A polymer field theory is a statistical field theory describing the statistical behavior of a neutral or charged polymer system. It can be derived by transforming the partition function from its standard many-dimensional integral representation over the particle degrees of freedom in a functional integral representation over an auxiliary field function, using either the Hubbard–Stratonovich transformation or the delta-functional transformation. Computer simulations based on polymer field theories have been shown to deliver useful results, for example to calculate the structures and properties of polymer solutions (Baeurle 2007, Schmid 1998), polymer melts (Schmid 1998, Matsen 2002, Fredrickson 2002) and thermoplastics (Baeurle 2006).
Canonical ensemble
Particle representation of the canonical partition function The standard continuum model of flexible polymers, introduced by Edwards (Edwards 1965), treats a solution composed of n {\displaystyle n} linear monodisperse homopolymers as a system of coarse-grained polymers, in which the statistical mechanics of the chains is described by the continuous Gaussian thread model (Baeurle 2007) and the solvent is taken into account implicitly. The Gaussian thread model can be viewed as the continuum limit of the discrete Gaussian chain model, in which the polymers are described as continuous, linearly elastic filaments. The canonical partition function of such a system, kept at an inverse temperature β = 1 / k B T {\displaystyle \beta =1/k_{B}T} and confined in a volume V {\displaystyle V} , can be expressed as
Z ( n , V , β ) = 1 n ! ( λ T 3 ) n N ∏ j = 1 n ∫ D r j exp ( − β Φ 0 [ r ] − β Φ ¯ [ r ] ) , ( 1 ) {\displaystyle Z(n,V,\beta )={\frac {1}{n!(\lambda _{T}^{3})^{nN}}}\prod _{j=1}^{n}\int D\mathbf {r} _{j}\exp \left(-\beta \Phi _{0}\left[\mathbf {r} \right]-\beta {\bar {\Phi }}\left[\mathbf {r} \right]\right),\qquad (1)}
where Φ ¯ [ r ] {\displaystyle {\bar {\Phi }}\left[\mathbf {r} \right]} is the potential of mean force given by,
Φ ¯ [ r ] = N 2 2 ∑ j = 1 n ∑ k = 1 n ∫ 0 1 d s ∫ 0 1 d s ′ Φ ¯ ( | r j ( s ) − r k ( s ′ ) | ) − 1 2 n N Φ ¯ ( 0 ) , ( 2 ) {\displaystyle {\bar {\Phi }}\left[\mathbf {r} \right]={\frac {N^{2}}{2}}\sum _{j=1}^{n}\sum _{k=1}^{n}\int _{0}^{1}ds\int _{0}^{1}ds'{\bar {\Phi }}\left(\left|\mathbf {r} _{j}(s)-\mathbf {r} _{k}(s')\right|\right)-{\frac {1}{2}}nN{\bar {\Phi }}(0),\qquad (2)}
representing the solvent-mediated non-bonded interactions among the segments, while Φ 0 [ r ] {\displaystyle \Phi _{0}[\mathbf {r} ]} represents the harmonic binding energy of the chains. The latter energy contribution can be formulated as
… excerpt ends here. Continue reading the full article.
