties of polymer materials by a variety of mechanisms. With fullerene, carbon nanotube, and clay or graphene sheet nanocomposites in mind, we investigate how particle shape influences the melt shear viscosity η and the tensile strength τ, which we determine via molecular dynamics simulations. Our simulations of compact (icosahedral), tube or rod‐like, and sheet‐like model nanoparticles, all at a volume fraction ? ≈ 0.05, indicate an order of magnitude increase in the viscosity η relative to the pure melt. This finding evidently can not be explained by continuum hydrodynamics and we provide evidence that the η increase in our model nanocomposites has its origin in chain bridging between the nanoparticles. We find that this increase is the largest for the rod‐like nanoparticles and least for the sheet‐like nanoparticles. Curiously, the enhancements of η and τ exhibit opposite trends with increasing chain length N and with particle shape anisotropy. Evidently, the concept of bridging chains alone cannot account for the increase in τ and we suggest that the deformability or flexibility of the sheet nanoparticles contributes to nanocomposite strength and toughness by reducing the relative value of the Poisson ratio of the composite. The molecular dynamics simulations in the present work focus on the reference case where the modification of the melt structure associated with glass‐formation and entanglement interactions should not be an issue. Since many applications require good particle dispersion, we also focus on the case where the polymer‐particle interactions favor nanoparticle dispersion. Our simulations point to a substantial contribution of nanoparticle shape to both mechanical and processing properties of polymer nanocomposites. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1882–1897, 2007     

Further properties of nonzero-sum differential games

The general nonzero-sum differential game hasN players, each controlling a different set of inputs to a single nonlinear dynamic system and each trying to minimize a different performance criterion. These general games have several interesting features which are absent in the two bestknown special cases (the optimal control problem and the two-person, zero-sum differential game). This paper considers some of the difficulties which arise in attempting to generalize ideas which are well known in optimal control theory, such as theprinciple of optimality and the relation betweenopen-loop andclosed-loop controls. Two types of solutions are discussed: theNash equilibrium and thenoninferior set. Some simple multistage discrete games are used to illustrate phenomena which also arise in the continuous formulation.This research was supported by Joint Services Electronics Contracts Nos. N00014-67-A-0298-0006, 0005, 0008 and by NASA Grant No. NGR 22-007-068.