Adriana I. Pesci

Lattice theory of polymer blends and liquid mixtures: Beyond the Flory–Huggins approximation

Lattice model calculations of corrections to the Flory–Huggins mean field approximation from the preceding paper are applied to the thermodynamic properties of polymer blends. We describe the variation of the noncombinatorial entropy of mixing with composition and monomer structure by considering an extended lattice model in which monomers extend over several lattice sites and therefore have differing sizes and shapes. Composition and temperature dependences of heats of mixing or the equivalent Flory χ parameters are in accord with the magnitude observed experimentally. It is only because we treat the idealized limit of an incompressible blend, for simplicity, that the heat of mixing and Flory χ parameter depend on one effective interaction parameter that is represented in terms of differences in van der Waals energies. The corrections to the Flory–Huggins approximation produce a much lower critical temperature in general agreement with recent Monte Carlo simulations by Sariban and Binder. Concentrated po...

Instabilities and singularities in Hele-Shaw flow

A mechanism by which smooth initial conditions evolve towards a topological reconfiguration of fluid interfaces is studied in the context of Darcy’s law. In the case of thin fluid layers, nonlinear PDEs for the local thickness are derived from an asymptotic limit of the vortex sheet representation. A particular example considered is the Rayleigh–Taylor instability of stratified fluid layers, where the instability of the system is controlled by a Bond number B. It is proved that, for a range of B and initial data “subharmonic” to it, interface pinching must occur in at least infinite time. Numerical simulations suggest that “pinching” singularities occur generically when the system is unstable, and in particular immediately above a bifurcation point to instability. Near this bifurcation point an approximate analytical method describing the approach to a finite-time singularity is developed. The method exploits the separation of time scales that exists close to the first instability in a system of finite ex...

Lattice models of polymer fluids: Monomers occupying several lattice sites. II. Interaction energies

Nearest neighbor nonbonded van der Waals interaction energies are appended to the description of lattice models of flexible polymers in which monomers have specific structures and may cover several lattice sites. A formally exact representation is derived for the free energy of multicomponent system of these structured self and mutually avoiding lattice polymers with nearest neighbor van der Waals interactions. Systematic expansions of the free energy are developed in powers of the inverse of the lattice coordination number and the van der Waals energies using the mean field Flory–Huggins approximation as the zeroth order reference. Diagrammatic rules are given for the evaluation of energies, and the free energy of a binary blend is calculated to second order beyond Flory–Huggins theory for polymers with monomers having differing sizes and shapes. The accompanying paper compares these results with experiment and applies them to analyze heats of mixing of blends, their temperature and composition dependenc...

Theory of the molecular origins of the entropic portion of the Flory χ parameter for polymer blends

The lattice model of self‐ and mutually avoiding polymers is used to compute corrections to the Flory–Huggins entropy of mixing for binary blends of polymers in which the monomers have specific geometric structures and therefore occupy several lattice sites. Ten examples provide an entropic χ of close to the magnitude observed experimentally, thereby giving the first molecular basis for understanding the previously enigmatic and ad hoc entropic χ parameter.

Antiphase Synchronization in a Flagellar-Dominance Mutant of Chlamydomonas

Groups of beating flagella or cilia often synchronize so that neighboring filaments have identical frequencies and phases. A prime example is provided by the unicellular biflagellate Chlamydomonas reinhardtii, which typically displays synchronous in-phase beating in a low-Reynolds number version of breaststroke swimming. We report the discovery that ptx1, a flagellar-dominance mutant of C. reinhardtii, can exhibit synchronization in precise antiphase, as in the freestyle swimming stroke. High-speed imaging shows that ptx1 flagella switch stochastically between in-phase and antiphase states, and that the latter has a distinct waveform and significantly higher frequency, both of which are strikingly similar to those found during phase slips that stochastically interrupt in-phase beating of the wild-type. Possible mechanisms underlying these observations are discussed.

Soap-film Möbius strip changes topology with a twist singularity

It is well-known that a soap film spanning a looped wire can have the topology of a Möbius strip and that deformations of the wire can induce a transformation to a two-sided film, but the process by which this transformation is achieved has remained unknown. Experimental studies presented here show that this process consists of a collapse of the film toward the boundary that produces a previously unrecognized finite-time twist singularity that changes the linking number of the film’s Plateau border and the centerline of the wire. We conjecture that it is a general feature of this type of transition that the singularity always occurs at the surface boundary. The change in linking number is shown to be a consequence of a viscous reconnection of the Plateau border at the moment of the singularity. High-speed imaging of the collapse dynamics of the film’s throat, similar to that of the central opening of a catenoid, reveals a crossover between two power laws. Far from the singularity, it is suggested that the collapse is controlled by dissipation within the fluid film surrounding the wire, whereas closer to the transition the power law has the classical form arising from a balance between air inertia and surface tension. Analytical and numerical studies of minimal surfaces and ruled surfaces are used to gain insight into the energetics underlying the transition and the twisted geometry in the neighborhood of the singularity. A number of challenging mathematical questions arising from these observations are posed.

Attracting manifold for a viscous topology transition.

An analytical method is developed describing the approach to a finite-time singularity associated with collapse of a narrow fluid layer in an unstable Hele-Shaw flow. Under the separation of time scales near a bifurcation point, a long-wavelength mode entrains higher-frequency modes, as described by a version of Hill`s equation. In the slaved dynamics, the initial-value problem is solved explicitly, yielding the time and analytical structure of a singularity which is associated with the motion of zeros in the complex plane. This suggests a general mechanism of singularity formation in this system. {copyright} {ital 1995} {ital The} {ital American} {ital Physical} {ital Society}.

Domain of convergence of perturbative solutions for Hele-Shaw flow near interface collapse

Recent work [Phys. Fluids 10, 2701 (1998)] has shown that for Hele-Shaw flows sufficiently near a finite-time pinching singularity, there is a breakdown of the leading-order solutions perturbative in a small parameter e controlling the large-scale dynamics. To elucidate the nature of this breakdown we study the structure of these solutions at higher order. We find a finite radius of convergence that yields a new length scale exponentially small in e. That length scale defines a ball in space and time, centered around the incipient singularity, inside of which perturbation theory fails. Implications of these results for a possible matching of outer solutions to inner scaling solutions are discussed.

Topological constraints and their breakdown in dynamical evolution

A variety of physical and biological systems exhibit dynamical behaviour that has some explicit or implicit topological features. Here, the term ‘topological’ is meant to convey the idea of structures, e.g. physical knots, links or braids, that have some measure of invariance under continuous deformation. Dynamical evolution is then subject to the topological constraints that express this invariance. The simplest problem arising in these systems is the determination of minimum-energy structures (and routes towards these structures) permitted by such constraints, and elucidation of mechanisms by which the constraints may be broken. In more complex nonequilibrium cases there can be recurring singularities associated with topological rearrangements driven by continuous injection of energy. In this brief overview, motivated by an upcoming program on ‘Topological Dynamics in the Physical and Biological Sciences’ at the Isaac Newton Institute for Mathematical Sciences, we present a summary of this class of dynamical systems and discuss examples of important open problems.

Topological Transitions in Hele-Shaw Flow

The processes which lead to topological reconfiguration of fluid interfaces are studied in Hele-Shaw flow. This is motivated by recent experiments of droplet forma tion through the osculation of two interfaces separating immiscible fluids. In the Hele-Shaw approximation, such a configuration reduces to interface dynamics through their representation as vortex sheets. To begin, we focus on thin fluid layers and develop an asymptotic theory which yields simplified, nonlinear equations for the local thickness. As particular examples, we consider the dynamics of gravity driven fluid jets, as well as the motion of unstably stratified fluid layers. In both cases, the bounding interfaces collide at a finite time, with an associated singularity in the fluid velocity. Some comparison is made with simulations of the full Hele-Shaw equations, and connections with experiments are discussed.