Constraint release in entangled binary blends of linear polymers: a molecular dynamics study

We present extensive molecular dynamics simulations of the dynamics of diluted long probe chains entangled with a matrix of shorter chains. The chain lengths of both components are above the entanglement strand length, and the ratio of their lengths is varied over a wide range to cover the crossover from the chain reptation regime to tube Rouse motion regime of the long probe chains. Reducing the matrix chain length results in a faster decay of the dynamic structure factor of the probe chains, in good agreement with recent neutron spin echo experiments. The diffusion of the long chains, measured by the mean square displacements of the monomers and the centers of mass of the chains, demonstrates a systematic speed-up relative to the pure reptation behavior expected for monodisperse melts of sufficiently long polymers. On the other hand, the diffusion of the matrix chains is only weakly perturbed by the diluted long probe chains. The simulation results are qualitatively consistent with the theoretical predictions based on constraint release Rouse model, but a detailed comparison reveals the existence of a broad distribution of the disentanglement rates, which is partly confirmed by an analysis of the packing and diffusion of the matrix chains in the tube region of the probe chains. A coarse-grained simulation model based on the tube Rouse motion model with incorporation of the probability distribution of the tube segment jump rates is developed and shows results qualitatively consistent with the fine scale molecular dynamics simulations. However, we observe a breakdown in the tube Rouse model when the short chain length is decreased to around N-S = 80, which is roughly 3.5 times the entanglement spacing N-e(P) = 23. The location of this transition may be sensitive to the chain bending potential used in our simulations.

New fast SCFT algorithm applied to binary diblock copolymer/homopolymer blends

We present an efficient strategy for mapping out the classical phase behavior of block copolymer systems using self-consistent field theory (SCFT). With our new algorithm, the complete solution of a classical block copolymer phase can be evaluated typically in a fraction of a second on a single-processor computer, even for highly segregated melts. This is accomplished by implementing the standard unit-cell approximation (UCA) for the cylindrical and spherical phases, and solving the resulting equations using a Bessel function expansion. Here the method is used to investigate blends of AB diblock copolymer and A homopolymer, concentrating on the situation where the two molecules are of similar size.

Feedback design for regularizing descriptor systems

This paper surveys numerical techniques for the regularization of descriptor (generalized state-space) systems by proportional and derivative feedback. We review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the Weierstraß canonical form. Three condensed forms display the controllability and observability properties of a descriptor system. The condensed forms are obtained through orthogonal equivalence transformations and rank decisions, so they may be computed by numerically stable algorithms. In addition, the condensed forms display whether a descriptor system is regularizable, i.e., when the system pencil can be made to be regular by derivative and/or proportional output feedback, and, if so, what index can be achieved. Also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback.

Duality, observability, and controllability for linear time-varying descriptor systems

A characterization of observability for linear time-varying descriptor systemsE(t)x(t)+F(t)x(t)=B(t)u(t), y(t)=C(t)x(t) was recently developed. NeitherE norC were required to have constant rank. This paper defines a dual system, and a type of controllability so that observability of the original system is equivalent to controllability of the dual system. Criteria for observability and controllability are given in terms of arrays of derivatives of the original coefficients. In addition, the duality results of this paper lead to an improvement on a previous fundamental structure result for solvable systems of the formE(t)x(t)+F(t)x(t)=f(tt).

Regularization of descriptor systems by output feedback

Conditions are given under which a descriptor, or generalized state-space system can be regularized by output feedback. It is shown that under these conditions, proportional and derivative output feedback controls can be constructed such that the closed-loop system is regular and has index at most one. This property ensures the solvability of the resulting system of dynamic-algebraic equations. A reduced form is given that allows the system properties as well as the feedback to be determined. The construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way.

Regularization of Descriptor Systems by Derivative and Proportional State Feedback

For linear multivariable time-invariant continuous or discrete-time singular systems it is customary to use a proportional feedback control in order to achieve a desired closed loop behaviour. Derivative feedback is rarely considered. This paper examines how derivative feedback in descriptor systems can be used to alter the structure of the system pencil under various controllability conditions. It is shown that derivative and proportional feedback controls can be constructed such that the closed loop system has a given form and is also regular and has index at most 1. This property ensures the solvability of the resulting system of dynamic-algebraic equations. The construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way. The problem of pole placement with derivative feedback alone and in combination with proportional state feedback is also investigated. A computational algorithm for improving the “conditioning” of the regularized closed loop system is derived.

Minimum norm regularization of descriptor systems by mixed output feedback

We study the regularization problem for linear, constant coefficient descriptor systems Ex' = Ax+Bu, y1 = Cx, y2 = Γx' by proportional and derivative mixed output feedback. Necessary and sufficient conditions are given, which guarantee that there exist output feedbacks such that the closed-loop system is regular, has index at most one and E+BGΓ has a desired rank, i.e., there is a desired number of differential and algebraic equations. To resolve the freedom in the choice of the feedback matrices we then discuss how to obtain the desired regularizing feedback of minimum norm and show that this approach leads to useful results in the sense of robustness only if the rank of E is decreased. Numerical procedures are derived to construct the desired feedback gains. These numerical procedures are based on orthogonal matrix transformations which can be implemented in a numerically stable way.

Eigenstructure assignment in descriptor systems

Coordinate free conditions are given for pole assignment by feedback in linear descriptor (singular) systems which guarantee closed-loop regularity. These conditions are shown to be both necessary and sufficient for assignment of the maximum possible number of finite poles. Transformation to special coordinates are not used and the results provide a robust algorithm for the computation of the required feedback.

Regularization of time-varying descriptor systems by the asvd

We study linear variable coefficient control problems in descriptor form. Based on a behaviour approach and the general theory for linear differential algebraic systems we give the theoretical analysis and describe numerically stable methods to determine the structural properties of the system.