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Journal of Statistical Physics

, Volume 54, Issue 3–4, pp 581–680 | Cite as

Statistical mechanics of polymer networks of any topology

  • Bertrand Duplantier
Articles

Abstract

The statistical mechanics is considered of any polymer network with a prescribed topology, in dimensiond, which was introduced previously. The basic direct renormalization theory of the associated continuum model is established. It has a very simple multiplicative structure in terms of the partition functions of the star polymers constituting the vertices of the network. A calculation is made toO2), whered=4−ε, of the basic critical dimensionsσL associated with anyL-leg vertex (L≥1). From this infinite series of critical exponents, any topology-dependent critical exponent can be derived. This is applied to the configuration exponent γ G of any networkG toO2), includingL-leg star polymers. The infinite sets of contact critical exponents θ between multiple points of polymers or between the cores of several star polymers are also deduced. As a particular case, the three exponents θ0, θ1, θ2 calculated by des Cloizeaux by field-theoretic methods are recovered. The limiting exact logarithmic laws are derived at the upper critical dimensiond=4. The results are generalized to the series of topological exponents of polymer networks near a surface and of tricritical polymers at theΘ-point. Intersection properties of networks of random walks can be studied similarly. The above factorization theory of the partition function of any polymer network over its constitutingL-vertices also applies to two dimensions, where it can be related to conformal invariance. The basic critical exponents σ L and thus any topological polymer exponents are then exactly known. Principal results published elsewhere are recalled.

Key words

Polymer networks star polymers self-avoiding walks multiplicative renormalization critical exponents ε expansion O(n) model conformal invariance two dimensions Θ-solvent surface critical behavior 

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Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • Bertrand Duplantier
    • 1
  1. 1.Service de Physique Théorique (Laboratoire de l'Institut de Recherche Fondamentale du Commissariat à l'Energie Atomique) de SaclayGif-sur-YvetteFrance

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