Variational bounds for the shear viscosity of gelling melts

We study shear stress relaxation for a gelling melt of randomly crosslinked, interacting monomers. We derive a lower bound for the static shear viscosity $\eta$, which implies that it diverges algebraically with a critical exponent $k\ge 2\nu-\beta$. Here, $\nu$ and $\beta$ are the critical exponents of percolation theory for the correlation length and the gel fraction. In particular, the divergence is stronger than in the Rouse model, proving the relevance of excluded-volume interactions for the dynamic critical behaviour at the gel transition. Precisely at the critical point, our exact results imply a Mark-Houwink relation for the shear viscosity of isolated clusters of fixed size.Comment: 5 pages; CHANGES: typos corrected, some references added; version as publishe

Hierarchical Graph Transformation

If systems are specified by graph transformation, large graphs should be structured in order to be comprehensible. In this paper, we present an approach for the rule-based transformation of hierarchically structured (hyper)graphs. In these graphs, distinguished hyperedges contain graphs that can be hierarchical again. Our framework extends the well-known double-pushout approach from at to hierarchical graphs. In particular, we show how pushouts and pushout complements of hierarchical graphs and graph morphisms can be constructed recursively. Moreover, we make rules more expressive by introducing variables which allow to copy and to remove hierarchical subgraphs in a single rule application

Shear viscosity of a crosslinked polymer melt

We investigate the static shear viscosity on the sol side of the vulcanization transition within a minimal mesoscopic model for the Rouse-dynamics of a randomly crosslinked melt of phantom polymers. We derive an exact relation between the viscosity and the resistances measured in a corresponding random resistor network. This enables us to calculate the viscosity exactly for an ensemble of crosslinks without correlations. The viscosity diverges logarithmically as the critical point is approached. For a more realistic ensemble of crosslinks amenable to the scaling description of percolation, we prove the scaling relation $k=\phi-\beta$ between the critical exponent $k$ of the viscosity, the thermal exponent $\beta$ associated with the gel fraction and the crossover exponent $\phi$ of a random resistor network.Comment: 8 pages, uses Europhysics Letters style; Revisions: results extende

A Port Graph Rewriting Approach to Relational Database Modelling

International audienceWe present new algorithms to compute the Syntactic Closure and the Minimal Cover of a set of functional dependencies, using strategic port graph rewriting. We specify a Visual Domain Specific Language to model relational database schemata as port graphs, and provide an extension to port graph rewriting rules. Using these rules we implement strategies to compute a syntactic closure, analyse it and find minimal covers, essential for schema normalisation. The graph program provides a visual description of the computation steps coupled with analysis features not available in other approaches. We prove soundness and completeness of the computed closure. This methodology is implemented in PORGY

Quantum Mechanical Aspects of Cell Microtubules: Science Fiction or Realistic Possibility?

Recent experimental research with marine algae points towards quantum entanglement at ambient temperature, with correlations between essential biological units separated by distances as long as 20 Angstr\"oms. The associated decoherence times, due to environmental influences, are found to be of order 400 fs. This prompted some authors to connect such findings with the possibility of some kind of quantum computation taking place in these biological entities: within the decoherence time scales, the cell "quantum calculates" the optimal "path" along which energy and signal would be transported more efficiently. Prompted by these experimental results, in this talk I remind the audience of a related topic proposed several years ago in connection with the possible r\^ole of quantum mechanics and/or field theory on dissipation-free energy transfer in microtubules (MT), which constitute fundamental cell substructures. Quantum entanglement between tubulin dimers was argued to be possible, provided there exists sufficient isolation from other environmental cell effects. The model was based on certain ferroelectric aspects of MT. In the talk I review the model and the associated experimental tests so far and discuss future directions, especially in view of the algae photo-experiments.Comment: 31 pages latex, 11 pdf figures, uses special macros, Invited Plenary Talk at DICE2010, Castello Pasquini, Castiglioncello (Italy), September 13-18 201

Dynamics of gelling liquids: a short survey

The dynamics of randomly crosslinked liquids is addressed via a Rouse- and a Zimm-type model with crosslink statistics taken either from bond percolation or Erdoes-Renyi random graphs. While the Rouse-type model isolates the effects of the random connectivity on the dynamics of molecular clusters, the Zimm-type model also accounts for hydrodynamic interactions on a preaveraged level. The incoherent intermediate scattering function is computed in thermal equilibrium, its critical behaviour near the sol-gel transition is analysed and related to the scaling of cluster diffusion constants at the critical point. Second, non-equilibrium dynamics is studied by looking at stress relaxation in a simple shear flow. Anomalous stress relaxation and critical rheological properties are derived. Some of the results contradict long-standing scaling arguments, which are shown to be flawed by inconsistencies.Comment: 21 pages, 3 figures; Dedicated to Lothar Schaefer on the occasion of his 60th birthday; Changes: added comments on the gel phase and some reference

Critical Dynamics of Gelation

Shear relaxation and dynamic density fluctuations are studied within a Rouse model, generalized to include the effects of permanent random crosslinks. We derive an exact correspondence between the static shear viscosity and the resistance of a random resistor network. This relation allows us to compute the static shear viscosity exactly for uncorrelated crosslinks. For more general percolation models, which are amenable to a scaling description, it yields the scaling relation $k=\phi-\beta$ for the critical exponent of the shear viscosity. Here $\beta$ is the thermal exponent for the gel fraction and $\phi$ is the crossover exponent of the resistor network. The results on the shear viscosity are also used in deriving upper and lower bounds on the incoherent scattering function in the long-time limit, thereby corroborating previous results.Comment: 34 pages, 2 figures (revtex, amssymb); revised version (minor changes