Polymer field theory
Encyclopedia
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).
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 considered 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 
and confined in a volume 
, can be expressed as
where
is the potential of mean force
given by,
representing the solvent-mediated non-bonded interactions among the segments, while
represents the harmonic binding energy of the chains. The latter energy contribution can be formulated as
where
is the statistical segment length and 
the polymerization index.
Using this definition, one can rewrite Eq. (2) as
Next, one converts the model into a field theory by making use of the Hubbard-Stratonovich transformation
or delta-functional transformation
where
is a functional and
is the delta
functional given by
with
representing the
auxiliary field function. Here we note that, expanding the field function in a Fourier series, implies that periodic boundary conditions are applied in all directions and that the
-vectors designate the reciprocal lattice vectors of the supercell.
where
can be interpreted as the partition function for an ideal gas of non-interacting polymers and
is the path integral of a free polymer in a zero field with elastic energy
In the latter equation the unperturbed radius of gyration of a chain
, where the space dimension 
. Moreover, in Eq. (6) the partition function of a single polymer, subjected to the field 
, is given by
where
is the chemical potential and 
is given by Eq. (6). Performing the sum, this provides the field-theoretic representation of the grand canonical partition function,
where
is the grand canonical action with
defined by
Eq. (8) and the constant
Moreover, the parameter related to the chemical potential is given by
where
is provided by Eq. (7).
(PE) solutions (Schmid 1998, Matsen 2002, Fredrickson 2002). There are, however, a multitude of cases for which SCFT provides inaccurate or even qualitatively incorrect results (Baeurle 2006a). These comprise neutral polymer or polyelectrolyte solutions in dilute and semidilute concentration regimes, block copolymers near their order-disorder transition, polymer blends near their phase transitions, etc. In such situations the partition function integral defining the field-theoretic model is not entirely dominated by a single MF configuration and field configurations far from it can make important contributions, which require the use of more sophisticated calculation techniques beyond the MF level of approximation.
confined PE solutions (Tsonchev 1999). However, in situations where the MF approximation is bad many computationally demanding higher-order corrections to the integral are necessary to get the desired accuracy.
, which has originally been devised to calculate functional integrals arising in quantum field theories (QFT's). In QFT's a standard approximation strategy is to expand the functional integrals in a power series in the coupling constant using perturbation theory
. Unfortunately, generally most of the expansion terms turn out to be infinite, rendering such calculations impracticable (Shirkov
2001). A way to remove the infinities from QFT's is to make use of the concept of renormalization (Baeurle 2007). It mainly consists in replacing the bare values of the coupling parameters, like e.g. electric charges or masses, by renormalized coupling parameters and requiring that the physical quantities do not change under this transformation, thereby leading to finite terms in the perturbation expansion. A simple physical picture of the procedure of renormalization can be drawn from the example of a classical electrical charge,
, inserted into a polarizable medium, such as in an electrolyte solution. At a distance 
from the charge due to polarization of the medium, its Coulomb field will effectively depend on a function 
, i.e. the effective (renormalized) charge, instead of the bare electrical charge, 
. At the beginning of the 1970s, K.G. Wilson further pioneered the power of renormalization concepts by developing the formalism of renormalization group
(RG) theory, to investigate critical phenomena
of statistical systems (Wilson 1971).
near the phase transition
and a self-avoiding random walk
of a polymer chain of infinite length on a lattice, to calculate the polymer excluded volume
exponents (de Gennes 1972). Adapting this concept to field-theoretic functional integrals, implies to study in a systematic way how a field theory model changes while eliminating and rescaling a certain number of degrees of freedom from the partition function integral (Wilson 1974).
-order MF contribution, to renormalize the model parameters and extract in a self-consistent way the dominant length scale of the concentration fluctuations in critical concentration regimes.
(MC) algorithms and to sample the full partition function integral in field-theoretic formulation. The resulting procedure is then called a polymer
field-theoretic simulation
. In a recent work, however, Baeurle demonstrated that MC sampling in conjunction with the basic field-theoretic representation is impracticable due to the so-called numerical sign problem
(Baeurle 2002). The difficulty is related to the complex and oscillatory nature of the resulting distribution function, which causes a bad statistical convergence of the ensemble averages of the desired thermodynamic and structural quantities. In such cases special analytical and numerical techniques are necessary to accelerate the statistical convergence (Baeurle 2003, Baeurle 2003a, Baeurle 2004).
, providing its so-called mean-field representation. This strategy was previously successfully employed by Baer et al. in field-theoretic electronic structure calculations (Baer 1998). Baeurle could demonstrate that this technique provides a significant acceleration of the statistical convergence of the ensemble averages in the MC sampling procedure (Baeurle 2002, Baeurle 2002a).
boost in the statistical convergence of the desired ensemble averages (Baeurle 2002).
Statistical field theory
A statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. In other words, the microstates of the system are expressed through field configurations...
describing the statistical behavior of a neutral or charged polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
system. It can be derived by transforming the partition function
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
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
Hubbard-Stratonovich transformation
The Hubbard–Stratonovich transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard...
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).
Particle representation of the canonical partition function
The standard continuum model of flexible polymers, introduced by Edwards (Edwards 1965), treats a solution composed of 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 considered 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 and confined in a volume , can be expressed aswhere
is the potential of mean forcePotential of mean force
The Potential of Mean Force of a system with N molecules is strictly the potential that gives the average force over all the configurations of all the n+1...N molecules acting on a particle at any fixed configuration keeping fixed a set of molecules 1...n...
given by,
representing the solvent-mediated non-bonded interactions among the segments, while
represents the harmonic binding energy of the chains. The latter energy contribution can be formulated aswhere
is the statistical segment length and the polymerization index.Field-theoretic transformation
To derive the basic field-theoretic representation of the canonical partition function, one introduces in the following the segment density operator of the polymer systemUsing this definition, one can rewrite Eq. (2) as
Next, one converts the model into a field theory by making use of the Hubbard-Stratonovich transformation
Hubbard-Stratonovich transformation
The Hubbard–Stratonovich transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard...
or delta-functional transformation
where
is a functional and is the deltafunctional given by
with
representing theauxiliary field function. Here we note that, expanding the field function in a Fourier series, implies that periodic boundary conditions are applied in all directions and that the
-vectors designate the reciprocal lattice vectors of the supercell.Basic field-theoretic representation of canonical partition function
Using the Eqs. (3), (4) and (5), we can recast the canonical partition function in Eq. (1) in field-theoretic representation, which leads towhere
can be interpreted as the partition function for an ideal gas of non-interacting polymers and
is the path integral of a free polymer in a zero field with elastic energy
In the latter equation the unperturbed radius of gyration of a chain
, where the space dimension . Moreover, in Eq. (6) the partition function of a single polymer, subjected to the field , is given byBasic field-theoretic representation of grand canonical partition function
To derive the grand canonical partition function, we use its standard thermodynamic relation to the canonical partition function, given bywhere
is the chemical potential and is given by Eq. (6). Performing the sum, this provides the field-theoretic representation of the grand canonical partition function,where
is the grand canonical action with
defined byEq. (8) and the constant
Moreover, the parameter related to the chemical potential is given by
where
is provided by Eq. (7).Mean field approximation
A standard approximation strategy for polymer field theories is the mean field (MF) approximation, which consists in replacing the many-body interaction term in the action by a term where all bodies of the system interact with an average effective field. This approach reduces any multi-body problem into an effective one-body problem by assuming that the partition function integral of the model is dominated by a single field configuration. A major benefit of solving problems with the MF approximation, or its numerical implementation commonly referred to as the self-consistent field theory (SCFT), is that it often provides some useful insights into the properties and behavior of complex many-body systems at relatively low computational cost. Successful applications of this approximation strategy can be found for various systems of polymers and complex fluids, like e.g. strongly segregated block copolymers of high molecular weight, highly concentrated neutral polymer solutions or highly concentrated block polyelectrolytePolyelectrolyte
Polyelectrolytes are polymers whose repeating units bear an electrolyte group. These groups will dissociate in aqueous solutions , making the polymers charged. Polyelectrolyte properties are thus similar to both electrolytes and polymers , and are sometimes called polysalts. Like salts, their...
(PE) solutions (Schmid 1998, Matsen 2002, Fredrickson 2002). There are, however, a multitude of cases for which SCFT provides inaccurate or even qualitatively incorrect results (Baeurle 2006a). These comprise neutral polymer or polyelectrolyte solutions in dilute and semidilute concentration regimes, block copolymers near their order-disorder transition, polymer blends near their phase transitions, etc. In such situations the partition function integral defining the field-theoretic model is not entirely dominated by a single MF configuration and field configurations far from it can make important contributions, which require the use of more sophisticated calculation techniques beyond the MF level of approximation.
Higher-order corrections
One possibility to face the problem is to calculate higher-order corrections to the MF approximation. Tsonchev et al. developed such a strategy including leading (one-loop) order fluctuation corrections, which allowed to gain new insights into the physics ofconfined PE solutions (Tsonchev 1999). However, in situations where the MF approximation is bad many computationally demanding higher-order corrections to the integral are necessary to get the desired accuracy.
Renormalization techniques
An alternative theoretical tool to cope with strong fluctuations problems occurring in field theories has been provided in the late 1940s by the concept of renormalizationRenormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
, which has originally been devised to calculate functional integrals arising in quantum field theories (QFT's). In QFT's a standard approximation strategy is to expand the functional integrals in a power series in the coupling constant using perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
. Unfortunately, generally most of the expansion terms turn out to be infinite, rendering such calculations impracticable (Shirkov
Dmitry Shirkov
Dmitry Vasil'evich Shirkov is a Russian theoretical physicist known for his contribution to quantum field theory and to the development of the renormalization group method.-Biography:...
2001). A way to remove the infinities from QFT's is to make use of the concept of renormalization (Baeurle 2007). It mainly consists in replacing the bare values of the coupling parameters, like e.g. electric charges or masses, by renormalized coupling parameters and requiring that the physical quantities do not change under this transformation, thereby leading to finite terms in the perturbation expansion. A simple physical picture of the procedure of renormalization can be drawn from the example of a classical electrical charge,
, inserted into a polarizable medium, such as in an electrolyte solution. At a distance from the charge due to polarization of the medium, its Coulomb field will effectively depend on a function , i.e. the effective (renormalized) charge, instead of the bare electrical charge, . At the beginning of the 1970s, K.G. Wilson further pioneered the power of renormalization concepts by developing the formalism of renormalization groupRenormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
(RG) theory, to investigate critical phenomena
Critical phenomena
In physics, critical phenomena is the collective name associated with thephysics of critical points. Most of them stem from the divergence of thecorrelation length, but also the dynamics slows down...
of statistical systems (Wilson 1971).
Renormalization group theory
The RG theory makes use of a series of RG transformations, each of which consists of a coarse-graining step followed by a change of scale (Wilson 1974). In case of statistical-mechanical problems the steps are implemented by successively eliminating and rescaling the degrees of freedom in the partition sum or integral that defines the model under consideration. De Gennes used this strategy to establish an analogy between the behavior of the zero-component classical vector model of ferromagnetismFerromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
near the phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
and a self-avoiding random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
of a polymer chain of infinite length on a lattice, to calculate the polymer excluded volume
Excluded volume
The concept of excluded volume was introduced by Werner Kuhn in 1934 and applied to polymer molecules shortly thereafter by Paul Flory.- In liquid state theory :...
exponents (de Gennes 1972). Adapting this concept to field-theoretic functional integrals, implies to study in a systematic way how a field theory model changes while eliminating and rescaling a certain number of degrees of freedom from the partition function integral (Wilson 1974).
Hartree renormalization
An alternative approach is known as the Hartree approximation or self-consistent one-loop approximation (Amit 1984). It takes advantage of Gaussian fluctuation corrections to the-order MF contribution, to renormalize the model parameters and extract in a self-consistent way the dominant length scale of the concentration fluctuations in critical concentration regimes.Tadpole renormalization
In a more recent work Efimov and Nogovitsin showed that an alternative renormalization technique originating from QFT, based on the concept of tadpole renormalization, can be a very effective approach for computing functional integrals arising in statistical mechanics of classical many-particle systems (Efimov 1996). They demonstrated that the main contributions to classical partition function integrals are provided by low-order tadpole-type Feynman diagrams, which account for divergent contributions due to particle self-interaction. The renormalization procedure performed in this approach effects on the self-interaction contribution of a charge (like e.g. an electron or an ion), resulting from the static polarization induced in the vacuum due to the presence of that charge (Baeurle 2007). As evidenced by Efimov and Ganbold in an earlier work (Efimov 1991), the procedure of tadpole renormalization can be employed very effectively to remove the divergences from the action of the basic field-theoretic representation of the partition function and leads to an alternative functional integral representation, called the Gaussian equivalent representation (GER). They showed that the procedure provides functional integrals with significantly ameliorated convergence properties for analytical perturbation calculations. In subsequent works Baeurle et al. developed effective low-cost approximation methods based on the tadpole renormalization procedure, which have shown to deliver useful results for prototypical polymer and PE solutions (Baeurle 2006a, Baeurle 2006b, Baeurle 2007a).Numerical simulation
Another possibility is to use Monte CarloMonte Carlo
Monte Carlo is an administrative area of the Principality of Monaco....
(MC) algorithms and to sample the full partition function integral in field-theoretic formulation. The resulting procedure is then called a polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
field-theoretic simulation
Field-theoretic simulation
A field-theoretic simulation is a numerical strategy to calculate structure and physical properties of a many-particle system within the framework of a statistical field theory, like e.g. a polymer field theory. A convenient possibility is to use Monte Carlo algorithms, to sample the full...
. In a recent work, however, Baeurle demonstrated that MC sampling in conjunction with the basic field-theoretic representation is impracticable due to the so-called numerical sign problem
Numerical sign problem
The numerical sign problem refers to the difficulty of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and negative contributions to the integral...
(Baeurle 2002). The difficulty is related to the complex and oscillatory nature of the resulting distribution function, which causes a bad statistical convergence of the ensemble averages of the desired thermodynamic and structural quantities. In such cases special analytical and numerical techniques are necessary to accelerate the statistical convergence (Baeurle 2003, Baeurle 2003a, Baeurle 2004).
Mean field representation
To make the methodology amenable for computation, Baeurle proposed to shift the contour of integration of the partition function integral through thehomogeneous MF solution using Cauchy's integral theoremCauchy's integral theorem
In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane...
, providing its so-called mean-field representation. This strategy was previously successfully employed by Baer et al. in field-theoretic electronic structure calculations (Baer 1998). Baeurle could demonstrate that this technique provides a significant acceleration of the statistical convergence of the ensemble averages in the MC sampling procedure (Baeurle 2002, Baeurle 2002a).
Gaussian equivalent representation
In subsequent works Baeurle et al. (Baeurle 2002, Baeurle 2002a, Baeurle 2003, Baeurle 2003a, Baeurle 2004) applied the concept of tadpole renormalization, leading to the Gaussian equivalent representationof the partition function integral, in conjunction with advanced MC techniques in the grand canonical ensemble. They could convincingly demonstrate that this strategy provides a furtherboost in the statistical convergence of the desired ensemble averages (Baeurle 2002).