Abstract the selfconsistent field theory scft based upon coarsegrained. Selfconsistent field theory scft for dense polymer melts, has been highly successful in describing complex morphologies in block copolymers. First, we present a pedagogical introduction to an improved version of the opensource polymer. Applications of the wormlike chain model in polymer physics. Applications of the wormlike chain model in polymer. The self consistent field approach solves the integrals over the fluctuating fields in saddlepoint approximation. Numerical selfconsistent field theory study of the. We study the phase behavior of diblock copolymer melts with one block possessing orientationdependent segmental interactions using selfconsistent field theory. E cient orderadaptive methods for polymer self consistent field theory hector d. We propose efficient pseudospectral numerical schemes for solving the selfconsistent, meanfield equations for inhomogeneous polymers.
A finite element approach to selfconsistent field theory. Particlepolymer interactions cavities in the polymer density enforced through the. We use results on polymer absorption obtained from a recent meanfield theory with two order parameters to interpret numerical results of the scheutjens and fleer approach. The challenges facing the incorporation of particles into the fieldbased model of polymers are how to treat the polymerparticle. Efficient orderadaptive methods for polymer selfconsistent. Numerical methods of polymer brushes using selfconsistent field theory tanya l. Analysis of interactions between finitesized particles. Numerical advances in self consistent field theory simulations of. The theoretical framework and numerical method of solving the selfconsistent equations are presented. We discuss different analytical and numerical approaches to studying such a theory. Ceniceros november 30, 2018 abstract a highly accurate and memorye cient approach for the solution of polymer selfconsistent eld theory scft is proposed. Numerical solution of polymer selfconsistent field theory article pdf available in siam journal on multiscale modeling and simulation 23.
David morse research group david morse research group. Therein, the fieldbased model for pncs 18,22,29,30 is pursued due to the power of the selfconsistent field theory scft in predicting the mesoscopic structures of multicomponent polymeric systems 34,35,36. The validity of the new method is analyzed by comparing with results from the spectral method. Selfconsistent field theory of inhomogeneous polymeric. Interactions, phase behavior and rheological properties of. Typical inhomogeneous polymeric materials are in the form of polymer blends. The selfconsistent field approach solves the integrals. Here, we develop a new fieldbased model which unifies the nanoparticle description with the polymer field within the selfconsistent field theory. In mft, the effect of all the other individuals on any given. In this paper, different structures of polymer can be considered, such as homopolymer, block copolymer, polydisperse polymer and charged polymer. Upon increasing the concentration of polymer, the coils will start to interpenetrate, the solution is then called. A generalized coarsegrained description is introduced based on the local polar orientational order parameter and k, an effective frank elastic constant for orientational gradients. Numerical solution of polymer selfconsistent field theory. Consequently, the size of an isolated polymer in solution scales as r0.
However, the traditional scft is based on the gaussian chain model, completely neglecting the chain rigidity effects, which is ascribed to one of the intrinsic properties of. In physics and probability theory, meanfield theory aka mft or rarely selfconsistent field theory studies the behavior of highdimensional random models by studying a simpler model that approximates the original by averaging over degrees of freedom. First, we present a pedagogical introduction to an improved version of the opensource polymer selfconsistent field pscf software package and of the. Phase behavior of rod coil diblock copolymer and homopolymer.
Numerical methods of polymer brushes using selfconsistent. Selfconsistent field theory and its applications in polymer. The selfconsistent field theory of polymers describes the thermodynamic properties of inhomogeneous polymeric systems such as polymer blends and block copolymers. However, the traditional scft is based on the gaussian chain model, completely neglecting the chain rigidity e. A new realspace numerical implementation of the self consistent field theory for semiflexible polymers is developed. We present a hybrid numerical method to introduce hydrodynamics in dynamic self consistent field scf studies of inhomogeneous polymer systems.
Thermodynamics of a compressible maiersaupe model based. Broadly accessible selfconsistent field theory for block polymer. Selfconsistent field theory scft particle based model. Introduction the morphology of multiblock polymers has been of interest for many years due to potential applications that depend on tailored microstructure. Sketches of a approximating a real polymer by a backbone wormlike chain. Results and analysis of the performance are presented in section 5. The self consistent field theory of polymers describes the thermodynamic properties of inhomogeneous polymeric systems such as polymer blends and block copolymers. Using a diblock copolymer melt as a model system, we show that complex langevin. Highresolution implementation of selfconsistent field theory eric w.
Selfconsistent field theory is a theoretical framework for the study of manybody systems. Masking method was used to model an a or battractive surface can compare the selfassembly behavior to a 3d ab diblock copolymer melt con. Physically, this approximation makes the assumption that there is a single mean. Numerical advances in self consistent field theory. Numerical selfconsistent field theory study of the response of strong polyelectrolyte brushes to external electric fields.
An extension of the analytical theory accounting for the local swelling of the polymer is also presented and discussed. Wickham a model of liquid crystalline homopolymers using selfconsistent. It can be derived by transforming the partition function from itsstandard manydimensional integral representation over the particle. Selfconsistent field theory for diblock copolymers. A hybrid multiscale approach is presented which implements polymer self consistent field theory in combination with the mcmillanmayer framework to deduce the polymermediated interactions between the particles. The influences of grafting density, average charge fraction, salt concentration, and. Numerical methods of polymer brushes can used to look at an ab binary blend brush grafted to a spherical nanoparticle. The selfconsistent field equations are solved with andersonmixing iterations using dynamical parameters and an alignment procedure to prevent angular drift of the solution.
Pscf is a fortran 90 program for numerically solving the polymer selfconsistent field theory scft for spatially periodic structures formed by block copolymer melts and mixtures of block copolymers with linear homopolymers andor small molecule solvents. Jul 26, 2006 2019 numerical implementation of pseudospectral method in self consistent mean field theory for discrete polymer chains. Numerical selfconsistent field theory study of the response. Our numerical methods can be used to perform a more extensive study of phase space. Instead of being ensembleaveraged continuous distribution, the particle density in the final morphology can represent individual particles located at preferred positions. Scft provides a method whereby the hamiltonian of a complex system may be transformed into a coarsegrained field theory description, whose meanfield solution is amenable to a battery of analytic and. Our work on self assembly in systems that contain block copolymers has relied heavily on the use of numerical self consistent field theory scft to predict equilibrium structures, and has often been motived by andor carried out in collaboration with experimental colleagues. The influences of grafting density, average charge fraction, salt concentration, and mobile ion size on the variation of the brush. Parallel algorithm for numerical selfconsistent field. The central idea is to combine spectral integration in the polymer chain contour variable with a spectral. Analysis of interactions between finitesized particles and terminally attached polymer using numerical selfconsistentfield theory by bradley michael steels b. Broadly accessible selfconsistent field theory for block.
This expression for particleparticle ev repulsion is appropriate only in t. In mean field theory, the mean field appearing in the singlesite problem is a scalar or vectorial timeindependent quantity. Parallel algorithm for numerical selfconsistent field theory. Self consistent field theory scft for dense polymer melts, has been highly successful in describing complex morphologies in block copolymers. We present a hybrid numerical method to introduce hydrodynamics in dynamic selfconsistent field scf studies of inhomogeneous polymer systems. Selfconsistent field theory is treated in detail, which is a collection of analytical and numerical techniques for obtaining solutions of polymer field theory models in the meanfield approximation. A polymer field theory within the framework of statistical mechanics is a statistical field theory, describing the statistical behavior of a neutral or charged polymer system within the fieldtheoretic approach. Selfconsistent field theory is treated in detail, which is a collection of analytical and numerical techniques for obtaining solutions of polymer field theory models in the mean field approximation. Selfconsistent field theory scft is a powerful tool for the design and interpretation of experiments on block polymer materials. Analysis of interactions between finitesized particles and terminally attached polymer using numerical self consistent field theory by bradley michael steels b. Different solutions of these selfconsistent field scf equations. Chantawansri complex fluids design consortium annual meeting february 2, 2008. Self consistent field theory equations for the diblock system described above, the hamiltonian of the system is given by.
Rasmussen and kalosakas 18 proposed a strang splitting 19 method. Selfconsistent field theory and its applications in. Selfconsistent field theory for smectic ordering of. The theoretical framework and numerical method of solving the self consistent equations are presented. Consistent field theory of inhomogeneous polymeric.
Selfconsistent field theory and its applications in polymer systems. E cient orderadaptive methods for polymer selfconsistent. Self consistent field theory is treated in detail, which is a collection of analytical and numerical techniques for obtaining solutions of polymer field theory models in the mean field approximation. Ceniceros november 30, 2018 abstract a highly accurate and memorye cient approach for the solution of polymer self consistent eld theory scft is proposed. E xact solution for qr,s computational cost scales as v3. New numerical implementation of selfconsistent field. Broadly accessible scft for block polymer materials discovery. Hybrid lattice boltzmanndynamic selfconsistent field. The resulting potential v h is then consistent with the orbitals that generate it, and it is for this reason called self consistent eld. Pdf numerical solution of polymer selfconsistent field theory. Selfconsistentfield theory for chain molecules wur edepot. Structure and properties of polydisperse polyelectrolyte.
The central idea is to combine spectral integration in the polymer chain contour variable with a spectral deferred correction technique to solve the scft modified diffusion equations with arbitrarily high order of accuracy. High order numerical simulations for the polymer self. In particular, we introduce a robust class of semiimplicit methods that employ asymptotic small scale information about the nonlocal density operators. Mar 26, 2014 self consistent field theory is a theoretical framework for the study of manybody systems. Self consistent field theory scft is a powerful tool for the design and interpretation of experiments on block polymer materials. A polymer field theory is a statistical field theory describing the statistical behavior of a neutral or charged polymer system. Applications of selfconsistent field theory in polymer systems. In this perspective, we lower the barrier to entry to the use of scft by experimental groups by two means. The rod blocks are modeled as wormlike chains and the corresponding scft equations are solved using a hybrid method, in which the orientationdependent functions are discretized on a unit sphere, while the positional spacedependent functions are treated using a spectral method. A demonstration of the algorithm is provided for thin films of diblock copolymer grafted to the surface of a spherical core, in which the sequence of equilibrium.
A new realspace numerical implementation of the selfconsistent field theory for semiflexible polymers is developed. Selfconsistent field theory and its applications by m. Scft provides a method whereby the hamiltonian of a complex system may be transformed into a coarsegrained field theory description, whose mean field solution is amenable to a battery of analytic and. We consider a system with n conformationally symmetric diblock copolymers and each has a and b arms joined together with a covalent bond. Self consistent field theory scft or mean field theory approximation has been a powerful tool to investigate and discover polymer phases see for example. The self consistent field theory scft has reveived a great success in prediction of the physical properties of a variety of polymeric systems in the recent two decades. Selfconsistent field theory for smectic ordering of semi. Model we consider a brush formed by polyelectrolyte chains with degree of polymerization i. In particular, polymeric scft has been successfully applied to inhomogeneous polymeric systems such as polymer blends and block copolymer melts. Highresolution implementation of selfconsistent field theory.
We consider a system with n conformationally symmetric diblock copolymers and each has a and b. An accurate numerical solution of field theoretic equations is employed to discern the role of particle curvature in governing the equilibrium phase behavior, gelation transitions and rheological characteristics of such mixtures, with particular emphasis on the nanoparticle regime. The relaxation schemes are further embedded in a multilevel strategy resulting in a method that can cut. A finite volume method for selfconsistent field theory of. We have made several improvements to the opensource polymer selfconsistent field code that allow for i computing the equilibrium composition for complicated phases, like the frankkasper. The selfconsistent field theory scft is a powerful framework for the study of the. E cient orderadaptive methods for polymer selfconsistent field theory hector d. A polymer field theory within the framework of statistical mechanics is a statistical field theory, describing the statistical behavior of a neutral or charged polymer system within the field theoretic approach. Thermodynamic averages like the average composition and the structure factor can be expressed exactly as averages of these fields.
Specifically, a finite volume algorithm on a unit sphere with an icosahedron triangular mesh is employed to describe the orientation degree of freedom of the wormlike chains. Consistent field theory of cylindrical polyelectrolyte brushes li. The response of strong polyelectrolyte pe brushes grafted on an electrode to electric fields generated by opposite surface charges on the pegrafted electrode and a second parallel electrode has been numerically investigated by self consistent field theory. An accurate numerical solution of field theoretic equations is. A new selfconsistent field model of polymernanoparticle. Our work on selfassembly in systems that contain block copolymers has relied heavily on the use of numerical selfconsistent field theory scft to predict equilibrium structures, and has often been motived by andor carried out in collaboration with experimental colleagues. Jian qu beijing national laboratory for molecular sciences bnlms, state key laboratory of polymer physics and chemistry, institute of chemistry, chinese academy of sciences, beijing 100190, china. It can be derived by transforming the partition function from its standard manydimensional integral representation over the particle degrees of freedom in a functional integral representation over an auxiliary field function, using either the hubbardstratonovich. A new selfconsistent field model of polymernanoparticle mixture. We propose efficient pseudospectral numerical schemes for solving the self consistent, mean field equations for inhomogeneous polymers. Orientational interactions in block copolymer melts. Computationally, polymer scft amounts to three problems. The resulting potential v h is then consistent with the orbitals that generate it, and it is for this reason called selfconsistent eld.
Pscf is a fortran 90 program for numerically solving the polymer self consistent field theory scft for spatially periodic structures formed by block copolymer melts and mixtures of block copolymers with linear homopolymers andor small molecule solvents. One widely used numerical scheme to solve these equations is the finite. Analysis of interactions between finitesized particles and. The selfconsistent field theory scft has reveived a great success in prediction of the physical properties of a variety of polymeric systems in the recent two decades. Here, i provide a numerical framework applying two dimensional finite volume method fvm to treat this conservation problem. Solving this vector equation subject to possible boundary conditions for r. A finite volume method for selfconsistent field theory of brush in cylindrical and spherical coordinate systems. The basic problem is that if particles interact, that. It can be derived by transforming the partition function from itsstandard manydimensional integral representation over the particle degrees of freedom in a functional integral. Numerical methods of polymer brushes can used to look at an ab binary blend. A highly accurate and memoryefficient approach for the solution of polymer self consistent field theory scft is proposed. Efficient orderadaptive methods for polymer selfconsistent field. New numerical implementation of selfconsistent field theory. Such models consider many individual components that interact with each other.
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