Segmental dynamics in entangled linear polymer melts

We present molecular dynamics (MD) and slip-springs model simulations of the chain segmental dynamics in entangled linear polymer melts. The time-dependent behavior of the segmental orientation autocorrelation functions and mean-square segmental displacements are analyzed for both flexible and semiflexible chains, with particular attention paid to the scaling relations among these dynamic quantities. Effective combination of the two simulation methods at different coarse-graining levels allows us to explore the chain dynamics for chain lengths ranging from Z ≈ 2 to 90 entanglements. For a given chain length of Z ≈ 15, the time scales accessed span for more than 10 decades, covering all of the interesting relaxation regimes. The obtained time dependence of the monomer mean square displacements, g1(t), is in good agreement with the tube theory predictions. Results on the first- and second-order segmental orientation autocorrelation functions, C1(t) and C2(t), demonstrate a clear power law relationship of C2(t) C1(t)m with m = 3, 2, and 1 in the initial, free Rouse, and entangled (constrained Rouse) regimes, respectively. The return-to-origin hypothesis, which leads to inverse proportionality between the segmental orientation autocorrelation functions and g1(t) in the entangled regime, is convincingly verified by the simulation result of C1(t) g1(t)−1 t–1/4 in the constrained Rouse regime, where for well-entangled chains both C1(t) and g1(t) are rather insensitive to the constraint release effects. However, the second-order correlation function, C2(t), shows much stronger sensitivity to the constraint release effects and experiences a protracted crossover from the free Rouse to entangled regime. This crossover region extends for at least one decade in time longer than that of C1(t). The predicted time scaling behavior of C2(t) t–1/4 is observed in slip-springs simulations only at chain length of 90 entanglements, whereas shorter chains show higher scaling exponents. The reported simulation work can be applied to understand the observations of the NMR experiments.

Plane wave approximation in linear elasticity

We consider the approximation of solutions of the time-harmonic linear elastic wave equation by linear combinations of plane waves. We prove algebraic orders of convergence both with respect to the dimension of the approximating space and to the diameter of the domain. The error is measured in Sobolev norms and the constants in the estimates explicitly depend on the problem wavenumber. The obtained estimates can be used in the h- and p-convergence analysis of wave-based finite element schemes.

Mozzarella-type curd made from buffalo, cows’ and ultrafiltered cows’ milk. 1. Rheology and microstructure

Rennet-induced curd was made from both natural buffalo and cows’ milk, and ultrafiltered cows’ milk (cows’ milk was concentrated such that it had a chemical composition approximately equivalent to that of the buffalo milk). These milk samples were compared on the basis of their rheology, physicochemical characteristics and curd microstructure. The ionic and soluble calcium contents were found to be similar in all milk samples studied. The total and casein bound calcium were higher in concentrated cows’ milk than in standard cows’ milk. Both cows’ milk types were found to have lower total and casein bound calcium than the buffalo milk. This is probably due to concentration of the colloidal part of milk (casein), during the ultrafiltration (UF) process. The rennet coagulation time was similar in UF cows’ and buffalo milk while both were shorter when compared with that of the cows’ milk. The dynamic moduli (G′, G″) values were higher in both the buffalo and UF cows’ milk than in the cows’ milk after 90 min coagulation. The loss tangent, however, was found to be similar in both the UF cows’ and buffalo milk curds and was lower than that observed for the cows’ milk (0.42, 0.42 and 0.48, respectively). The frequency profile of each type of curd was recorded 90 min after the enzyme addition (0.1–10 Hz); all samples were found to be “weak” viscoelastic, frequency dependent gels. The yield stress was also measured 95 min after the enzyme addition, and a higher value was observed in buffalo milk curd when compared with other curd samples made from both the natural cows’ milk and the UF cows’ milk. The cryo-scanning electron and confocal laser scanning micrographs showed that curd structure appeared to be more “dense” and less porous in buffalo milk than cows’ milk even after concentration to equivalent levels of protein/total solids to those found in the buffalo milk.

Optimal linearization trajectories for tangent linear models

We examine differential equations where nonlinearity is a result of the advection part of the total derivative or the use of quadratic algebraic constraints between state variables (such as the ideal gas law). We show that these types of nonlinearity can be accounted for in the tangent linear model by a suitable choice of the linearization trajectory. Using this optimal linearization trajectory, we show that the tangent linear model can be used to reproduce the exact nonlinear error growth of perturbations for more than 200 days in a quasi-geostrophic model and more than (the equivalent of) 150 days in the Lorenz 96 model. We introduce an iterative method, purely based on tangent linear integrations, that converges to this optimal linearization trajectory. The main conclusion from this article is that this iterative method can be used to account for nonlinearity in estimation problems without using the nonlinear model. We demonstrate this by performing forecast sensitivity experiments in the Lorenz 96 model and show that we are able to estimate analysis increments that improve the two-day forecast using only four backward integrations with the tangent linear model. Copyright © 2011 Royal Meteorological Society

Temporal constraints on linear BRDF model parameters

Linear models of bidirectional reflectance distribution are useful tools for understanding the angular variability of surface reflectance as observed by medium-resolution sensors such as the Moderate Resolution Imaging Spectrometer. These models are operationally used to normalize data to common view and illumination geometries and to calculate integral quantities such as albedo. Currently, to compensate for noise in observed reflectance, these models are inverted against data collected during some temporal window for which the model parameters are assumed to be constant. Despite this, the retrieved parameters are often noisy for regions where sufficient observations are not available. This paper demonstrates the use of Lagrangian multipliers to allow arbitrarily large windows and, at the same time, produce individual parameter sets for each day even for regions where only sparse observations are available.