Many practical applications of polymers and other soft condensed matter systems involve mixtures that through equilibrium self-assembly or non-equilibrium processing steps develop complex, multi-phase morphologies. The desirable and marketable properties of such materials, which include plastic alloys, block and graft copolymers, and polyelectrolyte solutions, complexes, and gels, depend critically on the ability to control and manipulate morphology by adjusting a combination of molecular and macroscopic variables. Furthermore, the property profiles are intimately connected to the relationship between the molecular parameters and the extent and type of nanoscale self-assembly that takes place within the materials. Unfortunately, such relationships are traditionally determined by trial and error experimentation that is both laborious and costly. In the design of new materials, it would obviously be highly desirable if theoretical methods could be used to anticipate nano and micro-scale self-assembly and further relate such morphological characteristics to properties of interest in specific applications.

Unfortunately, theoretical techniques for anticipating thestructure and equilibrium phase behavior of complex polymeric fluids are still in their infancy. Non-equilibrium methods for predicting structural evolution of multi-phase systems under realistic processing conditions are even less developed. Nevertheless, progress is being made and current theoretical tools have met with sufficient success to justify use in many of the R\&D organizations of leading polymeric materials suppliers.

Modern computer simulation methods for polymers and other soft materials can be grouped into three major categories: atomistic, coarse-grained particle-based, and field-theoretic. Fully atomistic methods typically involve building classical (as opposed to quantum descriptions of a polymeric or complex fluid with atomic resolution. Interactions in such models are described by some combination of bonded and non-bonded potential functions, typically parameterized at the two-body and/or three-body level. In principle these potentials can be obtained by quantum chemical calculations, so that first principles parameterization of new systems is possible. Determination of equilibrium or non-equilibrium properties involves carrying out a computer simulation, usually by employing Monte Carlo (MC) or molecular dynamics (MD) techniques. The major drawback of atomistic methods is that except in rare instances, it is very difficult to equilibrate sufficiently large systems of polymers at realistic densities in order to extract meaningful information about structure and thermodynamics. This limitation is particularly acute for multi-phase, inhomogeneous systems, which are often those of primary interest.

A reasonable alternative to a fully atomistic computer simulation is a coarse-grained, particle-based approach in which atoms or groups of atoms arelumped into larger ``particles''. At the lowest level this could simply amount to a ``united atom'' approach where, e.g., each unit in a polyethylene chain is replaced by a single effective particle. Interactions in such a model are then effective interactions between lumped $CH_2$ particles and standard MC or MD simulation methods can be employed. Often even more extensive coarse-graining is carried out. For example, bead-spring polymer chains are often employed in which each bead might represent the force center associated with 10 or more backbone atoms. A difficulty with such models is that the effective interactions between beads (particles) are often difficult to parameterize accurately. Moreover, they remain expensive to simulate, especially at melt densities and for heterogeneous systems that exhibit nanoscale or macroscale phase separation. One solution to speed up the simulations is to introduce artificially soft repulsive
inter-particle potentials, as is conventionally done in dissipative particle dynamics (DPD), but this has a
number of adverse effects including artificially high fluid phase compressibilities, loss of topological constraints between chains, and often loss of connection to the atomic/chemical details of the underlying complex fluid.

Coarse-grained {\it field theory models} have also proved extremely important in the development of modern polymer physics. Field theories  can be easily  derived for other multicomponent complex fluid systems such as polymer solutions, multiblock copolymers and blends. The integrals accompanying such theories have been attacked with a number of approximate analytical techniques including mean-field (saddle point) approximation, perturbation expansions, and renormalization group theory. While these analytical results provide a rather satisfactory understanding of polymers dissolved in good solvents\cite{doied}, they provide a less complete description of heterogeneous solutions and melts. Indeed, block copolymer melts are typically examined only by means of the mean-field approximation, commonly referred to in this context as self-consistent mean-field theory (SCFT).

Surprisingly, there has been little or no work on direct numerical sampling of functional integrals accompanying field theories that are relevant to polymer structure and thermodynamics. Such simulations should be particularly useful in elucidating the effects of fluctuations on  order-disorder and order-order phase transitions of block copolymers, as well as facilitating studies of phases, such as polymeric bicontinuous microemulsions, that owe their existence to thermal fluctuations. Furthermore, the same techniques should be equally applicable to other field-theoretic models of complex fluids, and for instance be employed to discern ``exactly'' the fluctuation effects accompanying highly charged electrolyte and polyelectrolyte systems.

In the present research, we developed two numerical procedures to effect such a sampling, and showed that these methods, subsequently referred to as ``field-theoretic polymer simulation'', provide an attractive alternative to more conventional ``particle-based'' simulations.
 

We demonstrated the technique by applying it to examine fluctuation effects on the order-disorder transition in symmetric diblock copolymer melts. Extensions to more complex polymer blends, copolymers, and solutions have also been achieved.