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 toO(ε2), whered=4−ε, of the basic critical dimensionsσ L 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 toO(ε2), 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.
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Duplantier, B. Statistical mechanics of polymer networks of any topology. J Stat Phys 54, 581–680 (1989). https://doi.org/10.1007/BF01019770
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DOI: https://doi.org/10.1007/BF01019770